Xiaojuan Zhao

Professeur, AgroParisTech, Rapporteur Francis Courtois

Professeur Khashayar Saleh

Henri Berthiaux

Michel Cabassud

Contribution to the implementation of a process control system for continuous powder mixing

Keywords: Continuous powder mixing, Implementation, Process control, Markov chain, PID controller, Closed-loop control Mélange de poudres en continu, Implémentation, Contrôle-commande, Chaînes de Markov, Contrôleur PID, Commande en boucle fermée

Powder mixing is an essential operation in various industrial fields, such as pharmaceuticals, agrofood, cements, etc. Continuous powder mixing, as an alternative to conventional batch mixing, has attracted a lot of interest mainly due to its capacity in handling high volume manufacturing. This work aims at the contribution to the implementation of process control applications for powder mixing in a pilot-scale continuous mixer. Prior to developing process control strategies, new developments have been presented for better understanding continuous mixing of two components. Hold-up weight and relative hold-up weight distribution of each component in the whole mixer have been experimentally investigated under different operating conditions. An improved Markov chain model has been finally presented to predict the mean concentration of the mixtures obtained at the mixer's outlet. The implementation of a proportional-integral-derivative (PID) controller has been experimentally performed as an initial attempt to real-time control the homogeneity of the mixture produced. The rotational speed of the stirrer, identified as an important deciding factor towards the mixer's efficiency, has been considered as the manipulated variable. The closed-loop control is based on either the mean concentration or the relative standard deviation. The performances of the proposed closedloops have been evaluated for continuous mixing subjected to step changes in feed rates of the mixer. Four case studies have been defined and presented. The main challenge in the process control system is the tuning of PID parameters. The performance of closed-loop control of either the mean concentration or the relative standard deviation has been compared to open-loop operation.

GENERAL INTRODUCTION

Powder mixing is a very common unit operation in many industries, including pharmaceuticals, food, cements, etc. This operation is intended to ensure a uniform distribution of all components in the end product. It can be either a batch or continuous process. Continuous mixing offers several advantages over traditional batch mixing, such as larger volume manufacturing, easier scale-up, and lower segregation risk due to the absence of the intermediate storage and handling. While for continuous process, continuous real-time quality control is indispensable. Like in the pharmaceutical and food processing, the implementation of closed-loop control system is strongly encouraged by important regulatory authorities (e.g., Food and Drug Administration, FDA and European Medicine Agency, EMA, etc.), hence ensuring the quality of each end product, in turn eliminating product waste due to manufacturing failure.

In recent years, intensive research efforts have mainly focused on the understanding of continuous powder mixing processes using both experimental and computational approaches.

The measurements of powder mixing homogeneity have been able to perform non-invasively or in real-time with the development of online/inline monitoring techniques (e.g. near infrared spectroscopy, Raman spectroscopy, etc.). However, continuous powder mixing processes are still difficult to be characterized or predicted due to complicated effects of operating conditions (feed rate, impeller speed, etc.) and geometric designs (mixer size, impeller types, etc.) on the mixing efficiency and throughput of mixers. To date, little work has been published on the development of control strategies for continuous powder mixing. This manuscript is in continuity with previous thesis defended in the laboratory [START_REF] Marikh | Mélange des poudres en continu dynamique et modélisation[END_REF][START_REF] Ammarcha | Mélange des poudres en continu : modèles dynamiques et caractérisation en ligne[END_REF]. This work aims to contribute to the implementation of a process control system for continuous powder mixing.

An effective design and implementation of continuous powder mixing typically consists of a feeding system, a continuous mixer and a monitoring and control system as follows. Chapter II describes the materials and the experimental setting used for continuous powder mixing. Two materials (component A and B, respectively), fine couscous and medium couscous are used. Critical material properties (e.g. particle size distribution, bulk density, flowability, etc.) have been measured experimentally. For the experimental set-up, the feeding system, the continuous mixer and the on-line image analysis for mixture homogeneity are first described as three individual units and then combined together by operating systems based on the Labview platform.

Chapter III presents a new development in the understanding of continuous powder mixing.

Hold-up weight and relative hold-up weight distribution in the whole mixer, virtually divided into 5 compartments, are experimentally investigated under different operating conditions (e.g. mixing formulation, total feed rate and rotational speed of the mixer). Based on the empirical expressions determined for hold-up weight and relative hold-up weight distribution, an improved Markov chain model is finally obtained to predict the process behaviour over a much wider range of operating conditions. The reliability of the model is also verified by additional mixing experiments.

Chapter IV presents the design an implementation of PID controllers in the continuous powder mixing process. The process is first analysed based on a "black-box" model. The rotational speed of the mixer is finally selected as the manipulated variable (or actuator). The controlled variables are the mean concentration and the relative standard deviation, respectively. A PID controller is designed for closed-loop control of either of two controlled variables. The implementation of PID controllers in the real plant is realised using PID Control Toolkit in Labview software package. Two performance criteria and the PID tuning procedure are proposed to obtain satisfactory control performance.

Chapter V presents the experimental results of the proposed PID control in four possible industrial cases, which correspond to four different step changes in flow rates of the two powder components.

CHAPTER I: Background on powder mixing process 1 Generality on powder mixing process

Powder mixing is an operation aiming to achieve a uniform distribution of different powder ingredients or powders of the same nominal composition but having different particles sizes.

In some industrial fields, it is called blending. The adverse effect tending to be present in powder mixing is segregation or demixing, which makes the production of an ideal or random mixture difficult. When segregation occurs, the final state reached in the mixer is an equilibrium between mixing and segregation. The final mixture quality will be determined by the relative importance of each mechanism but segregation should be minimized in favor of the mixing operation. Depending on the extent of mixing, the distribution of powders after the process can be perfectly mixed, or randomly mixed, or partially segregated, or even completely segregated [START_REF] Rhodes | Principles of Powder Technology[END_REF][START_REF] Rhodes | Introduction to Particle Technology[END_REF] as depicted in Figure I-1 picturing these possibilities for two components. A perfect mixture of two types of particles is one in which any group of particles taken from any position in the mixture will contain the same proportions of each particle, the nominal one. In practice, a perfect mixture cannot be obtained. Generally, the aim is to produce a random mixture, in which the probability of finding a particle of any component is the same whatever the location in the mixture.

Mixing mechanisms

Mixing of powders is accomplished within the flowing three types of particle transport mechanisms:

(a) Diffusive mixing. This refers to random motions of individual particles resulting in micromixing as individual particles randomly dispersed through the mixture.

(b) Convective mixing. Convection is related to the movement of groups of particles from one position to another within the mixture, resulting in macromixing.

(c) Shear mixing. In shear mixing, shear stress give rise to slip zones and mixing takes place by interchange of particles between layers within the slip zone.

Segregating mechanisms

In industrial practice, particles to be mixed almost always have some different physical properties and tend to segregate. Segregation occurs due to differences in particulate properties such as particle size, shape, density, etc. Of these, size difference is by far the most important factor driving to segregation. Four mechanisms of segregation according to size are identified [START_REF] Rhodes | Introduction to Particle Technology[END_REF][START_REF] Weinekötter | Mixing of solids[END_REF][START_REF] Harnby | Mixing in the process industries[END_REF], as illustrated in (b) Segregation by percolation. Finer particles trickle down through the gaps between the larger ones. These gaps act like a sieve. Percolation is the most commented segregation effect and percolation occurs even when there is a very small difference in particle size. Percolation of fine particles usually occurs in charging and discharging storage hoppers. (d) Elutriation segregation. When a powder is poured into the top of a hopper, a large volume of air is displaced. The upward velocity of air may exceed the terminal free-fall velocity of some of the finer particles, in turn causing the fines to elutriate while the coarse particles settle down.

Equipment for powder mixing

A large number of equipment is available on the market to suit the multiplicity of mixing tasks. An appropriate mixer tends to minimize the segregation effect. Most mixers for powder mixing can be broadly classified into the following categories [START_REF] Rhodes | Introduction to Particle Technology[END_REF][START_REF] Weinekötter | Mixing of solids[END_REF][START_REF] Harnby | Mixing in the process industries[END_REF][START_REF] Edward | Handbook of Industrial Mixing[END_REF] :

•

• Convective mixers

In convective mixers, mixing is strongly convective and segregation is far less pronounced than in tumbler mixers. Most convective mixers have a rotating impeller within a stationary vessel, and moves groups of particles from one location to another, thus creating mixing. The stationary mixing chamber may be conical, cylindrical, or trough-shaped, and impeller designs range from ribbons to paddles to screws. There are four common types of convective mixers: the ribbon mixer, the paddle mixer, the orbital screw (Nauta) mixer and the vertical screw mixer, as illustrated in Figure I-4. The greatest attraction is that many convective mixers can be designed to accommodate continuous rather than batch processing, further adding to their utility.

• Fluidized mixers

Powders can also be mixed by aeration using a fluidized mixer. Fluidization is caused by the passage of air through a bed of particles. Excellent mixing can be produced if there is considerable turbulence within the bed and the combination of turbulence and particle mobility. An important feature of the fluidized bed mixer is that several processing steps (mixing reaction, coating, drying, etc.) may be carried out in the same vessel [START_REF] Rhodes | Introduction to Particle Technology[END_REF], thus avoiding transportation and contamination problems.

• Other types of mixers

There are many other kinds of mixer, such as hopper mixers, high shear mixers, etc. Among these, the high shear mixers emphasize breaking down agglomerates of cohesive powders rather than breaking individual particles.

Batch and continuous powder mixing processes

Powder mixing can be carried out either in batch or continuous mode [START_REF] Weinekötter | Mixing of solids[END_REF].

• Batch (or discontinuous) powder mixing

In batch mode, the mixer is filled with ingredients then started and after a certain time ("mixing time") the mixture is discharged. The filling, mixing and discharging operations are performed one after another.

• Continuous powder mixing

In a continuous mixing process, the ingredients are continuously fed into the mixer, mixed, preparing the product for the next processing stage. The operations of feeding, mixing and discharging follow each other locally but contemporaneously. The mixers in process industries are generally selected based on different aspects, such as process requirements, mixture quality, mixing costs, etc.

Assessment of powder mixtures 2.1 Mixture homogeneity

The evaluation of the homogeneity of a powder mixture is key to ensure that the product is satisfactory for its end-use, which imposes a scale of scrutiny on the mixture.

Scale of scrutiny

Danckwets [START_REF] Danckwerts | The definition and measurement of some characteristics of Mixtures[END_REF] defined a "scale of scrutiny" for a mixture to describe the minimum size of the regions in a particular mixture which would cause it to be regarded as insufficient mixed. An identification of the scale of scrutiny for a product fixes the size of the sample to be taken from the mixture and assessed for mixture quality. In pharmaceutical applications, for a tablet or capsule formulation, the appropriate sample size is that of dosage form (tablet or capsule).

The homogeneity for a powder mixture can be determined once a suitable scale of scrutiny is set. 9

Scale and intensity of segregation

Two important concepts related to mixture homogeneity have been proposed by Danckwerts [START_REF] Danckwerts | The definition and measurement of some characteristics of Mixtures[END_REF] such as scale and intensity of segregation. •

Scale of segregation

The scale of segregation is a measure of the size of regions of segregation in the mixture, as depicted in Figure 1-5 (a). It is a function of space for batch mixing systems and a function of time for continuous mixing systems. The smaller the scale of segregation is, the better the mixture is.

• Intensity of segregation

The intensity of segregation is a measure of the spread of the concentration of the component of interest in the mixture, as depicted in Figure I-5 (b). The lower the intensity of segregation is, the better the mixture is. The role of a mixture is to reduce the scale and intensity of segregation within the mixture.

Homogeneity for batch or continuous mixing

• Homogeneity for batch mixing 10

In the case of a batch mixer, the mixture uniformity is usually measured at the end of the operation, known as the end-point. The homogeneity of the mixture can be determined by two methods: [START_REF] Marikh | Mélange des poudres en continu dynamique et modélisation[END_REF] the estimation of mixture homogeneity by taking randomly distributed samples across the whole mixture (2) the examination of mixture homogeneity by taking all samples from the mixture flow, which is formed by continuously discharging the mixture from the mixer.

• Homogeneity for continuous mixing

For continuous mixers, homogeneity must be defined at the level of the mixture flowing out of the mixer, and falling down to a conveyor or any other transportation equipment.

Statistical estimation of homogeneity

Typically, the degree of homogeneity of a mixed product can be measured by statistical analysis of a number of samples of the mixture. The statistics relevant to a mixture of two components are summarized below:

• Mean composition

The sample mean x s is given by

x s = 1 n x i i=1 n ! (I-1)
where x i is the composition of the key component in the ith sample and n is the number of samples. It is used for estimating the mean composition of a specified mixture.

• Variance, standard deviation, and relative standard deviation

The sample variance s 2 is defined as

s 2 = 1 n !1 (x i ! x s ) 2 i=1 n " (I-2)
The value of standard deviation s is an estimation of the actual value of the standard deviation of the mixture, ! . Another important measure of blend uniformity is the relative standard deviation (RSD), also known as the coefficient of variance (CV), defined to be

RSD = CV = s x s (I-3)
In the pharmaceutical industry, the content uniformity tests are used for proving the uniform distribution of the active ingredient in a production batch. The content uniformity test of United States Pharmacopeia (USP) is performed by measuring the contents of active ingredient in each of 10 dosage units taken at random. The USP acceptance criteria [START_REF] Sawoniak | Formulary Considerations Related to Warfarin Interchangeability[END_REF] for content uniformity are: [START_REF] Marikh | Mélange des poudres en continu dynamique et modélisation[END_REF] the mean content should fall within the range of 92.5% to 107.5% of the content stated on the label; (2) the content of active ingredient in each individual dosage unit should fall within the range of 85% to 115% of the content stated on the label; (3) RSD ! 6% .

• Theoretical limits of variance

The limiting variance values were calculated for a two components mix in which the particles are of the same size. where p and (1! p) are the proportions of the two components determined from samples and n p is the number of particles in the sample.

• Mixing indices

The quality of mixing of a real mixture would be between that of the completely segregated and that of the random mixed. Various definitions of mixing indices have been proposed using the three variances ! 2 , ! 0 2 and ! R 2 , as shown in Table I-1. For example, the Lacey index [START_REF] Lacey | Developments in the theory of particle mixing[END_REF] is the ratio of "mixing occurred" to "mixing could occur". A Lacey index of zero would represent completely segregated mixture and a value of unity would represent a completely randomized mixture. Practical values of this index, however, are found to be restricted in the range of 0.75 to 1.0, which makes the Lacey index nearly insensitive to mixture quality. Compared to Lacey index, the mixing index suggested by Poole et al. [START_REF] Poole | Mixing powders to fine-scale homogeneity : studies of batch mixing[END_REF] gives better discrimination for practical mixture and approaches unity for fully random mixtures.

Table I-1: Some definitions of mixing indices.

Mixing index definition Reference

M = ! 0 2 !! 2 ! 0 2 !! R 2 Lacey [10] M = 1! ! ! 0
Rose [START_REF] Rose | A suggested equation relating to the mixing of powders and its application to the study of the performance of certain types of machines[END_REF] M = ! ! R Poole et al. [START_REF] Poole | Mixing powders to fine-scale homogeneity : studies of batch mixing[END_REF] M = ln! 0 2 ! ln! 2 ln! 0 2 !! R

Ashton and Valentin [START_REF] Ashton | The mixture of powders and particles in industrial mixers[END_REF]

Mixture structure

Let us consider a powder mixture flow as a mono-dimensional layer divided into N consecutive "elementary" samples. The global (or macro) analysis of the homogeneity can be determined by the use of statistics (e.g. mean composition and variance). Thus, it is necessary to examine the mixing structure. As introduced by Danckwerts [START_REF] Danckwerts | Continuous flow systems[END_REF][START_REF] Danckwerts | Theory of mixtures and mixing[END_REF] in the mid-fifties and more recently reviewed by Gyenis [START_REF] Gyenis | Assessment of mixing mechanism on the basis of concentration pattern[END_REF], the mixing structure can be accounted for by the use of the autocorrelation function R(r). For n consecutive samples, if x i and x m are, respectively, the composition of the sample i in a certain component and the mean composition for the n values, the mathematical definition of R(r) is as flows:

R(r) = (x i ! x m )(x i+r ! x m ) i=1 n!r " (x i ! x m ) 2 i=1 n " (I-6)
It is used to characterize the interdependence of the composition of two samples separated by a distance (number of samples) r. If no correlation exists, then the value of R(r) is equal to zero, and the mixture can be considered as homogeneous at this scale.

Methods for measuring mixture quality

Conventional methods for measuring powder-blend uniformity are mainly based on invasive (or manual) sampling and off-line destructive analysis with HPLC or UV spectroscopy [START_REF]Guidance for Industry Guidance for Industry PAT -A Framework for Innovative Pharmaceutical[END_REF].

These, however, are often time-consuming, error-prone and may include artifacts, since segregation and contamination may be introduced [START_REF] Muzzio | Sampling practices in powder blending[END_REF]. Due to advance in effective and fast sensing technologies, at-/in-/on-line monitoring methodologies have been recently developed.

It aims at acquiring fast and reliable data with representative or non-invasive sampling and thus collecting data in real-time [START_REF] Wu | Quality-by-Design (QbD): An integrated multivariate approach for the component quantification in powder blends[END_REF]. Furthermore, this would facilitate real-time decisions and adjustments during processing by implementing appropriate data analysis and modeling strategy. For instance, in the pharmaceutical industry, the Process Analytical Technology (PAT) [20] system has been proposed by the Food and Drug Administration (FDA) as "a system for designing, analyzing, and controlling manufacturing through timely measurements (i.e., during processing) of critical quality and performance attributes of raw and in-process materials and processes with the goal of ensuring final product quality." The ultimate goal of the PAT initiative is to enhance process understanding, leading to improved control of the manufacturing process.

Image processing techniques

Studies for defining and characterizing homogeneity in powder mixing using image processing techniques have been published by Muerza et al. [START_REF] Muerza | A dynamic study of static mixing using on-line image analysis[END_REF], Realpe and Velazquez [START_REF] Realpe | Image processing and analysis for determination of concentrations of powder mixtures[END_REF],

Berthiaux et al [START_REF] Berthiaux | Principal component analysis for characterising homogeneity in powder mixing using image processing techniques[END_REF], Obregon and Velazquez [START_REF] Obregón | Discrimination limit between mean gray values for the prediction of powder concentrations[END_REF], and André et al [START_REF] André | Dimensional analysis of a planetary mixer for homogenizing of free flowing powders: Mixing time and power consumption[END_REF]. What they have in common is the use of components of different colors and the analysis of mixtures with a PCbased vision system mainly consisting of a charge-coupled device (CCD) camera, an illumination system, an image acquisition board integrated to a computer and software based on graphical programming. Demeyre [START_REF] Demeyre | Caractérisation de l'homogénéité de mélange de poudres et de l'agitation en mélangeur Triaxe[END_REF] have applied a threshold value to acquire gray-level images in order to distinguish couscous colored in black from the nearly yellow semolina. The couscous proportion was then obtained from the ratio of black pixels to white pixels and a calibration curve was established to convert pixel proportion into mass composition. The image processing was carried out at the emptying stage of a batch mixer Triaxe®.

Figure I-8:

Steps to determine homogeneity from images from Rosas and Blanco [START_REF] Rosas | A criterion for assessing homogeneity distribution in hyperspectral images. Part 1: homogeneity index bases and blending processes[END_REF].

Recently, Rosas and Blanco [START_REF] Rosas | A criterion for assessing homogeneity distribution in hyperspectral images. Part 1: homogeneity index bases and blending processes[END_REF] have studied either binary mixtures (or mixtures of two components) or ternary mixtures (or mixtures of three components), using colored sand (SiO 2 ) with three different particles size (small, medium and large). A miniature blender robot and both invasively and non-invasively monitoring systems were used. Labeled images (binary and ternary images) have been introduced by labeling every pixel in the image with its cluster obtained from the result of color segmentation. "Binary images" were denoted for labeled images from binary mixtures and ternary images for labeled images from ternary mixtures. The procedure for assessing image homogeneity is summarized by Figure I-8.

Spectroscopic techniques

Driven by the PAT initiative, spectroscopic techniques like Raman spectroscopy and near infrared (NIR) spectroscopy, were increasingly implemented for non-invasive monitoring of pharmaceutical processes. These spectroscopic methods provide physical as well as chemical information of the active and inactive components in composite mixtures.

