Mathematical modelling of vanadium redox...

IN DEGREE PROJECT ENGINEERING MATERIALS SCIENCE 120, SECOND CYCLECREDITS

, STOCKHOLM SWEDEN 2015

Mathematical modelling ofvanadium redox batteries

MILTON DE OLIVEIRA ASSUNÇÃO JUNIOR

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF INDUSTRIAL ENGINEERING AND MANAGEMENT

SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP

Data de Depósito: Assinatura:_______________________

Milton de Oliveira Assunção Junior

Modelagem matemática de baterias redox de vanádio

Dissertação apresentada ao Instituto de Ciências Matemáticas e de Computação - ICMC-USP, como parte dos requisitos para obtenção do título de Mestre em Ciências - Ciências de Computação e Matemática Computacional. VERSÃO REVISADA

Área de Concentração: Ciências de Computação e Matemática Computacional

Orientador: Prof. Dr. José Alberto Cuminato

USP – São Carlos Junho de 2015

Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassie Seção Técnica de Informática, ICMC/USP,

com os dados fornecidos pelo(a) autor(a)

Junior, Milton de Oliveira Assunção

J634m Modelagem matemática de baterias redox de vanádio

/ Milton de Oliveira Assunção Junior; orientador José

Alberto Cuminato. – São Carlos – SP, 2015.

69 p.

Master dissertation (Master student – Programa de

Pós-Graduação em Ciências de Computação e Matemática

Computacional) – Instituto de Ciências Matemáticas e de

Computação, Universidade de São Paulo, 2015.

1. modelagem matemática. 2. redução assintótica.

3. baterias redox de vanádio. I. Cuminato, José Alberto,

orient. II. Título.

Milton de Oliveira Assunção Junior

Mathematical modelling of vanadium redox batteries

Master dissertation submitted to the Instituto de Ciências Matemáticas e de Computação - ICMC-USP, in partial fulfillment of the requirements for the degree of the Master Program in Computer Science and Computational Mathematics. FINAL VERSION

Concentration Area: Computer Science and Computational Mathematics

Advisor: Prof. Dr. José Alberto Cuminato

USP – São Carlos June 2015

ACKNOWLEDGEMENTS

I would like to acknowledge the São Paulo Research Foundation – FAPESP for the

financial support provided to the this research in the form of a Master’s Degree grant identified

by the process n.2013/15875-2 and the grant conceded in the Program of Research Internships

Abroad (BEPE) n.2014/03787-4 from the same foundation.

Foremost, I would like to gratefully thank Dr. Michael Vynnycky, professor and resear-

cher from the Royal Institute of Technology (KTH), who supervised the progress of this research

throughout its development including the period of internship in that university. Without his

valuable contributions and guidance it would not have been possible to conduct this research.

Also, I thank the Royal Institute of Technology for granting me access to the housing

and research facilities during the period of internship.

RESUMO

ASSUNÇÃO JUNIOR, M. O.. Modelagem matemática de baterias redox de vanádio. 2015.69 f. Master dissertation (Master student em em Ciências – Ciências de Computação e Matemá-tica Computacional) – Instituto de Ciências Matemáticas e de Computação (ICMC/USP), SãoCarlos – SP.

A modelagem matemática por meio de equações diferenciais é uma importante ferramenta para

prever o comportamento de baterias redox de vanádio, pois ela pode contribuir para o aperfei-

çoamento do produto e melhor entendimento dos princípios da sua operação. Os estudos de

modelagem podem ser aliados à análise assintótica no intuito de promover reduções ou simplifica-

ções que tornem os modelos menos complexos, isso é feito a partir da observação da importância

que cada termo exerce sobre as equações. Tais simplificações são úteis neste contexto, visto que

os modelos geralmente abordam uma célula apenas - a menor unidade operacional da bateria

- enquanto aplicações reais exigem o uso de dezenas ou centenas delas implicando em uma

maximização do uso de recursos computacionais. Neste trabalho, foram investigadas múltiplas

formas de reduções assintóticas que empregadas na construção dos modelos puderam acelerar o

tempo de processamento em até 2,46 vezes ou reduzir os requisitos de memória principal em até

11,39%. As simulações computacionais foram executadas pelo software COMSOL Multiphysics

v. 4.4, e também por scripts desenvolvidos em ambiente de programação MATLAB. A valida-

ção dos resultados foi feita comparando-os a dados experimentais presentes na literatura. Tal

abordagem permitiu também validar as rotinas implementadas para a simulação dos modelos

comparando suas soluções com aquelas providas pelo COMSOL.

Palavras-chave: modelagem matemática, redução assintótica, baterias redox de vanádio.

ABSTRACT

ASSUNÇÃO JUNIOR, M. O.. Modelagem matemática de baterias redox de vanádio. 2015.69 f. Master dissertation (Master student em em Ciências – Ciências de Computação eMatemática Computacional) – Instituto de Ciências Matemáticas e de Computação (ICMC/USP),São Carlos – SP.

Mathematical modelling using differential equations is an important tool to predict the behavior

of vanadium redox batteries, since it may contribute to improve the device performance and lead

to a better understanding of the principles of its operation. Modelling can be complemented

by asymptotic analysis as a mean to promote reductions or simplifications that make models

less complex. Such simplifications are useful in this context, whereas these models usually

addresses one cell only – the smallest operating unit – while real applications demand tens

or hundreds cells implying on larger computational requirements. In this research, several

options for asymptotic reductions were investigated and, applied to different models, were able

to speed up the processing time in 2.46× or reduce the memory requirements up to 11.39%. The

computational simulations were executed by COMSOL Multiphysics v.4.4, also by in-house

code developed in MATLAB. The validation of results was done by comparing it to experimental

results available in literature. Additionally, correlating the results provided by COMSOL with

the ones arising from the implemented sub-routines allowed to validate the developed algorithm.

Key-words: mathematical modelling, asymptotic reduction, vanadium redox batteries.

LIST OF ILLUSTRATIONS

Figure 1 – Fundamental operation scheme for vanadium redox battery. Source: (VYNNYCKY,

2011) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Figure 2 – Internal operation in a VRB during charge cycle. Source: (VYNNYCKY,

2011) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Figure 3 – Current density, i, in absolute values as function of porosity, ε , for several

cell potentials, Ecell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 4 – Current density, i, as function of cell potential, Ecell, for different mesh

refinements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Figure 5 – Over-potential values at the positive electrode and at the negative electrode,

ηc and ηa, as function of cell potential, Ecell. . . . . . . . . . . . . . . . . . 34

Figure 6 – Unidimensional domain used in Model D, the measures hcoll, hf, and hm are

present in Table 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Figure 7 – Operation diagram of the quasi-steady model with unidimensional domain.

On the bottom lines can be seen the software used on each stage. . . . . . . 46

Figure 8 – Stencil for time discretization using BDF. . . . . . . . . . . . . . . . . . . 47

Figure 9 – Stencil for spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . 48

Figure 10 – Electrolyte velocity on Y -direction in transversal cut to the positive and

negative electrodes, at X = L/2. . . . . . . . . . . . . . . . . . . . . . . . . 52

Figure 11 – Electrolyte velocity on X-direction in longitudinal cut to the positive and

negative electrodes, at Y = hcoll +hf/2 e Y = hcoll +hf +hm +hf/2. . . . . . 53

Figure 12 – Comparison between charge and discharge curve for model B . . . . . . . . 54

Figure 13 – H+ concentration at the positive electrode after 6500s of charge for model B 55

Figure 14 – H+ concentration at positive electrode after 13000s of charge for Model B . 55

Figure 15 – H+ concentration at positive electrode after 5500s of discharge for Model B 56

Figure 16 – H+ concentration at positive electrode after 13000s of discharge for Model B 56

Figure 17 – V3+ concentration at negative electrode after 10s of charge for Model B . . 57

Figure 18 – V3+ concentration at negative electrode after 6500s of charge for Model B . 57

Figure 19 – V3+ concentration at negative electrode after 13000s of charge for Model B 58

Figure 20 – V3+ concentration at negative electrode after 5500s of discharge for Model B 58

Figure 21 – V3+ concentration at the negative electrode after 13000s of discharge for

Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Figure 22 – Comparison between charge-discharge curve for Model C against experimen-

tal data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Figure 23 – Comparison between charge-discharge curve for Models B and E . . . . . . 60

Figure 24 – Comparison between charge-discharge curve for the Model E implemented

in COMSOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Figure 25 – Comparison between charge-discharge curve for the Model E implemented

in MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

LIST OF SYMBOLS

ε — Porosity of carbon felt

ω — Volumetric electrolyte flux

φe — Liquid phase potential

φm — Membrane potential

φs — Solid phase potential

σcoll — Electrical conductivity of the current collector

σm — Electrical conductivity of the membrane

σs — Electrical conductivity of the carbon felt

A — Specific electroactive area

c0k — Initial concentration of specie k

df — Carbon felt fiber diameter

Deffk — Effective diffusion coefficient in electrode of specie k

Dmk — Effective diffusion coefficient in membrane k

Dk — Diffusion coefficient of specie k

F — Faraday constant

hcoll — Thickness of current collectors

hf — Thickness of the carbon felt

hm — Thickness of the membrane

icoll — Superficial current density vector in the collector

ie — Superficial current density vector in the liquid phase

is — Superficial current density vector in the solid phase

kφ — Electrokinetic permeability in membrane

km — Hydraulic permeability in membrane

L — Porous electrode length

Lt — Porous electrode width

Nk — Molar flux of specie k

R — Gas constant

Sk — Source term of specie k

T — Temperature in Kelvin

uin — Inlet electrolyte velocity

V — Electrolyte volume in each half-cell

zk — Charge number of specie k

SUMMARY

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1 Energy Storage Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 Operation and electrochemical principles of VRBs . . . . . . . . . . 16

1.3 Performance and applications of VRBs . . . . . . . . . . . . . . . . . 17

1.4 Research description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.1 General goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.2 Specific goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.3 Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1 Mass transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Butler-Volmer equation . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Mathematical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Porose carbon electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2 Proton-exchange membrane . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.3 Current collectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.4 Interfacial and initial conditions . . . . . . . . . . . . . . . . . . . . . . 29

2.3.5 Previous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 MATERIALS AND METHODS . . . . . . . . . . . . . . . . . . . . 35

3.1 Mathematical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Asymptotic reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.1.1 Electrolyte velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.1.2 Long and thin geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1.1.3 Quasi-steady nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.1.4 Boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.1.5 Cross-contamination by ions of vanadium . . . . . . . . . . . . . . . . . . 42

3.1.2 Computational simulations . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.2.1 Model A: bidimensional transient formulation . . . . . . . . . . . . . . . . 44

3.1.2.2 Model B: bidimensional transient formulation with constant velocity . . . . 45

3.1.2.3 Model C: bidimensional quasi-steady formulation with constant velocity . . 45

3.1.2.4 Model D: unidimensional quasi-steady formulation with constant velocity,

and long and thin geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1.2.5 Model E: quasi-steady formulation with consnta velocity, long and thin

geometry, and bidimensional domain. . . . . . . . . . . . . . . . . . . . . . 47

4 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . 51

4.1 Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Model C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4 Model D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5 Model E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.6 Computational resources usage . . . . . . . . . . . . . . . . . . . . . . 61

5 FINAL CONSIDERATIONS . . . . . . . . . . . . . . . . . . . . . . . 65

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

CHAPTER

1

INTRODUCTION

This section talks about the strategic role of energy storage systems in intelligent models

of consumption, provides an introduction to the operation of a vanadium redox batteries (VRBs),

describes its structure, chemical reactions involved in the process of energy transformation,

shows a performance analysis compared to other batteries and lists some of the applications that

already use this technology. Then the details of the research will be presented by listing its main

objectives and justification.

