Chapter
2.2.3 Semi-Batch Reactor Operation
3 Maximization of Reaction Rates and Fed-Batch Operation
3.1 Intuitive Maximization of a Single Reaction Rate
3.1.1 Optimum One-Reactor Operations
3.1.2 Optimum Two-Reactor Operations
3.1.3 One-Reactor Operation Mimicking a Two-Reactor Operation
3.1.3.1 The Optimal Operational Sequence, a Semi-Batch Operation
3.1.4 Rationale for Mimicking Optimal Two-Reactor Operations
3.2 Optimization of Multiple Reactions
3.2.1 Case of Constant Yields
3.2.2 Case of Variable Yields
3.3 Maximization of Cell Mass of a Simple Microbial Fed-Batch Culture
3.3.1 Feed Rate to Maintain the Substrate Concentration That Maximizes the Specific Growth Rate,
3.3.2 Optimal Feed Rate Sequence for a Fed-Batch Culture
4 Phenomena That Favor Fed-Batch Operations
4.1.1 Substrate Inhibition1-10
4.1.2 Glucose (Crabtree) Effect11-15
4.1.3 Catabolite Repression16-41
4.1.4 Utilization of Auxotrophic Mutants42-44
4.2.1 High Cell Density45-52,90
4.2.2 Extension of Operational Period53-55
4.2.3 Alleviation of High Broth Viscosity56,57
4.2.4 Makeup for Lost Water by Evaporation59-61
4.2.5 Better Plasmid Stability of Recombinant Cells64,68-86
4.3.1 Experimental Kinetic Studies63-67
4.3.2 Various Other Situations
5 Classification and Characteristics of Fed-Batch Cultures
5.1 Classification Based on Feeding Patterns
5.1.1 Mass Balance Equations
5.1.5 Overall Mass Balance
5.1.6 Fed-Batch Cultures with Constant Feed Rates
5.1.6.1 Early Growth Phase or Very Small Specific Growth Rate
5.1.6.2 Quasi Steady States
5.1.6.3 Rate Data Acquisition
5.1.7 Fed-Batch Cultures with Linearly Varying Feed Rates
5.1.8 Fed-Batch Cultures with Exponential Feed Rates
5.1.8.1 Constant Dilution Rate
5.1.8.2 Constant Substrate and Cell Concentrations
5.1.9 Extended Fed-Batch Cultures
5.1.10 Fed-Batch Cultures with Intermittent Feed Rates
5.1.11 Fed-Batch Cultures with Empirical Feed Rates
5.1.12 Fed-Batch Cultures with Optimal Feed Rates
5.2 Classification Based on Number of Operational Cycles
5.2.1 Single-Cycle Operations
5.2.2 Multiple-Cycle Operations, Repeated Fed-Batch Operations
6 Models Based on Mass Balance Equations
6.1 Mass Balance Equations
6.1.1 Total Mass Balance Equation
6.1.2 Component Mass Balances
6.1.2.2 Limiting Substrate Balance
6.1.2.3 Key Component Balances
Example 6.E.1: Simplest Fed-Batch Operation, Growing Cells
Example 6.E.2: Simplest Fed-Batch Operation, Metabolite Production
6.2.1 Specific Growth Rate of Cells,
6.2.1.1 Substrate Concentration-Dependent Forms,
6.2.1.2 Substrate and/or Cell Concentration-Dependent Forms,
6.2.1.3 Inhibition Forms,
6.2.1.3.1. Substrate Inhibition .
6.2.1.3.2. Product Inhibition .
6.2.1.3.3. Inhibitor Inhibition .
6.2.2 Specific Product Formation Rate,
6.2.2.1 Constant-Yield Coefficients
6.2.2.2 Variable-Yield Coefficients
6.2.3 Specific Substrate Consumption Rate,
6.2.5 Temperature and pH Effects on Specific Rates
6.2.5.1 Influence of Temperature on Specific Rates
6.2.5.2 pH Effects on Specific Rates
Example 6.E.3: A Structured Model49