(1) Principle of Raman spectroscopy [START_REF] Ferraro | Introductroy Raman Spectroscopy[END_REF] It is vibrational spectroscopic technique that is based on light scattering phenomena, where scattering occurs due to collision between photons and molecules. When the energy of the incident photon is unaltered after collision with a molecule, the scatter photon has the same frequency as the incident photon. This is elastic or Rayleigh scattering. When energy is transferred from the molecule to the photon or vice versa, the scatter photon has less or more energy than that of incident photon. This is inelastic or Raman scattering. The energy difference between incident photons and scattered photons is called "Raman shift". A Raman spectra is a plot of scattered intensity versus the frequency of the Raman shift. It usually consists of sharp peaks whose position and intensity can be used to characterize the functional chemical groups. This information can be interpreted to determine chemical structure and to identify the compounds present. Since excipients and APIs will exhibit difference in Raman peak intensities and shift positions, it is useful to determine the composition of a pharmaceutical powder mixture.

(2) Principle of NIR spectroscopy [START_REF] Siesler | Near-Infrared Spectroscopy: Principles, Instruments, Applications[END_REF] It is based on absorption. The entire near-infrared region (NIR) of the electromagnetic spectra encompasses light with wavelength ranging from 750 to 2,500 nm, lying between the visible light and the infrared light. NIR spectra predominantly correspond to harmonics of overtones and combination of fundamental vibration transitions more frequently associated with (mid-) infrared spectroscopy. Overton and combination absorptions are principally seen for C-H, O-H, and N-H molecular groups. A NIR spectra is a plot of these absorptions as a function of wavelength or wavenumber.

(3) Preprocessing and quantitative analysis of spectral data

Raman/NIR spectra could be affected by the physical properties of samples and other interferences or noises. Therefore, it is necessary to preprocess (or "correct") Raman/NIR spectral data to reduce, eliminate or standardize the above-mentioned effects. Common preprocessing methods include baseline correction [START_REF] Dobrowolski | Theory of the intensian method[END_REF], first and second derivative Savitzky

Golay [START_REF] Savitzky | Smoothing and Differentiation of Data by Simplified Least Squares Procedures[END_REF], Standard normal variate (SNV) [START_REF] Barnes | Standard Normal Variate Transformation and De-trending of Near-Infrared Diffuse Reflectance Spectra[END_REF], and multiple scattering correction (MSC) [START_REF] Isaksson | The Effect of Multiplicative Scatter Correction (MSC) and Linearity Improvement in NIR Spectroscopy[END_REF]. The preprocessed spectral data will be then quantitatively analyzed using of multivariate regression methods, such as multiple linear regression (MLR), principal component analysis/regression (PCA/PCR), and partial least squares regression (PLSR), etc. These quantitative models enable prediction of desired quality variable (e.g. API content). A detailed description of quantitative analysis of spectra can be found in [START_REF] Howard | Chemometrics in Spectroscopy[END_REF].

In the pharmaceutical industry, the use of Raman spectroscopy for process analysis is so far limited compared to the use of NIR spectroscopy [START_REF] Matousek | Emerging Raman Applications and Techniques in Biomedical and Pharmaceutical Fields[END_REF]. This is attributed to two aspects. On one hand, Raman spectroscopy is less sensitive to variations in physical properties (e.g. particle size), which directly influence the product performance. On the other hand, when dealing with small amounts of material and low level of API, low sensitivity of Raman may become an issue.

• Application examples for batch mixers

A binary powder mixture of acetyl salicylic acid (ASA) as an active pharmaceutical ingredient (API) and α-lactose monohydrate (LM) as an excipient in a four-blade mixer were characterized in-line using NIR spectroscopy by Koller et al [START_REF] Koller | Continuous quantitative monitoring of powder mixing dynamics by near-infrared spectroscopy[END_REF]. The optimal spectral range for API and LM was found to be 6000-10,000 cm -1 , including several specific peak regions of ASA (around 6000 cm -1 and 8500-9000 cm -1 ) and LM (6000-7000cm -1 ), as illustrated in It was also found that convective mixing is strongest for low filling levels, whereas diffusive mixing dominates the overall process for high fill levels. becomes plateaus around 0.02. In addition, mathematical model fitting for both showed that the contribution of the method error was negligible.

Near infrared chemical imaging (NIR-CI) technique

Near infrared chemical imaging (NIR-CI) adds spatial distribution information to the spectral information by combining traditional NIR spectroscopy with digital imaging. It can be used to visualize the spatial distribution of the chemical compounds in a sample (providing a chemical image). In NIR-CI, a NIR spectra is recorded in each pixel of the sample image resulting in a hyperspectral data cube, from which chemical concentrations can be estimated.

A detail description of NIR-CI on pharmaceutical solid dosage can be referred in [START_REF] Ravn | Near-infrared chemical imaging (NIR-CI) on pharmaceutical solid dosage forms-comparing common calibration approaches[END_REF]. The use of NIR-CI for the determination of powder blend and tablet homogeneity based on API concentrations has been previously described [START_REF] El-Hagrasy | Near-Infrared Spectroscopy and Imaging for the Monitoring of Powder Blend Homogeneity[END_REF][START_REF] Lewis | NIR Chemical Imaging as a Process Analytical Tool[END_REF][START_REF] Lyon | Nearinfrared spectral imaging for quality assurance of pharmaceutical products: analysis of tablets to assess powder blend homogeneity[END_REF][START_REF] Westenberger | Quality assessment of internet pharmaceutical products using traditional and nontraditional analytical techniques[END_REF].

Recently, Wu et al. [START_REF] Wu | Monitoring of a pharmaceutical blending process using near infrared chemical imaging[END_REF]

Characterization of continuous powder mixing

For a continuous mixing of powders, the mixing performance can be characterized by mainly two approaches: the determination of mixture homogeneity and the examination of flow behavior in the mixer. To achieve the latter, residence time distribution (RTD), variance reduction ratio (VRR), and powder hold-up weight in the mixer were widely utilized in the literature [START_REF] Marikh | Influence of stirrer type on mixture homogeneity in continuous powder mixing: A model case and a pharmaceutical case[END_REF][START_REF] Vanarase | Effect of operating conditions and design parameters in a continuous powder mixer[END_REF][START_REF] Ammarcha | Predicting bulk powder flow dynamics in a continuous mixer operating in transitory regimes[END_REF].

• Residence time distribution (RTD)

The residence time distribution (RTD) was previously proposed by Danckwerts [START_REF] Danckwerts | Continuous flow systems[END_REF]. It was measured using the pulse test method, where a large number of tracer particles were injected at the entrance of the mixer and the tracer concentration was measured at the mixer outflow.

The residence time distribution (RTD) function E(t) is given by:

E(t) = C(t) C(t)dt 0 ! " (I-7)
where C(t) is the concentration of the tracer in the outlet stream over time. It reflects the time spent by the particles within the mixer and captures the non-ideality associated with the flow.

Two important parameters derived from the RTD function are the mean residence time <t> as

< t >= tE(t)dt 0 ! " (I-8)
and the variance ! 2 as

! 2 = (t ! t m 0 " # ) 2 E(t)dt (I-9)
In continuous powder mixing processes, Sen et al. [START_REF] Sen | Multi-dimensional population balance modeling and experimental validation of continuous powder mixing processes[END_REF], Vanarase et al. [START_REF] Vanarase | Effect of operating conditions and design parameters in a continuous powder mixer[END_REF][START_REF] Vanarase | Real-time monitoring of drug concentration in a continuous powder mixing process using NIR spectroscopy[END_REF], Gao et al.

[57] experimentally investigated RTD as follows. Bulk material was fed in the mixer until steady state is reached. A known mass of (colored) tracer, usually being API particle (e.g.

APAP)

, was then inserted into the inflow stream as an "instantaneous" pulse. The concentration of the tracer at the outlet of the mixer was subsequently determined by collecting samples at various times.

• Variance reduction ratio (VRR)

The earliest variance reduction ratio (VRR) was introduced by Beaudry [START_REF] Beaudry | Blender efficiency[END_REF] for characterizing the fluctuation of continuous fluid mixers. VRR is the ratio of the variance of the concentration fluctuations at the inlet ! in 2 to that at the outlet ! out 2 . The higher the value of VRR, the better the performance of the mixer. Danckwerts, Williams and Rahman, and

Weinekötter and Reh [START_REF] Danckwerts | Continuous flow systems[END_REF][START_REF] Williams | Prediction of the performance of continuous mixers for particulate solids using residence time distributions Part I[END_REF][START_REF] Weineikötter | Continuous mixing of fine particles[END_REF] expressed VRR as a function of the auto-correlation coefficient from the feeder fluctuations and the residence time distribution (RTD), as given by Eq. (I-10) and Eq. (I-11).

1 VRR = ! out 2 ! in 2 = 2 R(r)E(t)E(t + r)dr 0 ! " (I-10) R(r) = cov(x t , x t+r ) var(x t ) (I-11)
where R(r) stands for the autocorrelation coefficient, x t is the concentration at input at time t, and r is the window of observation and E(t) is the residence time distribution (RTD) function.

• Hold-up

Powder hold-up in the mixer is important because it determines the geometrical residence time [hold-up (kg)/flow rate (kg.h -1 )]. As Vanarase and Muzzio [START_REF] Vanarase | Effect of operating conditions and design parameters in a continuous powder mixer[END_REF], Ammarcha et al. [START_REF] Ammarcha | Predicting bulk powder flow dynamics in a continuous mixer operating in transitory regimes[END_REF] illustrated, hold-up can be measured by simultaneously monitoring the weight of the powder collected at the mixer's outlet and that of powder being fed. According to the evolution of powder hold-up weight in the mixer, a common continuous mixing process can be divided into three phases: start-up, steady state and emptying. In preliminary measurements, hold-up is initially zero. In the period of start-up, hold-up increases with time, and finally reaches a plateau. The mixer operating under constant hold-up was considered to be operating at steady state in which the mixer has reached a constant mass inflow and outflow. In the emptying phase, the remaining material is discharged from the mixer.

For bulk medium couscous in a continuous mixer Gericke GCM 500, Ammarcha et al. [START_REF] Ammarcha | Predicting bulk powder flow dynamics in a continuous mixer operating in transitory regimes[END_REF] used a minimum hold-up weight " M min " below which no powder can flow out the mixer during start-up phase, and a maximum hold-up weight " M r "representing the constant value at steady-state, as shown in By means of RTD and hold-up measurements, the effects of various factors, such as particulate properties (size distribution, powder cohesion, etc.), operating conditions (feed rate, impeller rotation rate) and geometric designs (mixer size, blade configuration, etc.) have been examined.

The effects of particulate properties

As mentioned above, the physical properties of the particles are known to have an effect on the mixing and segregation of powders. The segregation of powders mainly occurs due to a particle size difference between the individual components of powder mixtures. Since powder flowability is dominantly influenced by the particle size, powders can be generally categorized into two groups: free-flowing powders and cohesive powders [START_REF] Weinekötter | Mixing of solids[END_REF]. In general, free flowing powders are ones relatively easy to mix, but the resultant mixture is prone to showed that the effect of powder cohesion depends geometric parameters of the blender, having a significant effect in the larger mixer. It was also pointed out that the effects of cohesion could be system-dependent. Since cohesion can affect the degree of variability in the flow rate delivered by a powder feeder, it should be expected that cohesion could have impacted the performance of the integrated system, including feeders, mixers, and downstream finishing equipment. The effect of cohesion on the particles paths lengths inside a continuous mixer has also been examined by Portillo et al. [START_REF] Portillo | Investigation of the effect of impeller rotation rate, powder flow rate, and cohesion on powder flow behavior in a continuous blender using PEPT[END_REF] using positron emission particle tracking (PEPT), where the impeller rotation speed was adjusted to three different rotational speeds (16, 75, and 170 rpm). It was found that the effect of cohesion was not significant at the highest rotation speed (170 rpm). This means the influence of cohesion will be undermined by higher energy imparted by the impeller on the particles.

The effects of operating conditions and design parameters

In the continuous mixers considered, feed rate and impeller rotational speed were found to be significant process parameters affecting mixing performance. As Weinekötter and Gericke [START_REF] Weinekötter | Mixing of solids[END_REF] illustrated, feed rate is a crucial process parameter in continuous powder mixing. It depends on the accuracy and capacity of the powder feeding system employed. Gao et al. [START_REF] Gao | Characterizing continuous powder mixing using residence time distribution[END_REF][START_REF] Gao | Characterization of Feeder Effects on Continuous Solid Mixing Using Fourier Series Analysis[END_REF] has experimentally investigated the influence of feed rate variability on the output variance ! out 2 .

Two loss-in-weight feeders provided by Schenck-Accurate were utilized to feed the mixer.

The feed rate variability of API and excipient were separately measured at the corresponding feed rates using a catch-scale that records the weight of powder discharged by the feeders as a In the literature, the most common continuous mixers studied are convective mixers, in which there is usually a stirring device consisting of several impeller blades attached to a rotating shaft. The stirring speed is a key factor that affects material dispersion throughout the mixer.

The effect of impeller rotational speed was generally investigated under various inflow rates, and design parameters (e.g. stirrer type, mixer configuration, etc.). In addition, this "All forward" blade configuration of stirring device has also been compared

with " Alternate" blade configuration consisting of alternate blades directing in forward and backward with blade angle of 20°. Figure I-21 concluded that the effect of blade configuration was not very significant at lower rotational speed, while the mean residence and hold-up were greater for the "Alternate" blade configuration at higher rotational speed. This implied that there is powder recirculation in the mixer when using the "Alternate" blade configuration.

Powder mixing models

Several modeling approaches exist in the literature for powder mixing process. The current modeling approaches can be mainly categorized into Markov chain models, discrete element method (DEM) models, convection-diffusion models, convection-dispersion models and population balance models (PBM).

Markov chain model

A Markov chain is a probabilistic model that exhibits a special type of dependence, that is, where the state of the system on the (n+1) th observation depends only on the state of the system on the n th observation. This means that, given the present state of the process, the future state is independent of the past. This property is usually referred as the Markov property [START_REF] Tamir | Applications of Markov Chains in Chemical Engineering[END_REF].

The theory of Markov chains has been early applied for batch mixing processes [START_REF] Oyama | Studies on the Mixing of Particulate Solids[END_REF]66], static mixing processes [START_REF] Wang | Axial Mixing of Grains in a Motionless Sulzer (Koch) Mixer[END_REF][START_REF] Wang | Stochastic modeling of segregation in a motionless mixer[END_REF] and powder mixing in fluidized beds and a hoop mixer [START_REF] Dehling | A stochastic model for mixing and segregation in slugging fluidized beds[END_REF][START_REF] Harris | Stochastic modelling of the particle residence time distribution in circulating fluidised bed risers[END_REF][START_REF] Aoun-Habbachea | An experimental method and a Markov chain model to describe axial and radial mixing in a hoop mixer[END_REF].

More recently, Markov chain-based models have been used to describe continuous convective The mixer has been presented by a series ( n !1) of interconnected cells. It was assumed that the exchange of material inside the mixer through zones are perfectly mixed. The last cell ( n th ) corresponds to the absorbing state: a collector of particles of outflow, which is not part of the mixer. The chain composed of these n cells is observed in discrete intervals, which are called system states. The set of probabilities to occupy all the cells forms, in the sample space, the state vector S with elements S i ( i =1, 2,..., n ), which is shown as a column vector of the size n !1 . The transition duration !t is chosen to be small enough so that the particles are allowed to transit only to neighboring cells. It is obvious that the current time is equal to k!t after k transition. If the state vector at the k th moment of time is S(k) , then after the current transition it will be S(k +1) . The correlation between S(k) and S(k +1) can be described by the matrix formula:

S(k +1)=PS(k ) (I- 12
)
where P is the matrix of transition probabilities -the basic operator of a Markov chain. Every transition has its own probability. p ij means the probability of particles in the cell j to move to the cell i. If i=j, then p ii means the probability of particles in the cell i to remain in the cell. The evolution of the vector of hold-up was expressed by Eq. (I-13), where M i (k +1) is the hold-up weight of particles in i th cell after k transitions; p ✐✐ (k) and p i+1, i (k) are respectively the probabilities of to stay in the i th cell and to move from the i th cell to the i +1 th cell for k th transition; p ii (k) = 1! p i+1, i (k) and p 66 (k) = 1 because there is no backward for the last cell.

M 1 (k +1) M 2 (k +1) M 3 (k +1) M 4 (k +1) M 5 (k +1) M 6 (k +1) ! " # # # # # # # = P 11 (
! " # # # # # # # $ M 1 (k) + Q't M 2 (k) M 3 (k) M 4 (k) M 5 (k) M 6 (k) ! " # # # # # # # $ (I-13)
It was assumed that the transition probability p i+1, i (k) during the start-up stage varies linearly with the measured hold-up. While it reaches its maximum value and remains constant at steady state and the value depends on the feeding rate Q and the rotational speed of the stirring device N . Thus, the transition probabilities can be calculated as follows:

p i+1, i (k) = 0, for M i < M i, min P i+1, i max M i, max ! M i, min (M i (k) ! M i, min ), for M i, min " M i < M i, max P i+1, i max , for M i = M i, min # $ % % % & % % % (I-14) P i+1, i max (k) = Q!t M i, max + Q!t , for i = 1 Q!t M i, max , for i = 2, 3, 4, 5 " # $ $ $ % $ $ $ (I-15) M i+1, i max = f 1 (Q, N ), for i = 1, 2, 3, 4 f 2 (Q, N ), for i = 5 ! " # $ # (I-16)
The Markov chain model presented above was used for predicting bulk powder flow dynamics in the continuous mixer. However, in this study, our aim is to model a continuous powder mixing process. For this purpose, it is proposed to modify this previously developed model for simulating a continuous mixing dynamics of two particulate components. The possible way is to determine the matrix of transition probabilities for each individual component.

Convection-dispersion model and Population balance model

•

Convection-dispersion model

The convection-dispersion model is based on the Fokker-Planck equations (FPEs) and it is used to describing mixing in the axial direction, as given by Eq. (I-17).

!C !t = "v !C !z + D ! 2 C !z 2 (I-17)
where C is the concentration of particles of one particulate component in a system, z is the axial direction of motion, v is related to particle velocity, and D is a dispersion coefficient.

Sommer [START_REF] Sommer | Continuous powder mixing[END_REF][START_REF] Sommer | Mixing of particulate solids[END_REF] introduced FPEs for the case of continuous mixers and solved for each component in the mixture. Weinekötter et al. [START_REF] Weineikötter | Continuous mixing of fine particles[END_REF] used this model for a low dosage mixture. Later Kehlenbeck and

Sommer [START_REF] Kehlenbeck | Modeling of the mixing process of very fine powders in a continuous dynamic mixer[END_REF] later used FPEs to model the dynamics of mixing in a continuous rotary powder blender. Particle streams with merely identical physical properties were used, so that it could be assumed that all the components in the mixture exhibited the same particle velocity and dispersion coefficient. Once FPEs were written, the important issue is to solve FPEs by determining parameters. Results showed that the ratio of the mean residence time to the time period of the fluctuation in the feeder is the most important parameter affecting the variance reduction ratio.

• Population balance model

Population balance models (PBMs) have been frequently applied to model particulate process such as crystallization and granulation but few for powder mixing till recently. It can be expressed as [START_REF] Ramkrishna | Population Balances[END_REF] :

! !t F(x, z, t) + ! !x F(x, z, t) dx dt " # $ % & ' + ! !z F(x, z, t) dz dt " # $ % & ' = ( formation ) ( depletion (I-18)
where F(x, z, t) is the population distribution function, x is the vector of internal co-ordinates used to characterize the internal distribution, z is the vector of external co-ordinates used to represent spatial position of the particles and t is the time. The term ! !x F x, z, t dx dt [ ] accounts for the rate of change of particle distribution due to change in particle size. The term

! !z F x, z, t dz dt [
] accounts for the rate of change of particle distribution with respect to spatial co-ordinates. ! formation and ! depletion stand for particles being formed and depleted respectively. A multi-dimensional population balance model have been formulated to elucidate the dynamics of a continuous powder mixing process by Sen et al. [START_REF] Sen | Multi-dimensional population balance modeling and experimental validation of continuous powder mixing processes[END_REF]. It has been assumed that particle size could not change, so the rate of particle formation and depletion are zero. The modified population balance equation is thus given as follows:

! !t F(z, t) + ! !z F(z, t) dz dt " # $ % & ' = 0 (I-19)
where z can be expressed by the spatial co-ordinate in the axial direction x and the spatial co- ordinate in radial direction y. The above equation can be written in multi-dimensional form as shown below:

! !t F(n, x, y, t) + ! !x F(n, x, y, t) dx dt " # $ % & ' + ! !y F(n, x, y, t) dy dt " # $ % & ' = Inflow ( outflow (I-20)
It represent an n multi-dimensional formulation where n=2 to represent the API and excipient components. The effect of important process parameters on flow and mixing (e.g. API composition, RSD and RTD) have been investigated by Sen and Ramachandran [START_REF] Sen | A multi-dimensional population balance model approach to continuous powder mixing processes[END_REF] using the PBM scheme presented above.