1.1 Energy Storage Systems

Ensuring the availability of clean energy for the future is not just about improving clean

energy production processes, or reducing energy consumption, but it also requires its efficient

transport from the production source to the end user. The transport and storage of electrical

energy are tasks directly related to intelligent energy consumption.

Very often the production of electricity is not constant: many renewable sources such

as wind or solar are not available all the time. Storing the energy provided by these sources

allows the storage systems to meet the demand by making the electricity created from solar

sources available during night and day, for example. Furthermore, the stored energy can assist

in providing power during periods of high demand, helping to reduce the load on generating

systems.

Combining different sources of power generation - including renewable - and reducing

the demand on generating systems in peak times, storage systems enhance the cost/benefit ratio,

reliability, efficiency and reduce environmental impact the processes of generation, transmission

and distribution of energy. (NREL. . . , 2014)

The storage can be done using redox flow batteries (Redox Flow Batteries - RFB) in

which two electrolytes are stored separately in external tanks and circulate through the cell as

CHAPTER 1. INTRODUCTION

needed. These batteries have some technical advantages over other technologies as the excellent

combination of energy efficiency, low capital cost and low cost per life cycle. (PARASURAMAN

et al., 2013).

Vanadium redox battery (Vanadium Redox Batteries - VRB) is one among the batte-

ries of this kind, first developed in the 80s by the team coordinated by Maria Skyllas Kaza-

cos researcher in the University of New South Wales (SUM; SKYLLAS-KAZACOS, 1985)

(SKYLLAS-KAZACOS; SUM; RYCHCIK, 1985) (SKYLLAS-KAZACOS et al., 1986) .

1.2 Operation and electrochemical principles of VRBs

A schematic showing the operation of the VRB is presented in Figure 1. The image

shows a single power cell while the battery is constituted by an assembly these units forming a

stack, in which two electrolytes flow at rates of 0.5-3 ml s−1mL through porous carbon felts that

act as electrodes and are separated by a membrane that allows the exchange of protons; the felt

and the membrane have about 3 to 10 cm in length, indicating a thin geometry.

Figure 1 – Fundamental operation scheme for vanadium redox battery. Source: (VYNNYCKY, 2011)

Both electrolytes are based on vanadium: the electrolyte in the positive half-cell, also

called the anode, contains ions VO +2 and VO2+, whilst the electrolyte in negative half-cell

(cathode) contains ions V 3+ and V2+. The electrolyte may be prepared by a number of proces-

ses, including the dissolution of vanadium pentoxide (V2O5) in sulfuric acid (H2SO4). In the

vanadium redox batteries, both half-cells are connected to tanks and pumps, so that very large

volumes of electrolyte can circulate through the cell. At the moment when the vanadium battery

is being charged, ions VO2+ in the positive half-cell is converted into VO+ ions, and electrons are

removed from the positive terminal of the battery. Likewise, in the negative half-cell, electrons

are introduced by converting ions V3+ into ions V2+; during discharging, the reverse process

occurs.

16

1.3. PERFORMANCE AND APPLICATIONS OF VRBS

Charge and discharge can be summarized by:

charge

V3++e− ⇋ V2+ negative electrode,

VO2++H2O ⇋ VO+2 +e−+2H+ positive electrode,

discharge

(1.1)

Figure 2 – Internal operation in a VRB during charge cycle. Source: (VYNNYCKY, 2011)

What happens inside a VRB during charge cycle is shown in Figure 2. Five regions can

be observed: the proton-exchange membrane, two electrodes of porous carbon felts and two

metal plates that act as current collectors. In the carbon electrode located on the anode ions VO2+

react with water according to the equation (1.1), producing ions VO+, one electron is transported

along the outer circuit and one ion H+ is transported through the membrane to the cathode. In

the cathode carbon felt on the other hand, ions V3+ are reduced to V2+.

1.3 Performance and applications of VRBs

The technology of the vanadium redox batteries, while still under improvement, is

employed in many applications such as:

∙ In the Portuguese city of Evora, the company Renewable Energy Dynamics Technology

Ltd. was responsible for the installation of a 5kW rated power system with 12 hours of

storage capacity (60kWh). Connected to a photovoltaic system in the farming campus

owned by University of Evora, in the northern part of the country the storage system will

maximize the efficiency of the 6.6kW of produced by the system installed on the hospital

roof aiming to reduce energy costs. The installation is able to supply 2kW of power during

2 or 3 days of low insolation periods. (REDT, 2014)

∙ With the support from the Australian Greenhouse Office grant, the company Pinnacle VRB

installed a storage system with capacity of 200 kW for Hydro Tasmania on King Island,

17


Tasmania. The main objective was to reduce spending on diesel fuel. Hydro Tasmania

applied this technology as a first experience on the construction and operation of VRB at

the commercial level.

(SKYLLAS-KAZACOS, 2004)

∙ Winafrique Technologies Ltd. from Kenya, implemented two 5kW system with 4 hours of

capacity each. The system was applied to two telecommunication stations in the country

as a partnership with mobile phone company Safaricom Limited. Usually the power supply

is held by diesel generators and lead acid batteries, but the rising cost of fuel, concerning

about pollutants emission and maintenance costs aroused interest on the combined use of

wind and solar energy with a system capable of withstanding numerous cycles of charge,

with high reliability, long lifespan and low maintenance cost, as the VRBs. The system

ensures the power supply to the stations up to 14 hours a day, reducing the consumption of

diesel fuel and increasing the lifespan of the entire station. (RICHMOND, 2014)

∙ In Oxnard, California, a storage system based on VRB with three 200kW modules capable

of providing power for 6 hours was implemented. Developed and operated by Prudent

Energy Corporation on behalf of the company Gills Onions, the main motivation was to

reduce costs involved in the shift of the generation system during the peak times. The

system may still be required to operate during short periods of high demand with 600kW

capacity available 24 hours a day or, if necessary, provide 50 % of additional load working

at 900kW for 10 minutes every hour. The estimated savings for the company is on hundreds

of thousands of dollars a year. (ESA, 2014)

Nowadays, the VRB may reach performance suitable for application on large scale

energy storage systems, with a durability around 10-20 years and other advantages presented in

table below.

Tabela 1 – Comparison of vanadium redox batteries (VRG) among other technologies: zinc-bromine (ZBB), sodium-sulfur (NaS) and lead-acid.(HUANG et al., 2008)

Technology Cycle Oscillations Durability Operating Effective energy Green O&Mefficiency (%) (MWh) in cycles temperature(oC) density costs(KWh)

VRB 78 0.5-100 13.000 30 24 Yes 0.001ZBB 68 0.05-1 2500 50 33 Yes 0.015NaS 65-70 0.15-1 3000 350 40 No 0.02

Lead-acid 45 0.001-40 1000 Room 12 No 0.02

The VRB, as well as all RFB have electrolyte storage tanks and may offer greater amount

of energy according to the application needs by simply using larger tanks, after discharged the

tanks may be substituted continuing the power supply, the electrolyte that circulates through all

the battery cells have cooling action and reduces problems related to temperature management.

But unlike other RFB, in the VRB both electrolytes are composed of the same chemical element,

so there is no risk of contamination due the electrolyte mixing. Pumping the solution through the

a number of cells VRB means, according to (SKYLLAS-KAZACOS, 2004), that all the cells are

18

1.4. RESEARCH DESCRIPTION

in the same state of charge making monitoring and maintenance simpler since it is not necessary

to watch or individually adjust each cell .

However, there are some disadvantages about the VRB: the low rate of energy/volume

which is about 25Wh/kg of electrolyte, while the lead-acid battery has 30-40Wh/kg ration and

a lithium-ion has 80-200Wh/kg; and the high complexity of the system, compared to other

batteries. (VYNNYCKY, 2011)

1.4 Research description

In this situation, this research is developed in order to meet the objectives categorized

below.

1.4.1 General goal

Using the theoretical framework presented in (VYNNYCKY, 2011), it is proposed the

development of mathematical models based on multidimensional time-dependent differential

partial equations as a complement to existing experimental work. Such models would be delinea-

ted by an approach that emphasizes the use of asymptotic reductions to define a set of reduced

equations that describe from first principles the operation of VRBS, in the point of view of

electrochemistry and fluid mechanics.

1.4.2 Specific goals

1. Continue obtaining general knowledge about electrochemistry in (TICIANELLI; GON-

ZALEZ, 2005), (NAG, 2002), (NEWMAN; THOMAS-ALYEA, 2004), and others, as

well as recent papers published in the area as (CHEN; YEOH; CHAKRABARTI, 2014),

(PARASURAMAN et al., 2013), (KEAR; SHAH; WALSH, 2011), (SHARMA et al.,

2014) and others;

2. Using the theoretical framework to develop a multidimensional time-dependent models

compatible with experimental data found in literature;

3. With the software Comsol Multiphysics and based on existing implementation for stationary

1D model, develop and run computational simulation for the following models:

∙ Bi-dimensional transient model;

∙ Bi-dimensional quasi-steady model;

∙ Uni-dimensional quasi-steady model;

4. Validate the results for each model according to experimental data found in literature.

5. Compare different models according to its use of computational resources.

19


1.4.3 Justification

Altough numerical models for this system already exist in literature [(AL-FETLAWI;

SHAH; WALSH, 2009), (AL-FETLAWI; SHAH; WALSH, 2010), (SHAH; AL-FETLAWI;

WALSH, 2010), (SHAH; WATT-SMITH; WALSH, 2008) ], they are time-consuming and unwi-

eldy, particularly with respect to ultimate use an optimization tool.

The present approach emphasizes the use of asymptotic reduction, hand-in-hand with the

development of numerical algorithms to solve the resulting reduced model of equations.

The computational simulations are conducted using the software COMSOL Multiphysics

- an interactive environment for modeling and simulating scientific and engineering problems.

The software has already been successfully used by other researchers on similar projects.

20

CHAPTER

2

REVIEW

Mathematical modeling of a power device such as VRB includes several electrochemical

concepts, some of which are briefly presented in order to facilitate the understanding of the

reader less familiarized. In addition, the main strategies adopted in papers published in the area

that have significant contributions in modelling vanadium redox batteries will be presented.

2.1 Mass transfer

An electrochemical process may be viewed as the association of three steps: the transport

of the reactant species present in the electrolyte towards the electrode surface; the transfer

reaction at the electrode/solution interface characterized by the electron shift from electrode

to the reactant (cathodic reaction) or from reactant to the electrode (anode reaction); and the

transport of the resulting product away from the electrode.

The concentration of the reactant species near the electrode surface is thus one of the

factors that can affect or even dominate the overall rate of reaction. The events directly related to

the concentration are transport phenomena. In general the flow or movement of ions and neutral

molecules in a solution occurs by three different ways: diffusion, convection and migration,

described below. The terms of the equations that describe these movements will be explained

later, in the section that introduces the Nernst-Planck equation.

∙ Diffusion

Diffusion occurs in a solution due to the concentration gradient or chemical potential

gradient. This form of transport of ions and neutral molecules is significant in an electrolytic

experiment when the electrolytic reaction occurs only in the electrode surface, hence there

will be less reagent concentration near the electrode and a highest concentration within the

solution, the reverse is true for the concentration of the reaction products. According to

CHAPTER 2. REVIEW

(NEWMAN; THOMAS-ALYEA, 2004) the diffusive flux of chemical species is given by:

Nk,diffusion =−Dk∇ck. (2.1)

∙ Convection This transport phenomena occurs is due to the agitation of the solution caused

among other factors by pumping action, gas flow, temperature gradients, or gravity. The

equation that describes it is given by:

Nk,convection = ckvk. (2.2)

∙ Migration

According to (TICIANELLI; GONZALEZ, 2005), migration is the movement of ionic

species owing to the action of electric fields or electric potential gradients, responsible for

the conduction of electricity in electrolytes.

