New Paradigms in Flow Battery Modelling

Contents
1 Introduction
	1.1 Motivation and Outline of this Book
	1.2 Why Energy Storage?
	1.3 Types of Energy Storage
		1.3.1 Chemical Energy Storage
		1.3.2 Electrical Energy Storage
		1.3.3 Mechanical Energy Storage
		1.3.4 Thermal Energy Storage
		1.3.5 Electrochemical Energy Storage
	1.4 Summary
	References
2 Electrochemical Theory and Overview  of Redox Flow Batteries
	2.1 Introduction
	2.2 Properties of Redox Flow Batteries
	2.3 Fundamental Electrochemical Principles of Flow Batteries
		2.3.1 Redox Reactions at the Electrodes
		2.3.2 Faraday's Law
		2.3.3 Thermodynamics and Nernst's Equation
		2.3.4 Charge-Transfer Reaction
		2.3.5 An Electrode Surface Under Equilibrium Conditions
		2.3.6 An Electrode Surface Under Non-equilibrium Conditions
		2.3.7 Mass Transport
		2.3.8 Migration
		2.3.9 Diffusion
		2.3.10 Convection-Diffusion
	2.4 Brief Overview of Redox Flow Battery Developments
	2.5 Types of Flow Batteries
		2.5.1 Systems with Energy Stored on the Electrodes
		2.5.2 Hybrid Flow Batteries
	2.6 Design Considerations and Components of Flow Batteries
		2.6.1 Construction Materials
		2.6.2 Electrode Materials
		2.6.3 Carbon-Based Electrodes
		2.6.4 Metal-Based Electrodes
		2.6.5 Composite Electrodes
		2.6.6 Membranes
		2.6.7 Commercially Available Membranes
		2.6.8 Modified and Composite Membranes
		2.6.9 Flow Distributor and Turbulence Promoter
	2.7 Current Developments in Flow Batteries
		2.7.1 Electrolyte Formulation
		2.7.2 Improvement in Battery Efficiencies
		2.7.3 Electrical Distribution System
	2.8 Prototypes of Redox Flow Batteries
	2.9 Applications of Redox and Hybrid Flow Batteries
	2.10 Summary
	References
3 Modelling Methods for Flow Batteries
	3.1 Introduction
	3.2 Overview of Available Physics-Based Modelling Approaches
	3.3 Macroscopic Modelling
		3.3.1 Eulerian and Lagrangian Descriptions
		3.3.2 Conservation Laws
		3.3.3 Conservation of Multiple Charged and Neutral Species
		3.3.4 Flow in Porous Media
		3.3.5 Transport of Water and Ions in Membranes
		3.3.6 Charge Balances
		3.3.7 The Volume-of-Fluid Method
		3.3.8 The Level-Set Method
		3.3.9 Arbitrary Lagrangian Eulerian Methods
		3.3.10 Immersed Boundary Methods
	3.4 Mesoscopic Models
		3.4.1 Phase-Field Models
		3.4.2 Kinetic Theory Models
		3.4.3 The Lattice-Boltzmann Model
	3.5 Molecular Dynamics Simulations
		3.5.1 Interatomic Potentials
		3.5.2 Force Fields and Molecular Mechanics
		3.5.3 Ensembles and Statistical Averages
		3.5.4 The Micro-canonical Ensemble and Macroscopic Observables
		3.5.5 Solving the Hamiltonian System
		3.5.6 Thermostats and Other Ensembles
	3.6 Quantum Mechanical Calculations
		3.6.1 Background in Many-Body Quantum Theory
		3.6.2 Hartree-Fock, Semi-empirical and Post-Hartree-Fock Methods
		3.6.3 Hohenberg-Kohn and Levy-Leib Formulations and Functionals
		3.6.4 Kohn-Sham Density Functional Theory
		3.6.5 Exchange-Correlation Functional Hierarchy
	3.7 Data Driven or Machine Learning Approaches
		3.7.1 Surrogate Models
		3.7.2 Design-of-Experiment and Data Generation
		3.7.3 Data Normalisation
		3.7.4 Basic Framework for Supervised Machine Learning