Discrete element model

Till now, considerable reports from the literature have focused on a numerical technique -the discrete element methods (DEM) for modeling the mixing of granular materials [START_REF] Bertrand | DEM-based models for the mixing of granular materials[END_REF]. The DEM approach has been originally developed by Cûdall and Strack [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF] and has become a useful tool that can provide detailed microscopic information about a given particulate system: particle location, velocity and force acting on each individual particle at any time during the process. (

1) the geometries of particles

For the computational simplicity, the majority of DEM models were realized using the simplest geometrical shapes for particles, such as disc-shaped and spherical elements. Some complex shapes were also invented such as ellipse, ellipsoid, polygon, polyhedron and superquadric particles. However, the implementations of these geometries encounter limitations when a large-scale calculation is essential.

(

2) the use of a suitable contact model to resolve any forces acting on a particle in the system

A model is required to evaluate contact force term F c, ij between particles i and j , which is the sum of normal (head-on and tangential (shear) contributions. Many different contact models have been proposed in DEM simulations and the reader can referred to Schafer et al.

[82] and Bertrand et al. [START_REF] Bertrand | Simulation of the consolidation of paper coating structures: probabilistic versus deterministic models[END_REF] for more details. The most common representation used in a 40 DEM simulation is a spring-dashpot model, such as the linear-spring-dashpot (LSD) model of Cûndall and Strack [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF] and the non-linear-spring-dashpot model based on the work of Hertz [START_REF] Hertz | Über die berührung fester elastischer körper[END_REF]. Recently, a combination of these two models has been proposed by Zhou et al. [START_REF] Zhou | Rolling friction in the dynamic simulation of sandpile formation[END_REF].

(3) the application of Newton's second law of motion to calculate changes in the motion of particles resulting from any unbalanced forces.

The simulation advance using small incremental time steps, and the total force on each particle is determined at every instant of time. The total force is the sum of all mechanical contact and body forces. These forces are then integrated over time to obtain the position and the velocity of each individual particle using Newton's second law of motion as follows:

m i dv i dt = ( j=1 k i ! F c, ij + F d, ij ) + m i g + F i (I-21)
where m i is the mass of particle i , v i is its translation velocity, and k i is the number of particles in contact with i at time t . The right-hand side of Eq. (I-21) represents the total force acting on the particle, including the contact force F c, ij and contact damping force F d, ij between particles i and j , the gravitational force m i g , any other forces F i such as fluid drag or cohesion arising from Van der Waals interactions, liquid bridging, or electrostatic or magnetic effects.

Until now, for mixing processes the DEM has been mainly applied for modeling granular flows in various batch mixers [START_REF] Bertrand | DEM-based models for the mixing of granular materials[END_REF], such as modeling of a V-blender by Lemieux [START_REF] Lemieux | Largescale numerical investigation of solids mixing in a V-blender using the discrete element method[END_REF], a double cone mixer by Manickam [START_REF] Manickam | Investigating mixing in a multi-dimensional rotary mixer: Experiments and simulations[END_REF] Each section is comprised of two sets of two blades and four periodic slice models were used to simulate each of the four full mixers sections. Although DEM can be used as efficient tool, considerable limitations and challenges exist.

The primary limitation is that DEM models are computationally intensive and require weeks/months of simulation, even if parallel computers are used. The computational complexity varies linearly with the number of particles and it is difficult to simulating full populations of particles (millions and billions) using conventional DEM. Recently, a hybrid DEM-Compartment model proposed by Portillo [START_REF] Portillo | Hybrid DEM-Compartment Modeling Approach for Granular Mixing[END_REF], hybrid DEM-PBM models proposed by Freireich et al. [START_REF] Freireich | Incorporating particle flow information from discrete element simulations in population balance models of mixer-coaters[END_REF] and Sen et al. [START_REF] Sen | Multi-scale flowsheet simulation of an integrated continuous purification-downstream pharmaceutical manufacturing process[END_REF] and hybrid DEM-Markov models proposed by Doucet et al. [START_REF] Doucet | Modeling of the mixing of monodisperse particles using a stationary DEM-based Markov process[END_REF], Tjakra et al. [START_REF] Tjakra | Modeling collective dynamics of particulate systems under time-varying operating conditions based on Markov chains[END_REF] and Trabelsi [START_REF] Trabelsi | Simulation numérique de l'écoulement et mélange granulaires par des éléments discrets ellipsoïdaux[END_REF], demonstrated the potential for overcoming the defects of conventional DEM.

Studies on control strategies for particulate processes

The aim of control strategies is to ensure the stability of the process, to minimize the influence of disturbances and to optimize the overall performance. For developing a control strategy for an existing process, it is essential to analyze the process from the control engineer's viewpoint, so as to identifying three important types of variables [START_REF] Seborg | Process Dynamics and Control[END_REF].

• Process variables (PVs): variables that are measured, monitored, and controlled. The desired value of a controlled variable is referred to as its set-point.

• Manipulated variables (MVs): variables that can be adjusted in order to keep the controlled variable at or near their set-points.

• Disturbance variables (DVs): variables that affect the controlled variables but cannot be manipulated. Disturbances generally are related to changes in the operating environment of the process.

The specifications and selections of PVs, MVs and DVs should be based on process knowledge, experiences, and control objectives. To design control systems, two general approaches have been developed [START_REF] Nof | Springer Handbook of Automation[END_REF]:

(

1) Traditional approach

The control strategy is selected based on knowledge of the process, experience, and insight.

After the control system is installed in the plant/process, the controller settings are adjusted.

(

2) Model-based approach

A dynamic model of the process is first developed that can be helpful in at least three ways: (a)

it can be used as the basis for model-based controller design methods, (b) the dynamic model can be incorporated directly in the control law, and (c) the model can be used in a computer simulation to evaluate alternative control strategies and to determine preliminary values of the controller settings.

PID control and Model-based predictive control theories

Proportional-integral-derivative (PID) control

The PID controller has been the most popular and is by far the most applied form of feedback in use. More than 90% of all control loops are PIDs [START_REF] Åström | The future of PID control[END_REF]. This predominance of PID control is due to its simplicity and its effectiveness in a wide range of applications. available [START_REF] Lequin | Iterative feedback tuning of PID parameters: comparison with classical tuning rules[END_REF], such as the Ziegler-Nichols tuning method, the internal model control method, and the integral square error method, etc. PID control performance can generally be improved by careful tuning, and performance may be unacceptable with poor tuning. Moreover, in order to enhance the single loop PID feedback control performance, a number of special control configurations such as cascade control, feedforward control and inferential control are often used and details can be found in the book of Spitzer [START_REF] Spitzer | Advanced Regulatory Control: Applications and Techniques, First Edit[END_REF].

Model-based predictive control (MBPC) strategy

Model based predictive control (MBPC) is also named model predictive control (MPC).

Besides PID control, MPC is well known as the second most used advanced control technique [START_REF] Jelali | Control performance management in industrial automation assessment, diagnosis and improvement of control loop performance[END_REF] in the process industry, in which there is plant/process model to predict the future behavior of a process. A block diagram of a MPC system is shown in At the current sampling instant k, the MPC strategy calculates a set of M values of the input.

The set consist of the current input and M-1 future inputs. The input is held constant after the M control moves. The inputs are calculated so that a set of P predicted outputs reaches the set point in an optimal manner. The control calculations are based on optimizing an objective function. The number of predictions P is referred to as the prediction horizon while the number of control moves M is called the control horizon.

Although the development and application of MPC technology was driven by industry, the main disadvantage of MPC is that it cannot be able of explicitly dealing with plant model uncertainties. A model with good predictive capability is not always available and has to be derived. The model accuracy determines the performance of MPC. On the other hand, the model was required as simple as possible to make on-line optimization feasible. It is obvious that accuracy and simplicity are often contradictory requirements.

Several recent publications provide a good introduction to applications of control technology in particulate processes. Christofides et al. [START_REF] Christofides | Control of particulate processes: Recent results and future challenges[END_REF][START_REF] Christofides | Model-based control of particulate processes[END_REF] Ramanchandran et al. [START_REF] Suresh | OntoMODEL: Ontological Mathematical Modeling Knowledge Management in Pharmaceutical Product Development[END_REF]. So far, few studies have focused on control loops for continuous powder mixing processes. In the following, we will present two examples.

Examples of model-based control

• Model predictive control of a dynamic powder mixing

VRR(t) = ! i 2 (t) ! o 2 (t) = exp ! 2w(t) W (t) " # $ % & ' "(t !1) 1! exp ! w(t) W (t) " # $ % & ' "(t !1) ( ) * + , - 2 . exp !2 j w(t) W (t) " # $ % & ' "(t ! j) ( ) * + , - j=1 K / + 2 exp ! w(t) W (t) " # $ % & ' ( j"(t ! j) +L L = j+1 K / !(t !L ))r jL j=1 K!1 / 0 1 2 3 2 4 5 2 6 2 !1 (I-22) dW (t) dt = w(t) ! 0.3!(t)W d HD cos" (I-23) !(t) = 0.126" 2 (t) ! 2.25"(t) +11.16 t > 0 0 t " 0 # $ % & % (I-24)
where the variables listed are defined as follow: VRR , the variance reduction ration; W , the mass in the mixer; W 1 , the lower mass limit in the mixer; ! , the screw speed; w , the inlet mass flow; W d , the pitch of the screw; H , the gap between the screw and the barrel; D , the diameter of the barrel; ! , the helix angle of the screw; ! , the measure of blender efficiency;

jL , the input serial correlation coefficient; and k , the number of samples of input.

Using the dynamic mixing model presented above, a model predictive control (MPC) was implemented. The VRR and the mass in the mixer W were used as controlled variables. The inlet flow rate w(t) and the rotational speed !(t) were manipulated variables. It was assumed that there are no disturbance variables in the operation. The change in the manipulated variables were modeled using a first order actuator dynamics, as given by:

dw dt = w d ! w ! w , d" dt = " d !" ! " (I-25)
where parameters w d and ! d are the set-points, ! w and ! " are the time constants for the inlet flow rate and the rotational speed, respectively. The application of the MPC was performed using OntoMODEL, which is an ontological tool for mechanistic mathematical model management that facilitates systematic and standardizable methods for model storage, In the feeding operation, three feeders were implemented for the active pharmaceutical ingredient (API), excipient and lubricant, respectively. The feed rates were controlled by adjusting the screw speeds of feeders. In the mixing operation, the rotational speed and feed rates of the mixer were identified as manipulated variables; the exiting total flow rate, API composition and relative standard deviation (RSD) were identified as controlled variables.

The proposed control approach mainly consists of three steps : (1) to obtain control (or process) models via system identification from dynamic models, experimental data, and/or literature data, (2) formulate a suitable control structure, (3) optimal tuning of controller 50 parameters via a suitable control-loop performance criteria, e.g. a scaled integral time error (ITAE) defined as follows:

ITAE( p) = y reponse,m (t z , p) ! y setpoint, m (t z ) y setpoint, m (t z ) t z z=1 n z " m=1 n m " 1 (I-26)
where p represents the vector of tuning parameters for the various m controllers, z represents the number of time points and y represents the controlled variables of interest.

The mixer was modeled via a dynamic population model (PBM). The transfer function, as process models for continuous powder mixing, were obtained via system identification from PBM, experimental data, and/or literature data. Due to uncertainties present in the data, the process models will always be an inaccurate representation of the true process. For instance, a different set of transfer functions were obtained due to uncertainties in model parameters.

Therefore, cases with three intensities (mild, moderate and high) of multivariable process interactions were analysed. was performed using a proportional-integral (PI) controller for manipulating the mixing rotational speed (rpm_mix). The control loop of RSD (y 2 ) consisted of two PI controllers (master and slave), where a slave loop controlled the lubricant feed rate (y 4 ) set point generated through a master loop that controlled by manipulating the lubricant feeder screw speed (rpm_lub). The control loop of the exit total flow rate (y 3 ) involved three PI controllers (one master and two slave), where a master loop was implemented to provide the feed rate set-points of API (y 5 ) and excipient (y 6 ) while two slave loops tracked y 5 and y 6 set-points by manipulating respective feeder screw speed (rpm_API and rmp_exp).

For each case, the performance of the proposed control system for set-point tracking was evaluated. In particular, the step changes in the set point have been made for analyzing the ability of the controller to track the provided set point. Simulation results showed that for mild process interactions, the set-point tracking ability of the control strategy and tuning parameters was robust. In addition, the disturbance rejection ability of the proposed control strategy was also investigated by setting stochastic disturbances (e.g. random noise) and results revealed that the control system couldn't lighten the effect of the disturbances.

Conclusion

Powder mixing process depends on many factors like particles properties, mixer designs, operating conditions, and mixture formulations, etc. In comparison with batch mixing, continuous powder mixing is an attractive alternative for large volume processing in industry.

Studies on powder mixing in continuous mixers were reviewed, involving at-/in-/on-line measurements of mixture homogeneity by various monitoring techniques and characterizations of powder flow behavior using VRR, RTD, and hold-up. In addition, modeling approaches (e.g. Markov chain, DEM), were introduced to predict the behavior of powder mixing in mixers. The ultimate goal of all these studies is to understand how the critical process parameters influence the performance of continuous powder mixing. Although several control strategies were proposed for continuous particulate process, control system loops have not yet been introduced for continuous mixing in experimental set-up. Thus, this calls for further research on the developpement of control strategies for continous powder mixing process in order to ensure good performance and stability.

CHAPTER II: Materials and experimental setting 1 Materials

The two granular materials used in the experiments are fine couscous and medium couscous, provided by RICCI and FERRERO. Fine couscous was colored in black with a 10% Iodine solution and then air-dried under a fume hood. It is noted that the fine couscous particles become slightly larger and tend to form agglomerates during color processing. A sieve with a mesh opening of 1250 µm was used for pre-screening both the colored fine-grain couscous and medium-grain couscous. For the colored fine couscous, the pre-screening permits the removal of agglomerates, and the powder with particles less than 1250 µm was used as component A. For the medium-grain couscous, the powder with particles larger than 1250 µm was utilized as component B for mixing experiments. The bulk behavior of powders of each component is characterized in terms of particle size distribution, density, surface state, porosity, flowability and floodability properties. These properties play an important role in powder handling and processing operations, such as feeding, transportation, mixing, compression, etc.

Particle size distribution

The particle size distribution of each component is assed by sieve analysis with a series of RESTCH sieve shakers (AS 200), for which the particles are separated on sieves of different sizes by setting vibration amplitude and sieving time.

54

From each cumulative curve, it is easy to determine particle size distribution parameters such as the d 10 , d 50 , and d 90, respectively, representing 10%, 50%, and 90% of the particles finer than these sizes. In other words, the values of d 10 , d 50 , and d 90 correspond to the cumulative distribution at 10%, 50% and 90%. In addition, the width of the particle size distribution can be evaluated by the "Span", which is calculated by:

Span = d 90 ! d 10 d 50 (II-1)
The values of the key particle size distribution parameters of component A and B are illustrated in Table II

Density and Porosity

• True (particle) density

The true density of a solid is the average mass per unit volume, exclusive of all voids that are not fundamental part of the particle packing arrangement. Therefore, the true density of a solid is an intrinsic property. In this study, the true density of each component is determined experimentally using a helium pycnometer, for which the true volume of a solid is determined by measuring the pressure difference when a known quantity of helium under pressure is allowed to flow from a precisely known reference volume into a sample cell containing the solid material.

• Aerated bulk density

Aerated bulk density is, in practical terms, the density at which the powder is in its most loosely packed form. The apparatus, for determining aerated bulk density, usually consists of a screen cover, a screen sieve, a spacer ring and a stationary chute attached to a vibrator. A Hosokawa Powder tester is used for measuring the aerated bulk density of each component studied. The powder sample is poured by a vibrating sieve and allowed to fall from a fixed height (about 25 cm) through the stationary chute into a test cup (100ml). This allows to calculate an aerated volume, and thus the aerated bulk density, given the powder mass.

• Tapped density

The tapped density of a powder is the ratio of the mass of the powder to the volume occupied by the powder after tapping. The tapped density of a powder represents its random dense packing. Tapped density is measured by first gently pouring a specified mass of powder into a graduated cylinder and then tapping mechanically the cylinder against a hard surface, from a standard height (typically 15mm), by a defined number of times. In this study, the tapped density of each granular component is measured after 1500 taps.

• Porosity

Porosity is also an important granular characteristic and it is defined as the fraction of the void volume over the total volume of the powder sample. Porosity can be derived directly from powder density as: of the two components are the same.

! aerated = " true ! " aerated " true or ! tapped = " true ! " tapped " true (II-2)

Flowability

Powder flowability refers to the ability of a powder to flow under given conditions and it is an important property in powder handling and processing operations. Therefore, evaluation of the flowability of any powder is a very important issue for many industries. To evaluate the flowability, three tests are carried out to measure:

• Hausner ratio (H) and Carr's index (CI)

Hausner ratio and Carr's index (compressibility) are two closely related, empirically derived methods for assessing flowability. The two indexes are determined based on the aerated bulk density and tapped density mentioned above. Hausner ratio (H) is defined as the ratio of tapped density to aerated bulk density, given by Eq. (II-3).

H = ! tapped ! aerated (II-3)
For a Hausner ratio of 1.0-1.1, the powder is classified as free-flowing; 1.0-1.25, mediumflowing; 1.25-1.4, difficult-flowing; and >1.4, very-difficult-flowing.

Carr's compressibility index (CI) is defined as follows:

CI(%) = ! tapped ! ! aerated ! aerated "100 (II-4)
The smaller the Carr's compressibility index, the more fluidity the powder behaves. Thus, a free-flowing powder should have a low compressibility index, because the inter-particle forces are not as significant as in a poorly-flowing powder, which implies the value of aerated bulk density is close to that of tapped density.

• Angle of repose ( R )

Angle of repose is defined as the angle of the free surface of a pile of powder to the horizontal plan. Depending on the conditions under which the pile has been poured and how the angle is measured, somewhat different values of the angle can be obtained for the same powder. The major factors influencing the angle of repose are size distribution, surface roughness of the particles, void fraction of the powder bed, and moisture content. An angle of repose of less than 38° is considered a free-flowing powder, whilst more than 55° is considered a cohesive powder.

• Angle of spatula ( S )

Angle of spatula provides an indication of the internal friction between particles. It is determined by inserting a flat blade into a pile of granular material and lifting vertically. The new angle of repose that the material forms relative to the horizontal blade surface is know as the angle of spatula ( S ). For a material to be considered free-flowing, its angle of spatula should be below 40°. The flowabiltiy evaluation of two granular materials is given by Table II-

Floodability

Floodability is related to the tendency of powder to flood due to interstitial gas overpressure.

It assigns an estimated value to the behavior of a bulk product when it is moved after an initial state of rest (feeding, transporting, emptying containers, etc.). Two additional tests are carried out to evaluate the floodability, by measuring:

• Angle of fall ( F )
The angle of fall is obtained as the new angle of repose when the powder cone is mechanically shocked.

• Angle of difference ( D )

The angle of difference is:

! D = ! R !! F (II-5)
This parameter is related to the internal cohesion of the granular particles. A powder with high angle of difference is more likely to have an unsteady floodable flow.

Table II-

Experimental set-up

The experimental set-up shown in Figure II-4 includes three parts: a feeding system, a continuous mixing equipment, and an on-line analysis system for powder mixture homogeneity. The feeding system is used to feed continuously the powder directly into the mixer inlet chute. This study is related to the mixing of two powder ingredients, therefore two powder feeders are used. The continuous mixer is a Gericke GCM 500, for which where the two powder ingredients fed at the inlet are continuously mixed and removed from the outlet.