(NEWMAN; THOMAS-ALYEA, 2004) explains the phenomenona with the assumption of

two electrodes immersed in a solution in which an electric field is applied. That field will

cause the movement of charged species: the cations are taken toward the cathode electrode,

and anions are dragged to the anode electrode, whilst the cations move in the opposite

direction of the potential gradient. The velocity of ions k in response to the electric field is

called migration velocity:

vk,migration = zkMkF∇φ . (2.3)

The Nernst-Planck equation is a mass conservation equation relating the effects listed above to

describe the flow of chemical species in a fluid taking into account both the gradient of the ionic

concentration and the electric field. Based on the mass conservation equation for incompressible

fluids as (BAZANT, 2014), we have:

∂ck

∂ t+u∇ck +∇F = 0. (2.4)

Where the first term represents the change in concentration of species k, the second term

represents the convection phenomena and can be determined as the solvent velocity in a certain

solution, the latter term in turn, refers to flux density as expressed by:

Fk = Mk∇µkck. (2.5)

Mk is the particle mobility coefficient, that is, if a force G acts on a particle it acquires average

velocity v(G) due to this force. (RODENHAUSEN, 1989). By Einstein’s relation:

Mk =Dk

kbT, (2.6)

22

2.2. BUTLER-VOLMER EQUATION

to kb meaning the Boltzmann constant, T is the absolute temperature, and D is the diffusion

coefficient, ∇µk is the gradient of the chemical potential of the solution given by :

∇µk =kbT

ck∇ck + zke∇φ . (2.7)

The chemical potential is the quantity that allows to describe the state of balance of each

component of a chemical system in the definition above zk is the charge number of specie k, e

is the elementary charge, i.e., electric charge from a single proton and φ refers to the electric

potential. Substituting equations (2.7) and (2.6) in relation (2.5):

Fk = Dk

(

∇ck +zkeck∇φ

kbT

)

. (2.8)

Replacing in (2.6) the Boltzmann and Avogrado constants respectivelly by:

kb =R

Na,

Na =F

e.

Results in:

Fk =−Dk

(

∇ck +zkck∇φ

RT

)

. (2.9)

Finally, replacing (2.9) in (2.4), leads to equação de Nernst-Planck:

∂ck

∂ t+u∇ck +∇

[

−Dk

(

∇ck +zkck∇φ

RT

)]

= 0. (2.10)

2.2 Butler-Volmer equation

The Butler-Volmer equation describes the reaction rate compared to the overpotential:

i = i0

{

exp

(

−βFη

RT

)

− exp

[

(1−β )Fη

RT

]}

. (2.11)

According to (TICIANELLI; GONZALEZ, 2005), the concept of overpotential is linked

to the following situation: given an electrochemical system (in the scope of this work is the redox

battery of vanadium), the interface between the electrode and the solution has an intense electric

field, i.e. an electric potential difference between the electrode and the electrolyte represented

by∆φ . Such potential difference can be changed by a potentiostat, to this positive or negative

external contribution is given the name of overpotential expressed by the letter η . This value plus

the existing potential difference in the system in equilibrium ∆φe results in the total potential

difference:

∆φ = ∆φe +η . (2.12)

23

CHAPTER 2. REVIEW

The symmetry factor β represents the fraction of applied potential which promotes the

cathodic reaction. Likewise (1−β ) is the fraction of applied potential which promotes anodic

reaction. It is often assumed that β = 12 , according to (NEWMAN; THOMAS-ALYEA, 2004).

F , R and T are, respectively, the Faraday constant, the universal gas constant and the absolute

temperature, i0 refers to the exchange current density and depends on the composition of the

solution adjacent to the electrode and its temperature. Considering that at equilibrium state the

overpotential and current density are zero:

i0 = FkcCOexp

(

−∆−→G=o +Fβ∆φe

RT

)

,

i0 = FkaCRexp

(

−∆←−G=o −F(1−β )∆φe

RT

)

, (2.13)

where kc and ka are the rate constants of cathodic reactions (reduction - gain of electrons) and

anodic reaction (oxidation - loss of electrons), while CO and CR are the concentrations of reduced

and oxidized species respectively in the electrochemical system. In this study, the cathode process

takes place in the formation of V 2+ from V 3+ and VO2+ from VO+2 and the anodic process is

the reverse: oxidation of V 2+ in V 3+ and VO2+ in VO+2 , as expressed in reaction 1.1. The terms

∆←−G=o and ∆

−→G=o represent the activation energy of the anodic and cathodic reactions, respectively,

when there is no electric field (at equilibrium).

2.3 Mathematical modelling

This subsection shows the main strategies adopted by most researchers to model the

VRBs, the model is discussed separately for each of the battery components illustrated in Figure

1.

2.3.1 Porose carbon electrodes

Assuming that the transport of charged species occurs by hydrodynamic dispersion,

electrokinetic effects and convection. The molar flux Nk is expressed by the modified Nernst-

Planck equation, as (SHAH; WATT-SMITH; WALSH, 2008), (VYNNYCKY, 2011) and (AL-

FETLAWI; SHAH; WALSH, 2009):

Nk = ckva/c−zkckD

e f fk F

RT∇φe−D

e f fk ∇ck, (2.14)

where ck is the concentration of species k, φe is the electric potential in electrolyte, va/c is

the electrolyte velocity in the anode (a) and the cathode (c) zk is the number of charge. The

24

2.3. MATHEMATICAL MODELLING

modification of the equation is given when considering the additional effect of material porosity

ε on the effective diffusion coefficient Dk originally given by Bruggeman relation:

Deffk = ε

32 Dk. (2.15)

The electrolyte velocities vpos/neg on both electrodes are usually modeled by Darcy’s law:

vpos/neg =d2

f

KµH2O

ε3

(1− ε)2 ∇ppos/neg (2.16)

where ppos/neg are the pressures of the fluid in the positive and negative electrodes, µ is the

dynamic viscosity d f is the average diameter of the carbon fibers, and K is the constant Kozeny-

Karman.

Another modification resulting from the inclusion of porosity occurs in equation that

describes the material balance in the porous carbon electrode for species k, given by:

∂

∂ tεck +∇ ·nk =−Sk. (2.17)

Here Sk is the source term of the species k. The model presented in (VYNNYCKY, 2011)

distinguished from the others by assuming a constant concentration of water, so the species

represented by k do not include H2O, this simplification was based on the diluted electrolyte

theory and on low variation in the concentration of water in earlier numerical simulations.

In the equation 2.14, the first term models the ion transport phenomena due to convection,

the second is due to electromigration, and the third, in turn, is the hydrodynamic dispersion.

Some previous work as (CHEN; YEOH; CHAKRABARTI, 2014) and (YOU; ZHANG; CHEN,

2009) simplify this equation by neglecting the effect of electromigration on the transport of ionic

species, even for no apparent reason, resulting in:

Nk = ckva/c−Deffk ∇ck. (2.18)

The transfer current densities for the reactions that occur on the surface of porous carbon

electrodes are conveniently described by the Butler-Volmer expressions:

jc = εAFkc(csV 3+)

α−,c(csV 2+)

α+,c

{

exp

(

α+,cFηc

RT

)

−exp

(

−α−,cFηc

RT

)}

,

ja = εAFka(csVO2+)

α−,c(csVO+

2)α+,c

{

exp

(

α+,aFηa

RT

)

−exp

(

−α−,aFηa

RT

)}

, (2.19)

where jc e ja represents the current density transfer rate to the cathode and the anode, respectively,

ε is the porosity, A is the electroactive area, F the Faraday constant, kc and ka are constant

25

CHAPTER 2. REVIEW

reaction rates, α±,c and α±,a represent the transfer rates for reactions shown in Equation 1.1.

The overpotentials are defined by the letter η as:

ηa/c = φs−φe−Ea/c. (2.20)

csV 3+ , cs

V 2+ , csVO2+ and cs

VO+2

represent the vanadium concentrations of these species at

the electrode/solution interface, this value is usually different from those concentrations present

within the solution due to the additional resistance to the transport of species from the interior

bulk to the interfaces . (VYNNYCKY, 2011), (SHAH; AL-FETLAWI; WALSH, 2010) and

(CHEN; YEOH; CHAKRABARTI, 2014) assume the following expressions that replaced in 2.19

eliminate the concentrations in the surface substituting them with the values given as functions

of concentrations in the bulk:

csVO2+ =

cVO2+ + εkae−F(ψ−φ−E0,a)/2RT (cVO2+/γVO+2+ cVO+

2/γVO2+)

1+ εka

(

e−F(ψ−φ−E0,a)/2RT/γV + eF(ψ−φ−E0,a)/2RT/γVO2+

) ,

csVO+

2=

cVO+2+ εkae−F(ψ−φ−E0,a)/2RT (cVO2+/γVO+

2+ cVO+

2/γVO2+)

1+ εka

(

e−F(ψ−φ−E0,a)/2RT/γV + eF(ψ−φ−E0,a)/2RT/γVO2+

) ,

where γk is the ratio between the diffusion coefficient of the species k in the solution and the

average distance between the fibers of the carbon felt.

In the equation 2.20, Epos/neg are the potential balance described by the Nernst equation

brought below where E0,neg and E0,pos represent these potential for standard conditions.

Eneg = E0,neg +RT

Fln

(

aV3+

aV2+

)

,

where ak denotes the activity of the species k and is given by the concentration multiplied

by their activity coefficient. In the work presented in (CORCUERA; SKYLLAS-KAZACOS,

2012), it was assumed that this coefficient would be constant for each chemical species, while in

(KJEANG et al., 2007), activity coefficients of all redox species should be equal, both approaches

lead to neglect such coefficients of the equation resulting in:

Eneg = E0,neg +RT

Fln

(

cV3+

cV2+

)

, (2.21)

similarly sets up Epos, the equilibrium potential for the positive electrode, given by:

Epos = E0,pos +RT

Fln

(

aVO2+(aH+)2

aVO+2

)

,

or

Epos = E0,pos +RT

Fln

(

cVO2+(cH+)2

cVO+2

)

; (2.22)

26


The porosity represented by the term ε indicates the portion of the ε not occupied by

carbon fibers, a space which is filled by the electrolyte during the battery operation. Thus,

two different currents are reported. The ionic currentie, and the electronic current is belonging

respectively to the felt portion occupied by the electrolyte, and to that occupied by the solid

portion of the porous felt and follow the equilibrium relation:

∇ · ie +∇ · is = 0 (2.23)

The divergent of ionic current results in the current density transfer rate defined in

equation 2.19, this adding to the equation 2.23 implies on:

∇ · ie = jpos/neg =−∇ · is (2.24)

Being ie defined as:

ie = ∑k

zkFNk (2.25)

where Nk is the molar flux from equation 2.14, resulting in:

ie = F ∑k

zkckva/c−z2

kckDeffk F

RT∇φe− zkDeff

k ∇ck (2.26)

Assuming the electroneutrality for the electrolyte, i.e., ∑k zkck = 0 eliminates the first

and the third terms of the equation 2.26 corresponding to the convective and diffusive transport,

respectively, leaving:

ie = F ∑k

−z2

kckDeffk F

RT∇φe (2.27)

The electronic current, in turn, is defined by Ohm’s law as:

is =−σ effs ∇φs, (2.28)

where φs is the potential in the solid phase, andσ effs relates the effective coefficient of conductivity,

since it takes into account the solid portion of the porous felt, it is calculated as σ effs = (1−ε)1.5σs

for conductivity coefficient σs.

