Wavelet based dynamic modelling of simulated moving bed chromatographic processes
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Simulated moving bed chromatography process (SMBCP) is the technical realisation of a countercurrent adsorption process through the cyclic port switching. SMB technology reduces the cost of packing material with high loading capacity and provides high purity and high recovery in a very short time. Major commodity applications have been found in the petroleum, food, biotechnology, pharmaceutical and fine chemical industries. The industrial applications bring an emergent demand to improve the SMBCP operation for higher product quality, productivity, efficiency and robustness. However, for this particular process, we encounter several challenges. Firstly, the interplay of the effects of strong nonlinearities, competition of solutes, mass transfer resistance and fluid dynamic dispersion produces steep concentration fronts. Mathematical model accounted for this particular property constitutes a serious difficulty for the solution procedure. Secondly, a dynamic SMB model consists of a set of partial differential, ordinary differential and algebraic equations, which are highly coupled. The large size is a problem due to its intensive computation when on-line optimisation and real-time control are necessary. Thirdly, the SMB unit operation exhibits complex dynamics. Process metrics for design and operation can be determined only when a cyclic steady state is reached after a certain number of switching. Achieving this steady state by solving the PDE models cycle after cycle involves expensive calculation. Studies have been carried out to solve these problems through process analysis, investigation on spatial discretisation techniques, and development of an accelerated integration scheme.Through a systematic study on the advances of SMB modelling, design and control, a set of functionally equivalent models for SMBCP are identified and summarized for their practical applications. The limitations of the existing modelling techniques in industrial applications are also identified. Furthermore, structural analysis of the existing models is conducted for a better understanding of the functionality and suitability of each model. Suggestions are given on how to choose an appropriate model with sufficient accuracy while keeping the computational demand reasonably low for real time control.Effort is made on to the systematic investigation of different numerical methods for the solution of PDEs to circumvent the steep gradients encountered in chromatographic separation. Comprehensive studies are conducted on a single column chromatographic process represented by a transport-dispersive-equilibrium linear model. Numerical solutions from the upwind-1 finite difference, wavelet-collocation, and high resolution methods are evaluated by quantitative comparisons with the analytical solution for a range of Peclet numbers. It reveals that for a PDE system with a low Peclet number, all existing numerical methods work well, but the upwind finite difference method consumes the most time for the same degree of accuracy of the numerical solution. The high resolution method provides an accurate numerical solution for a PDE system with a medium Peclet number. The wavelet collocation method is capable of catching up steep changes in the solution, and thus can be used for solving PDE models with high singularity.The advantages and disadvantages of the wavelet based approaches are further investigated through several case studies on real SMBCP system. A glucose-fructose separation process is firstly chosen with its relatively simple isotherm representations. Simulations are conducted using both wavelet collocation and upwind finite difference methods. For more complicated applications, an enantiomers separation process is selected. As the PDEs model exhibit a certain degree of singularity, wavelet collocation and high resolution methods are adopted for spatial discretisation. It is revealed that both the wavelet based approaches and high resolution methods are good candidates in terms of computation demand and prediction accuracy on the steep front. This is the first time that these two frontier numerical methods are used for such a complex SMB system models and our results are encouraging for the development of model-based online control scheme.In developing a new scheme to rapidly obtain the solution at steady state for any arbitrary initial condition, the concept of Quasi-Envelope (QE) is adopted under the consideration that a SMBCP can be treated as a pseudo-oscillatory process because of a large number of continuous switching. The scheme allows larger steps to be taken to predict the slow change of starting state within each switching. Combined with previously developed wavelet-based technique, this method is successfully applied to the simulation of a SMB sugar separation process. Investigations are also carried out on the location of proper starting point for the algorithm and on the effect of changing stepsize to the convergence of iteration method. It is found that if the starting state of Quasi-Envelope is chosen to be the same as the original function, the multivalue algorithm would require similar computational effort to achieve the steady state prediction, regardless of the integration stepsize. If using constant stepsize, launching QE later is helpful when quasi-envelope displays steep change at the start-up period. A changing stepsize produces slow convergence compared to the constant stepsize strategy, thus increasing the work load where the stepsize change is occurring. Other iteration method is required to be imposed to achieve faster convergence right from the beginning. Potential applications can be seen for other chemical engineering processes with inherent cyclic behaviour.
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