At the outlet of the mixer, the mixed powder is discharged through a chute and fall on a moving belt. The homogeneity of the powder mixture on the belt will be directly measured on line.

Figure II-4: Experimental set-up.

Feeding system

In the feeding system, two loss-in-weight feeders are used for the experiments to ensure a high accuracy feed rate control. As shown in Figure II-5, each feeder mainly consists of a hopper, a screw feeder, and a weighing unit. To prevent powder caking and bridging during feeding, two types of agitators (vertical and horizontal) with separate drives are respectively installed in the hopper and feeder. The material to be fed is stored in the hopper, which has an upper and lower fill level. That implies that the hopper should be refilled when the minimum fill level is reached and the refilling should be stopped when the maximum fill level is reached. In this study, the hopper is pre-filled to its upper level before starting each experiment. The feeder under the hopper has a rotating spiral screw that moves material from the hopper to the outlet of the feeder through a discharge tube. The rotational screw is driven by a separate electric motor that is controlled for adjusting the rotation speed of the screw through a variable-speed drive (VSD) model VLT 5000. The weighing unit permits the measurement of weight reduction with time during discharge, which is also instantaneous mass flow rate (or feed rate). In the feeding system, a closed-loop control is applied for continuously modifying the screw's rotational speed to compensate for the difference between the setpoint and the current measured value of the mass flow rate. This allows to ensure that the actual flow rate is maintained as close as possible to its setpoint. The empirical ratio between N m (rpm) and N (Hz) is 2.6. In addition, the rotational speed of the motor is up to 60 Hz in order to avoid particulate attrition in the process.

On-line image analysis system

The on-line analysis system is for on-line monitoring the composition and the homogeneity of the powder discharged from the mixer. Actually, the implementation of the on-line analysis system has been presented by Demeyre [START_REF] Demeyre | Caractérisation de l'homogénéité de mélange de poudres et de l'agitation en mélangeur Triaxe[END_REF] and Ammarcha [START_REF] Ammarcha | Mélange des poudres en continu : modèles dynamiques et caractérisation en ligne[END_REF]. It mainly consists of a green conveyor belt of 30 cm width and a digital line scan camera, as depicted in This camera has a 1-D (line) CCD image sensor with a resolution of 5000 pixels. In order to exclude external light interference, the camera is set in an opaque wooden box, in which an illumination system is provided for increasing the image contrast and resolution. This helps improve the overall performance of the system. In this study, the camera is fixed in an appropriate height to ensure that the entire 30 cm width of the conveyor belt lays in the photosensitive area of 5000 pixels in width. This implies that each pixel represents 60 µm x 60 µm area under inspection. Because the camera scans one line at a time, the line rate of the camera should be synchronized with the moving speed of the conveyor belt in order to ensure that consecutive lines can be captured without overlap nor skip space. Every 200 consecutive lines 65 create an image (or frame), for which each line is composed of 5000 pixels in width. The images are continuously acquired at a rate close to 14 images per second.

Experimental operating system

The experimental set-up is controlled using a Labview software, which is a system design

Operating platform for the on-line image analysis system

The NI PCI-1422 board is implemented to acquire images from the digital line scan camera.

This digital image acquisition board is directly controlled by NI-IMAQ driver and it performs high-speed, large-image, high-resolution digital image capture, and can capture up to 16 bits of data at a frequency of 40 MHz for a total acquisition rate of 80 MB/s. The image processing and analysis is then accomplished using the NI Vision development module in Labview. All these procedures are key to achieve the ultimate objective of determining the concentration of the component interest and the homogeneity of powder mixtures in real time.

Image processing and analysis

As shown in (1) Each acquired image is first saved as an 8-bit (from 0 to 255) gray-scaled image and transferred to a memory buffer with a separate number in the buffer list. It can then be extracted from the buffer list for further image processing and analysis. The buffer list follows a first-in first-out principle, outputting data in the order it arrived.

68

(2) In order to differentiate the object (the material to be measured) from the others being background, each gray-scaled image is then converted to a binary image in which the object has a pixel intensity of 1 (white) and the background has a pixel intensity of 0 (black). The binarization is realized using an IMAQ thresholding function, for which each pixel value in the image is compared with two specified threshold intensities (or values), the lowest pixel value and the highest pixel value. Pixel values that lay within the range of two threshold values will be set to 1 (white), while others will be set to 0 (black).

surface fraction of A = %S A %S A + %S B (II-6) Or surface fraction of B = %S B %S A + %S B = 1! surface fraction of A (II-7) 69
Once the composition of powder mixture in surface is determined, the calibration is required to estimate the corresponding mass composition. Samples of powder mixture of different compositions, 0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, and 100% of

Computational approach for characterizing continuous mixing performance

As mentioned in Section 2. For each "window", the composition of the final mixture is estimated by calculating the mean composition of component A (X A ) and the mixing homogeneity is measured by calculating the relative standard deviation (RSD). The equations are as follows:

X A (k) = 1 n x i (k) i=1 n ! (II-8) ! (k) = 1 n x i (k) X A (k) [ ] 2 i=1 n " (II-9) RSD(k) = ! (k) X A (k) (II-10)
Where n is the number of images containing in one "window", x i is the mass fraction of component A monitored in each image, k is the order number of the "window" acquired at time instant k, and σ is the standard deviation. In this work, n will be fixed at 14 and this means that the mean composition X A and RSD are examined every second, because the image acquisition rate is about 14 images per second.

Conclusion

The first section introduces the two granular materials for mixing: component A (fine couscous colored in black) and component B (medium couscous). The materials have been characterized by physical properties, such as particle size distribution, density, flow, etc. The main difference between the two materials is found in particle size distribution. The second section gives the mechanical descriptions of the experimental equipment for the continuous powder mixing, including the feeders, the continuous mixer and the on-line image analysis system. In this work, the overall experimental set-up is operated in an automatic mode. The last section demonstrates how all instruments have been connected to a computer and controlled by the Labview software. The experimental operating system has been summarized in the two parts: the operating platform for the feeders and the mixer, and the operating platform for the on-line image analysis system. The former is related to the operating conditions of the feeders and the mixer. The latter is for the on-line measurement of the mixing performance in terms of the mean composition of component A (X A ) and the relative standard deviation (RSD).

CHAPTER III: New developments in the understanding of continuous powder mixing

The continuous mixer is the same as used and reported previously by Marikh [START_REF] Marikh | Mélange des poudres en continu dynamique et modélisation[END_REF] and Ammarcha [START_REF] Ammarcha | Mélange des poudres en continu : modèles dynamiques et caractérisation en ligne[END_REF]. In this work, a substantial number of experiments have been performed to At the start of each experiment, the mixer is empty. At the mixer's inlet, powder is continuously fed. At the mixer's outlet, a balance is used to weigh the exiting powder thus deducing the mass outflow rate. The process is stopped when the mass outflow rate nearly reaches a constant value, indicating a steady state.

For bulk handling, the mixer is emptied at one time to measure the hold-up weight. For powder mixing, the mixer's cells are successively emptied from cell 5 to cell 1 in sequence in order to measure the hold-up in each cell. For the powder retrieved from each cell, the powder weight as well as the mixture composition is measured. When considering a mixing of two components (A and B), the mixture composition can be determined by separating the two components through a sieve of 1250 µm.

2 Experimental design and results

Experiments with bulk powder

Bulk powder flow in the continuous mixer has been extensively studied under different rotational speeds [START_REF] Marikh | Influence of stirrer type on mixture homogeneity in continuous powder mixing: A model case and a pharmaceutical case[END_REF][START_REF] Ammarcha | Predicting bulk powder flow dynamics in a continuous mixer operating in transitory regimes[END_REF]. For a given bulk powder feed rate, an increase in rotational speed leads to a decrease in the hold-up weight as well as an acceleration to steady state. Thus, prior to the mixing experiments, the effect of bulk feed rate on the mixer's hold-up has been studied at a relatively high rotational speed (50 Hz) for each component. Experiments were done with bulk material of fine couscous (component A) or medium couscous (component B)

for four different feed rates (1.67, 2.78, 4.17 and 4.86 g.s -1 ).

Powder mixing experiments

Thirty experiments of continuous mixing of component A and B were performed under various operating conditions, such as mixing formulation, total feed rate and rotational speed, as given in Table III-1.

Table III-1: Mixing experiments proceeding conditions.

In brief, the continuous mixer was studied over the following operating ranges:

 three mixing formulations: 30% A -70% B, 50% A -50% B, and 70% A -30% B;

 five total feed rates: 5.56 g.s -1 , 6.94 g.s -1 , 8.33 g.s -1 , 9.72 g.s -1 and 11.11 g.s -1 ;

 two rotational speeds : 30 Hz (medium value) and 50 Hz (high value).

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In each mixing experiment, powder flow and mixing behaviour is investigated through holdup in the whole mixer and hold-ups in the five axial cells of the mixer.

Hold-up in the whole mixer

Hold-up weight in the whole mixer is examined for the whole mixture as well as for each component in the mixture.

• Hold-up of mixture For the same total feed rate, at either 30 or 50 Hz, hold-up is the highest for the mixing formulation of 70% A -30% B, indicating the highest average residence time. This corresponds to higher bulk residence time for component A than for component B under the same operating condition. However, a lower mass fraction of component A in feed stream does not necessarily imply a smaller hold-up. For example, at 30 Hz, smaller hold-ups are found for 50% A -50% B than for 30% A -70% B, while at 50 Hz, very close hold-ups are found.

• Hold-up weight has been empirically expressed as a function of rotational speed at different flow rates (or feed rates) [START_REF] Marikh | Influence of stirrer type on mixture homogeneity in continuous powder mixing: A model case and a pharmaceutical case[END_REF][START_REF] Ammarcha | Predicting bulk powder flow dynamics in a continuous mixer operating in transitory regimes[END_REF]. However, this function is more often focused on understanding powder bulk flow rather than powder mixing. In this study, based on experimental data of continuous mixing of the components A and B, the following empirical relationships are proposed to link the hold-up of each component with the studied process parameters, such as mixing formulation, total feed rate and rotational speed of the mixer.

(

1) Empirical expression of hold-up of component A

The hold-up weight of component A in the entire mixer, denoted by M (A) , has been expressed as follows:

M (A) = a 1 x A b 1 Q c 1 N d 1 (III-1)
where x A is the mass fraction of component A in the feed stream, Q is the total feed rate of the mixer (g. (

2) Empirical expression of hold-up of component B

The hold-up weight of component B in the whole mixer, denoted by M (B) , has been expressed as follows:

M B = a 2 1! x A b 2 Q c 2 N d 2 (III-2)
where In conclusion, Eq. (II-1) and Eq. (III-2) have been proposed to express hold-up weight of component A and B, respectively, as a function of mixing formulation, total feed rate and rotational speed. The parameters of the two equations have been experimentally estimated using least square fitting.

Hold-ups in the five cells of the mixer

In the following, the hold-up profiles of component A and B in the mixer will be discussed from the subdivision into five axial cells. The process is observed in discrete moments of time with the sampling time Δt. The evolutions of the state (or mass) vectors for component A and B are described by the following matrix formulas:

M (A) = M i (A) i=1 ✺ ! (III-3) M (B) = i (B)
S 1 (n +1) S 2 (n +1) S 3 (n +1) S 4 (n +1) S 5 (n +1) S 6 (n +1) ! " # # # # # $ = p 11 (n) 0 0 0 0 0 p 21 (n) p 22 (n) 0 0 0 0 0 p 32 (n) p 33 (n) 0 0 0 0 0 p 43 (n) p 44 (n) 0 0 0 0 0 p 45 (n) p 55 (n) 0 0 0 0 0 p 65 (n) p 66 (n) ! " # # # # # $ % & & & & & S 1 (n) + Q A 't S 2 (n) S 3 (n) S 4 (n) S 5 (n) S 6 (n) ! " # # # # # $ (III-7) K 1 (n +1) K 2 (n +1) K 3 (n +1) K 4 (n +1) K 5 (n +1) K 6 (n +1) ! " # # # # # $ = q 11 (n) 0 0 0 0 0 q 21 (n) q 22 (n) 0 0 0 0 0 q 32 (n) q 33 (n) 0 0 0 0 0 q 43 (n) q 44 (n) 0 0 0 0 0 q 45 (n) q 55 (n) 0 0 0 0 0 q 65 (n) q 66 (n) ! " # # # # # $ K 1 (n) + Q B 't K 2 (n) K 3 (n) K 4 (n) K 5 (n) K 6 (n) ! " # # # # # $ (III-8)
In Eq. (III-7), Q A is the mass flow rate of component A; S i (n +1) is the mass of component A in cell i after n transitions; p ii (n) and p (i+1)i (n) are respectively the probabilities of component A to stay in cell i and to move from cell i to cell i+1 during the n th transition;

p ii (n) = 1! p (i+1)i (n) and p 66 (n) = 1 because there is no backward flow for the last cell.

In Eq. (III-8), Q B is the mass flow rate of component B; K i (n +1) is the mass of component B in cell i after n transitions; q ii (n) and q (i+1)i (n) are respectively the probabilities of component B to stay in cell i and to move from cell i to cell i+1 during the n th transition;

q ii (n) = 1! q (i+1)i (n) and q 66 (n) = 1 .
Since the model is used to predict the continuous mixing process during two phases: (1) start stage and (2) steady state, the transition probabilities p (i+1)i (n) as well as q (i+1)i (n) should be identified for the two phases. During start stage, it is assumed that the transition probabilities for each component vary linearly with the corresponding powder mass in the mixer as evidenced by Ammarcha [START_REF] Ammarcha | Mélange des poudres en continu : modèles dynamiques et caractérisation en ligne[END_REF]. The transition probability p (i+1)i (n) or q (i+1)i (n) is zero when the mass of corresponding powder component is zero in cell i. While, at steady state, the transition probabilities are kept constant, denoted as p (i+1)i max and q (i+1)i max , because the mass of each component in each cell equals to its stationary hold-up weight.

For component A, the transition probability p (i+1)i (n) can be expressed as a function of S i (n)

representing the mass of component A in cell i = 1, 2, 3, 4 or 5 at every transition. As given in Eq. (III-9), M i (A) is the hold-up mass of component A in cell i and p (i+1)i max is the transition probability at steady state, which depends on cell i, Q A , !t and M i (A) , and can be calculated from Eq. (III-10).

(i+1)i ( ) = 0, for i ( ) = 0 (i+1)i max M i ( ) i ( ), for 0 < i ( ) < M i ( ) (i+1)i max , for i ( ) = M i ( ) ! " # # $ # # (III-9) (i+1)i max = Q ❆ !t M i ( ) + Q ❆ !t , for i = 1 Q ❆ !t M i ( ) , for i = 2, 3, 4, 5 " # $ $ % $ $
(III-10)

For component B, the transition probability q (i+1)i (n) can be expressed as a function of (

)i ( ) = 0, for K i ( ) = 0 (i+1)i max M i ( ) K i ( ), for 0 < K i ( ) < M i ( ) (i+1)i max , for K i ( ) = M i ( ) ! " # # $ # # (III-11) (i+1)i max = Q ❇ !t M i ( ) + Q ❇ !t , for i = 1 Q ❇ !t M i ( ) , for i = 2, 3, 4, 5 " # $ $ % $ $ (III-12) i+1
The above equations indicate that the determinations of transition probabilities are fundamentally linked to M i (A) and M i (B) . Thus, the prediction performance of the Markov chain model mainly depends on whether M i (A) and M i (B) can be estimated for different operating conditions. As stated in Section 2, the prediction of M i (A) can be performed using the empirical expression of M (A) and the empirical linear relationships between M i (A) and M (A) . The similar prediction can be made for M i (B) . Implementing these empirical expressions, the algorithm for the improved Markov chain model can be summarised as

x(n) = S 6 (n +1) ! S 6 (n) S 6 (n +1) ! S 6 (n) [ ] + K 6 (n +1) ! K 6 (n) [ ] (III-13)
The simulations for continuous mixing of the two powder components were performed in Matlab (see the program in Annex 1). The transition time !t has been set at 0.1 second. To compare simulation results with experimental data, the Markov chain model has been used to predict four process outputs:

(1) Mean concentration X of component A

X A (k) = 1 1 x i (k) i=1 1✵ ! (III-14)
X A at time instant k is the mean value of x for 10 consecutive samples.

(2) Relative standard deviation RSD

RSD(k) = 1 10 [x i (k) X A (k)] 2 i=1 10 " X A (k) (III-15)
The RSD is calculated to quantify the homogeneity over 10 consecutive samples.

(3) Whole powder mass in 6 th cell (or absorbing state) M 6 (g)

M 6 (n) = S 6 (n) + K 6 (n) (III-16)
(4) Outflow rate of the mixer Q out (g.s -1 )

Q out (n) = M 6 (n +1) ! M 6 (n) "t (III-17)
Figure III-14: General structure of the continuous mixer simulator.

To summarise, the process simulator contains three inputs and four outputs, as shown in The outputs are the mean concentration X A , the relative standard deviation RSD, the whole powder mass obtained at the outlet M 6 and the mass outflow rate of the mixer Q out .

Reliability study of the model

The reliability of the model has been examined by experimental tests of continuous mixing of component A and B under three different operating conditions as follows:

(1) Constants N, Q A and Q B

(2) Constants flow rate Q A , Q B with a step change in rotational speed N

(3) Constants N and Q B with a step change in flow rate Q A

There is no powder in the mixer at the start of each experiment. It is also worth noting that experimental measurements of X A , RSD, M 6 , and Q out were performed not tightly next to the mixer's outlet but at a distance of 84 cm. The time delay is about 5 s, because the moving speed of the conveyor belt at the outlet is 16.8 cm.s -1 . However, simulation results were obtained tightly next to the mixer's outlet. Measurements during the first 5 s were removed from experimental data in order to be consistent with those obtained from the simulations. show good agreement between measured and predicted M 6 and Q out . After about 80 s, the 91 mixing reached the steady state because the mass outflow rate of the mixer Q out nearly equals to the total feed rate of 9 g.s -1 .

An example for constant operating conditions

An example for step change in rotational speed of the mixer

The mixing was started at a fixed rotational speed (50 Hz). Both flow rates Q A and Q B were set at 4.17 g.s -1 . At the moment of 100 s, the rotational speed was changed to 30 Hz by a negative step. show that roughly same trends were found between experimental and predicted X A , except for the instantaneous decline of predicted X A at 85 s. Two possible analyses can be listed here trying to explain: (1) If the error is come from the model, it may be attributed to the model assumptions, the non-considered process parameters, and unmeasurable even unavoidable disturbances in the real plant. (2) If the error is come from the measurements, it implies that the instantaneous change in X A may not be detected by the monitoring technique used in the study, or the instantaneous change may be weaker or even disappeared due to powder transport from the mixer's outlet to the moving belt. At least, from about 95 s, the same increasing of X A has been obtained from both experimental and simulation results. From

An example for step change in flow rate

Simulation studies

Since the prediction of hold-up mass for each component, as an important part in the Markov chain model, has been presented as a function of mixing formulation, total feed rate and rotational speed, various simulations were performed to study the influence of the three operating parameters.

Simulation studies were classified into two categories:

(1) Influence of rotational speed and total feed rate at a constant mixing formulation;

(2) Influence of rotational speed and mixing formulation at a constant feed rate. 94

Influence of rotational speed and total feed rate at a constant mixing formulation

The mixing formulation of components A and B was fixed at 50% -50%. The simulations were performed under two different rotational speeds (30 an 50 Hz) for three total feed rates (7, 9, and 11 g.s -1 , respectively). The influence has been analysed from the predicted mean concentration of component A X A and the predicted outflow of the mixer Q out , respectively. The total feed rate was fixed at 11 g.s -1 . The simulations were performed under two different rotational speeds (30 an 50 Hz) for three mixing formulations (30% A -70% B, 50% A -50% B, and 70% A -30% B, respectively). The influence has been also analysed from the predicted mean concentration X A and the predicted outflow of the mixer Q out , respectively.