2.3.2 Proton-exchange membrane

The membrane is a critical component of redox flow batteries, it is crucial for both

performance and economic viability of the battery. It acts separating two electrolyte to prevent

its mixture, and must allow positive ions to pass closing the circuit during the flow of electric

current. (PRIFTI et al., 2012)

The electroneutrality condition included in electrodes modelling, is not valid for the

liquid present in the membrane, which composed by water and protons. However, (VYNNYCKY,

27

CHAPTER 2. REVIEW

2011) and (SHAH; AL-FETLAWI; WALSH, 2010) argue that it can be used when fixed charge

sites are assumed resulting , as well as (AL-FETLAWI; SHAH; WALSH, 2009), in:

zH+cH+ + z f c f = 0, (2.29)

where z f is the charge number of fixed sites and c f is its concentration, while cH+ refers to the

concentration of the species H+, whose charge number zH+ is equal to 1. Hence it follows that

cH+ is a constant.

The velocity of the fluid within the membrane is described by the Schlögl equation:

v =−kφ

µH2O

FcH+∇φ −km

µH2O

∇pm, (2.30)

where v is the velocity, p is the pressure of the liquid, kφ is the electrokinetic permeability, km is

the hydraulic permeability, φ represents the electric potential µH2O is the viscosity of water, F

Faraday’s constant, and cH+is the concentration of protons. The current density is given by:

im = zH+FNH+ , (2.31)

the current conservation ∇im = 0, leads to:

∇

(

−F2

RTDm

H+cH+

)

= 0, (2.32)

with DmH+ as the proton diffusion coefficient in the membrane.

Following the electrolyte dilute theory over the concentrate electrolyte theory, (VYNNYCKY,

2011) considers that the concentration of water on the membrane will be the same constant value

as that on the porous electrode. While (SHAH; AL-FETLAWI; WALSH, 2010) incorporates the

next mass balance equation to the model:

∂cH2O

∂ t+∇ · (cH2Ov) = ∇

(

DeffH2O∇cH2O

)

. (2.33)

2.3.3 Current collectors

The current collectors are relatively simple components, its modeling usually relies on

Ohm’s law and the conservation of current, thus:

icoll =−σcoll∇φcoll,

−σcoll∇2φcoll = 0,

where icoll is the current density, φcoll is the electric potential, and σcoll is the electrical conducti-

vity of the current collectors plates.

28


2.3.4 Interfacial and initial conditions

The boundary conditions are presented based on the geometry shown in Figure 2. n is

the vector normal for the interface.

∙ Horizontal interfaces

– Positive collector interface [y = 0]

A condition to electrical grounding must be imposed:

φcoll = 0 (2.34)

Charge or discharge current:

icoll =±I (2.35)

positive sign for charge cycle, negative for discharge.

– Coletor/positive electrode interface [y = hcoll]

Continuity of solid electrical potential between collector and electrode:

φcoll = φs (2.36)

Continuity of solid current density between collector and electrode:

icoll ·n = ie ·n (2.37)

Null molar flux for k = cH+,pos, cHSO−4 ,pos, cV 2+ , cV 3+:

Nk ·n = 0 (2.38)

– Positive electrode/membrane interface [y = hcoll +hf]

There is a potential difference for the liquid phase between the membrane and the

electrode represented by the Donnan potential:

φm = φe−RT

Fln

(

aH+,pos

aH+,mem

)

(2.39)

Continuity of electrolyte velocity between electrode and membrane:

vpos ·n = vm ·n (2.40)

Continuity of current density between membrane and electrode:

im ·n = ie ·n (2.41)

Null current density for solid phase:

is ·n = 0 (2.42)

Null molar flux for k = cHSO−4 ,pos, cV 2+ , cV 3+:

Nk ·n = 0 (2.43)

29

CHAPTER 2. REVIEW

– Membrane/negative electrode interface [y = hcoll +hf +hm]

Potential difference for the liquid phase between the membrane and the electrode

accounting for Donnan potential:

φm = φe−RT

Fln

(

aH+,neg

aH+,m

)

(2.44)

Continuity of electrolyte velocity between electrode and membrane:

vneg ·n = vm ·n (2.45)

Continuity of current density between membrane and electrode:

im ·n = ie ·n (2.46)

Null current density for liquid phase:

is ·n = 0 (2.47)

Null molar flux for k = cHSO−4 ,neg,cVO2+ ,cVO+2

:

Nk ·n = 0 (2.48)

– Interface eletrodo negativo/coletor [y = hcoll +2hf +hm]

Continuity of electric potential for solid phase:

φcoll = φs (2.49)

Continuity of electric current for solid phase:

icoll ·n = ie ·n (2.50)

Null molar flux for k = cH+,neg,cHSO−4 ,neg,cVO2+ ,cVO+2

:

Nk ·n = 0 (2.51)

– Negative collector interface[y = 2hcoll +2hf +hm]

Charge or discharge current:

icoll =±Inegativesign f orchargecycle, positive f ordischarge. (2.52)

∙ Vertical interfaces

– Positive collector interface [x = 0 or x = L;0≤ y≤ hcoll]

Electrical insulation:

icoll ·n = 0 (2.53)

30


– Positive electrode inlet interface [x = 0;(hcoll)≤ y≤ (hcoll +h f )]


ie ·n = 0, is ·n = 0 (2.54)

Electrolyte injection of tanks into the battery k= cH+,pos, cHSO−4 ,pos, cV 2+ , cV 3+:

ck = cink (t) (2.55)

– Positive electrode outlet interface [x = L;(hcoll)≤ y≤ (hcoll +h f )]


ie ·n = 0, is ·n = 0 (2.56)

Outlet flux for k= cH+,pos, cHSO−4 ,pos, cV 2+ , cV 3+:

−Deffk ∇ck ·n = 0 (2.57)

– Membrane interface [x = 0 ou x = L;(hcoll +h f )≤ y≤ (hcoll +h f +hm)]


im ·n = 0 (2.58)

– Negative electrode inlet interface [x = 0;(hcoll +h f +hm)≤ y≤ (hcoll +2h f +hm)]


ie ·n = 0, is ·n = 0 (2.59)

Electrolyte injection of tanks into the battery k=cH+,neg,cHSO−4 ,neg,cVO2+ ,cVO+2

:

ck = cink (t) (2.60)

– Negative electrode outlet interface [x = L;(hcoll+h f +hm)≤ y≤ (hcoll+2h f +hm)]


ie ·n = 0, is ·n = 0 (2.61)

Outlet flux for k=cH+,neg,cHSO−4 ,neg,cVO2+ ,cVO+2

:

−Deffk ∇ck ·n = 0 (2.62)

– Negative collector interface[x = 0 ou x = L;(hcoll +2h f +hm)≤ y≤ (2hcoll +2h f +

hm)]


icoll ·n = 0 (2.63)

The conditions 2.55 and 2.60, cink (t) indicates the concentration of the chemical species k

in the battery tanks, where it is assumed instantaneous mixing between the liquid leaving

the battery and the electrolyte stored there with the following mass balance equation:

dcink

dt=

ω

V

(

coutk − cin

k

)

(2.64)

31

CHAPTER 2. REVIEW

in this equation, V is the total volume of electrolyte, ω is the volumetric flow, and coutk

indicates the average concentration in the electrolyte that leaves the battery defined for the

positive and negative electrodes respectively as:

coutk =

1

h f

∫ hcoll+h f

hcoll

(ck(t))x=Ldy (2.65)

coutk =

1

h f

∫ hcoll+2h f+hm

hcoll+h f+hm

(ck(t))x=Ldy (2.66)

2.3.5 Previous results

This section brings graphs of results obtained from computer simulation for stationary

one-dimensional model, the file that implements the model was generated with Comsol Mul-

tiphysics software and provided by Prof. Dr. Michael Vynnycky, the transferred file was used to

reproduce some of the results.

Figure 3 is an example of how modeling and computer simulation of VRBs could

propose optimizations for battery construction. For different potential values of the cell, there is

a maximum value of the current density related to the determined porosity, allowing to conclude

that a porosity ε ≥ 0.5 would be advantageous for the potential range tested.

Figure 3 – Current density, i, in absolute values as function of porosity, ε , for several cell potentials, Ecell.

32


Figure 4 – Current density, i, as function of cell potential, Ecell, for different mesh refinements.

33

CHAPTER 2. REVIEW

The graph of Figure 4 represents the current density obtained for each defined cell poten-

tial with various refinements of the mesh. Mesh independence is observed for the calculations,

so it is possible to adopt the less refined mesh in the simulations, this strategy reduces the

computational resources needed for execution.

Figure 5 shows the overpotential at the cathode (y<0) and the anode (y>0). The curves

defined by 0 ≤ Ecell ≤ Ea-Ec refer to the discharge process, while the others describe the battery

charging process. This simulation indicates much higher overpotentials at the anode compared to

the cathode, i.e., modeled events occur more slowly at the anode, which contradicts some results

presented in (ORIJI; KATAYAMA; MIURA, 2004) and (ORIJI; KATAYAMA; MIURA, 2005),

however, the portion of the curves valid for the discharge process is qualitatively similar to the

experimental results for vanadium fuel cell shown in (KJEANG et al., 2007).

Figure 5 – Over-potential values at the positive electrode and at the negative electrode, ηc and ηa, as function ofcell potential, Ecell.

34

CHAPTER

3

MATERIALS AND METHODS

This section describes the procedures used to build the mathematical models that will be

presented later as well as the tools employed on its computational implementation in order to

obtain the results that would be compared with experimental data found in the literature.

3.1 Mathematical modelling

In the next lines will be presented the equations and boundary conditions that characterize

the mathematical models for VRB. But first, some important assumptions valid for all models in

this research must detail.

As was assumed in (SHAH; AL-FETLAWI; WALSH, 2010), (VYNNYCKY, 2011),

(KNEHR et al., 2012), (CORCUERA; SKYLLAS-KAZACOS, 2012), (SHARMA et al., 2014),

among others, the cell is isothermal, i.e. not taking into account the Joule heating or ohmic

heating - heat generation process caused by passing electric current through a conductive material.

However, as was placed in (VYNNYCKY, 2011), it is argued that neglecting this phenomenon

does not imply that this model have significant losses in its accuracy, since the model for the fuel

cell studied in (VYNNYCKY et al., 2009) showed that the observed temperature gradient has

little influence in performance.

It is considered that the membrane is always humidified in order to allow the ions to

cross, assuming that H+ are the ones that pass through the membrane. A different approach

was taken, for example, in (KNEHR et al., 2012), in which was considered the flow vanadium

ions through the membrane. This phenomena implies on a change in vanadium concentrations,

however it can be disregarded in a single cycle, as will be shown later.

Also the validity of diluted electrolyte theory presented in Section 2.3.1 was assumed.

CHAPTER 3. MATERIALS AND METHODS

3.1.1 Asymptotic reductions

From equations presented in Section 2.3 and parameters noted on Tables 2, 3, 4, 5, and

6 different analysis may be conducted in order to study qualitatively some features initially

included in the mathematical model, besides, the correct definition of the scales for each variable

in building a dimensionless formulation, according to (HOWISON, 2005) allows to evaluate the

relevance of each term which combined with the correct interpretation of the problem can lead

to simplification of the least significant terms. Such simplifications or asymptotic reductions are

adopted this model in several aspects.