6.4.1 All State Variables Are Measurable
6.4.1.2 Estimation Methods
6.4.2 Some State Variables Are Not Measurable but Are Observable
6.4.2.1 Extended Kalman Filter
Example 6.E.4 Estimation by EKF
Appendix: Some Models Proposed in Literature
6.A.1 Cell Mass Fermentation39,40
6.A.2 Lysine Fermentation: A Model of Ohno et al.41
6.A.3 Alcohol Fermentation: A Model of Aiba et al.42
6.A.4 Penicillin Fermentation: A Model of Bajpai and Reuss43
6.A.5 Differential State Model for Penicillin Fermentation by Cagney49
6.A.6 Chitturs Model for Penicillin Fermentation31
6.A.7 Modak-Patkar Model for Invertase Fermentation60
6.A.8 Modified Modak-Patkar Model44 for Invertase Fermentation: Inhibition by Ethanol of Both Invertase Formation and Cell Growth on Glucose
6.A.10 -Amylase Fermentation46
6.A.11 A Poly--hydroxybutyric Acid Model47
6.A.12 Monoclonal Antibodies by Hybridoma Cells48
7 Non-Equation-Based Models
7.1.1 Basic Architecture of Neural Networks
7.1.2 Back-Propagation Training Algorithm
7.2 Neural Networks in Fed-Batch Fermentation
7.2.1 Yeast Fed-Batch Fermentation
7.2.2 Hybrid Neural Networks
8 Specific Rate Determination
8.1 Determination of Specific Rates by Classical Methods
8.1.1 Specific Rates by Shake Flask Cultures
8.1.1.1 Specific Cell Growth Rate,
8.1.1.2 Specific Substrate Consumption Rate,
8.1.1.2.1. Equal-Area Differentiation.
8.1.1.2.2. Numerical Differentiation Formula.
8.1.1.2.3. Polynomial Fit and Analytical Differentiation.
8.1.1.3 Specific Product Formation Rate,
8.1.2 Specific Rates by Batch Cultures
8.1.2.1 Differential Method
Example 8.E.1 Estimation of Specific Rates by Differential Methods
8.1.3 Specific Rates by Continuous Cultures
8.1.4 Specific Rates by Fed-Batch Cultures
8.2 A New Method of Determining Specific Rates Using Fed-Batch Cultures
8.2.1 Constant-Feed Fed-Batch Cultures
Example 8.E.2 Monod Model with Constant Cell Yield Coefficient
Example 8.E.3 Substrate-Inhibited Model with Constant Cell Yield Coefficient
Example 8.E.4 Substrate Inhibition Model with Variable Cell Yield
8.2.2 Utilization of Quasi Steady State
Appendix: Equal-Area Graphical Differentiation of Discrete Experimental Data
9 Optimization by Pontryagins Maximum Principle
9.1 Impulse and Parameter Optimizations
9.2 Optimization Criteria
9.2.1 Performance Indices
9.2.2 Free and Fixed Final Times
9.2.3 Free and Fixed Initial and Final States
9.2.4 Various Constraints
9.3 Choice of Manipulated Variables
9.4 Feed Rate Problem Formulation and Solution
9.4.1 Pontryagins Maximum Principle
9.4.2 Boundary Conditions on Adjoint Variables
9.4.2.1 A Summary of Pontryagins Maximum Principle
9.4.3.1 Nonlinear in Manipulated Variables
9.4.3.2 Linear in Manipulated Variables
9.5 Handling of Problems in Nonstandard Forms
9.5.1 Nonautonomous Processes
9.5.2 Performance Indices Depend Explicitly on the Final Time
9.5.3 Other Forms of Performance Index
9.5.4 Constraints on State Variables
9.5.5 Constraints on Control Variables
9.6 Optimization of Initial Conditions
9.7 Generalized Legendre-Clebsch Condition
9.8 Transformation to Nonsingular Problem36-38