Influence of rotational speed and mixing formulation at a constant total feed rate

In

Conclusion

The first section has proposed a method of powder hold-up measurement, for which the continuous mixer is divided into five cells in the axial direction and powder hold-up mass is detailed for each cell. The second section focuses on the experimental results. Firstly, bulk experiments for both components A and B were performed to study bulk powder flow in the continuous mixer. The results show that under the same operating condition component A has a higher hold-up mass in the mixer than component B. Secondly, mixing experiments of the two components were carried out by varying a set of operating conditions, such as mixing formulation, total feed rate and rotational speed of the mixer. For each experiment, powder flow and mixing behaviour was examined from the entire mixer and the individual cells of the mixer. For the entire mixer, the whole hold-up masses of component A and B, denoted as M (A) and M (B) , were empirically expressed as a function of the operating parameters. For the individual cells of the mixer, the hold-up masses of component A and B, denoted as M i (A) and M i (B) , were described as a linear function of M (A) and M (B) , respectively. In the last section, continuous mixing of component A and B has been modelled using two Markov chains, one for each component. The model has been improved by using the empirical functions related to hold-ups of each component, such as M (A) , M (B) , M i (A) and M (B) . The improved model was validated by comparing simulation results with experimental data in terms of mean concentration of component A X A , the relative standard deviation RSD, the whole powder mass obtained at the outlet M 6 and the mass outflow rate of the mixer Q out . The results of experimental validation show that the model can be used to predict X A , M 6 and Q out , except for RSD, which quantifies the homogeneity of samples per second. In addition, several simulations were performed to study the effects of operating conditions, such as rotational speed, total feed rate and mixing formulation. The simulation results of X A and Q out show that: (1) for a constant composition mixing, increasing in either rotational speed or total feed rate allows achieving steady state more rapidly. However, the impact of rotational speed is greater. ( 2) for a constant total feed rate, the influence of rotational speed is also more important than that of mixing formulation. Mixing achieves steady state more rapidly at higher rotational speed. A decrease in mass fraction of component A in feed stream reduces the time required to reach steady state.

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CHAPTER IV: The design and implementation of PID controllers in a continuous mixing process

In powder mixing process that requires a high degree of accuracy in product composition, such as pharmaceutical and food industries, the continuous processing can be a troublesome proposition in practice because of a lack of knowledge in both process understanding and control. Therefore, deeper process understanding and real-time control systems are crucial to reduce or even avoid off-specification products due to manufacturing failure.

For the continuous powder mixing process presented in this work, the following issues will be discussed.

(1) Why the process needs to be controlled? What are the process variables that need to be controlled?

(2) What are the process variables important to achieve good control?

(3) Is there any control strategy available for this process?

1 Process analysis for continuous powder mixing from a control point of view In summary, the poor or unstable responses of the outputs X A and RSD show that powder mixing in the continuous mixer should be controlled in order to ensure satisfactory mixture quality. Accordingly, the next step refers to the identification of the manipulated variable(s).

In this work, the rotational speed N has been considered as the manipulated variable. The following three points are listed to confirm the choice.

(1) As part of the process inputs, the flow rates of the two components can be well controlled by the feeding system, as shown in Figure IV-2. In addition, the two flow rates directly affect the setpoint of the mean concentration.

( (3) As illustrated in chapter II, for given flow rates of the two components, a higher rotational speed gives rise to an increase of mean concentration more rapidly (until to the desired value) during mixer start. However, the rotational speed N could not always keep at the highest value, due to high energy use, attrition phenomena, fluidized flow regime, etc.

Closed-loop control design for the process

During the last few years, both PID and model-predictive control systems for continuous powder-based processing have been suggested and discussed. However, no attempt has been made to implement a control system in a real plant continuous mixer. In this work, the implementation of a proportional-integral-derivative (PID) controller in the pilot continuous mixer has been realised using the PID Control Toolkit in Labview software package. PID control has been proposed as a first step of experimental control strategy, due to its easily implementation in practice and wide-use in the process industries. In the following, a single loop PID feedback control scheme is designed for either the mean concentration X A or the relative standard deviation RSD. The actuator V ao, mixer setting is calculated on the basis of the deviation from the setpoint of the mean concentration of component A, using a PID algorithm included in Labview.

Mean concentration control

e X A (k) = X A, setpoint ! X A (k) (IV-1) V ao (k) = K c e X A (k) + 1 T i e X A (i) + e X A (i !1) 2 " # $ % & ' i=1 k ( )t ! T d )t (X A (k) ! X A (k !1) " # $ % & ' (IV-2)
Where ∆t is the sampling time that will be set to 1 s in the following, since the controlled variable mean concentration X A has been measured every second. e X A is the deviation from the setpoint of the mean concentration of component A. K c is the controller gain, T i is the integral time in minutes and T d is the derivative time in minutes. The three parameters are the basic parameters of a PID controller.

Relative standard deviation control

V ao (k) = K c e RSD (k) + 1 T i e RSD (i) + e RSD (i !1) 2 " # $ % & ' i=1 k ( )t ! T d )t (RSD(k) ! RSD(k !1) " # $ % & ' (IV-4)
Where ∆t is the sampling time that will also be set to 1 s in the following. RSD is the deviation from the setpoint of RSD. K c , T i and T d are three parameters of the PID controller.

Implementation of PID in the pilot plant continuous mixer

As presented in Chapter II, the process contains two operating platforms: one for the feeding system and the mixer, another for the on-line image analysis system. The two operating platforms have been integrated into one programme through two while loops in parallel. The measurement results of X A and RSD obtained from the loop of on-line image analysis have been introduced into the running loop of the mixer by creating local variables for X A and RSD.

The detailed information can be found in Annexe 2 including two block diagrams.

105

Focused on the control of the mixer, there are three alternative regulations for the rotational speed of the engine N.

(1) The rotational speed of the engine N keeps at the initial value, if there is no PID control for the mixer. This is referred to as an open-loop control.

(2) The rotational speed of the engine N is regulated by the closed-loop control of the mean concentration X A .

( As seen from the figure, the operating procedure can be summarized. First of all, the initial process operating conditions including Q A , Q B and N should be defined. Once the process nearly attains steady state, the Q A and Q B can be redefined and the rotational speed of the engine N can be manipulated by one of the three regulation choices. In addition, an appropriate limit should be addressed to the rotational speed when there is a closed-loop control.

Performance criteria

The impact of proposed control loops on the continuous mixer's efficiency has been evaluated using the mean concentration and the RSD as performance criteria.

Mean concentration criteria

Prior to mixing homogeneity, the mean composition of powder mixtures at the outlet should be examined to evaluate the continuous mixer performance. The control objective is to rapidly attain the desired mean composition, which also means the error between the desired and measured mean composition should be minimized in time. Therefore, the integral absolute error (IAE), the integral square error (ISE) and the integral of time-weighted absolute error (ITAE) are used to evaluate the mixer's efficiency in terms of mean concentration X A . They are calculated as follows:

ISE = e(k) [ ] 2 !t k=1 n " (IV-5) IAE = e(k) k=1 n ! "t (IV-6) ITAE = t k e(k) !t k=1 n " (IV-7)
Where Δt is the sampling time and e(k) is the error between the desired and actual measured values of mean composition X A , at time instant k.

107

The three indices quantify the errors with respect to time through different ways. The ISE is similar to the IAE, but with more weight given to the magnitude of large errors, while the ITAE puts a heavy penalty on errors that persist for long periods of time.

RSD criteria

In the continuous process, an effective control strategy can significantly increase the number of qualified products on a continuous production line. In terms of RSD, the number of samples of the RSD less than 6% will be quantified during control stage.

Tuning of PID parameters

In order to obtain satisfactory control performance, PID parameters (K c , T i and T d ) in control loop of either X A or RSD need to be tuned.

PID tuning for mean concentration X A control

The commonly used method for PID tuning such as Ziegler-Nichols is a method based on the step response. The step response reveals how the controlled variable reacts to a step change in the manipulated variable. In this context, the controlled variable refers to the mean concentration X A and the manipulated variable refers to the rotational speed of the engine N. The detailed information about X A control based on the model can be found in Annex 3.

As seen from

109

As an example, the performance of X A control using a PID controller setting of K c =35, T i =0.08 min and T d =0.008 min has been examined both experimentally and via Markov chain simulations. In this example, the initial operating conditions were Q A =Q B =4.5 g.s -1 and N=30

Hz. At 85 s, Q A was changed to 5.68 g.s -1 and Q B was changed to 3.32 g.s -1 , which in turn decreased the setpoint of X A from 50% to 63.11%. The closed-loop control of X A was then used for set-point tracking. However, the mass flow rates have not yet been studied from a modelling viewpoint. The actual fluctuations in the flow rates have been added to the model using the experimentally measured flow rates.

PID tuning for relative standard deviation RSD control

For RSD control, the tuning of PID parameters have been directly accomplished by control experiments of RSD, since the presented model cannot be used to predict the variability in the concentration over 14 consecutive samples experimentally. In particular, the RSD control is based on a closed-loop while the performance evaluation of RSD is directly related to a pharmaceutical limit less than 6% rather than the RSD setpoint, which should be set to a value smaller than 6%.

In summary, for the closed-loop of X A , different PID settings are firstly tested via simulation and the performances are then evaluated and compared. PID settings giving optimal performance are finally tested in the closed-loop experiements. Possible adjustment of PID settings may be made according to experimental results. While, for the closed-loop control of RSD, different PID settings are directly tested in experiments.

Conclusion

In this chapter, we have analysed the continuous powder mixing process from a control point of view. The process control demand has been illustrated by several experimental results. The mean concentration X A and the relative standard deviation RSD have been identified as the important controlled variables for the assessment of the continuous mixer's efficiency. The rotational speed of the mixer N has been identified as the important manipulated variable and this decision has also been justified. A simplified PID control has been proposed to control either the mean concentration or the relative standard deviation. The implementation of a PID controller in a pilot scale continuous mixer has finally been realised using Labview. The performance criteria in terms of the mean concentration X A and the relative standard deviation The mixing composition 50% A -50% B is selected as the starting point, because this mixture is the most commonly studied in previous work of Marikh [START_REF] Marikh | Mélange des poudres en continu dynamique et modélisation[END_REF] and Ammarcha [START_REF] Ammarcha | Mélange des poudres en continu : modèles dynamiques et caractérisation en ligne[END_REF] using the continuous mixer. In all control experiments, the maximum rotational speed of engine is set at 55 Hz, since higher rotational speeds lead to particle breakage/attrition on component B, particles of diameter larger than 1250 µm. This remark is outlined in the present work as well as in [START_REF] Ammarcha | Mélange des poudres en continu : modèles dynamiques et caractérisation en ligne[END_REF].

As shown in Table V-1, in case 1 and 2, the magnitudes of step change in set-points of mean composition in component A are +13.11% and -13.11%, respectively, whereas total flow rates at the mixer inlet are kept constant. In case 3 and 4, the setpoints of X A are kept constant, whereas total flow rates are changed due to the same magnitude increase or decrease of both components. Two main reasons that can be listed to explain these designs are: (1) a tolerance of ±7.5% is acceptable for mean content in pharmaceutical industry, and the magnitudes of ±13.11% facilitates enlarging control effects; (2) mixtures after introducing step changes in 115 flow rates are still in the range of 30%-70%, in which experiments were performed for modelling the process, as given Table III-1.

Case study 1

In all experimental studies, the flow rates of both component A and B at the mixer's inlet have been investigated, because well controlled flow rates are prerequisites for obtaining the desired mixing efficiency. The step changes in Q A and Q B result a step change in the mean composition X A from 50% to 63.11% and the total feed rate Q A + Q B keeps at 9 g.s -1 the same as for the starting stage. In addition, small random fluctuations around the desired values can be observed in both flow rates, although loss-in-weight feeders are used to feed the continuous mixer.

Closed-loop control of mean concentration X A

First, the simulated closed-loop of X A has been used to tune PID parameters. The theoretical step changes in Q A and Q B have been considered in the model. The rotational speed of the engine N has been limited in the range of 25-55 Hz.

( The mixing performance has been quantified as shown in Table V

Closed-loop control of RSD

Since RSD could not be accurately predicted from the presented model so far, the PID controller in the closed-loop of RSD has been directly tuned by experiments. The RSD setpoint is an important starting point in the closed-loop control of RSD. For powder mixing, values less than 6% need to be used as RSD setpoints, as RSD is required to be less than 6% in the pharmaceutical industry. In this case, the closed-loop performance of RSD has been studied for the RSD setpoints of 4% and 3%, respectively. As can be seen from Table V-6, similar mixer performance has been obtained from the openloop and the RSD setpoint of 3%. The RSD setpoint of 4% has provide a higher ratio of RSD < 6% but large deviations in the mean concentration X A . In summary, the above-mentioned control parameters (e.g. RSD setpoint and PID parameters) need to be modified in order to be effective.

As previously observed from RSD setpoint of 3%, the setting of controller parameters caused poorly control of the mixer performance in terms of RSD as well as X A , since the rotational speeds approximately the minimum limit generated. Therefore, a negative value of the controller gain (K c ) has been proposed to reverse the regulation direction of the rotational speed of the engine. Then, a PID setting of K c = -28, T i = 0.05 min, T d = 0.008 min has been tested in the closed-loop of RSD with the setpoint of 3%. settings also show that when there is a step change in X A from 50% to 63.11% due to the step changes in Q A and Q B , the overall process can be adequately controlled by the rotational speeds rapidly increased at the beginning and then varied between certain bounds.

Case study 2

In case 2, the desired mean composition X A is changed from 50% to The performance assessment results are summarized in Table V As shown in Figure V-10, for the retuned 2, due to the generated higher rotational speeds, the response of the mean composition at the mixer's outlet is delayed more severely whereas the mixing homogeneity is improved, which can be seen from the ratios of RSD < 6% listed in

Conclusion

None of the settings used for closed-loop control of X A or RSD allow controlling the mixing performance in terms of X A and RSD at the same time. The outputs of the manipulated variable N from all settings show that when the setpoint of X A is changed from 50% to 32.89% due to the step changes in Q A and Q B , the rotational speeds consistently adjusted above 30 Hz lead to negative effects on X A while the rotational speeds consistently adjusted below 30 Hz lead to negative effects on RSD.

Case study 3

In The RSD setpoints for the three settings are 5%, 4% and 3%, respectively. The rotational speeds of the engine for setting 1 and 2 have the same constraints: N ∈ [START_REF] André | Dimensional analysis of a planetary mixer for homogenizing of free flowing powders: Mixing time and power consumption[END_REF][START_REF] Sen | Multi-dimensional population balance modeling and experimental validation of continuous powder mixing processes[END_REF], while the upper N constraint for setting 3 is set at 48 Hz.

The outputs of the process X A and RSD and the rotational speed N are plotted versus time in

Conclusion

When there is a same positive step change in Q A and Q B , the continuous mixing process is slightly controlled by the last three settings under closed-loop control of RSD. And the corresponding outputs of the actuators show that the mixing homogeneity RSD can be better improved by the rotational speeds consistently adjusted above 30 Hz while the mean concentration setpoint can be potentially tracked by the rotational speeds consistently adjusted below 30 Hz.

Case study 4

In case 4, the flow rates Q A and Q B are simultaneously decreased from 4.5 to 3.32 g.s -1 . Figure

Closed-loop control of mean concentration X A

The implementation of closed-loop control of X A is discussed for the control settings as follows:

Setting As shown in Table V

Closed-loop control of RSD

The closed-loop control of RSD is discussed for the two settings as follows:

As given in

Conclusion

When there is a same negative step change for In summary, the influence of the starting rotational speed on the mixing performance depends on the implemented control scheme and the effect will not be very significant if the control parameters are well tuned under the corresponding control loop.

Conclusion

In this chapter, the proposed control loop has been studied in four different cases, which correspond to different step changes in flow rates of the two components at the mixer's inlet.

From an industrial point of view, case 1 and 2 correspond to a change in the mixture composition without changing the total production volume, while case 3 and 4 correspond to an increase/decrease in production volume. Continuous mixing has been controlled in terms of either mean concentration X A or relative standard deviation RSD. The performance of the closed-loop control of either X A or RSD has been evaluated using both X A and RSD performance criteria, such as ISE, IAE, ITAE and ratio of RSD < 6%. In each case, control efficiency has been determined by comparing the performance of closed-loop control of either X A and RSD with that of open-loop operation, for which the rotational speed was kept constant at the initial value 30 Hz. In case 1, a PID controller for closed-loop control of either X A or RSD has been found to be well tuned to improve the mixer's performance in terms of both mean concentration and mixing homogeneity. The outputs of the manipulated variable show that a rapid increase in the rotational speed at the beginning of control contributes to achieve a new steady state rapidly when there is an increase in mass fraction of component A in the mixture. In other case, different control settings (PID controller parameters, the RSD setpoint, and the constraints of the manipulated variable N) have also been experimentally tested and discussed. The results show that since either X A or RSD has been considered the controlled variable in closed-loop control, most of the tested PID settings allow improving mixing performance in terms of either the X A or the RSD and the improvement of one output variable (X A or RSD) often accompanies with a negative impact on another output variable.

Hence, the coupling of the two control loops should be considered in further studies.

GENERAL CONCLUSION

This For what concerns control process, the motivation to develop a control system for the presented continuous mixer has been first illustrated by several experimental examples. The results show that an effective control system is highly needed due to the instability problems and strict regulatory constraints in terms of mixture quality. The control can be performed by adjustment of the important input variables looking at the powder mixtures flowing out from the continuous mixer, due to an on-line image analysis technique available for the measurement of mixture quality. Since the two flow rates as process inputs can be experimentally well controlled by the loss-in-weight mechanism applied in the two feeders and, on the other hand, in the literature the rotational speed of the mixer is commonly considered as a very important process variable, the rotational speed of the mixer has been considered as the manipulated variable in the process control system. This decision has also been justified by several step change experiments. As an initial step towards the experimental control of a pilot continuous mixer, a simple proportional-integral-derivative (PID) control has been proposed and designed to control either the mean concentration or the relative standard deviation. The implementation of a PID controller in a pilot scale continuous mixer has been realised using Labview, by which the feeding system and the measuring system for mixture quality have been incorporated in a single platform to enable data communication with the PID controller for the mixer. Two performance criteria in terms of both mean concentration and relative standard deviation have been proposed to evaluate the performance of closed-loop control of either the mean concentration or the relative standard deviation. For instance, in term of the mean concentration, the integral square error (ISE), the integral absolute error (IAE), and the integral of time-weighted absolute error (ITAE) have been used

to evaluate the mixer's performance. In term of the RSD, the number of samples of the RSD less than 6% has been quantified during the control.

The control performance is strongly influenced by the tuning of PID parameters. In this work, the commonly used tuning based on step response method couldn't be applied due to the special process responses to step changes in the manipulated variable. control with the open-loop operation for which there is no adjustment on the rotational speed of the engine. In case 1, for either the mean concentration control or the relative standard deviation control, a proper PID controller setting has been found to be efficient for controlling the mean concentration as well as the RSD. While in other cases, PID controller settings have been found to be efficient for only the controlled variable in the closed-loop control. For instance, in closed-loop control of mean concentration, mean concentration will be controlled but not RSD, while in closed-loop control of RSD, RSD will be controlled but not mean concentration. This indicates that further control study should be carried out to couple the two closed-loop controls.

In conclusion, several remarks can be outlined.

(1) For the proposed control, the PID tuning is a critical step to obtain a good control performance. To date, we have not yet found a systematic approach for the tuning of PID (2) The control performance for this process has been influenced not only by the PID parameters but also by many other process parameters, such as the lower and upper limits of the manipulated variable, the sampling time, the applied control type (for mean concentration or RSD), the load disturbance type, the type of changes in flow rates (step change, impulsion…), etc. The four cases studied in this work correspond to several different step changes in flow rates. Clearly, more work is needed to study the influences of these process parameters on the control loop.