Tabela 2 – Source terms for equation (2.17)

Source term Positive electrode Negative electrode

SH+ −2 j+/F 0SHSO−4

0 0

SSO2−4

0 0

SV2+ - j−/F

SV3+ - − j−/F

SVO2+ j+/F -SVO+

2− j+/F -

Tabela 3 – Initial conditions

Parameter Value Unit

c0H+,neg 4447.5 mol m−3

c0HSO−4 ,neg

2668.5 mol m−3

c0H+,pos 5097.5 mol m−3

c0HSO−4 ,pos

3058.5 mol m−3

c0V2+ 156 mol m−3

c0V3+ 884 mol m−3

c0VO2+ 884 mol m−3

c0VO+

2156 mol m−3

T 300 K

3.1.1.1 Electrolyte velocity

This reduction was first discussed by (VYNNYCKY, 2011) and can be verified for the

geometry here defined from equations 2.16 and 2.30 to electrolyte velocity on the electrode and

membrane, and boundary conditions 2.40 and 2.45:(

d2f

KµH2O

ε3

(1− ε)2 ∇ppos/neg

)

·n =

(

−kφ

µH2O

FcH+∇φm−kp

µH2O

∇pm

)

·n (3.1)

d2f

KµH2O

ε3

(1− ε)2

∂ ppos/neg

∂y=−

kφ

µH2O

FcH+∂φm

∂y−

kp

µH2O

∂ pm

∂y(3.2)

36


Tabela 4 – Parameters related to the battery geometry, compatible with experiments conducted in (KNEHR et al.,2012)


A 2×106 m−1

d f 1×10−5 mhcoll 6×10−3 mhf 4×10−3 mhm 2.03×10−4 mL 0.035 mLt 0.0285 mω 30 cm3 min−1

V 60 mluin 4.7161×10−3 m s−1

ε 0.93 -

Tabela 5 – Electrochemical constant

Parameter Value Unit Reference

cf 1990 mol m−3 (BERNARDI; VERBRUGGE, 1991)E0,pos 1.004 V (POURBAIX, 1974)E0,neg -0.255 V (POURBAIX, 1974)

K 5.55 - (SHAH; WATT-SMITH; WALSH, 2008)kpos 2.5×10−8 ms−1 (KNEHR et al., 2012)kneg 7×10−8 ms−1 (KNEHR et al., 2012)zf -1 - -

zH+ 1 - -zHSO−4

-1 - -

zSO2−4

-2 - -

zV2+ 2 - -zV3+ 3 - -

zVO2+ 2 - -zVO+

21 - -

α±,c 0.5 - (SHAH; WATT-SMITH; WALSH, 2008)α±,a 0.5 - (SHAH; WATT-SMITH; WALSH, 2008)

37


Tabela 6 – Constants related to transport of charge and mass.

Parameter Value Units Reference

DH+ 9.31×10−9 m2s−1 (NEWMAN; THOMAS-ALYEA, 2004)Dm

H+ 1.4×10−9 m2s−1 (SHAH; WATT-SMITH; WALSH, 2008)DHSO−4

1.39×10−9 m2s−1 (LIDE, 2003)

DSO2−4

1.07×10−9 m2s−1 (LIDE, 2003)

DV2+ 2.4×10−10 m2s−1 (YAMAMURA et al., 2005)DV3+ 2.4×10−10 m2s−1 (YAMAMURA et al., 2005)

DVO2+ 3.9×10−10 m2s−1 (YAMAMURA et al., 2005)DVO+

23.9×10−10 m2s−1 (YAMAMURA et al., 2005)

km 1.58×10−18 m2 (SHAH; WATT-SMITH; WALSH, 2008)kφ 1.13×10−19 m2 (SHAH; WATT-SMITH; WALSH, 2008)µ 10−3 Pa s -

σcoll 1000 S m−1 -σs 66.7 S m−1 (KNEHR et al., 2012)

Non-dimensionalizing via:

Ypos =y

h f

, Ym =y

hm, Ppos,mem =

ppos,neg

[ppos,neg], Φm =

φm

[φm], V =

v

uin, (3.3)

where uin = ω/(Lt *hf *ε) is the electrolyte input speed, assumed to be equal for both electrodes.

The positive electrode To this leads to

ε3

(1− ε)2

∂Ppos

∂Ypos

[ppos]

h f

=Kµ

d2f

(

kφ FcmH+

µH2O

∂Φm

∂Ym

[φm]

hm+

km

µ

∂Pm

∂Ym

[pm]

hm

)

(3.4)

ε3

(1− ε)2

∂Ppos

∂Ypos=

Kh f km

d2f hm

(

kφ FcmH+ [φm]

km[ppos]

∂Φm

∂Ym+

∂Pm

∂Ym

)

(3.5)

The scale [ppos] can be deduced from the velocity equation 2.16 and results in [ppos] =LµH2OKuind−2f .

Applying the parameters of Table 5 and adopting [φm]∼ 10−1, results in:

[ppos]∼ 9100,ε3

(1− ε)2 ∼ 102,Kh f km

d2f hm

∼ 10−7,kφ Fcm

H+ [φm]

km[ppos]∼ 150 (3.6)

Similar relationships can be obtained for the negative electrode, and they imply that

the component∂Ppos/neg

∂Ypos/negfrom the equation 2.16 is approximately zero in terms explained above,

leading to vpos,neg ≡ (uin,0).

That is, the convective transport of chemical species in the electrodes - represented by

the first term of the right side of the equation 2.14 - can be considered valid only in the X-axis

with a constant speed.

3.1.1.2 Long and thin geometry

Presented in (VYNNYCKY, 2011) and adopted in (SHARMA et al., 2014), the correla-

tion between width and thickness of the porous carbon felt h f /L being small (∼ 10−1) indicate

38


that the second order derivatives with respect to Y are more significant than those taken in

relation to X. Hence, including reductions presented in Section 3.1.1.1 the equation 2.14 turns

into:

ε∂ck

∂ t+uin

∂ck

∂x=

∂

∂y

(

zkckDeffk F

RT

∂φe

∂y+Deff

k

∂ck

∂y

)

−Sk (3.7)

3.1.1.3 Quasi-steady nature

From the equation 3.7 it is evaluated the relevance of the derivative of concentration

with respect to time dimension t, as shown in (SHARMA et al., 2014) and (VYNNYCKY;

ASSUNçãO, in prep.), from the non-dimensionalization through the terms presented in the

equation 3.3 and also:

X =x

L, τ =

t

[t], Ck =

ck

[ck], Sk =

Sk

[Sk](3.8)

resulting in:

ε∂Ck

∂τ

[ck]

[t]+uin

∂Ck

∂X

[ck]

L=

1

hf

∂

∂Y

(

zkDeffk F

RT hfCk

∂Φe

∂Y

[φe][ck]

hf+Deff

k

∂Ck

∂Y

[ck]

hf

)

+[Sk]Sk. (3.9)

ε∂Ck

∂τ+

uin[t]

L

∂Ck

∂X=

∂

∂Y

(

zkDeffk F [φe][t]

RT h2f

Ck

∂Φe

∂Y+

Deffk [t]

h2f

∂Ck

∂Y

)

+

{

[Sk][t]

[ck]

}

Sk. (3.10)

To characterize the quasi-stationary nature of the problem (SHARMA et al., 2014) and

(VYNNYCKY; ASSUNçãO, in prep.) state that must be verified that:

[t]max

(

uin

L,zkDeff

k F [φe]

RT h2f

,Deff

k

h2f

,[Sk]

[ck]

)

≫ 1. (3.11)

Where the [φe] can be deduced from the equation 2.27 as:

[φe]∼IRT hf

F2[cmax]Deffk

(3.12)

where I is the current density imposed on the boundary conditions 2.35 and 2.52. The usual

concentration values indicate that [ck] ∼ 102− 103 mol m−3, i.e. [cmax] ∼ 103 mol m−3, and

implies that [φe]∼ 10−1.

Since the scale of the source term is obtained by relation exposed in Table 3 which implies

that: [Sk]∼ I/(hfF)∼ 1 mol m−3 s−1. Continuing the evaluation of terms in 3.11 according to

the parameters in Table 3, leads to:

39


uin

L∼ 0.1 s−1, (3.13)

zkDeffk F [φe]

RT h2f

∼ 10−4−10−3 s−1, (3.14)

Deffk

h2f

∼ 10−4−10−5 s−1, (3.15)

[Sk]

[ck]∼ 10−3−10−2 s−1. (3.16)

Replacing the higher value among the terms discussed in equation 3.11, it is obtained:

[t]uin

L≫ 1 (3.17)

It follows therefore that the process is quasi-stationary for [t]≫ 10 s. It means that it is

possible to excluding the time derivative of the concentration without major losses for the model

keeping time dependent terms solely in the boundary conditions 2.64, resulting in:

uin∂ck

∂x=

∂

∂y

(

zickDeffk F

RT

∂φe

∂y+Deff

k

∂ck

∂y

)

−Sk (3.18)

3.1.1.4 Boundary layer

The boundary layer was first found in (VYNNYCKY; ASSUNçãO, in prep.) and indicates

the existence of regions close to the borders between the membrane and the electrodes where

concentrations of chemical species vary quickly.

Performing the non-dimensionalization from the equation 3.18:

uin∂Ck

∂X

[ck]

L=

1

hf

∂

∂Y

(

zkDkF

RTCk

∂Φe

∂Y

[φe][ck][Deffk ]

hf+Dk

∂Ck

∂Y

[ck][Deffk ]

hf

)

+[Sk]Sk, (3.19)

∂Ck

∂X=

LDk[Deffk ]

h2f uin

∂

∂Y

(

zkF [φe]

RTCk

∂Φe

∂Y+

∂Ck

∂Y

)

+[Sk]L

[ck]Sk, (3.20)

or∂Ck

∂X=

Dk

Pe

∂

∂Y

(

zkΠCk

∂Φe

∂Y+

∂Ck

∂Y

)

+ΛSk, (3.21)

Pe =uinh2

f

L[Deffk ]

, Π =F [φe]

RT, Λ =

[Sk]L

[ck]uin(3.22)

Now taking [Deffk ] = Deff

min so that Dk ∼ O(1), it is obtained:

Pe∼ 10000, Π∼ 10, Λk ∼ 10−3−10−2 (3.23)

40


Back to equation 3.21, the magnitude order of analyzed expressions allows to affirm that:

∂Ck

∂X∼ 0 (3.24)

That is, the concentrations of chemical species does not change significantly across the

electrode starting from the electrolyte input interface to the output. In addition, the boundary

conditions 2.55 and 2.60 indicate that:

ck ≡ cink (t) (3.25)

The next step adopted in (VYNNYCKY; ASSUNçãO, in prep.) is to expand to the regular

perturbation of concentrations Ck, of liquid phase potential Φe, and source term Sk, with that

expansion for Π/Pe that has the same magnitude order as Λk:

Ck =Ck,0 +

(

Π

Pe

)

Ck,1 +O

(

Π2

Pe2

)

; (3.26)

Φe = Φe,0 +

(

Π

Pe

)

Φe,1 +O

(

Π2

Pe2

)

; (3.27)

Sk = Sk,0 +

(

Π

Pe

)

Sk,1 +O

(

Π2

Pe2

)

; (3.28)

Restricting 3.26 to the order of O(Π/Pe):

∂Ck

∂X=

∂Ck,0

∂X+

(

Π

Pe

)

∂Ck,1

∂X(3.29)

accounting for the condition 3.24,

∂Ck,0

∂X+

(

Π

Pe

)

∂Ck,1

∂X= 0 (3.30)

remembering that, by condition 3.25, ck,0 ≡ cink,0(t)

(

Π

Pe

)

∂Ck,1

∂X=−

(

Dk

Pe

∂

∂Y

(

zkΠCink,0

∂

∂YΦe,0 +

∂Cink,0

∂Y

)

+ΛkSk,0

)

(3.31)

Replacing θk = ΛkPe/Π, as constant O(1)

∂Ck,1

∂X+Dk

∂

∂Y

(

zkCink,0

∂

∂YΦe,0

)

=−θkSk,0 (3.32)

Multiplying by zk and doing the sum for all k:

∑k

(

zk∂Ck,1

∂X+Dk

∂

∂Y

(

zkCink,0

∂

∂YΦe,0

))

=−∑k

zkθkSk,0 (3.33)

41


using the eleusando o princípio da eletroneutralidade:

∑k

z2kDkCin

k,0∂ 2

∂Y 2 Φe,0 =−∑k

zkθkSk,0 (3.34)

This equation can be satisfied at the interface between electrode and current collector (Y =

hcoll/hf), however it can not be verified at the interface between electrode and membrane

(Y = (hcoll +hf)/hf), since the condition 3.25 is assumed.