9.8.1 Transformation of Singular Problems into Nonsingular Problems
9.8.2 Substrate Concentration as Single Manipulated Variable
9.8.3 Singular Problems with Multiple Manipulated Variables
10 Computational Techniques
10.1 Computational Techniques for Processes with Known Mathematical Models
10.1.1 Boundary Condition Iterations (Simple Shooting Method)1
10.1.1.1 Iterations on Guessed Initial Values of Adjoint Variables (a Simple Shooting Method)
10.1.1.2 Iterations on Guessed Final Values of State Variables (a Simple Shooting Method)
10.1.2 Multiple Shooting Method
10.1.3 Control Vector Iterations3
10.1.4 Nonlinear Programming
10.1.4.1 Penalty Function Method4
10.1.4.2 Square Quadratic Programming5
10.1.5 A Special Transformation to Convert Singular to Nonsingular Problems7
10.2 Numerical Techniques for Processes without Mathematical Models
10.2.2 Genetic Algorithm10
11 Optimization of Single and Multiple Reactions
11.1 Single Reactions with a Single Feed Rate
11.1.1 Optimal Feed Rate Profile
11.2 Single Reactions with Both Feed and Withdrawal Rates
11.2.1 Solution via Pontryagins Maximum Principle2,3
11.2.2 Switching Space Analyses3
11.3 Optimization of Multiple Reactions with a Feed Rate
11.3.2.1 Formulation and Solution
12 Optimization for Cell Mass Production
12.1 Optimization by Pontryagins Maximum Principle
12.2 Maximization of Cell Mass at Fixed and Free Final Times
12.2.1 Problem Formulation
12.2.2 Solution by Pontryagins Maximum Principle
12.2.2.1 Optimal Feed Rate Profile
12.2.2.2 Factors Influencing Optimal Feed Rate Profile
12.2.3 Constant-Yield Coefficients
12.2.3.1 Free Final Time,
12.2.3.2 On Interior Singular Arc
12.2.3.3 Optimal Feed Rate Profile
12.2.3.4 Cell Lyses or Decay
12.2.3.5 Performance Indices and Optimization
12.2.3.6 Fixed Final Time,
12.2.3.7 Optimal Singular Feed Rate on the Singular Arc
12.2.3.8 The Optimal Singular Feed Rate Is in Feedback Control Form and Is Exponential
12.2.3.9 Optimal Feed Rate Profiles
12.2.3.10 A Special Case of Constant Substrate Concentration
12.2.4 Effects of Operating Parameters
12.2.4.1 Effect of Constraint on Maximum Feed Rate
12.2.4.2 Effect of Final Time
12.2.4.3 Effect of Initial Substrate Concentration
12.2.4.2 Effects of Initial Conditions and Final Time on the Substrate Feed Rate
12.2.5 Variable-Yield Coefficients
12.2.5.1 Singular Feed Rate
12.2.5.3 Feed Rate on the Singular Arc
12.2.5.4 Optimal Feed Rate Profiles
12.2.5.5 Summary for Maximum Cell Production at Free Final Time and for Variable-Yield Coefficient
12.2.5.6 Fixed Final Time
12.2.5.7 Feed Rate on Singular Arc
12.2.6 Assessment of Singular Regions
12.2.6.1 Substrate Concentration Profile during the Singular Feed Rate Period for the Variable-Yield Case
12.2.6.2 Sufficient Conditions for Admissible Singular Feed Rates for Cell Mass Maximization at Fixed Final Time
12.2.7 Singular Regions Characterized by Kinetic Parameters,
12.2.7.1 Summary for Maximum Cell Mass Production at Fixed Final Time for the Case of a Variable-Yield Coefficient