(3) In particular, for the closed-loop control of the relative standard deviation, the RSD setpoint has a direct influence on the process control. As in the pharmaceutical industry, the RSD is required to be less than 6%, the sample will be considered as satisfactory if its RSD is less than 6%, while from a control point view, the RSD will be controlled looking at the error between the RSD setpoint and the measured RSD. Therefore, there is still a doubt that if it is reasonable to control RSD by a classical closed-loop control.

if S(j)==0 p(j)=0; elseif S(j)<Smax(j) p(j)=P_A_max(j+1,j)/Smax(j)*S(j); else p(j)=P_A_max(j+1,j); end if K(j)==0 q(j)=0; elseif K(j)<Kmax(j) q(j)=P_B_max(j+1,j)/Kmax(j)*K(j); else q(j)=P_B_max(j+1,j);

1) 0 0 0 0 0 q(1) 1-q(2) 0 0 0 0 0 q(2) 1-q(3) 0 0 0 0 0 q(3) 1-q(4) 0 0 0 0 0 q(4) 1-q(5) 0 0 0 0 0 q(5) 1];

(i)=M_6_A(i)/(M_6_A(i)+M_6_B(i)); else c_A(i)=(M_6_A(i)-M_6_A(i-1))/((M_6_A(i)-M_6_A(i-1))+(M_6_B(i)-M_6_B(i-1))); end

LISTS OF TABLES

LISTS OF FIGURES

Figure 1 :

 1 Figure 1: General framework for a well-controlled continuous powder mixing process.

Figure I- 1 :

 1 Figure I-1: Types of mixture adapted from [3,4]: (a) Perfectly mixed; (b) Randomly mixed; (c) Completely segregated.

Figure I- 2 .Figure I- 2 :

 22 Figure I-2: Illustration of the mechanisms of segregation, adapted from [4-6]: (a) Trajectory segregation; (b) Percolation segregation; (c) Vibration segregation; (d) Elutriation segregation.

( c )

 c Segregation by vibration. If a mixture of particles of different size is vibrated, the coarser particles moves upwards and collect near the open surface area.

  Tumbler mixers Tumbler mixers are the simplest and the most common type of batch mixers. A tumbler mixer comprises a closed vessel rotating around an axis. It is generally attached to a drive shaft and supported on one or two bearings. Mixing is achieved dominantly by diffusion. Three of the most common geometries used are the V-mixer, the double cone mixer and the bin blender, sketched in Figure I-3. One of the main disadvantages of this type of mixer is the demixing possibility due to the funnel flow pattern formation that occur upon discharging.

Figure I- 3 :

 3 Figure I-3: Tumbler mixers, adapted from [4]: (a) V-mixer; (b) Double cone mixer; (c) Bin mixer.

Figure I- 4 :

 4 Figure I-4: Convective mixers, adapted from [4,7]: (a) Ribbon mixer; (b) Paddle mixer; (c) Nautamix; (d) Vertical screw mixer.

Figure I- 5 :

 5 Figure I-5: Schematic representation of scale and intensity of segregation: (a) reduction in scale of segregation; (b) reduction in intensity of segregation.

Figure I- 6

 6 illustrates two different mixture structures may be observed for the same state of macromixing.

Figure I- 6 :

 6 Figure I-6: Two different mixture structures observed in two mixtures of same macromixing state.

  Figure I-7 presents an experimental set-up described by Berthiaux et al [23] for powder mixtures flowing out of a continuous mixing or at the discharge of a batch process.

Figure I- 7 :

 7 Figure I-7: Experimental set-up for monitoring of powder mixture quality using image analysis from Berthiaux et al. [23].

Figure I- 9 .

 9 Figure I-9.

Figure I- 10

 10 Figure I-10 presented the experimental set-up. The fiber optical probe of the spectrometer was placed close to the rotor and at the powder top surface. The spectral data were finally transferred to a computer for data processing and interpretation of the powder composition with validated models. It was proved that the filling level and the filling protocol have strong impact on mixing performance, as powder flow and the blending dynamics depends on both.

Figure I- 9 :

 9 Figure I-9: (a) Raw spectra of pure LM and ASA (b) NIR spectra after data preprocessing for the calibration sets containing 11 samples ranging from 0-100% API content from Koller et al. [36].

Figure I- 10 :

 10 Figure I-10: In-line quantitative monitoring of powder mixing dynamics in a four-blade mixer via NIR from Koller et al. [36].

Figure I- 11 :

 11 Figure I-11: Homogeneity is reached after about 300 s: (a) CI plot for NIR data; (b) CI plot for Raman data from De Beer et al. [37].

Figure I- 12 :

 12 Figure I-12: NIR with a fiber optical probe from Martinez et al. [39].

Figure I- 13 :

 13 Figure I-13: Predicted NIR values (♦) with RSD (continuous line) for the four trials at different feeding rates and stirring rates from Martinez et al. [39].

Figure I- 14 :

 14 Figure I-14: Schematic of the multipoint NIR measurement system consisting of a fiber-optic lights source, 5 fiber-optic probes and a fiber-optic spectral camera by Aditya et al.[START_REF] Vanarase | Development of a methodology to estimate error in the on-line measurements of blend uniformity in a continuous powder mixing process[END_REF].

Figure I- 15 :

 15 Figure I-15: Blend uniformity (RSD) as a function of sample size measured by on-line (NIR) and off-line (UV) methods from Aditya et al. [42].

Figure I- 16 :

 16 FigureI-16 (b). The term macropixel refers to a square cluster of pixels whose intensity value equals to the average value of all included pixels. It was found that the RSD was dependent on

Figure I- 17

 17 (a). On the other hand, it was observed that the holdup weight obtained at steady state depends experimentally on the feed rate and the rotation speed, as shown in Figure I-17 (b).

Figure I- 17 :

 17 Figure I-17: Hold-up weight of bulk powder in the mixer: (a): minimum hold-up weight with inflow rates at 10, 20, 30, 40, 50, 60, 80 and 100 kg.h-1 (from bottom to top); (b): hold-up weight at steady state from Ammarcha [54].

Figure I- 18 :

 18 Figure I-18: Continuous mixers in two different geometries: (a): 1 st continuous mixer; (b) 2 end continuous mixer from Portillo et al. [61].

Figure I- 19 :

 19 Figure I-19: (a) Feed rate analysis of API; (b) Statistics of feed rate fluctuation of API at the investigated feed rates from Gao et al. [63].

  Figure I-20: Effect of rotation rate on RTD with flow rate-30 kg.h -1 and blade configuration -All forward from Vanarase and Muzzio [53]: (a) residence time distribution; (b) mean residence time; (c) relative standard deviation.

   showed that the mean residence time <t> decreased as the rotational speed increased. This indicated that higher rotational speed decreases the time available for mixing (lower residence time).

Figure I- 21 :

 21 Figure I-21: Effect of blade configuration on (a) mean residence time; (b) hold-up at steady stage from Vanarase and Muzzio [53].

Figure I- 22 :

 22 Figure I-22: Two stirrers design studied by Marikh et al. [52]: (a) stirrer A; (b) stirrer B.

Figure I- 23 :

 23 Figure I-23: Hold-up measured as a function of mass flow rate and mixture types for: (a) stirrer A; (b) stirrer B under dense phase flow regime; (c) stirrer B under fluidized flow regime from Marikh et al. [52].

  bulk flow and mixing of powders. Berthiaux et al. [72] have first introduced a onedimensional (or 1D) Markov chain model to describe continuous convective mixing of two components A and B. The basic of one-dimensional Markov chain is show in Figure I-24.

Figure I- 24 :

 24 Figure I-24: One-dimensional Markov chain proposed by Berthiaux et al. [72] for modeling the particle flow inside a continuous powder mixer: (a) graphical presentation; (b) state vector S ; (c) P matrix of transition probabilities.

Further,

  Figure I-25) for taking into account more features of real powder flow, such as crosswise mixing in the sections of a mixer, which is supposed to be perfect for one-dimensional models.

Figure I- 25 :

 25 Figure I-25: Two-dimensional Markov chain with two parallel chains proposed by Marikh et al. [73] for continuous powder mixing.

Figure I- 26 :

 26 Figure I-26: A simple Markov chain model for describing the macroscopic behavior of bulk powder flow in a continuous mixer from Ammarcha et al. [54].

  Figure I-28 described a continuous mixing of two components based on the convection and dispersion mechanisms.

Figure I- 28 :

 28 Figure I-28: Schematic of the axial mixing of two components in a continuous mixer.

Figure I- 29 :

 29 Figure I-29: Principe of DEM analysis from Bharadwaj [81].

Figure I- 30 :

 30 Figure I-30: Some DEM examples for batch mixers (a): V-blender simulated by Lemieux [86]; (b): Double cone mixer by Manickam [87]; (c): a paddler mixer by Hassanpour [87]; (d): Turbular mixer by Marigo et al. [89,90].

Figure I- 31 :

 31 Figure I-31: Schematic of the full mixer design and corresponding periodic slice models from Sarkar and Wassgren [94].

Figure I- 32 :

 32 Figure I-32: Block diagram of a PID controller in a feedback loop.

Figure I- 32

 32 Figure I-32 illustrates a block diagram of a single loop PID controller. It attempts to minimize the error (difference between the process output and the set-point) by adjusting the input of the process. The PID controller involves three actions: proportional action (denoted by P), integral action (denoted by I) and derivative action (denoted by D). The proportional action depends on the present error and the constant of proportionality is referred to the proportional gain, K p . The integral action is for handling error based on learning from the past and the integral gain is K i . The derivative action is a prediction of future errors and the derivative gain is K d . The determination of K p , K i and K d is well known as PID tuning. But PID tuning is usually a difficult problem, even though there are several different methods of PID tuning

  Figure I-33. A process model is used to predict the current values of the process output variables. The residuals, the differences between the actual and predicted outputs, serve as the feedback signal to the prediction block. The predictions are used in two types of MPC calculations: set-point calculations and control calculations. The set points for the control calculations, also called target, are calculated from an economic optimization based on a steady-state model of the process. The set points are typically calculated each time the control calculations are performed.

Figure I- 33 :

 33 Figure I-33: Block diagram for model predictive control (MBP) [102].

Figure I- 34 :

 34 Figure I-34: Basic concept for model predictive control (MPC) [102].

Figure I- 35 :

 35 Figure I-35: Schematic of a continuous dry mixer from Suresh et al. [112].

  usage and solution and effective as the multifunctional platform for mathematical modelbased application in pharmaceutical process and product development. The MPC controller simulation for the dynamic mixing operation described above is illustrated in Figure I-36.

Figure I- 36 :

 36 Figure I-36: Dynamic blender MPC controller instance from Suresh et al. [112].

Figure I- 38 :

 38 Figure I-38: Cascade control system proposed by Ramachandran et al. [114].

Figure II- 1 :

 1 Figure II-1: Granular material samples: (a) component A (colored fine couscous); (b) component B (medium couscous).

Figure II- 2 :

 2 Figure II-2: Particle size distribution: (a) colored fine couscous (Component A); (b) medium couscous (Component B).

1. 2

 2 Figure II-3: Particle images captured by an optical microscope: (a) particles of component A; (b) particles of component B.

  4, where the angle of repose and angle of spatula are measured by a Hosokawa Powder tester. All characteristic values of flowability indicate that the two components used in experiments can be classified as free-flowing materials.

5 :

 5 Granular material floodability evalaution. floodabilty for the two granular materials are shown in Table II-5. It is found that component A and B have similar floodability, since relatively close values of F or D are obtained from the two components.

  Figure II-5: Feeding system.

2. 2

 2 Mixing systemThe continuous mixing system used in this work is shown FigureII-6. It mainly comprises a horizontal convective mixer and a mixing tool in the mixer.

Figure II- 6 :

 6 Figure II-6: Gericke GCM 500 continuous mixer.

Figure II- 7 :

 7 Figure II-7: Mixing tool.

  Figure II-8: On-line image analysis system at the mixer's outlet.

Figure II- 9 :

 9 Figure II-9: Digital line scan camera Lord DVL 5000 T.

  platform and development environment for a visual programming language from National Instruments (NI). The overall experimental operating system is illustrated in Figure II-10.

Figure II- 10 : 3 . 1

 1031 Figure II-10: Experimental operating system layout.

Figure II- 11 :

 11 Figure II-11: Principles for driving the feeders and the mixer.

Figure II- 12 :

 12 Figure Correlation curve between the analog output voltage (V) and the mixer's motor frequency (Hz).

Figure II- 8 ,

 8 we observe three different colours with naked eyes: the green of the conveyor belt, the black of component A and the light yellow of component B. The procedure for determining the percentages of surfaces occupied by component A and B, respectively, can be summarized in the following steps:

Figure II- 13 :

 13 Figure II-13: Image processing and analysis for determination of the percentages of surfaces occupied by component A and B, respectively.

  Figure II-14, a third order polynomial function has been fitted to the data points of powder mixtures for which the mass fraction of component A varies from 10% to 80%, while the inclusion of the data points of 0%, 90%, and 100% samples will cause a larger estimation error.

Figure II- 14 :

 14 Figure II-14: Calibration curve for component A: mass fraction (%) versus surface fraction (%).

3 ,

 3 each image of powder mixtures represents a 30cm x 1.2cm inspection area and the image acquisition rate is 14 images/second. In this work, the size of one image is considered as the scale of scrutiny for characterizing continuous mixing performance. In addition, the weight of the powders captured in each image depends on the mass flow rate at the mixer's outlet, which also depends on the mass inflow rate and the rotational speed of the mixer. As schematized in Figure II-15, every n consecutive images (or samples) are inspected as one "window". The continuous mixing performance is evaluated by quantification of powder mixtures in successive "windows" in terms of the composition as well as the homogeneity.

Figure II- 15 :

 15 Figure II-15: Schematic representation of the scale of scrutiny and the "window" for characterizing the continuous mixing performance.

  deeper understand the continuous powder mixing in this equipment. The experiments are focused on studying powder flow and mixing in the mixer. An improved Markov chain model is finally obtained to predict process behaviour over a much wider range of operating conditions. In this chapter, the first section presents the method for measuring hold-ups in the mixer figured by five cells in the axial direction. Experimental results are presented and discussed in the next section. The new model development, the experimental validation and simulation studies are demonstrated in the last section.1 Hold-up measurementPowder hold-up weight in the continuous mixer is important because it determines the average residence time (hold-up/feed rate) of the powder, in turn affecting the quality of the mixtures[START_REF] Vanarase | Effect of operating conditions and design parameters in a continuous powder mixer[END_REF][START_REF] Marikh | Flow Analysis and Markov Chain Modelling to Quantify the Agitation Effect in a Continuous Powder Mixer[END_REF]. In this study, the hold-up measurement is extended to investigate the axial mixing or dispersion of powders in the mixer. The mixer has been virtually divided into five axial cells of same length 0.1 m. As shown in Figure III-1, the stirring only lays in the first 4 cells. Cells 2, 3 and 4 each have one pair of blades allocated at two sides, and cells 1 and 5 have two pairs of blades.

Figure III- 1 :

 1 Figure III-1: Virtual division of the mixer into 5 compartments.

Figure III- 2 : 74 For

 274 Figure III-2: Comparison of effects of bulk feed rate of component A and that of component B on (a) hold-up weight; (b) bulk residence time.

Figure III- 3 :

 3 Figure III-3: Hold-up weight of mixture measured under different operating conditions.

Figure III- 4 :

 4 Figure III-4: Composition analysis of hold-up in the whole mixer: (a) 30% A -70% B; (b) 50% A -50% B; (c) 70% A -30% B.

5 and d 1 =

 51 s -1 ), and N is the rotational speed (Hz). The unknown parameters a 1 , b 1 , c 1 , and d 1 have been determined by least square fitting. The minimization of the square error between the predicted and experimental data of M (A) was achieved with a 1 = 1124, b 1 = 1, c 1 = 0.-0.75.

Figure III- 5 :

 5 Figure III-5: Quality of prediction of Eq. (III-1) for the hold-up of component A in the whole mixer.

  (1x A ) is the mass fraction of component B in the feed stream, Q is the total feed rate of the mixer (g.s -1 ), and N is the rotational speed (Hz). The unknown parameters a 2 , b 2 , c 2 and d 2 have also been determined by least square fitting. The minimization of the square error between the predicted and experimental data of M (B) was achieved with a 2 = 294, b 2 = 1, c 2 = 0.5, and d 2 = -0.5.

Figure III- 6 :

 6 Figure III-6: Quality of prediction of Eq. (III-2) for the hold-up of component B in the whole mixer.

Figure III- 6

 6 Figure III-6 shows a good linear relationship (R 2 = 0.938) between the predicted and experimental M (B) .

Figure III- 7 :

 7 Figure III-7: Mass distribution of powders in the five virtual cells of the mixer.

  Figure III-8: Relative hold-up weight distribution of component A in the continuous mixer divided into 5 cells: (1) mixing experiments of 30% A -70% B; (2) mixing experiments of 50% A -50% B; (3) mixing experiments of 70% A -30% B.

Figure III- 9 :

 9 Figure III-9: Evidence of an empirical linear relationship between M 5 (A) and M (A) .

M 5 (

 5 Figure III-10: Relative hold-up weight distribution of component B in the continuous mixer divided into 5 cells: (1) mixing experiments of 30% A -70% B; (2) mixing experiments of 50% A -50% B; (3) mixing experiments of 70% A -30% B.

Figure III- 11 : 3 Improved Markov chain modelling and experimental validation of continuous powder mixing 3 . 1

 11331 Figure III-11: Empirical linear relationship between M 5 (B) and M (B) .

Figure III- 12 :

 12 Figure III-12: A two Markov chain scheme for describing flow dynamics of component A and B in the continuous mixer.

  K i (n) representing the mass of component B in cell i = 1, 2, 3, 4 or 5 at every transition. As given in Eq. (III-11), M i (B) is the hold-up mass of component B in cell i and q (i+1)i max is the transition probability at steady state, which depends on cell i, Q B , !t and M i (B) , and can be calculated from Eq. (III-12).

Figure III- 13 .

 13 Figure III-13.

Figure III- 13 :

 13 Figure III-13: Algorithm of iterative calculation for the two Markov chains.

Figure III- 14 .

 14 Figure III-14. The inputs are the rotational speed of the mixer N, the flow rates of Q A and Q B .

Figure III- 15 :

 15 Figure III-15: Comparison of simulated results with experimental data for constant experimental conditions Q A =Q B =4.5 g.s -1 and N=30 Hz: plots of (a) X A , (b) RSD, (c) M 6 and (d) Q out .

Figure III- 16 :

 16 Figure III-16: Validation result for a step change in rotational speed: plots of predicted vs. experimentally measured (a) X A , (b) RSD, (c) M 6 and (d) Q out .

Figure III- 17 :

 17 Figure III-17: Validation result for a step change in flow rate: plots of predicted vs. experimentally measured (a) X A , (b) RSD, (c) M 6 and (d) Q out .

Figure III- 17

 17 Figure III-17 (b), relatively small values of predicted RSD show that the model cannot be used to estimate the variability in the concentration between samples taken per second. In Figure III-17 (c) and (d), little difference was found between experimental and simulation results in terms of M 6 and Q out . The profiles of Q out show that the mixing has been achieved at steady state at 85 s and the Q out increased and finally became stable due to the increase in Q A .

Figure III- 18 :

 18 Figure III-18: Influence of rotational speed and total feed rate: (a) plots of X A ; (b) plots of Q out .

Figure III- 19 :

 19 Figure III-19: Influence of rotational speed and mixing formulation: (a) plots of X A ; (b) plots of Q out .

  Figure III-19, both X A and Q out are plotted as a function of time. The three colours correspond to the three mixing formulations. The dash-dotted lines and solid lines correspond to 30 and 50 Hz, respectively. As shown Figure III-19 (a), for either mixing formulation, the mean concentration of component A rises faster at high rotational speed (50 Hz). As seen from Figure III-19 (b), for either mixing formulation, Q out rises more rapidly at high rotational speed (50 Hz), indicating that X A reaches the expected concentration faster at 50 Hz. The time required to reach steady state is more influenced by rotational speed, since steady state was achieved at around 40 s for N=50 Hz while around 60 s for N=30 Hz. Under the same rotational speed and total feed rate, the mixing formulation has little influence on the time required to reach steady state.

Figure IV- 1 :

 1 Figure IV-1: The continuous powder mixing process described by a "black box" model.

Figure IV- 2 :

 2 Figure IV-2: Experimental observations of the flow rates of the two components.

Figure IV- 3 :

 3 Figure IV-3: Experimental results of the mean concentration X A .

Figure IV- 4 :

 4 Figure IV-4: Enlargement in region around 50% of the measured mean concentration X A .

Figure IV- 5 :

 5 Figure IV-5: An example of experimental results of the relative standard deviation RSD.

Figure IV- 6 :

 6 Figure IV-6: An example of measurements of (a) X A and (b) RSD.

Figure IV- 7 :

 7 Figure IV-7: Closed-loop control of the mean concentration at the mixer's outlet.

Figure IV- 8 :

 8 Figure IV-8: Closed-loop control of RSD at the mixer's outlet.

Figure IV- 9 : 106 Figure

 9106 Figure IV-9: Graphic user interface of Labview: (a) control panel; (b) display panel for measurement results.

Figure IV- 10 :( 1 )( 2 )

 1012 Figure IV-10: Simulated step response test: (a) positive step change in rotational speed N from 30 to 45 Hz (b) responses to the step change for the mixing 50% A -50% B with Q A =Q B =4.5 g.s -1 and the mixing 36.89% A -63.11% B with Q A =3.32 g.s -1 and Q B =5.68 g.s -1 .