Being Pe≫ 1, as in (HOWISON, 2005), it serves to prove the existence of a boundary

layer with thickness Pe−1/2 from the membrane, from this situation new changes in the equation

3.18 are proposed to the interior of this region:

Y = Pe−1/2(Y +(hcoll +hf)/hf) (3.35)

Φe = Φbulke +Π−1Φe (3.36)

where Φbulke is the potential outside the boundary layer and is depending on the position X and

time, while Φe is the potential therein varying with time, and the positions X , Y . This leads to

the equation:∂Ck

∂X= Dk

∂

∂Y

(

zkCk∂

∂YΦe +

∂

∂YCk

)

+ Jk (3.37)

3.1.1.5 Cross-contamination by ions of vanadium

In the approach presented in (KNEHR et al., 2012), Knehr et al. obtained an agreement

with the experimental data with a model that includes the passage of vanadium ions through the

membrane, however, this compliance is possible even without the inclusion of this phenomenon.

For the above-mentionend article, in the region near the interface between the electrode

and the membrane there are two distinct phases: the inner region in the membrane with thickness

δ m, and the inner region in the electrode with thickness δ e, both measuring 1% of the thickness

of the membrane, i.e., about 2x10−4 m.

The ions of vanadium present on those regions near the interfaces would be subject to the

transport between the two electrodes. The phenomenon was modeled by replacing the conditions

2.43 and 2.48 by:

Nerk =−Deff

k

(cek− c

junck )

δ e−

zkDeffk F

RT

(cek + c

junck )

2

Kφ (φe−φm)

δ e(3.38)

Nmrk =−Dm

k

(cjunck − cm

k )

δ m−

zkDeffk F

RT

(cjunck + cm

k )

2

(1−Kφ )(φe−φm)

δ m(3.39)

where Nerk and Nmr

k represent the molar flux of species k = cV 2+ , cV 3+ ,cVO2+ ,cVO+2

in the interfacial

regions in the electrodes and membrane, respectively. Kφ is a parameter representing the potential

42


difference that occurs at the interface being taken to 0.25 (that means a difference of 25%). cjunck

is the concentration of the species k at the junction between membrane and electrode. Deffk is the

effective diffusion coefficient for the electrode, while Dmk is the coefficient valid for the species k

inside the membrane given by Table 7.

Tabela 7 – Parameters involved in the cross-contamination of vanadium ions through the membrane. Source:(KNEHR et al., 2012)


Kφ 0.25 1Dm

V2+ 3.125x10−12 m2s−1

DmV3+ 5.93x10−12 m2s−1

DmVO2+ 5.0x10−12 m2s−1

DmVO+

21.17x10−12 m2s−1

The scale [Nerk ] can be obtained from equation (3.38) and non-dimensionalized with

the expressions defined in 3.3 and 3.8. Whereas [ce]∼ [c junc]∼ 102−103mol m−3, and [φe]∼

[φm]∼ 10−1 V results in:

[Nerk ]∼−

Deffk

δ e ([ce]− [cjunc])−zkDeff

k F([ce]− [cjunc])

2RT·

Kφ ([φe]− [φm])

δ e(3.40)

Deffk

δe∼ 10−6 m s−1, (3.41)

Deffk FKφ

2RT δe∼ 10−6 V−1 s−1. (3.42)

Similarly, for [Nmrk ]:

[Nmrk ]∼−

Dmk

δ m ([cjunc]− [cm])−zkDm

k F([cjunc]+ [cmk ])

2RT·(1−Kφ )([φe]− [φm])

δ m(3.43)

Dmk

δe∼ 10−8 m s−1, (3.44)

Dmk F(1−Kφ )

2RT δe∼ 10−7 V−1 s−1. (3.45)

Therefore, [Nerk ] ∼ 10−4− 10−3 mol m−2 s−1 and [Nmr

k ] ∼ 10−6− 10−5 mol m−2 s−1,

are factors with little importance for usual concentrations of these chemical species.

The asymptotic reductions discussed in this section are incorporated in different mathe-

matical models. In total, 5 different models are specified, with the reductions summarized in

Table 8.

43


Tabela 8 – The columns refer to asymptotic reductions incorporated to each model and are specified on Sections3.1.1.1, 3.1.1.3, and 3.1.1.2, respectively. ∘ - adopted reduction, x - unadopted reduction.

Model ID. Constant velocity Quasi-steadiness Long and thing geometry

A x x xB ∘ x xC ∘ ∘ xD ∘ ∘ ∘E ∘ ∘ ∘

3.1.2 Computational simulations

3.1.2.1 Model A: bidimensional transient formulation

The model was implemented using the software COMSOL Desktop. This programming

environment allows the user to include interfaces (meaning components, not borders) containing

pre-programmed functions, letting the user to adapt them with more appropriate expressions

to its problem. The following lines will be listing the components that are used to solve the

equations.

The component Tertiary Current Distribution, Nersnt-Planck was used to model the

chemical species concentrations by means of the equations 2.17 and 2.14, the electric potential

in the liquid phase, and in the solid phase given by the equations 2.24, 2.26, and 2.14 in each

of the electrodes with the respective boundary conditions. The component Secondary Current

Distribution, in turn include the equation of liquid membrane potential 2.31. The expressions

that calculate the input concentrations 2.64 are included in the Boundary ODEs and DAEs.

Electrolyte velocity in both electrodes and in the membrane, defined by equations 2.16 and

2.30 are inserted through the Darcy’s Law, the component flexible enough to implement both

equations of Darcy and Schlögl.

COMSOL Multiphysics uses the finite element method with Lagrangian elements for the

approximation of partial differential equations. By default, the Tertiary Current Distribution,

Nersnt-Planck employs linear order elements in the discretization of electrical potential equation

for the solid phase, and quadratic elements for the others dependent variables. According

(COMSOL, 2013), this approach is less prone to introduce spurious oscillations and thus

increases the numerical robustness. The method used for the temporal discretization is the

backwards differences formula of second order (BDF2) , adopting the method of first order only

for the first iteration. To speed up the simulation adaptive steps are applied one a scheme that

allows to choose longer intervals when possible. Such a choice is made, according to (COMSOL,

2013) by adjusting the time step and verifying the validity of the following expression:

(

1

M∑

j

1

N j∑

i

(

|EiUi|

Aus,i +R|Ui|

)2)1/2

< 1 (3.46)

where Aus,i represents the absolute tolerance for i-th mesh element, R is the relative tolerance, M

44


is the number of variables computed, N j is the number elements in the domain of the dependent

variable j, E is the estimated error for the solution U committed during the tested time step. The

new solution is accepted if the above expression is true, otherwise the time step is reduced and a

new evaluation follows.

Newton’s method is employed by COMSOL for the solution of the nonlinear system, to

the intrinsic linear system the algorithm MUMPS (Massively Parallel Sparse Direct Multifrontal

Solver) is used, that is a routine incorporated into the software that allows to solve linear systems

with parallelization. The mesh used for spatial discretization of the domain has 40 elements

along the X axis, along the Y axis there are 40 elements for each electrode elements, 20 for each

current collector, and 10 elements for the membrane. In total 5200 elements are distributed with

smaller spacing near the interfaces between collector/electrode/membrane.

3.1.2.2 Model B: bidimensional transient formulation with constant velocity

The distinction between Model A and Model B is in the inclusion of asymptotic reduction

refered in Section 3.1.1.1 taken by adoption of constant velocity of the electrolyte across the

electrodes. The strategy for implementation also resembles the previous model, but does not

make use of the component Darcy’s Law to compute the velocity.

3.1.2.3 Model C: bidimensional quasi-steady formulation with constant velocity

The Model C results from the incorporation of reductions present in Sections 3.1.1.1 and

3.1.1.3 the original model. Also implemented making use of the software COMSOL Multiphysics,

the model counted on the flexibility of the component Tertiary Current Distribution, Nersnt-

Planck to vanish the time derivative of concentration in the equation 2.17 in order to solve for its

reduced form.

3.1.2.4 Model D: unidimensional quasi-steady formulation with constant velocity, and long

and thin geometry.

This model simplifies the previous one when considering the reduction seen in Section

3.1.1.2 about the long and thin geometry of the electrode leading to the equation 3.18.

As suggested in (VYNNYCKY, 2011) during the implementation the spatial dimension

X can exchange by the time dimension t, this enables the use of a one-dimensional domain as

in Fig.6 that preserves the dimension Y , since only the equations 2.64 to compute the input

concentrations are time dependent. More strictly, the equation inserted in the software is:

uin∂ck

∂ρ=

∂

∂y

(

zckDeffk F

RT

∂φe

∂y+Deff

k

∂ck

∂y

)

−Sk (3.47)

45


where ρ is solved by COMSOL as the temporal dimension, but the result should be interpreted

as the spatial dimension x. This means that the method to step in time employee in the software

is actually solving the equation 3.47 for different cross sections of the battery (0≤ ρ ≤ L).

Figure 6 – Unidimensional domain used in Model D, the measures hcoll, hf, and hm are present in Table 4.

As a result the inlet conditions in which the time dependence must be resolved outside the

COMSOL environment and re-inserted into the model at every time step. To accomplish this task

it was necessary to complement operations performed by COMSOL with scripts developed in

MATLAB to which are dedicated the function of collecting output concentrations and calculate

the new inlet concentrations according to the equation 2.64 so the cycle may be repeated, see

Fig. 7.

Figure 7 – Operation diagram of the quasi-steady model with unidimensional domain. On the bottom lines can beseen the software used on each stage.

Communication between the two platforms is implemented through the LiveLinkTMfor

MATLAB, this tool allows to manipulate COMSOL models in the MATLAB prompt by es-

tablishing a client/server connection with COMSOL server (COMSOL, 2013). Developed in

Java R○, the interface was used to the modification, execution, and information retrieval from the

model file in COMSOL.

On the side developed in MATLAB, the time derivative in the equations that define the

inlet concentrations are approximated by a second-order backward differences scheme (BDF2),

with a similar scheme (BDF1) but with first-order of accuracy was used in the first iteration, as

shown in Figure 8, and equations below:

46


U ′n =1

h(Un−Un−1),or

U ′n =(l2−h2)

lh(l−h)Un−

l

h(l−h)Un−1 +

h

l(l−h)Un−2,

were l = h+m, h is the interval between tn and tn−2, and m is the interval between tn−1 and tn−2.

Applying those approximations results in:

ckin,n =

V

Va1 +ω(−a2ck

in,n−1−a3ckin,n−2 +

ω

Vck

out,n), (3.48)

a1 =(l2−h2)

lh(l−h), a2 =−

l

h(l−h), a3 =

h

l(l−h)(3.49)

Figure 8 – Stencil for time discretization using BDF.

3.1.2.5 Model E: quasi-steady formulation with consnta velocity, long and thin geometry, and

bidimensional domain.

This model was developed in order to investigate the impact of adopting a bidimensional

domain over the onedimensional model as done in Model D. Therefore, it was needed to

implement a formulation similar to the previous model in all other aspects: constant velocity,

long and thin geometry, and quasi-stationary nature.

However, for the components COMSOL Multiphysics Tertiary Current Distribution,

Nersnt-Planck and Secondary Current Distribution determining a bidimensional geometry

implies on the use of spatial derivatives in both directions. To work around this difficulty such

components have been replaced by a more flexible option: General Form PDE.

