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1 Membrane Processes
1.1 Membrane Separation
1.1.1 Pore Sizes
1.1.2 Operation Modes
1.1.3 Modules
1.1.4 Configurations
1.1.5 Fouling of Membranes
1.2 Mathematical Modelling of Membrane Processes
1.3 Diafiltration Process
1.3.1 Process Model
1.3.2 Fouling Models
1.3.3 Operational Modes of Diafiltration
1.3.4 Optimisation of Diafiltration Process
{section}{References}{25}{section*.95}
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2 Optimal Control Problem
2.1 Objective Functional
2.1.1 Typical Optimal Control Tasks
2.2 Constraints
2.3 Process Model
2.3.1 Linear Time-Invariant System
2.3.2 Input Affine System
2.4 Summary of Problem Definition
{section}{References}{34}{section*.145}
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3 Solution of Optimal Control Problems
3.1 Necessary Conditions for Optimality
3.2 Analytical Methods
3.2.1 Calculus of Variations
3.2.2 Dynamic Programming
3.2.3 Pontryagin's Minimum Principle
3.3 Numerical Methods
3.3.1 Control Vector Iteration
3.3.2 Boundary Condition Iteration
3.3.3 Complete Discretisation
3.3.4 Control Vector Parametrisation
3.3.5 Direct Multiple Shooting
3.4 Methods for Computing Gradients
3.5 Feedback Strategies for Optimal Control
3.5.1 Model Predictive Control
{section}{References}{53}{section*.228}
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4 Operation at Limiting Flux
4.1 Process Model and Definition of Optimisation Problem
4.1.1 Filtration Modes
4.1.2 Optimisation Problem
4.2 Optimal Operation
4.2.1 Numerical Results
4.2.2 Theoretical Results
4.2.3 Discussion
4.3 Case Studies
4.3.1 Example 1
4.3.2 Separation of Pectin from Sugar
4.3.3 Purification of Soybean Water Extracts
4.4 Models Derived from Limiting Flux
4.4.1 Viscosity Dependent Mass Transfer Coefficient
4.4.2 Osmotic Pressure Model
{section}{References}{79}{section*.374}
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5 Perfect Rejection of Both Solutes
5.1 Optimal Operation
5.2 Case Studies
5.2.1 Separation of Lactose from Proteins
5.2.2 Albumin -- Ethanol Separation
{section}{References}{105}{section*.511}
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6 Perfect Rejection of Macro-Solute
6.1 Optimal Operation
6.1.1 Minimum Time Operation -- Alternative Derivation
6.2 Case Studies
6.2.1 Dye -- Salt Separation
6.2.2 Radiopaque -- Ethylene Glycol Separation
6.2.3 Sucrose -- Sodium Chloride Separation
{section}{References}{125}{section*.654}
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7 Constant Incomplete Rejection of Solutes
7.1 Optimal Operation
7.2 Case Studies
7.2.1 Extended Limiting Flux Model
7.2.2 Three Component Separation
{section}{References}{137}{section*.728}
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8 General Membrane Model
8.1 Optimal Operation
8.1.1 Singular Control
8.2 Case Studies
8.2.1 Radiopaque -- Ethylene Glycol Separation
8.2.2 Separation of Peptide from Trifluoroacetic Acid
{section}{References}{149}{section*.782}
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9 Conclusions and Future Research
9.1 Discussion
9.2 Conclusions
{section}{References}{153}{section*.786}
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Index