Figure IV- 11 :

 11 Figure IV-11: Comparison of model results with experimental results for X A control: (a) mean concentration X A ; (b) rotational speed of the engine N.

Figure IV- 12 :

 12 Figure IV-12: Theoretical values and experimental data of the flow rates Q A and Q B .

Figure IV- 13 :

 13 Figure IV-13: Performance comparison of simulated and experimental results by calculating ISE, IAE and ITAE.

RSD

  have been defined to evaluate closed-loop control performance. The tuning of PID parameters, as an important step towards good control, has been discussed. For the closedloop control of mean concentration X A , a PID tuning based on the process model has been proposed as the preliminary guidance for further PID tuning through control experiments. While for the closed-loop control of relative standard deviation RSD, a number of RSD control experiments have been required for PID tuning, since the presented process model has not yet been enough accurate for the RSD prediction. when the corresponding PID controller starts working. In addition, the duration of 100 s is chosen because the time is long enough to ensure the achievement of a new steady state under open-loop operation. Moreover, the closed-loop performance of either X A or RSD is compared against open-loop operation, which refers to the mixer running in "manual mode". In openloop operation, the rotational speed of engine is kept at 30 Hz.

Figure V- 1 :

 1 Figure V-1: The design of control procedure.

  Figure V-2 gives an example of experimentally obtained mass flow rates in response to step changes in Q A and Q B .

Figure V- 2 :

 2 Figure V-2: Flow rate responses to a positive step change in Q A and a negative step change in Q B without changing the total feed rate.

  PID controller (K c = 35, T i = 0.08 min, T d = 0.02 min) 0.295 3.815 122.993 In summary, from previous P/PI/PID tuning based on the simulated closed-loop of X A , it can be seen that K c = 35 and T i = 0.08 min are proper values if a PI controller is used, and K c = 35, T i = 0.08 min and T d = 0.008 min are proper values if a PID controller is used. Therefore, the two sets PI(D) parameters have been tested in the experimental closed-loop of X A .

Figure V- 3

 3 Figure V-3 (a) shows the PI/PID closed-loop responses of mean composition in component A at the mixer's outlet, and they are also compared to the open-loop response of X A . Clearly, the PI/PID controller drives the controlled X A to its new setpoint in a significantly shorter time than the one required in an open-loop operation. Slightly smaller oscillations at the final steady state can be observed in the proposed PID control. As shown in Figure V-3 (b), the PI/PID closed-loop of mean concentration X A provide lower RSD values, which means better mixing homogeneity, as compared to the open-loop. For Figure V-3 (b), the percentage of samples of RSD value less than 6% is quantified as the ratio of RSD < 6%.Figure V-3 (c)

Figure V- 3 :

 3 Figure V-3: Experimental results from PI/PID closed-loop control of mean concentration X A in case 1: (a) mean composition X A , (b) relative standard deviation RSD, (c) rotational speed of the engine N.

  IAE and ITAE can be observed between simulation and experiments. This is because the fluctuations in flow rates have not been taken into account in the model which has been used for P/PI/PID tuning. The corresponding PI/PID control simulations including the actual measured flow rates have also been performed.

Figure V- 4 :

 4 Figure V-4: Comparison of simulated values with measured values for the mean concentration performance criteria ISE, IAE and ITAE.

  Simulation_1 and simulation_2 refer to control simulation without and with actual flow rate fluctuations, respectively. As can been seen from Figure V-4, for either the PI or the PID control, the IAE and ITAE obtained from simulation_2 are more closer to the experimental data compared to those obtained from simulation_1. This indicates that flow rate fluctuations, even if they are very small, play an important role in the reliability and accuracy of the modelbased control study.

A

  PID setting of K c = 28, T i = 0.08 min, T d = 0.008 min has been tested and the experimental results have been depicted in Figure V-5. In Figure V-5 (a), the RSD setpoint of 4% leads to a negative effect on the mean concentration X A , as can be seen from the measured X A , which has finally exceeded the desired value. While from Figure V-5 (b), it seems that the RSD profile obtained from the RSD setpoints of 4% is better. For the RSD setpoint of 4%, the final excessive X A and better RSD profile can be attributed to a large increase in the rotational speed at later times, as shown in Figure V-5 (c).

Figure V- 5 :

 5 Figure V-5: Experimental results from closed-loop of RSD with the RSD setpoints of 4% and 3% in case 1: (a) mean composition X A , (b) relative standard deviation RSD, (c) rotational speed of the engine N.

Figure V- 6 :

 6 Figure V-6: Experimental results from closed-loop of RSD with negative controller gain in case 1: (a) X A , (b) RSD, (c) N.

(Kc = 28 ,- 28 ,

 2828 Ti = 0.05 min, Td = 0.008 min) Ti = 0.05 min, Td = 0.008 min) negative controller gain of -28 has been tested again by setting the upper limit of the rotational speed at 48 Hz. As can be seen from Figure V-7 (a), the upper limit of N at 48 Hz provides a fast X A response (as compared to the open-loop) without too much overshoot (as compared to the control under the upper limit of N at 55 Hz). The RSD responses and the regulation of N are shown in Figure V-7 (b) and (c), respectively.

Figure V- 7 :

 7 Figure V-7: Experimental results from closed-loop of RSD under different limits of the manipulated variable N in case 1: (a) mean composition in component X A , (b) relative standard deviation RSD, (c) rotational speed of the engine N.

  36.89% due to a step change in Q A from 4.5 to 3.32 g.s -1 and a step change in Q B from 4.5 to 5.68 g.s -1 . The total feed rate Q A + Q B keeps at 9 g.s -1 . Figure V-8 shows an example of experimentally obtained flow rates at the mixer's inlet in response to the step changes in both Q A and Q B .

Figure V- 8 :

 8 Figure V-8: Flow rate responses to a negative step change in Q A and a positive step change in Q B without changing the total feed rate.

3. 1

 1 Closed-loop control of mean concentration X A Two experiments are firstly performed for two different settings of PI/PID controller parameters as follows: Setting 1: K c = 35, T i = 0.08 min, T d = 0 min. Setting 2: K c = 35, T i = 0.08 min, T d = 0.008 min. The two controller settings are selected because they have been previously studied in case 1 and some improvements have been obtained. In case 2, the open-loop responses are also illustrated and compared to closed-loop responses, as shown in Figure V-9.Figure V-9 (a) shows the responses of mean concentration X A at the mixer's outlet. As shown in the figure, similar results are obtained from the three experiments. Long lag time is present before X A starts to decrease.Figure V-9 (b) shows the evolutions of RSD. The regulation of the rotational speed N through the PI/PID controllers is shown in Figure V-9 (c). The rotational speed of the engine N is mainly regulated below 30 Hz under the controller setting 1 and it is varied between 25 and 38 Hz under the controller setting 2.

Figure V- 9 :Figure V- 10 :

 910 Figure V-9: Experimental results from closed-loop of X A with controller settings 1 and 2 in case 2: (a) mean concentration X A , (b) relative standard deviation RSD, (c) rotational speed of the engine N.

3. 2 Setting 1 :Figure V- 11 :

 2111 Figure V-11: Experimental results from closed-loop of RSD with the RSD setpoint of 3% in case 2: (a) mean concentration X A , (b) relative standard deviation RSD, (c) rotational speed of the engine N.

Figure V- 11 Retuned 1 :Retuned 2 :

 1112 Figure V-12 (c), retuned 1 provides greater and more frequent variations in the rotational speed of the engine compared to retuned 2. From Table V-12, smaller values of the ISE, IAE

Figure V- 12 :

 12 Figure V-12: Experimental results from closed-loop of RSD with the RSD setpoint of 5% in case 2: (a) mean concentration X A , (b) relative standard deviation RSD, (c) rotational speed of the engine N.

  case 3, the Q A and Q B are simultaneously changed from 4.5 to 5.68 g.s -1 . From the experimental measurements of Q A and Q B at the mixer's inlet, as shown in Figure V-13, it can be seen that the step changes in Q A and Q B are well tracked by the loss-in-weight feeders. In this case, the process control aims to maintain the mixing composition as well as to improve the mixing homogeneity.

Figure V- 13 : 4 . 1 Figure V- 14 :

 134114 Figure V-13: Flow rate responses to a positive step change in both Q A and Q B without changing the mixture composition.

4. 2 Setting 1 :Setting 2 :Setting 3 :

 2123 Closed-loop control of RSD The following control settings are experimentally examined under closed-loop control of RSD. RSD setpoint = 5%, K c = 28, T i = 0.05 min, T d = 0.008 min, N ∈ [25, 55] RSD setpoint = 4%, K c = 38, T i = 0.05 min, T d = 0.008 min, N ∈ [25, 55] RSD setpoint = 3%, K c = -28, T i = 0.05 min, T d = 0.008 min, N ∈ [25, 48]

Figure V- 15 Figure V- 15 :

 1515 Figure V-15 (a), (b) and (c), respectively. The rotational speed of the engine is differently

V- 16

 16 shows experimental results of Q A and Q B at the mixer's inlet after the step changes.

Figure V- 16 :

 16 Figure V-16: Flow rate responses to a negative step change in both Q A and Q B without changing the mixture composition.

1 :Figure V- 17 :

 117 Figure V-17: Experimental results from closed-loop of X A in case 4: (a) mean concentration X A , (b) relative standard deviation RSD, (c) rotational speed of the engine N.

Setting 1 :Setting 2 :Figure V- 18 :

 1218 Figure V-18: Experimental results from closed-loop of RSD in case 4: (a) mean concentration X A , (b) relative standard deviation RSD, (c) rotational speed of the engine N.

  Q A and Q B , a PID setting used for closed-loop control of X A shows positive effects on X A but little effects on RSD, while a PID setting used for closed-loop control of RSD shows positive effects on RSD but little effects on X A . The outputs of the manipulated variable N show that the mean concentration setpoint can be better tracked by the rotational speeds adjusted in large amplitudes (around 30 Hz) while the mixing homogeneity RSD can be better improved by the rotational speeds adjusted between 25 and 35 Hz. 6 Study of the influence of the starting rotational speed N As presented above, the rotational speed of the engine N is initially fixed at 30 Hz before introducing the load disturbances. An alternative initial rotational speed of 40 Hz will now be used to exercise a possible influence of this value. The subjected step changes in the flow rates Q A and Q B are the same as those in case 1, which in turn leads to a step change in X A from 50% to 63.11%. Like in case 1, good control results are obtained from the control settings as follows: Setting for closed-loop of X A : K c = 35, T i = 0.08 min and T d = 0.008 min, N ∈ [25, 55]. Setting for closed-loop of RSD: RSD setpoint =3%, K c = 25, T i = 0.08 min and T d = 0.008 min, N ∈ [25, 48]. So the two control settings are respectively examined under N initial = 40 Hz. The experimental results are also compared to those obtained from the case 1 with N initial = 30 Hz.

Figure V- 19 :

 19 Figure V-19: Plots of X A : (1) closed-loop control of X A starting with N=30 Hz and N=40 Hz, respectively; (2) closed-loop control of RSD starting with N=30 Hz and N=40 Hz, respectively.

Figure V- 20 :

 20 Figure V-20: Plots of RSD: (1) closed-loop control of X A starting with N=30 Hz and N=40 Hz, respectively; (2) closed-loop control of RSD starting with N=30 Hz and N=40 Hz, respectively.

Figure V- 21 :

 21 Figure V-21: Plots of the rotational speed N: (1) closed-loop control of X A starting with N=30 Hz and N=40 Hz, respectively; (2) closed-loop control of RSD starting with N=30 Hz and N=40 Hz, respectively.

  work has been concerned with a new development in understanding of a pilot continuous powder mixing process from a practical control point of view. The implementation of a simplified regulatory control system (PID) in a pilot scale continuous mixer has been presented and considered as an initial step toward the development of an efficient real-time control of continuous powder mixing processes. The mixing of powders of two components with different size distributions has been considered. The pilot process mainly consists of two loss-in-weight feeders, a Gericke GCM 500 continuous mixer and a previously developed online monitoring system of mixture quality. The synchronization control of the three individual parts has been realised using Labview. Without giving much importance to the interactions of the material in the blender, the continuous mixer has been analysed from a "black box" model, for which the flow rates of each component and the rotational speed of the mixer are the input variables, while the mean concentration and the relative standard deviation of the mixture are output variables indicating the mixing efficiency. Due to the complex effects of the inputs, the process control study has been started with new developments in the understanding of continuous powder mixing. The continuous mixer has been virtually divided into 5 compartments. Hold-up weight and relative hold-up distribution in the whole mixer have been experimentally investigated for each component under different operating conditions, including the mixing formulation, the total feed rate and the rotational speed. Empirical correlations for each component have been proposed and determined to describe the hold-up weight and the relative hold-up distribution in the whole mixer as a function of operating conditions. The model involves two Markov chains, for which one chain corresponds to one component. The model has been completed based on the investigations on hold-ups. The comparison of experimental and simulation results demonstrate that the improved process model could be used to predict the profiles of the mean concentration and the mass outflow rate of the mixer, but is mainly non-predictive for the relative standard deviation.

  Thereby, a PID tuning based on the process model has been proposed to avoid extensive experiments. The improved process model has then been re-programmed in Labview in order to use a PID controller, which can be equally used in real control experiments. For closed-loop control of mean composition, PID parameters have been preliminary tunned based on the process model and then tested or retuned in mean composition control experiments. While for closed-loop control of relative standard deviation, PID parameters have been directly tuned by relative standard deviation control experiments, since the presented model has not yet been enough accurate to predict RSD profiles. The closed-loop control of either the mean concentration or the relative standard deviation has finally been studied in four different cases. The four cases include different step changes in flow rates of the two components. In general, they correspond to two possible industrial demands during the process such as a change in mixture composition without changing the production volume and an increase/decrease in production volume. For either closed-loop control in each case, the control efficiency has been evaluated by comparing the closed-loop

  parameters due to special responses to step changes in the manipulated variable. As presented above, in order to avoid extensive experiments, the PID tuning for closed-loop control of the mean concentration has been preliminarily performed with a model-based PID control. The comparison of control simulations of the mean concentration with and without the actual fluctuations in flow rates has shown that the reliability of the presented model can be improved by taking into account the fluctuations. However, the model taking into account the actual flow rate fluctuations still couldn't be used to predict the RSD, mainly due to unmeasured disturbances. Therefore, a further improvement on the model is required in order to provide the possibility of the RSD prediction.

  end end %===============Matrix of transition probabilities for component A========= %**************************P_A******************************************** * P_A=[1-p(1probabilities for component B========= %**************************P_B******************************************** * P_B=[1-q

Figure I- 1 : 4 Figure I- 2 : 5 Figure I- 3 : 6 Figure I- 4 : 7 Figure I- 5 : 9 Figure I- 6 : 13 Figure I- 7 : 14 Figure I- 8 : 15 Figure I- 9 : 18 Figure I- 10 : 18 Figure I- 11 : 19 Figure I- 12 :Figure I- 13 : 21 Figure I- 15 : 21 Figure I- 16 : 23 Figure I- 17 : 26 Figure I- 18 : 27 Figure I- 19 : 29 Figure I- 21 : 30 Figure I- 22 : 30 Figure I- 23 : 31 Figure I- 24 : 33 Figure I- 25 : 34 Figure I- 26 : 34 Figure I- 27 : 35 Figure I- 28 : 37 Figure I- 29 : 39 Figure I- 30 : 41 Figure I- 31 : 42 Figure I- 32 : 44 Figure I- 33 : 45 Figure I- 34 : 45 Figure I- 35 : 46 Figure I- 36 : 48 Figure I- 37 : 49 Figure I- 38 : 50 Figure II- 1 : 52 Figure II- 2 : 53 Figure II- 3 : 55 Figure II- 4 : 60 Figure II- 5 : 61 Figure II- 6 : 62 Figure II- 7 : 63 Figure II- 8 : 63 Figure II- 9 : 64 Figure II- 10 : 65 Figure II- 11 : 66 Figure II- 12 : 67 Figure II- 13 : 68 Figure II- 14 : 69 Figure II- 15 : 70 Figure III- 1 : 72 Figure III- 2 : 73 Figure III- 3 : 75 Figure III- 4 : 76 Figure III- 5 : 77 Figure III- 6 : 78 Figure III- 7 : 79 Figure III- 8 : 80 Figure III- 9 : 81 Figure III- 10 : 82 Figure III- 11 : 84 Figure III- 13 : 87 Figure III- 14 : 89 Figure III- 15 : 90 Figure III- 16 : 91 Figure III- 17 : 92 Figure III- 18 : 94 Figure III- 19 : 95 Figure IV- 1 : 98 Figure IV- 2 : 99 Figure IV- 3 : 99 Figure IV- 4 : 100 Figure IV- 5 : 100 Figure IV- 6 : 104 Figure IV- 9 : 105 Figure IV- 10 : 108 Figure IV- 11 : 109 Figure IV- 12 : 110 Figure IV- 13 : 110 Figure V- 1 : 114 Figure V- 2 :Figure V- 3 :Figure V- 4 :Figure V- 5 :Figure V- 6 :Figure V- 7 :Figure V- 8 :Figure V- 9 :Figure V- 11 :Figure V- 12 :Figure V- 14 :Figure V- 15 :Figure V- 16 :Figure V- 17 :Figure V- 19 :Figure V- 21 : 2 variance of a mixture ! 0 2

 14253647596137148159181018111912132115211623172618271929213022302331243325342634273528372939304131423244334534453546364837493850152253355460561662763863964106511661267136814691570172273375476577678779880981108211841387148915901691179218941995198299399410051006104910510108111091211013110111423456789111214151617192120 Figure I-1: Types of mixture adapted from [3,4]: (a) Perfectly mixed; (b) Randomly mixed; (c) Completely segregated. ................................................................................................. 4 Figure I-2: Illustration of the mechanisms of segregation, adapted from [4-6]: (a) Trajectory segregation; (b) Percolation segregation; (c) Vibration segregation; (d) Elutriation segregation. ........................................................................................................................ 5 Figure I-3: Tumbler mixers, adapted from [4]: (a) V-mixer; (b) Double cone mixer; (c) Bin mixer. .................................................................................................................................. 6 Figure I-4: Convective mixers, adapted from [4,7]: (a) Ribbon mixer; (b) Paddle mixer; (c) Nautamix; (d) Vertical screw mixer. .................................................................................. 7 Figure I-5: Schematic representation of scale and intensity of segregation: (a) reduction in scale of segregation; (b) reduction in intensity of segregation. .......................................... 9 Figure I-6: Two different mixture structures observed in two mixtures of same macromixing state. .................................................................................................................................. 13 Figure I-7: Experimental set-up for monitoring of powder mixture quality using image analysis from Berthiaux et al. [23]. .................................................................................. 14 Figure I-8: Steps to determine homogeneity from images from Rosas and Blanco [27]. ........ 15 Figure I-9: (a) Raw spectra of pure LM and ASA (b) NIR spectra after data preprocessing for the calibration sets containing 11 samples ranging from 0-100% API content from Koller et al. [36]. ......................................................................................................................... 18 Figure I-10: In-line quantitative monitoring of powder mixing dynamics in a four-blade mixer via NIR from Koller et al. [36]. ........................................................................................ 18 Figure I-11: Homogeneity is reached after about 300 s: (a) CI plot for NIR data; (b) CI plot for Raman data from De Beer et al. [37]. ......................................................................... 19 Figure I-12: NIR with a fiber optical probe from Martinez et al. [39]. .................................... 20 Figure I-13: Predicted NIR values (♦) with RSD (continuous line) for the four trials at different feeding rates and stirring rates from Martinez et al. [39]. ................................. 20 Figure I-14: Schematic of the multipoint NIR measurement system consisting of a fiber-optic lights source, 5 fiber-optic probes and a fiber-optic spectral camera by Aditya et al. [42]. .......................................................................................................................................... 21

Table II -

 II

	From
	Density [g.cm -3 ]	Component A	Component B
	! true	1.437	1.444
	! aerated	0.840	0.737
	! tapped	0.875	0.756
	Table II-3: Inter-particle porosity of component A and B
	Porosity	Component A	Component B
	! aerated	0.415	0.490
	! tapped	0.391	0.476
		57

2: Different types of densities of component A and B.