	3.8 Summary
	References
4 Numerical Simulation of Flow  Batteries Using a Multi-scale Macroscopic-Mesoscopic Approach
	4.1 Introduction
	4.2 Macroscopic Modelling Approaches
		4.2.1 Conservation of Momentum and Fluid Flow
		4.2.2 Conservation of Mass
		4.2.3 Conservation of Charge
		4.2.4 Equations Specific to the Membrane
		4.2.5 Conservation of Thermal Energy
		4.2.6 Electrochemical Kinetics
		4.2.7 Reservoirs and Inlet Conditions
	4.3 Lattice-Boltzmann Models
	4.4 Pore Structure of Electrode
	4.5 Boundary Conditions
	4.6 Validation and Numerical Details
	4.7 Analysis of the Performance of a Vanadium-Iron Flow Battery
		4.7.1 Influence on Flow Field
		4.7.2 Influence on Electrochemical Performance
		4.7.3 Effect of Electrode Structures and Feeding Modes
	4.8 Summary
	References
5 Pore-Scale Modelling of Flow Batteries and Their Components
	5.1 Introduction
	5.2 Pore-Scale Modelling: Averaging Over Space
	5.3 Transport Phenomena
	5.4 Mathematical Framework
		5.4.1 Multiphase Model and Closure
	5.5 Numerical Procedure for Pore-Scale Simulations
		5.5.1 Geometry Reconstruction
		5.5.2 Numerical Reconstruction
		5.5.3 Size of Representative Elementary Volume
		5.5.4 Pore-Scale Models
		5.5.5 Multiple Relaxation Time Lattice-Boltzmann Model
		5.5.6 Solid Mechanics
	5.6 Results and Discussion
		5.6.1 Reconstruction
		5.6.2 Explicit Dynamics Simulation of Compression
		5.6.3 Computed Effective Transport Properties
		5.6.4 Combining Models at Different Scales
	5.7 Summary
	References
6 Machine Learning for Flow Battery Systems
	6.1 Introduction
	6.2 Linear Regression
	6.3 Regularised Linear Regression
	6.4 Locally Linear and Locally Polynomial Regression
	6.5 Bayesian Linear Regression
		6.5.1 The Evidence Approximation for Linear Regression
	6.6 Kernel Regression
	6.7 Univariate Gaussian Process Models
	6.8 Approximate Inference for Gaussian Process and Other Bayesian Models
		6.8.1 Laplace's Method
		6.8.2 Mean Field Variational Inference
		6.8.3 Markov Chain Monte Carlo
	6.9 Support Vector Regression
	6.10 Gaussian Process Models for Multivariate Outputs
		6.10.1 Intrinsic Coregionalisation Model
		6.10.2 Dimensionally Reduced Model
	6.11 Other Approaches to Modelling Random Fields
		6.11.1 Tensors and Multi-arrays
		6.11.2 Tensor-Variate Gaussian Process Models
		6.11.3 Tensor Linear Regression
	6.12 Neural Networks and Deep Learning for Regression and Classification
		6.12.1 Multi-layer Perceptron
		6.12.2 Convolutional Networks
		6.12.3 Recurrent Networks
		6.12.4 Bi-directional Recurrent Networks
		6.12.5 Encoder-Decoder Models
		6.12.6 The Attention Mechanism
	6.13 Linear Discriminant Classification and Support Vector Machines
	6.14 Linear Dimension Reduction
		6.14.1 Principal Component Analysis and the Singular Value Decomposition
		6.14.2 Multidimensional Scaling
		6.14.3 Reduced Rank Tensor Decompositions
	6.15 Manifold Learning and Nonlinear Dimension Reduction
		6.15.1 Kernel Principal Component Analysis
		6.15.2 Isomap
		6.15.3 Diffusion Maps
		6.15.4 Local Tangent Space Alignment
		6.15.5 The Inverse Mapping Problem in Manifold Learning
