Jet schemes of complex plane branches and equisingularity

For $m \in \mathbb{N}$, we determine the irreducible components of the $m$-th Jet Scheme of a complex branch $C$ and give formulas for their number $N(m)$ and for their codimensions, in terms of $m$ and the generators of the semigroup of $C$. This structure of the Jet Schemes determines and is determined by the topological type of $C$.Comment: 22 page

ADVANCED MODELLING OF EMULSION TERPOLYMERISATION FOR ONLINE OPTIMISATION AND CONTROL

Polymer manufacturing is a major worldwide industry, attracting the attention of numerous industrial units and research institutes. Increasing demands on polymer quality, process safety and cost reduction are the main reasons for growing interest in the design and control of emulsion polymerisation. Emulsion polymerisation process implemented with free radical polymerisation has significant advantages over other processes, such as the production of polymer of higher molecular weights at high conversion rates, easier temperature control due to the low viscosity of the reaction media, high degree of selectivity and more friendly to environment due to the use of an aqueous medium. It allows for the production of particles with specially-tailored properties, including size, composition, morphology, and molecular weights. Introducing two or more different monomers to the polymerisation process (named multi-polymerisation) can lead to the synthesis of an almost unlimited number of new polymers types. Emulsion polymers are products by process, meaning that the manner in which the polymerisation is carried out is perhaps more important than the raw materials in determining the form of the final product. This highlights the significance of the systematic approach in online process control which requires thorough understanding of the process phenomena as a prerequisite for development of a mathematical description of the process as the model. It is thus evident and based on research observations that process control for emulsion terpolymerisation is a particularly difficult task because of the lack of validated models and the lack of online measurements of most of polymer properties of interest. Therefore, a well validated model is crucial for optimising and controlling the emulsion terpolymerisation operations allowing for design of the polymer product properties. In this study, a framework for process design and control of emulsion terpolymerisation reactors was developed. This framework consisted of three consecutive stages, dynamic modelling of the process, optimising the process for finding the optimal operating strategies and final online controlling the obtained optimal trajectories through multivariable constrained model predictive control. Within this framework, a comprehensive dynamic model was developed. Then a test case of emulsion terpolymerisation of styrene, methyl methacrylate and methyl acrylate was investigated on state of the art facilities for predicting, optimising and control end-use product properties including global and individual conversions, terpolymer composition, the average particle diameter and concentration, glass transition temperature, molecular weight distribution, the number- and weight-average molecular weights and particle size distribution. The resulting model was then exploited to understand emulsion terpolymerisation behavior and to undertake model-based optimization to readily develop optimal feeding recipes. The model equations include diffusion-controlled kinetics at high monomer conversions, where transition from a ‘zero-one’ to a ‘pseudo-bulk’ regime occurs. Transport equations are used to describe the system transients for batch and semi-batch processes. The particle evolution is described by population balance equations which comprise of a set of integro-partial differential and nonlinear algebraic equations. Backward finite difference approximation method is used to discretise the population equation and convert them from partial differential equations to ordinary differential equations. The model equations were solved using the advanced simulation environment of the gPROMS package. The dynamic model was then used to determine optimal control policies for emulsion terpolymerisation in a semi-batch reactor using the multiobjective dynamic optimisation method. The approach used allows the implementation of constrained optimisation procedures for systems described by complex mathematical models describing the operation of emulsion terpolymerisation reactors. The control vector parameterisation approach was adopted in this work. Styrene monomer feed rate, MMA monomer feed rate, MA monomer feed rate, surfactant feed rate, initiator feed rate and the temperature of reactor were used as the manipulating variables to produce terpolymers of desired composition, molecular weight distribution (MWD) and particle size distribution (PSD). The particle size polydispersity index (PSPI), molecular weight polydispersity index (MWPI) and the overall terpolymer composition ratios were incorporated as the objective functions to optimise the PSD, MWD and terpolymer composition, respectively. The optimised operational policies were successively validated with experiments via one stirred tank polymerisation reactor. Due to the lack of online measurements of key process product attributes for emulsion terpolymerisation, an inferential calorimetric soft sensor was developed based on temperature measurements. The calorimetric soft sensor obtains online measurements of reactor temperature, jacket inlet and outlet temperatures helped estimate the rate of polymerisation. The model includes the mass and energy balance equations over the reactor and its peripherals. Energy balance equations include the heat of reaction, internal and external heat transfer effects, as well as external heat losses. An online multivariable constrained model predictive control was formulated and implemented for online control of the emulsion terpolymerisation process. To achieve this implementation, a novel generic multilayer control architecture for real-time implementation of optimal control policies for particulate processes was developed. This strategy implements the dynamic model for the emulsion terpolymerisation as a real-time soft sensor which is incorporated within the implemented MPC. The methodology was successively validated using six case studies within the on-line control of terpolymerisation reactors. The cases were online controlled the composition of terpolymers, PSD and Mn with specific constraints for the operation conversion and particle average radius. An advanced Supervisory Control Architecture named ROBAS was used in this work. It provides a completely automated architecture allowing for the real time advanced supervisory monitoring and control of complex systems. The real time control application strategy was developed within MATLAB, Simulink, gPROMS and Excel Microsoft softwares and implemented on line through ROBAS Architecture. The manipulated variables are measured using on-line measurements connected to the DCS system through Honeywell. These measurements were sent to MATLAB and then to the dynamic model in gPROMS through an excel spread-sheet interface. Then the dynamic model used them to estimate the controlled variables of the MPC. The estimated values of the controlled variables obtained from the dynamic model, were then sent to the Simulink and fed through the DCS system to the MPC developed in MATLAB. The MPC would then calculate optimal trajectories, which are then sent as set point signals through the DCS system to the regulatory controller. The MPC formulation was found to be robust and handles disturbances to the process. The result showed that the online multivariable constrained MPC controller was able to control the desired composition and Mn as specified set points with great accuracy. The MPC algorithm succeeded under constrained conditions, in driving the PSD to the desired target. Although some offset was observed with a certain degree of model mismatch, the experimental results agreed well with predictions