Controlling crystal size distribution (CSD) is important to downstream processing and to product quality. It is well-recognized that selective removal functions can be used to influence CSD, for example by manufacturing a product with a larger dominant size or narrower distribution. Early work on the use of feedback control to manipulate the residence time distribution functions of fines in a continuous crystallizer demonstrated the utility of such an approach in handling process upsets and cycling that resulted from system instability. These efforts were extended to batch crystallization, although there remained significant difficulty associated with on-line analysis of the size distribution. The development of new technologies, such as Focused Beam Reflectance Measurement (FBRM), provides a methodology for on-line monitoring of a representation of the crystal population in either batch or continuous crystallization systems. The FBRM technology is based on laser light scattering; properly installed, it allows on-line determination of the chord length distribution (CLD), which is statistically related to the CSD and depends on the geometry of the crystal. The purpose of the present study is to use the FBRM to monitor the evolution of CSD characteristics and to implement a feedback control scheme that provides the flexibility to move the CSD in a preferred direction. Cooling batch crystallizations of paracetamol has been chosen to investigate implementation of the control scheme. The work will show how fines removal and varying cooling rates provide reliable and practical control of crystal size distribution.

Batch crystallization is used widely in the production of high-value added species. It is widely recognized that product properties, some of which may be related directly to the utility of the drug, and downstream processes, such as tableting, are influenced by crystal morphology, size, and shape. The ability to observe on-line the evolution of the population density and detect a polymorphic transformation would constitute a major asset in understanding crystallizer operation and the phenomena that influence product quality. Focused-beam reflectance measurement (FBRM) is among the process analytical technologies (PAT) that hold promise for enhanced monitoring of pharmaceutical crystallization. It is based on scattering of laser light and provides a methodology for on-line monitoring of a representation of the crystal population in either batch or continuous crystallization systems. Properly installed, the FBRM allows on-line determination of the chord-length density, which is a complex function of crystal geometry and is statistically related to the population density. A model based on the geometry of the crystal was therefore established to relate both densities and thus enable computation of the population density from a measured chord length density. The evolution of the population density as a function of time leads to the estimation of the supersaturation and therefore allows the determination of the systems kinetics. From there, the population balance can be solved. Paracetamol is a common substance which exhibit polymorphism and is mainly used as an analgesic and antipyretic drug. The developed model was here applied to batch cooling crystallization of paracetamol from ethanol solutions; this system was used to explore the utility of FBRM data in detection of the polymorphic transformations. As different shapes generate different chord length densities, a transition from one polymorphic form with one specific crystal habit to another can be tracked through the FBRM. The purpose of the present study is to use the FBRM to monitor the evolution of the crystallization process, develop a model describing the evolution of the process, and monitor polymorphic transformation. The end results would be the possibility to implement a better control of the crystallization process that would ensure that downstream processing and product quality meet expectations.

Cette thèse consiste en trois parties que j'ai faites pendant ces trois ans.La première partie va être constituée de l'étude de la distribution de la longueur de corde sur le plan hyperbolique. Nous montrons l'identité de Pleijel pour le plan hyperbolique. En utilisant cette identité, nous remontrons l'identité de formule de Crofton et l'inégalité isopérimétrique pour le plan hyperbolique, et puis nous calculons la distribution de la longueur de corde associée à un triangle idéal et celle associée à un quadrilatère idéal. Ensuit, nous montons les résultats analogues pour les surfaces riemannienne simplement connexes avec la courbure constante. La seconde partie va contribuer aux études de la métrique de pression sur l'espace de Teichmüller d'un tore privé d'un disque. En étudiant la dégénération du tore quand la longueur du bord va à l'infini, nous trouvons la relation de cette métrique avec la métrique de pression sur l'espace modulaires des graphes métriques. Nous montrons ensuite que la fonction de l'entropie n'est pas constante sur les feuilles symplectique de l'espace Teichmüller d'une surface à bord.Finalement, la troisième partie concerne la quantification de l'espace de Teichmüller d'une surface avec les piqûres. nous montrons. Dans ce chapitre, nous étudions l'extension centrale du groupe modulaire via la quantification de Chekhov-Fock et calculons sa classe de cohomologie qui est 12 fois la classe de Meyer plus les classes d'Euler associées aux piqûres. / This thesis consists of three parts corresponding to the three subjects that I have studied during the last three years.The first part contains the study of the chord length distribution associated to a compact (or non-compact) domain in the hyperbolic plane. We prove the hyperbolic Pleijel identity. By using this identity, we find new approaches to the Crofton's formula and the isoperimetric inequality, and then compute the chord length distribution associated to an ideal triangle and that associated to an ideal quadrilateral. Then we prove the analogue results for the simply connected Riemannian surface with constant curvature.The second part of this thesis (Chapter 5) consists of the study of the pressuremetric on the Teichmüller space of one-holed torus. By studying the degeneration of the torus when the boundary length goes to infinity, we find the relation of this metric to the pressure metric on the moduli space of metric graphs. Then we study the entropy function and prove that it is not constant on the symplectic leaf of the Teichmüller space of a bordered surface.Finally, the third part concerns the quantization of the Teichmüller space of a punctured surface. In this chapter, we study the central extension of the mapping class group coming from the quantization and compute its cohomology class which is 12 times the Meyer class plus the Euler classes associated to punctures.