Table

  II-2 it can be seen that the true density of component A is very close to that of component B, which is logical since these components are of the same nature. However, component B has lower aerated and tapped densities than component A, thereby showing a higher inter-particle porosity, as listed in Table II-3. The higher porosity of component B is mostly due to high irregular shape and surface roughness of the particles, because the "Span"

Table II -

 II

	Measures of flowability	Component A	Component B
	H	1.042	1.026
	CI (%)	4.2	2.6
	R (°)	30.5	33.6
	S (°)	18.4	20.3

4: Granular material flowability evaluation.

Table II -

 II

	Number of pixels	5000 pixels
	Size of pixel	7 µm x 7µm
	Photo-sensitive area	35 mm x 7 µm
	Sensitivity	300 000 lsb/lux.s
	Maximum pixel frequency	20 MHz
	Maximum line frequency	3.77 KHz
	Output video format	16 bits RS422
	Lens mount	F-Nikon 24 x 36

6: Principle optoelectronic characteristics of camera DVL 5000 T.

1) Tuning of P controller

  Prior to PID controller, Labview PID controller has been used as P controller by appropriate setting of T i → ∞ and T d = 0. Three values of the proportional gain K c were tested. The performances of the three P controllers have been evaluated by calculating ISE, IAE and ITAE, and compared with that of open-loop operation. As shown in Table V-2, P controller of K c = 35 performs better than two other P controllers, whereas X A controlled by P controllers seems to be worse than that obtained in open-loop operation.Table V-2: Results of P tuning based on the model.

		ISE	IAE	ITAE
	Open-loop	0.497	3.818	53.837
	P controller (K c = 25)	0.541	4.53	96.88
	P controller (K c = 35)	0.392	4.255	98.86
	P controller (K c = 45)	0.471	4.452	115.407
	(2)

Tuning of PI controller Then

  , Labview PID controller has been used as PI controller by setting of T d = 0. Keeping K c at 35, three values of T i were examined. The performances of PI controllers have been evaluated as given in Table V-3. It is found that the PI setting of K c = 35 and T i = 0.08 min gives the best performance. Using K c = 35 and T i = 0.08 min, Labview PID controller has been tuned by varying T d . Smaller errors were obtained from PID setting of K c = 35, T i = 0.08 min, T d = 0.008 min, as shown in Table V-4.

	(3) Tuning of PID controller
	Table V-4: Results of PID tuning based on the model
		ISE	IAE	ITAE
	Open-loop	0.497	3.818	53.837
	PID controller (K c = 35, T i = 0.08 min, T d = 0.002 min)	0.197	1.897	16.489
	PID controller (K c = 35, T i = 0.08 min, T d = 0.008 min)	0.194	1.86	15.82
	Table V-3: Results of PI tuning based on the model
		ISE	IAE	ITAE
	Open-loop	0.497	3.818	53.837
	PI controller (K c = 35, T i = 0.02 min)	0.193	1.852	15.645
	PI controller (K c = 35, T i = 0.08 min)	0.187	1.807	15.107
	PI controller (K c = 35, T i = 0.14 min)	0.208	2.111	21.508

  -5. By comparing the ISE, IAE, ITAE values of the PI/PID closed-loop to the open-loop, it can be found that the PID closed-loop of X A exhibits the best mixing performance in terms of both X A and RSD.When we turn back to the simulation results of ISE, ITAE and ITAE, significant differences of

	Table V-5: Performance assessment results from experiments of closed-loop control of X A in
	case 1.
		ISE	IAE	ITAE	Ratio of RSD < 6%
	Open-loop	0.329	3.783	87.614	56 / 101
	PI closed-loop	0.241	3.001	74.204	80 / 101
	PID closed-loop	0.239	2.687	54.815	87 / 101

Table V -

 V 6: Performance assessment results from experiments of closed-loop control of RSD with the RSD setpoints of 4% and 3% in case 1.

		ISE	IAE	ITAE	Ratio of RSD < 6%
	Open-loop	0.329	3.783	87.614	56 / 101
	RSD setpoint = 4%	0.306	4.316	162.717	83 / 101
	RSD setpoint = 3%	0.411	4.272	94.25	61 / 101

Table V -

 V 7: Performance assessment results from experiments of closed-loop control of RSD with negative controller gain in case 1.

		ISE	IAE	ITAE	Ratio of RSD < 6%
	Open-loop	0.329	3.783	87.614	56 / 101
	RSD setpoint = 3%

Table V -

 V 8: Performance assessment results from experiments of closed-loop control of RSD under different limits of the manipulated variable N in case 1.

		ISE	IAE	ITAE	Ratio of RSD < 6%
	No control loop	0.3294	3.7825	87.6144	56 / 101
	N ∈ [25, 55] (Kc = -28, Ti = 0.05 min, Td = 0.008 min)	0.311	4.272	131,6188	86 / 101
	N ∈ [25, 48]	0.3155	3.3953	82.9138	76 / 101
	(Kc = -28, Ti = 0.05 min, Td = 0.008 min)

Table V -

 V [START_REF] Lacey | Developments in the theory of particle mixing[END_REF]. In conclusion, it is found that the feedback control purpose is not yet reached by using the retuned controller settings in the closed-loop control of X A .

	Table V-10: Performance assessment results from experiments of closed-loop control of X A
	with controller retuned 1 and 2 in case 2.
		ISE	IAE	ITAE	Ratio of RSD < 6%
	Open-loop	0.460	4,981	131,940	33 / 101
	Retuned 1	0,460	5,207	154,959	29 / 101
	Retuned 2	0,625	6,339	196,771	55 / 101

Table V -

 V 11: Performance assessment results from experiments of closed-loop control of RSD with the RSD setpoint of 3% in case 2.None of the three settings used in the closed-loop control of RSD is able to decrease the ISE, IAE and ITAE as well as to increase the ratio of RSD < 6%. Since the RSD curves in

		ISE	IAE	ITAE	Ratio of RSD < 6%
	Open-loop	0.460	4.981	131.940	33 / 101
	Setting 1	0.414	4.655	128.240	29 / 101
	Setting 2	0.745	7.730	290.590	47 / 101
	Setting 3	0.491	5.229	147.777	32 / 101

Table V -

 V Table V-12, smaller values of the ISE, IAE and ITAE obtained from retuned 1 indicate an improvement for the X A setpoint tracking, while smaller values of the ratio of RSD < 6% obtained from returned 1 and 2 indicate poorer mixing homogeneity at the mixer's outlet. 12: Performance assessment results from experiments of closed-loop control of RSD with the RSD setpoint of 5% in case 2.

		ISE	IAE	ITAE	Ratio of RSD < 6%
	Open-loop	0.460	4.981	131.940	33 / 101
	Retuned 1	0.358	3.904	89.180	22 / 101
	Retuned 2	0.590	5.955	178.463	29 / 101

  Table V-13. For the tested control setting, larger values of the ISE, IAE and ITAE indicate a worse control of X A , while a higher ratio of RSD < 6% indicates an improvement in RSD.

	Table V-13: Performance assessment results from experiments of closed-loop control of X A in
	case 3.
		ISE	IAE	ITAE	Ratio of RSD < 6%
	Open-loop	0.026	1.407	72.258	35 / 101
	PID closed-loop	0.05	1.789	89.37	53 / 101

  The mixing performance under each control setting is evaluated and compared to that of open-loop operation, as shown in Table V-14. In comparison to the open-loop, each control setting brings smaller values of the ISE, IAE and ITAE and a higher ratio of RSD < 6%. The overall process performance is slightly improved by any one of the three control settings. The control of X A is more significant by setting 1 and 2 and the control of RSD is more efficient by

	setting 3.
	Table V-14: Performance assessment results from experiments of closed-loop control of RSD
	in case 3.
		ISE	IAE	ITAE	Rate of RSD < 6%
	Open-loop	0.026	1.407	72.258	35 / 101
	Setting 1	0.023	1.237	56.707	40 / 101
	Setting 2	0.025	1.245	54.451	43 / 101
	Setting 3	0.028	1.385	60.17	55 / 101

  -15, compared to the open loop operation, setting 1 and 2 provide smaller values of the ISE, IAE, and ITAE, whereas little different values of the ratio of RSD < 6%. These results indicate that setting 1 and 2 for closed-loop control of X A present notable improvements in the mean composition rather than the RSD.

	Table V-15: Performance assessment results from experiments of closed-loop control of X A in
	case 4.
		ISE	IAE	ITAE	Rate of RSD < 6%
	Open-loop	0.109	3.004	158.345	25 / 101
	Setting 1	0.07	2.239	120.68	28 / 101
	Setting 2	0.049	1.8	93.988	29 / 101

Table V -

 V 16, the mixing performance for each experiment is quantified and compared to open-loop operation. Values of the ISE, IAE and ITAE of both settings are greater than those of the open-loop, which indicate that the applied control settings cannot reduce the effect of the flowrate disturbances on the mean concentration. However, both settings under closed-loop control of RSD show improvements in the mixing homogeneity, indicated by higher ratios of RSD < 6% compared to the open-loop.

	Table V-16: Performance assessment results from experiments of closed-loop control of RSD
	in case 4.
		ISE	IAE	ITAE	Ratio of RSD < 6%
	Open-loop	0.109	3.004	158.345	25 / 101
	Setting 1	0.136	3.362	184.204	51 / 101
	Setting 2	0.114	3.157	168.267	34 / 101

Table V -

 V 17: Comparison of process performances obtained from closed-loop control of X A starting with N=30 Hz and N=40 Hz. From Table V-17, in closed-loop control of X A , the mixing performance obtained under N initial = 40 Hz is little less than that obtained under N initial = 30 Hz, since larger values of the ISE, IAE and ITAE and a lower ratio of RSD < 6% are observed under N initial = 40 Hz. It is possible due to the notable decreases in the rotational speed during the time t=70 and 80 second. Table V-18: Comparison of process performances obtained from closed-loop control of RSD starting with N=30 Hz and N=40 Hz.

		ISE	IAE	ITAE	Rate of RSD < 6%
	Closed-loop of X A (N initial = 30 Hz)	0.239	2.687	54.815	87 / 101
	Closed-loop of X A (N initial = 40 Hz)	0.231	3.145	90.838	79 / 101
		ISE	IAE	ITAE	Rate of RSD < 6%
	Closed-loop of RSD (N initial = 30 Hz)	0.316	3.395	82.914	76 / 101
	Closed-loop of RSD (N initial = 40 Hz)	0.297	3.249	69.039	88 / 101
	From

Table V -

 V [START_REF] Muzzio | Sampling practices in powder blending[END_REF], in closed-loop control of RSD, the N initial = 40 Hz shows a better mixing performance than the N initial = 30 Hz, since smaller values of the ISE, IAE and ITAE and a higher ratio of RSD < 6% are observed under N initial = 40 Hz.

Table I -

 I 1: Some definitions of mixing indices. ........................................................................ 12Table II-1: Particle size distribution parameters of component A and B. ................................ 54 Table II-2: Different types of densities of component A and B. .............................................. 56 Table II-3: Inter-particle porosity of component A and B ....................................................... 56 Table II-4: Granular material flowability evaluation. .............................................................. 58 Table II-5: Granular material floodability evalaution. ............................................................. 59 Table II-6: Principle optoelectronic characteristics of camera DVL 5000 T. .......................... 64 Table III-1: Mixing experiments proceeding conditions. ......................................................... 74 Table V-1: The four cases studied. ......................................................................................... 113 Table V-2: Results of P tuning based on the model. .............................................................. 116

Table V -

 V

3: Results of PI tuning based on the model .............................................................. 116 Table V-4: Results of PID tuning based on the model ........................................................... 117 Table V-5: Performance assessment results from experiments of closed-loop control of X A in case 1. ............................................................................................................................. 119 Table V-6: Performance assessment results from experiments of closed-loop control of RSD with the RSD setpoints of 4% and 3% in case 1. ............................................................ 120 Table V-7: Performance assessment results from experiments of closed-loop control of RSD with negative controller gain in case 1. .......................................................................... 123 Table V-8: Performance assessment results from experiments of closed-loop control of RSD under different limits of the manipulated variable N in case 1. ..................................... 123

Table V -

 V 9: Performance assessment results from experiments of closed-loop control of X A with controller settings 1 and 2 in case 2. ...................................................................... 126 Table V-10: Performance assessment results from experiments of closed-loop control of X A with controller retuned 1 and 2 in case 2. ...................................................................... 129 Table V-11: Performance assessment results from experiments of closed-loop control of RSD with the RSD setpoint of 3% in case 2. .......................................................................... 131 Table V-12: Performance assessment results from experiments of closed-loop control of RSD

with the RSD setpoint of 5% in case 2. .......................................................................... 131

Table V -

 V 13: Performance assessment results from experiments of closed-loop control of X A in case 3. ............................................................................................................................. 135 Table V-14: Performance assessment results from experiments of closed-loop control of RSD in case 3. ......................................................................................................................... 137 Table V-15: Performance assessment results from experiments of closed-loop control of X A in case 4. ............................................................................................................................. 140 Table V-16: Performance assessment results from experiments of closed-loop control of RSD in case 4. ......................................................................................................................... 142 Table V-17: Comparison of process performances obtained from closed-loop control of X A starting with N=30 Hz and N=40 Hz. ............................................................................. 145 Table V-18: Comparison of process performances obtained from closed-loop control of RSD starting with N=30 Hz and N=40 Hz. ............................................................................. 145

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Control system programmed in Labview -User graphical interface (2)

Control system programmed in Labview -Block program (1)

Control system programmed in Labview -Block program (2)

Mots clés: Mélange de poudres en continu, Implémentation, Contrôle-commande, Chaînes de Markov, Contrôleur PID, Commande en boucle fermée

ACKNOWLEDGEMENTS

My first and sincerely acknowledgement goes to both my supervisors Henri Berthiaux and Michel Cabassud for their guidance, continuous advice and encouragement during my PhD study. I'm grateful to Henri for his patience, motivation, knowledge and words that inspire and bring me to finish my thesis step by step. I thank Michel for his travels from Toulouse for my thesis meetings in Albi and his inspiring suggestions to my thesis research. I would like to express my special thank to my tutor Jean-Louis Dirion for his help, support and encouragement especially at the end of my thesis. I would also like to thank my tutor Cendrine Gatumel for her help, support and kindness during my PhD study. I would like to give gratitude to the technicians Denis, Laurent and Philippe of the laboratory Rapsodee who have helped me. I would like to thank my best friends Sarah, Nibal, and David for their spiritual support. I would like to thank Madame Fabienne Espitalier, Madame Anne-Marie Fontes, Madame Claudine Sablayrolles and Madame Rita Franco for their help and kindness to me. I would also like to thank the director of laboratory Rapsodee, Ange Nzihou, who has offered me extra two months' finance to let me finish my thesis. I would like to thank Fanny, William, Graciela, Jacqueline, Pierre-Marie, Baptiste and all other PhD students with whom I've had the pleasure to work.

Last but not least, I would like to thank my parents and sisters for their love and support.

CHAPTER V: Control case studies 1 Case study design

We focus on the performance assessment of closed-loops subjected to step changes in flow rates Q A and Q B . Four case studies have been designed according to possible step changes in

Table V-1: The four cases studied. From an industrial viewpoint, the study of case 1 and 2 aims at contributions to control the process when requiring a change in mixture composition during the production while no change in the total production volume. Case 3 corresponds to an increase in continuous production volume and case 4 correspond to a decrease in continuous production volume.

The overall control procedure is summarized in Figure V-1. In each case, the continuous mixing process is experimentally started at 50%, with component A and B, both flow rates at 4.5 g.s -1 . The rotational speed of the engine N is initially set to 30 Hz. Once the continuous mixing process reaches steady state, the closed-loop of either X A or RSD is implemented and the defined step changes in flow rates are performed simultaneously. Each experiment is totally run for 185 s with a step changes implemented at t=85 s after steady state is achieved.

Consequently, the control time lasts 100 s because it should be counted from the moment ---------------------------- for i=1:n(lp)/dt if lp==1 t(i)=i*dt; else t(n(1)/dt+i)=(n(1)/dt+i)*dt; end p=zeros(5,1); q=zeros(5,1); %=========Determination of transition probabilities for component A and B========== for j=1:5 if S(j)==0 p(j)=0; elseif S(j)<Smax(j) p(j)=P_A_max(j+1,j)/Smax(j)*S(j); else p(j)=P_A_max(j+1,j); end if K(j)==0 q(j)=0; elseif K(j)<Kmax(j) q(j)=P_B_max(j+1,j)/Kmax(j)*K(j); else q(j)=P_B_max(j+1,j); (1) 0 0 0 0 0 q(1) 1-q(2) 0 0 0 0 0 q(2) 1-q(3) 0 0 0 0 0 q(3) 1-q(4) 0 0 0 0 0 q(4) 1-q(5) 0 0 0 0 0 q(5) 1];

)); end else c_A(n(1)/dt+i)=(M_6_A(n(1)/dt+i)-M_6_A(n(1)/dt+i-1))/((M_6_A(n(1)/dt+i)-M_6_A(n(1)/dt+i-1))+(M_6_B(n(1)/dt+i)-M_6_B(n(1)/dt+i-1))); end end end

ABSTRACT

Powder mixing is an essential operation in various industrial fields, such as pharmaceuticals, agrofood, cements, etc. Continuous powder mixing, as an alternative to conventional batch mixing, has attracted a lot of interest mainly due to its capacity in handling high volume manufacturing. This work aims at the contribution to the implementation of process control applications for powder mixing in a pilot-scale continuous mixer. Prior to developing process control strategies, new developments have been presented for better understanding continuous mixing of two components. Hold-up weight and relative hold-up weight distribution of each component in the whole mixer have been experimentally investigated under different operating conditions. An improved Markov chain model has been finally presented to predict the mean concentration of the mixtures obtained at the mixer's outlet. The implementation of a proportional-integral-derivative (PID) controller has been experimentally performed as an initial attempt to real-time control the homogeneity of the mixture produced. The rotational speed of the stirrer, identified as an important deciding factor towards the mixer's efficiency, has been considered as the manipulated variable. The closed-loop control is based on either the mean concentration or the relative standard deviation. The performances of the proposed closedloops have been evaluated for continuous mixing subjected to step changes in feed rates of the mixer. Four case studies have been defined and presented. The main challenge in the process control system is the tuning of PID parameters. The performance of closed-loop control of either the mean concentration or the relative standard deviation has been compared to open-loop operation.

RESUME

Le mélange de poudres est une opération essentielle dans divers domaines industriels, tels que les produits pharmaceutiques, agro-alimentaires, ciments, etc. Le mélange de poudres en continu, comme alternative intéressante au mélange conventionnel en batch, suscite beaucoup d'intérêt surtout en raison de sa capacité à gérer de grands volumes de production. Ce travail vise à contribuer à l'implémentation des applications de contrôle-commande. Avant de développer des stratégies de contrôle-commande, de nouveaux développements ont été présentés pour mieux comprendre le mélange continu de deux composants. La masse retenue et la répartition relative de la masse retenue de chaque composant dans le mélangeur ont été étudiées et prédites sous les différentes conditions opératoires. L'amélioration d'un modèle basé sur les chaînes de Markov a été finalement présentée pour prédire la concentration moyenne des mélanges obtenus à la sortie du mélangeur. L'implémentation d'un contrôleur PID a été expérimentalement réalisée comme une première tentative pour contrôler en temps réel l'homogénéité du mélange produit. La vitesse de rotation du mobile d'agitation, identifiée comme un facteur important influençant l'efficacité du mixer, a été considérée comme la variable manipulée. La commande en boucle fermée est basée soit sur la concentration moyenne, soit sur le coefficient de variation. Les performances des boucles fermées proposées ont été évaluées pour le mélange continu subi à des changements d'échelon dans les débits d'alimentation du mélangeur. Quatre études de cas ont été définies et présentées. Le défi principal dans le système de contrôle-commande est le réglage des paramètres PID. La performance de commande en boucle fermée soit de la concentration moyenne, soit du coefficient de variation a été comparée à l'opération en boucle ouverte.