In addition, as a mean to evaluate the results provided by the COMSOL Multiphysics,

was made an implementation of the same model using solely the MATLAB programming

environment. In this environment, the finite difference method is adopted for the approximation

of spatial derivatives. It used a second-order centered differences formula for points internal to

the domain, and an one-sided second-order formula for the approximation of Neumann boundary

conditions, as illustrated below in Figure 9 and equations below.

∙ Right boundary

U ′j =1

2∆x(3U j−4U j−1 +1U j−2)

U ′′j =1

∆x2 (2U j−5U j−1 +4U j−2−1U j−3)

47


(a) Right boundary (b) Left boundary

(c) Interior point

Figure 9 – Stencil for spatial discretization

∙ Left boundary

U ′j =1

2∆x(−3U j +4U j+1−1U j+2)

U ′′j =1

∆x2 (2U j−5U j+1 +4U j+2−1U j+3)

∙ Interior points

U ′j =1

2∆x(U j+1−U j−1)

U ′′j =1

∆x2 (U j−1−2U j +U j+1)

For the time derivative in the equations that define the inlet concentrations was employed

a second-order backwards differences formula (BDF2), with a similar first-order scheme (BDF1)

for the first iteration.

The discretization of the set of equations and boundary conditions results in a non-

linear system of equations for which solution was implemented the Newton-Raphson method as

described in (BURDEN; FAIRES, 2008), and Algorithm 1.

The linear system mentioned on line 5 from the Algorithm 1 is solved with the MATLAB

operator ‘\’ which will apply, in this case, the LU factorization to solve the system.

48


Algoritmo 1: Newton-Raphson method for non-linear systems

Input: x0 - initial guess; TOL - tolerance.Output: approximated solution x.

1 x[k] = x0;2 k = 0;3 repeat

4 Calculates F(x[k]) and the Jacobian matrix J(x[k]);

5 Solves the linear system J(x[k])δ [k] =−F(x[k]);

6 Updates the approximated solution x[k+1] = x[k]+δ [k];7 k = k+1;

8 until ||x[k]|| ≤ TOL;9 return x.

The Jacobian matrix below represented is calculated analytically from que discretized

equations.

J(x) =

∂ f1∂x1

(x) ∂ f1∂x2

(x) . . . ∂ f1∂xn

(x)∂ f2∂x1

(x) ∂ f2∂x2

(x) . . . ∂ f2∂xn

(x)...

......

∂ fn∂x1

(x) ∂ fn∂x2

(x) . . . ∂ fn∂xn

(x)

Where n corresponds to the number of dependent variables times the number of elements in the

computational domain. There are 23 dependent variables:

∙ Electric potential for liquid and solid phases:

φposcoll , φ

pose ,φ pos

s ,

φnegcoll , φ

nege ,φ neg

s ,

φ meme

∙ Species concentrations:

cH+,pos, cHSO−4 ,pos, cV 2+ , cV 3+ ,

cH+,neg,cHSO−4 ,neg,cVO2+ ,cVO+2

∙ Inlet concentrations:

cINH+,pos,cINHSO−4 ,pos,cINV 2+ ,cINV 3+ ,

cINH+,neg,cINHSO−4 ,neg,cINVO2+ ,cINVO+2

49

CHAPTER

4

RESULTS AND DISCUSSION

The next section is dedicated to showing the results obtained from computational si-

mulations associated to its comparison to experimental data obtained from (KNEHR et al.,

2012).

4.1 Model A

The results from this model allows to verify the electrolyte velocities across the porose

carbon felts, as calculated by Darcy’s law 2.16. Fig. 10 shows the velocity on Y -direction in

a cut transversal to the membrane/electrode interface and placed at the center of the battery.

Fig. 11, in turn, brings the velocity relative to the X-direction in a cut parallel to the interface

membrane/electrode throughout the porous felt.

Both graphics confirm the result predicted in the analysis from Section 3.1.1.1, showing

that its plausible to admit that vpos,neg ≡ (uin,0).

4.2 Model B

For this simulation the percentage error in the cell potential predicted by the transient

two-dimensional model was verified as a comparison with the experimental data. The comparison

also shown in Fig. 12 demonstrate an average difference of 1.13% and maximum of 1.82% for

charging cycle, and mean difference of 1.13% and maximum of 1.39% during the discharge

process. The duration of the charging procedure, determined by experiment in 3.5203 hours, was

here estimated by computational simulation in approximately 3.5566 hours, an error of 1.03 %

in the total charge time.

To discuss the asymptotic analysis presented in Section 3.1.1.4, there will be discussed

graphics showing the concentration profiles of certain chemical species along three sections

CHAPTER 4. RESULTS AND DISCUSSION

Figure 10 – Electrolyte velocity on Y -direction in transversal cut to the positive and negative electrodes, at X = L/2.

transversal to the membrane/electrode interface and placed at three different locations: near to

the electrolyte inlet (X = 1x10−3); at the center of the battery (X = L/2); and near the electrolyte

outlet (X = L−1x10−3).

The Figures 13, 14, 15, and 16 represent the concentration of ion H− during the charge

cycle, at the end of charge cycle, during the discharge cycle, and at the end of charge cycle,

respectively. Figures 18, 19, 20, and 21 show the concentrations of V3+ ion at the same time

periods, in addition Figure 17 shows its concentrations after 10 seconds of charge.

The boundary layer was measured as the distance from the membrane in which the

concentration difference of specie k starts to be smaller than 1x10−2 mol m−3, i.e., when it is

verified the expression: |c jk− c

j+1k | ≤ 1x10−2, for the j-th element of the computational mesh.

For the hydrogen protons, the boundary layer predicted from expression 3.22 would be

approximately 2.6288x10−4 m. The result presented for this model shows that the boundary

layer obtained in computational simulations at the horizontal line x = L/2 was 5.2380×10−4

m after 6500s of charge, e kept unchanged after 13000s, 18500s, and 26000s of the battery

operation.

For the concentration of V 3+ ions, the effects of boundary layer are less accurate. A

thickness of 4.22x10−5 m was predicted, however a value of 7.4285x10−4 m was verified for the

horizontal line x = L/2 after 10s of operation, unchanged after 6500s, growing to 8.6667x10−4

52

4.2. MODEL B

Figure 11 – Electrolyte velocity on X-direction in longitudinal cut to the positive and negative electrodes, atY = hcoll +hf/2 e Y = hcoll +hf +hm +hf/2.

m at the same position after 18500s, and back to 7.4285×10−4 m at the end of battery operation.

Moreover, differently from what happens to H+ concentrations, greater concentration variations

are observed out of the boundary layer and near the interface between electrode and current

collector, unlike the behavior expected by equation 3.24. In addition, at the end of charge cycle,

as shown in Figure 19, narrow variations inherent to boundary layers could not be found.

Such behavior can be referred back to equation 3.21, in which it is observed that the

expression multiplying the source term S depends on the concentration:

Λ =L

[ck]uin(4.1)

in fact, when the magnitude order [ck]∼ 102−103 is taken, the expression result will have order

Λ∼ 10−3−10−2 indicating that the source term contribution can be measured by:

1

(102−103)S (4.2)

In spite of that, because of the parameters L and uin, it results precisely in Λ∼ 7.4214x10−3−

7.4214x10−2, that leads to:

1

(1.3475x101−1.3475x102)S (4.3)

It means that when the concentration ck reaches values in the order of 103, the denominator that

divides the source term S is greater than 134.7, but when the concentration has the order of 102,

53


the same denominator is about 13.47 implying on a bigger significance of this term in equation

3.21. That is precisely the case for evaluated results: for the H+ ion, which concentration is

always greater than 5000 mol m−3, the source term plays a minor role and could be neglected in

the equation; nevertheless, for the specie V 3+ the source term mischaracterizes the relation 3.24

by the time when the ion reaches its lowest concentration.

Figure 12 – Comparison between charge and discharge curve for model B

4.3 Model C

The percentage error for this model during the charge process is around 1.25%, while

the maximum error reaches 3.14%. For the discharge the mean error is 2.47%, and the maximum

error is 3.82%. The estimated time for the charge cycle to complete is 3.3921 hours, representing

a difference of 3.64% on the real cycle duration.

4.4 Model D

The Figure 23 shows the curve of charge and discharge using the time steps provided

by the adaptive scheme from Model E. In other words, the same steps chosen by COMSOL

during the execution of Model E were saved and then employed in the time marching for Model

D. Were compared the predictions done through the uni-dimensional quasi-steady formulation

(Model D) and bi-dimensional quasi-steady (Model E).

54

4.4. MODEL D

Figure 13 – H+ concentration at the positive electrode after 6500s of charge for model B

Figure 14 – H+ concentration at positive electrode after 13000s of charge for Model B

55


Figure 15 – H+ concentration at positive electrode after 5500s of discharge for Model B

Figure 16 – H+ concentration at positive electrode after 13000s of discharge for Model B

56

4.4. MODEL D

Figure 17 – V3+ concentration at negative electrode after 10s of charge for Model B


57



Figure 20 – V3+ concentration at negative electrode after 5500s of discharge for Model B

58

4.4. MODEL D

Figure 21 – V3+ concentration at the negative electrode after 13000s of discharge for Model B

Figure 22 – Comparison between charge-discharge curve for Model C against experimental data

59


Figure 23 – Comparison between charge-discharge curve for Models B and E

The mean percentage error between both models is around 0.041% during the charge

cycle, never being greater than 0.05%. During the discharge cycle this value decreases to

0.0405%, and the maximum percentage error increases to 0.0587%.

4.5 Model E

The charge-discharge curve simulated by COMSOL can be seen in Figure 24 and shows

a mean error of 1.22%, and maximum error of 3.47% during the charge cycle, mean error of

2.55% and maximum of 3.93% for the the discharge process. The time for the charge to complete

is predicted as 3.3973 hours for this implementation, representing an error of 3.49% compared

to the experimental data.

For this same model, the charge and discharge resulting from the simulation described in

25 shows an mean error of 1.10% and maximum of 3.14% during the charge cycle, mean error of

0.99% and maximum of 1.46% for the discharge cycle. The total charge time for this simulation

is 3.4886 hours, differing by 0.89% from the experimental data. The time steps adopted in this

execution were as well provided by the adaptive scheme from Model E.

According to the documentation available in (COMSOL, 2013), in certain cases the

Newton method implemented by COMSOL for the numerical solution of non-linear systems

may, aiming at the economy of computational resources, apply the same Jacobian matrix used

60

4.6. COMPUTATIONAL RESOURCES USAGE

Figure 24 – Comparison between charge-discharge curve for the Model E implemented in COMSOL

in Algorithm 1 in several iterations, i.e., without having to update it after each new approxima-

tion. That may explain the slight better fitting of the results provided by MATLAB with the

experimental data.

4.6 Computational resources usage

The computer used to execute the simulations presented here has the following configu-

rations: Intel R○ CorelTMi7-3610QM CPU @ 2.30GHz processor, 8 GB of RAM memory, and

Windows 8-64bits operating system.

With respect to the usage of computational resources there were made comparisons to

the CPU time and the allocation of RAM memory. The data about the simulations executed

with COMSOL were provided by the log generated by the software, the term Physical Memory

corresponds to the consumption of main memory. The data about simulations executed with

MATLAB were obtained from the built-in profiler.

According to the Table 9, the simulations for Model B were executed 1.75× faster than

the Model A, the Model C in turn ran 2.00× faster. The RAM memory usage was reduced in

11.39% for the Models B and C, in comparison the the Model A.

For the Model D, the execution was 1.69× slower than the Model A, and the main

61


Figure 25 – Comparison between charge-discharge curve for the Model E implemented in MATLAB

Tabela 9 – Usage of computational resources for each model.