12.2.7.2 Comparison of Free and Fixed Time Problems for Cell Mass Maximization
12.3 Maximization of Cellular Productivity,
12.3.1 Constant Cell Mass Yield Coefficient
12.3.1.1 Optimal Feed Rate on Interior Singular Arc
12.3.1.2 The Optimal Feed Rate Profile for Cellular Productivity Maximization, Constant Yield
12.3.2 Variable Cell Mass Yield Coefficient
12.3.2.1 Feed Rate Profile on Singular Arc
12.4 Cell Mass Productivity Maximization through Time Optimal Formulation
12.4.1 Constant Cell Mass Yield Coefficient and Fixed Final Conditions
12.4.1.1 Feed Rate Profile on Singular Arc
12.4.1.2 Optimum Feed Rate Profile
12.4.1.3 A Target Point versus a Target Set
12.4.2 Variable Cell Mass Yield Coefficient and Fixed Final Conditions
12.4.2.1 Feed Rate Profile on a Singular Arc
12.5 Specific Rates as Functions of Substrate and Cell Concentrations
12.5.2 Constant Cell Mass Yield Coefficient,
12.5.3.1 Optimal Feed Rate Profile
12.5.3.2 Feed Rate on the Singular Arc
13 Optimization for Metabolite Production
13.1 Product Formation Models
13.2 General Optimization Problem for Metabolites
13.2.1 Choice of Manipulated Variables
13.2.2 Substrate Feed Rate as Manipulated Variable
13.2.3 Optimization Problem Formulation
13.3 Necessary Conditions for Optimality for Metabolite Production
13.3.1 Hamiltonian and Adjoint Vector
13.3.2 Optimal Feed Rate for Boundary Arc,
13.3.3 Optimal Feed Rate for Interior Singular Arc,
13.4 Substrate Concentration-Dependent Specific Rates
13.4.1 Constant-Yield Coefficients and No Maintenance Requirement
13.4.1.1 Performance Index Independent of Final Time,
13.4.1.1.1 Free (Unspecified) Final Time, Not in the Performance Index.
13.4.1.1.2 Fixed Final Time, .
13.4.1.1.3 Optimal Feed Rate Profile for Fixed , Constant Yield, .
13.4.1.2 Performance Index Dependent on Free Final Time,
13.4.1.2.1 Minimum Time Problems.
13.4.1.2.2 Optimal Feed Rate Profile for Minimum Time Problem.
13.4.1.2.3 Maximum Productivity Problems.
13.4.2 Variable-Yield Coefficients and Maintenance Requirement
13.4.2.1 Performance Index Independent of Final Time,
13.4.2.2 Free Final Time,
13.4.2.2.1 Case I Performance Index Depends on Only .
13.4.2.2.2 Case II Performance Index Depends on .
13.4.2.3 Optimal Feed Rate Profile
13.4.2.4 Fixed Final Time
13.4.2.4.1 Numerical Determination of the Switching Time .
13.4.2.5 Performance Index Dependent on Final Time
13.4.2.5.1. Minimum Time Problems.
13.4.2.5.2. Maximum Productivity Problems.
13.5 Substrate and Product Concentration-Dependent Specific Rates,
13.6 Recombinant Cell Products
13.6.1 Recombinant Cells with Plasmid Instability
13.6.1.1 Problem Formulation
13.6.1.2 Constant Yields with Growth-Associated Product Formation,
13.6.1.2.1 Singular Feed Rate.
13.6.1.2.2. Free Final Time Not in the Performance Index.
13.6.1.2.3. Free Final Time in the Performance Index.
13.6.1.2.4 Singular Regions in Terms of Specific Rates.
13.6.1.2.4.1 Optimal Feed Rate Sequences for Various Initial Conditions and Variations in Subsrate Concentrations in Singular Feed Period.