		6.15.6 A General Framework for Gaussian Process Latent Variable Models and Dual Probabilistic PCA
	6.16 K-means and K-Medoids Clustering
	6.17 Machine Learning-Assisted Macroscopic Modelling
	6.18 Machine Learning-Assisted Mesoscopic Models
	6.19 Machine Learning Models for Material Properties
		6.19.1 Introduction to Quantitative Structure-Activity Relationship Models
		6.19.2 Examples of Redox Potential and Solubility Estimation
	6.20 Summary
	References
7 Time Series Methods and Alternative Surrogate Modelling Approaches
	7.1 Introduction
	7.2 Multi-fidelity Models
		7.2.1 Multi-fidelity Data
		7.2.2 Autoregressive Models Based on Gaussian Processes
		7.2.3 Residual Gaussian Process Model
		7.2.4 Stochastic Collocation for Multi-fidelity Modelling
	7.3 Reduced Order Models
		7.3.1 Discretisations and Galerkin Projections onto a Subpsace
		7.3.2 Proper Orthogonal Decomposition via Karhunen-Loeve Theory
		7.3.3 Generalisations of POD Based on Alternative Hilbert Spaces
		7.3.4 Temporal Autocovariance Function and the Method of Snapshots
		7.3.5 Parameter Dependence
		7.3.6 Nonlinearity and the Discrete Empirical Interpolation Method
	7.4 Time Series Methods
		7.4.1 Basic Approaches and Data Embedding
		7.4.2 Autoregressive Integrated Moving Average Models
		7.4.3 Nonlinear Univariate Gaussian Process Autoregression
		7.4.4 Autoregression Networks
		7.4.5 Gaussian Process Dynamical Models
		7.4.6 Adjusting for Deterministic Trends and Seasonality
		7.4.7 Tests for Stationarity
		7.4.8 Autocorrelation and Partial Autocorrelation Analyses
	7.5 Multi-fidelity Modelling for Electrochemical Systems
	7.6 Summary
	References
8 Summary and Outlook
Appendix A Solving Linear Systems
A.1  Linear Systems
A.2  Gauss Elimination
A.3  Ill-Conditioned Systems and Pivoting
A.4  Gauss-Jordan Method
A.5  LU Decomposition
A.6  Solving Linear Systems with LU Decomposition
A.7  Iterative Methods
Appendix B Solving Ordinary Differential Equations
B.1  Ordinary Differential Equations
B.2  Error Analysis
B.3  Predictor-Corrector Methods
B.4 Runge-Kutta Methods
B.5 Second-Order Runge-Kutta Methods
B.6 Third-Order and Higher Runge-Kutta Methods
B.7 Adaptive Runge-Kutta
B.8 Multi-step Methods
B.9 Predictor-Corrector Methods Based  on Adams-Bashforth and Adams-Moulton
Appendix C Solving Partial Differential Equations
C.1 Partial Differential Equations
C.2 Finite Difference Method
C.3 Finite Difference Method for a 1D Hyperbolic Equation
C.4 von Neumann Stability Analysis
C.5 The Lax-Friedrichs and Leapfrog Methods
C.6 Finite Difference Method for Parabolic Equations
C.7 The θ Method
C.8 Other Boundary Conditions
C.9 Solving the Linear System
C.10 Finite Difference Method for Elliptic PDEs
C.11 Successive Over-Relaxation
C.12 The Finite-Volume Method
Appendix D Gradient-Based Methods for Optimisation
D.1 Outline of Optimisation
D.2 Golden Section Search
D.3 Newton's Method
D.4 General Gradient-Based Methods
D.5 Steepest Descent
D.6 Barzilai and Borwein Method
D.7 Backtracking
D.8 Newton-Raphson and Damped Newton Methods
D.9 Quasi-Newton Methods
D.10 Symmetric Rank 2 Updates
	References