Residue curve maps have been developed for membrane permeation systems. A "thought experiment" was conducted whereby a fluid mixture was allowed to permeate through a diffusion membrane in a batch still. The differential equations which describe any residue curve are derived from mass balances, implying that membrane residue curve maps (M-RCM's) can be used for any type of membrane, provided the appropriate permeation (flux) model is used. In this paper, a simple constant relative permeability model is used for demonstration purposes, and as an example, an M-RCM is plotted for methanol/butene/methyl tertiary butyl ether (MTBE) using the appropriate flux data. The curves in an M-RCM trace the change in composition of the residual retentate in the batch still. At any time during batch permeation, the permeate composition is related to that of the retentate at that particular time, thus allowing a membrane separation Vector to be defined as the vector difference between the two compositions.

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Abstractiii ABSTRACT This work applies the concepts of the attainable region for process synthesis in comminution. The attainable region analysis has been successfully applied for process synthesis of reactor networks. The Attainable Region is defined as the set of all possible output states for a constrained or unconstrained system of fundamental processes . A basic procedure for constructing the attainable region for the fundamental processes of reaction and mixing has been postulated in reaction engineering . This procedure has been followed in this work to construct the candidate attainable region for size reduction processes as found in a size reduction environment.A population balance model has been used to characterise the evolution of particle size distributions from a comminution event. postulated the dependency of grinding on the specific energy. A specific energy dependent population balance model was used for the theoretical simulations and for the fitting of experimental data.A new method of presenting particle size distributions as points in the Euclidian space was postulated in place of the traditional cumulative distribution. This allows successive product particle size distributions to be connected forming a trajectory over which the objective function can be evaluated. The curve connects products from successive batch grinding stages forming a pseudo-continuous process.Breakage, mixing and classification were identified as the fundamental processes of interest for comminution. Agglomeration was not considered in any of the examples. Mathematical models were used to describe each fundamental process, i.e. breakage, mixing and classification, and anThe application of the attainable region analysis in comminution Abstract iv algorithm developed that could calculate the evolution of product particle size distributions. A convex candidate attainable region was found from which process synthesis and optimisation solutions could be drawn in two dimensional Euclidian space. As required from Attainable Region Theory, the interior of the bounded region is filled by trajectories of higher energy requirements or mixing between two boundary optimal points.Experimental validation of the proposed application of the attainable region analysis results in comminution was performed. Mono-sized feed particles were broken in a laboratory ball mill and the products were successfully fitted using a population balance model. It was shown that the breakage process trajectories were convex and they follow first order grinding kinetics at long grind times. The candidate attainable region was determined for an objective function to maximise the mass fraction in the median size class 2. It was proved that the same specific energy input produces identical products. The kinematic and loading conditions are supposed to be chosen as a subsequent event after the required specific energy is identified.Finally the fundamental process of classification was added to the system of breakage and mixing. The attainable regions analysi...

Membrane residue curve maps (M-RCMs) [Peters et al. Ind. Eng. Chem. Res. 2006. 45, 9080] plot the change, over time, of the retentate composition in a batch still. In this paper, the relevance of M-RCMs to both batch and continuous processes is investigated. A constant relative permeability model is used for demonstration purposes, and the theory is then extended to a real system of methanol/butene/methyl tertiary butyl ether (MTBE). It has been shown that the differential material balances over a continuously operated membrane unit are mathematically equivalent to those which describe the M-RCM. The time variable in the batch setup is analogous to the spatial variable in continuous units. The retentate composition in a nonreflux continuous unit, for example, will follow the residue curve that passes through the initial feed composition. The M-RCM, in conjunction with the necessary flux equations, allows a designer to determine permeation area (or time) required in a continuous (or batch) process. Membrane columns operating at total and infinite reflux are discussed. A novel approach in synthesizing and designing hybrid distillation-membrane processes emerges: using the M-RCM in conjunction with column profile maps (CPMs) allows one to graphically interpret hybrids in an efficient manner. The method generates the attainable region (AR) for a chosen configuration and informs the designer of parameters like membrane area and number of distillation stages required.

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