Model ID. Software CPU time [s] RAM usage[GB]

A COMSOL 1078 2.02B COMSOL 614 1.79C COMSOL 538 1.79D COMSOL+MATLAB 1821 0.859E COMSOL 439 2.14

memory consumption was reduced in 57.58%. The processing time is longer because this

formulation implies on a non-linear system being solved for each one of the 40 transversal

sections of the battery cell for each time step. The reduction in memory usage is due the uni-

dimensional battery, from which 866 MB were occupied by COMSOL and the other 14 MB by

MATLAB.

The COMSOL implementation for Model E shows a processing time 2.46× faster than

the Model A, and a memory usage 5.94% bigger. The increasing in the storage expenses can

be related to the growth in the number of components employed in the programming: there are

needed three other generic components General Form PDE to replace each Tertiary Current

Distribution, Nersnt-Planck and one to replace the Secondary Current Distribution. That changes

increase the space allocated in memory to configure the problem.

For the implementation of the Model E using solely MATLAB, it is not reasonable to

62

4.6. COMPUTATIONAL RESOURCES USAGE

make comparisons about the computational performance, since it does not employ a parallel

algorithm to solve the linear system, unlike what is done by the software COMSOL Multiphysics.

This execution was planned and executed as a mean to verify the numerical results provided

by COMSOL. The addition of a parallel solver to the MATLAB script may be effectuated in a

future work, as it is goes beyond the scope of this research.

63

CHAPTER

5

FINAL CONSIDERATIONS

In this research the use of asymptotic reduction in the mathematical modeling of vana-

dium redox battery was investigated, as well as the changes that its incorporation may have on

the usage of computational resources, and on the agreement with experimental data existing

in literature. Thus, it is built a reference that summarizes a variety of models discussed from

the point of view of asymptotic analysis and numerical solution, discussing formulations which

performance diversify in the computational requirements as RAM memory and CPU time.

Moreover, the comparison between the solutions provided by COMSOL and that one

resulting from the MATLAB script implemented in this research represents a mutual numerical

validation for the computational simulations executed by the referred software.

REFERENCES

AL-FETLAWI, A.; SHAH, A. A.; WALSH, F. Modelling the effects of oxy-gen evolution in theall-vanadium redox flow battery. Electrochimica Acta, p. 55:3192–3205, 2010. Cited on page20.

AL-FETLAWI, A.; SHAH, A. A.; WALSH, F. C. Non-isothermal modelling of the all-vanadiumredox-flow battery. Electrochimica Acta, p. 55:78–89, 2009. Cited 3 times on pages 20, 24e 28.

BAZANT, M. Lecture 24: Diffusive Charge in Electrolytes. 2014. No-tas de aula. Disponível em: <http://ocw.mit.edu/courses/chemical-engineering/10-626-electrochemical-energy-systems-spring-2011/lecture-notes/MIT10_626S11_lec24.pdf>. Acesso em: 25 jan. 2014. Cited on page 22.

BERNARDI, D. M.; VERBRUGGE, M. W. Mathematical model of a gas diffusion electrodebonded to a polymer electrolyte. AIChe Journal, v. 37, p. 1151–1163, 1991. Cited on page 37.

BURDEN, R. L.; FAIRES, J. D. Análise numérica. [S.l.]: Cengage Learning, 2008. Cited onpage 48.

CHEN, C. L.; YEOH, H. K.; CHAKRABARTI, M. H. An enhancement to vynnycky’s modelfor the all-vanadium redox flow battery. Electrochimica Acta, Elsevier, p. 120:167–179, 2014.Cited 3 times on pages 19, 25 e 26.

COMSOL MULTIPHYSICS. Reference Manual: Version 4.4. Rio de Janeiro, 2013. 1270 p.Cited 3 times on pages 44, 46 e 60.

CORCUERA, S.; SKYLLAS-KAZACOS, M. State-of-charge monitoring and electrolyte reba-lancing methods for the vanadium redox flow battery. European Chemical Bulletin, v. 1, n. 12,p. 511–519, 2012. Cited 2 times on pages 26 e 35.

ESA. Peak shaving and demand charge avoidance: prudent

energy vanadium redox battery storage system (VRB-ESS). 2014.Disponível em: <http://energystorage.org/energy-storage/case-studies/peak-shaving-and-demand-charge-avoidance-prudent-energy-vanadium-redox>. Acessoem: 19 jan. 2014. Cited on page 18.

HOWISON, S. Practical Applied Mathematics - Modelling, Analysis, Approximation. [S.l.]:Cambridge University Press, 2005. Cited 2 times on pages 36 e 42.

HUANG, K.-L.; LI, X. gang; LIU, S. qin; TAN, N.; CHEN, L. quan. Research progress hofvanadium redox flow battery for energy storage in china. Renewable Energy, p. 33:186–192,2008. Cited on page 18.

KEAR, G.; SHAH, A. A.; WALSH, F. C. Development of the all-vanadium redox flow battery forenergy storage: a review of technological, financial and policy aspects. International journal

of energy research, Wiley Online Library, p. 36:1105–1120, 2011. Cited on page 19.

References

KJEANG, E.; PROCTOR, B. T.; BROLO, A. G.; HARRINGTON, D. A.; DJILALI, N.; SINTON,D. High-performance microfluidic vanadium redox fuel cell. Electrochimica Acta, p. 52:4942–4946, 2007. Cited 2 times on pages 26 e 34.

KNEHR, K. W.; AGAR, E.; DENNISON, C. R.; KALIDINDI, A. R.; KUMBUR, E. C. Atransient vanadium flow battery model incorporating vanadium crossover and water transportthrough the membrane. J. Electrochem. Soc., v. 159, n. 9, p. A1446–A1459, 2012. Cited 6times on pages 35, 37, 38, 42, 43 e 51.

LIDE, D. R. Handbook of Chemistry and Physics. 84th. ed. [S.l.]: CRC Press, Inc. BocaRaton, Florida, 2003. Cited on page 38.

NAG, P. K. Basic and applied thermodynamics. First edition. [S.l.]: Tata McGraw-Hill, 2002.Cited on page 19.

NEWMAN, J.; THOMAS-ALYEA, K. E. Electrochemical Systems. Third edition. [S.l.]: JohnWiley & Sons, 2004. Cited 4 times on pages 19, 22, 24 e 38.

NREL - National Renewable Energy Laboratory. Learning About Renewable Energy. 2014.Disponível em: <http://www.nrel.gov/learning/>. Acesso em: 13 jan. 2014. Cited on page 15.

ORIJI, G.; KATAYAMA, Y.; MIURA, T. Investigation on v(iv)/v(v) species in vanadium redoxflow battery. Eletrochimica Acta, p. 49:3091–3095, 2004. Cited on page 34.

. Investigation on v(iv)/v(v) and v(ii)/v(iii) redox reactions by various electrochemicalmethods. Journal of Power Sources, p. 139:321–324, 2005. Cited on page 34.

PARASURAMAN, A.; LIM, T. M.; MENICTAS, C.; SKYLLAS-KAZACOS, M. Review ofmaterial reserch and development for vanadium redox flow battery applications. Electrochimica

Acta, Elsevier, p. 101:27–40, 2013. Cited 2 times on pages 16 e 19.

POURBAIX, M. Atlas of electrochemical equilibria in aqueous solutions. 2nd. ed. [S.l.]:NACE, Houston, TX, 1974. 234 p. Cited on page 37.

PRIFTI, H.; PARASURAMAN, A.; WINARDI, S.; LIM, T. M.; SKYLLAS-KAZACOS, M.Membranes for redox flow battery applications. Membranes, p. 2:275–306, 2012. Cited onpage 27.

REDT. Sucessful installationg of REDT’S storage system in collaboration with FP7 and the

european commission. 2014. Disponível em: <http://www.redtenergy.com/news/fp7>. Acessoem: 17 jan. 2014. Cited on page 17.

RICHMOND, B. C. VRB power sells two 5kW x 4hr VRB energy sto-

rage systems (TM) to Winafrique Technologies for Deployment with Safari-

com, in Kenya. 2014. Disponível em: <http://www.newswire.ca/fr/story/97801/vrb-power-sells-two-5kw-x-4hr-vrb-energy-storage-systems-tm-to-winafrique-technologies\-for-deployment-with-safaricom-in-kenya>. Acesso em: 17 jan. 2014. Cited on page 18.

RODENHAUSEN, H. Einstein’s relation between diffusion constant and mobility for a diffusionmodel. Journal of Statistical Physics, Kluwer Academic Publishers-Plenum Publishers, p.55(5–6):1065–1088, 1989. Cited on page 22.

68

References

SHAH, A. A.; AL-FETLAWI, A.; WALSH, F. Dynamic modelling of hydrogen evolution effectsin the all-vanadium redox flow battery. Electrochimica Acta, p. 55:1125–1139, 2010. Cited 4times on pages 20, 26, 28 e 35.

SHAH, A. A.; WATT-SMITH, M. J.; WALSH, F. A dynamic performance model for redox-flowbatteries involving soluble species. Electrochimica Acta, p. 53:8087–8100, 2008. Cited 4 timeson pages 20, 24, 37 e 38.

SHARMA, A. K.; VYNYCKY, M.; LING, C.; BIRGERSSON, E.; HAN, M. The quasi-steadystate of all-vanadium redox flow batteries: A scale analysis. Electrochimica Acta, v. 147, p.657–662, 2014. Cited 4 times on pages 19, 35, 38 e 39.

SKYLLAS-KAZACOS, M. The vanadium redox battery and fuel cell for large-scale energystorage. In: 19th World Energy Congress. [S.l.: s.n.], 2004. Sydney, Australia. Cited on page18.

SKYLLAS-KAZACOS, M.; ROBINS, R. G.; FANE, A. F.; GREEN, M. A.; RYCHCIK, M. Newall-vanadium redox flow cell. Journal of Electrochemical Society, p. 133:1057–1058, 1986.Cited on page 16.

SKYLLAS-KAZACOS, M.; SUM, E.; RYCHCIK, M. Investigation of the v(v)/v(iv) system foruse in the positive half-cell of a redox battery. Journal of Power Sources, p. 16:85–95, 1985.Cited on page 16.

SUM, E.; SKYLLAS-KAZACOS, M. A study of the v(ii)/v(iii) redox couple for redox flow cellapplications. Journal of Power Sources, p. 15179–190, 1985. Cited on page 16.

TICIANELLI, E. A.; GONZALEZ, E. R. Eletroquímica: princípios e aplicções. Segundaedição. [S.l.]: Edusp, 2005. Cited 3 times on pages 19, 22 e 23.

VYNNYCKY, M. Analysis of a model for the operation of a vanadium redox battery. Energy, p.36:2242–2256, 2011. Cited 12 times on pages 9, 16, 17, 19, 24, 25, 26, 28, 35, 36, 38 e 45.

VYNNYCKY, M.; ASSUNçãO, M. Asymptotic and numerical modelling of vanadium redox

flow batteries. in prep. Cited 3 times on pages 39, 40 e 41.

VYNNYCKY, M.; SHUGAI, G.; YAKUBENKO, P.; MELLGREN, N. Asymptotic reduction fornumerical modeling of polymer electrolyte fuel cells. SIAM J. Appl. Math., v. 70, p. 455–487,2009. Cited on page 35.

YAMAMURA, T.; WATANABE, N.; YANO, T.; SHIOKAWA, Y. Electron transfer kinetics ofNp3+/Np4+, NpO+

2 /NpO2+2 , V2+/V3+, and VO2+/VO+

2 at carbon electrodes. J. Electrochem.

Soc., v. 152, p. A830–A836, 2005. Cited on page 38.

YOU, D.; ZHANG, H.; CHEN, J. A simple model for the vanadium redox battery. Electrochi-

mica Acta, p. 54:6827–6836, 2009. Cited on page 25.

69

www.kth.se

Mathematical modelling of vanadium redox...

Documents

Transcript of Mathematical modelling of vanadium redox...