13.6.1.2.5. Fixed Final Time Problems.
13.6.1.3 Constant Yields and General Product Formation Rate,
13.6.1.3.1 Free Final Time Not in the Performance Index.
13.6.1.3.2 Final Time in the Performance Index.
13.6.1.4 Variable-Yield Coefficients,
13.6.1.4.1. Free Final Time Not in the Performance Index.
13.6.1.4.2. Free Final Time That Appears in the Performance Index.
13.6.2 Recombinant Cells with Plasmid Instability and Subject to Cell Death
13.6.2.1 Problem Formulation
13.6.2.2 Constant Cell Mass Yield Coefficients, and
13.6.2.2.1 Performance Index Dependent on Free Final Time.
13.6.2.2.2 Performance Index Independent of the Amount of Substrate at the Final Time.
13.6.2.2.3 Analysis of Singular Arc.
13.6.2.2.4. Free Final Time Not in the Performance Index.
13.6.2.2.5 A Special Case, .
13.6.2.3. Feed Rate Policy for Maximizing PBC and Plasmid Stability
13.7.1 Animal Cell Cultures
13.7.2 Transformation of Singular Problems to Nonsingular Problems
13.7.3 Transformation of Singular Problems with Multiple Feed Rates
14 Simple Adaptive Optimization
14.1 Off-Line Cycle-to-Cycle (Sequential) Optimization
14.1.1 Off-Line Cycle-to-Cycle Optimization of Penicillin Production
14.1.1.1 Simulation Studies
14.1.1.2 Experimental Studies
14.1.2 Experimental Off-Line Cycle-to-Cycle Optimization of Invertase Production
14.2 On-Line Adaptive Optimization
14.2.1 Simulation Studies of On-Line Adaptive Optimization of Penicillin Production
14.2.2 Experimental On-Line Adaptive Optimization of Invertase Production
15 Measurements, Estimation, and Control
15.1 Measurements of Process Variables and Parameters
15.1.1 Physical Properties
15.1.2 Chemical Properties
15.1.2.1 Carbon Dioxide Evolution Rate
15.1.2.2 Off-Gas Analyses
15.1.2.5 Conductivity and Ionic Probes
15.1.3 Culture Conditions
15.1.3.1 Enzyme and Microbial Electrodes
15.1.3.2 Biomass Measurements
15.1.3.3 Gas-Liquid Oxygen Transfer
15.2 Estimation Techniques
15.2.1 Macroscopic Balances
15.2.2 Mathematical Estimation Techniques
15.2.2.1 Extended Kalman Filter
15.3 Feedback Control Systems
15.3.1 Single-Loop Control
15.3.1.1 Flow Rate Control Loop
15.3.1.2 Gas Pressure Control
15.3.1.3 Temperature Control
15.3.1.4 pH Control Systems
15.3.1.5 Dissolved Oxygen Control
15.3.2 Controller Selection and Tuning Methods
15.3.2.1 Controller Type Selections
15.3.2.1.1. Controller Tuning Methods.
15.3.3 Multiple-Loop Control
15.3.3.1 Feedforward-Feedback Control
15.3.3.3 Adaptive Control
15.4 Indirect Feedback Control
15.4.1 Carbon Dioxide Evolution Rate
15.4.2 Specific Growth Rates
15.5.1 Optimal Open-Loop Control
15.5.2 Optimal Closed-Loop (Feedback) Control
16 Feasibility Assessment and Implementable Feed Rates
16.1 Estimation of Specific Rates
16.1.1 Shake-Flask and Batch Experiments
16.1.2 Fed-Batch Operations
16.2 Sequential Approach to Feasibility Assessment
16.3 Implementable Optimal-Suboptimal Feed Rates
16.3.1 Cell Mass as Product
16.3.1.1 Specific Growth Rate Optimization
16.3.1.2 Yield Coefficient Optimization
16.3.1.3 Optimization of Specific Growth Rate and Yield Coefficient
16.3.2 Metabolites as Product
16.3.2.1 Constant-Yield Coefficients, , and without Maintenance,
16.3.2.1.1 Performance Index Independent of Free Final Time, .
16.3.2.1.2 Performance Index Dependent on Free Final Time, , Minimum Time Problem.
16.3.2.1.3 Maximum Productivity Problem.
16.3.2.2 Variable-Yield Coefficients and without Maintenance
16.3.2.2.1 Free Final Time.
16.3.2.2.2 Fixed Final Time.
16.3.2.3 Variable-Yield Coefficients and with Maintenance Requirement
16.3.2.3.1 Free Final Time, .
16.3.2.3.2 Maximum Productivity.
16.3.2.4 Intuitive Suboptimal Policy for Nonmonotonic Specific Rates
16.3.3 Recombinant Cell Products
16.3.3.1 Constant Yields and Growth-Associated Product Formation
16.3.3.1.1 Free Final Time.
16.3.3.1.2 Fixed Final Time.
Fed-Batch Cultures
：Principles and Applications of Semi-Batch Bioreactors
( Cambridge Series in Chemical Engineering )
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