MATHEMATICAL MODELING OF THE INITIAL DISCHARGE OF ALKALINE …m... · 2019. 9. 25. · Abstract A...

MATHEMATICAL MODELING OF THE INITIAL DISCHARGE OF ALKALINE

ZINC-MANGANESE DIOXIDE BATTERIES

A Thesis Presented

By

Zhicheng Lu

to

The Department of Chemical Engineering

in partial fulfillment of the requirements

for the degree of

Master of Science

in the field of

Chemical Engineering

Northeastern University

Boston, Massachusetts

July, 2019

Acknowledgement I would like to thank my advisor, Prof. Joshua Gallaway, for providing me the opportunity to

pursue this study, and also for his great guidance to me during my time as a master student. Prof.

Gallaway is very nice to me. He was always ready when I needed helps during the study.

I would also like to thank my colleagues, Matt Kim, Benjamin Howell, and Alyssa Stavola, for

their helps to me.

I would like to thank all my teachers for their knowledge they provided.

Special thanks to my parents. Without their love and support, this thesis would not have been

possible.

Zhicheng Lu

Abstract A simplified model is proposed to simulate discharge behavior of alkaline Zn-MnO2 batteries.

The alkaline Zn-MnO2 battery system analyzed here consists of a zinc anode, an inert separator, and

an MnO2 cathode. This simple model is based on macrohomogeneous porous electrode theory

including the considerations of potential drop in the electrolyte, porosity change, composition

change due to electrochemical or chemical reactions, charge-transfer effects, and particle-scale

transport effects including a core-shell model of MnO2 discharge. The thesis focuses on the

discharge behavior of MnO2 cathodes. Thus, the zinc anode is assumed as a reversible,

nonpolarizable electrode with uniform current distribution and a mixed-reaction model is applied to

describe the anode discharge. Newman's BAND method is used to solve the model numerically in

Python.

First, the model is developed in cylindrical coordinates to analyze the discharge behavior of

primary AA battery cathodes. The effect of initial electrolyte concentration and discharge rate is

investigated. Also, a secondary current distribution is evaluated in order to determine the reaction

distribution of an annular porous electrode at the initial discharge stage. The results are considered

as initial conditions for the numerical model. Conclusively, thinner electrode, high electrode

conductivity, and slow kinetics lead to a more uniform current distribution at the initial discharge

stage.

Second, the model is developed in Cartesian coordinates to investigate the initial discharge

behavior of prismatic, rechargeable Zn-MnO2 batteries. Prismatic Zn-MnO2 batteries are good

candidates for grid-scale energy storage due to the low fabrication cost, high safety, high energy

density, and environmental friendliness. Maintaining reversibility of cathode is an important task

for rechargeable Zn-MnO2 batteries, as irreversible materials are prone to form in the MnO2

cathodes.. Local DOD (depth of discharge) y is an important parameter in this regard, as it has been

found that when y is higher than 0.79 proton equivalents per Mn atom, Mn3O4 and ZnMn2O4 form,

destroying MnO2 rechargeability. The local DOD (depth of discharge) y is evaluated at various

discharge conditions (different cathode thicknesses & discharge rates) in order to map the design

space where Zn-MnO2 batteries can maintain rechargeability during initial discharge.

Content Chapter 1 Introduction .................................................................................................................................. 1

1.1 Development of Zn-MnO2 alkaline battery................................................................. 1

1.2 Structure of primary Zn-MnO2 cylindrical cells ........................................................ 2

1.3 Cathode of the Zn-MnO2 system .................................................................................. 3

1.4 Anode of the Zn-MnO2 system ..................................................................................... 4

Chapter 2: Porous electrode theory and its application to MnO2 cathodes ........................................... 5

2.1 Porous electrode theory ........................................................................................................ 5

2.2 Macroscopic description of porous electrode .................................................................... 5

2.3 A simple porous electrode model ........................................................................................ 6

2.4 A simple model in cylindrical coordinates ...................................................................... 11

2.5 Method used for numerical solution ................................................................................ 17

2.6 Conclusion ........................................................................................................................... 18

Chapter 3 Simplified model describing discharging behavior of cathode ......................................... 19

3.1 Introduction ......................................................................................................................... 19

3.2 Electrode Reactions............................................................................................................ 20

3.3 Assumptions adopted in the model .................................................................................. 21

3.4 Method used to solve the model ....................................................................................... 21

3.4 Model formulation.............................................................................................................. 23

3.5 Results of the model ........................................................................................................... 32

3.6 Conclusions ......................................................................................................................... 41

Chapter 4 Simplified model describing prismatic Zn-MnO2 battery cathode discharge behavior.. 42

4.1 Introduction ......................................................................................................................... 42

4.2 Prismatic Zn-MnO2 battery model ................................................................................... 43

4.3 Results of the model ........................................................................................................... 47

4.4 Conclusion ........................................................................................................................... 55

Chapter 5 Conclusion & future work ...................................................................................................... 56

Reference ..................................................................................................................................................... 58

Appendix ..................................................................................................................................................... 60

1

Chapter 1 Introduction

1.1 Development of Zn-MnO2 alkaline battery

The development of zinc-manganese batteries has undergone a long evolution. As early as

1868, the French engineer Georges Leclanche used manganese dioxide and carbon mixture as

positive electrode, which was pressed into the cylindrical body of porous ceramics. A zinc rod is

partially inserted into the solution as the negative electrode. The electrolyte is made of 20%

ammonium chloride aqueous solution, and the battery container is made of glass bottle. Thus, the

first Zn-MnO2 wet battery was invented. In 1886, Gass changed the ammonium chloride electrolyte

to a paste of ammonium chloride, zinc chloride, gypsum and water, made the zinc sheet into a

cylindrical container for the battery, and sealed it with paraffin to make the original primary Zn-

MnO2 battery. Shortly thereafter, flour and starch were used as gelling agents for the electrolyte

solution, which greatly improved the portability of the zinc-manganese battery and laid a good

foundation for the industrial production and widespread use of the battery. [1]

In 1870, amalgamated zinc powder was used as anodes for the Leclanche Cell, preventing self-

discharge phenomena for zinc anodes. In 1877, the carbon rod current collector was treated with

dipping wax to prevent mixing with the electrolyte. After these improvements were made, the

Leclanche cell was put into wide industrial production in 1890. The further development of the

Leclanche cell mainly focused on the following four points.

• Increasing MnO2 mass loading of cathode

• Adding ZnO powder into the anode to lessen anode erosion by the alkaline electrolyte

• Using of NaOH or KOH as the electrolyte instead of NH4Cl and ZnCl2

After 50 years of development, around the 1950s, the alkaline Zn-MnO2 battery was invented.

In alkaline Zn-MnO2, electrolytic manganese dioxide was used as the cathode, a Zn & ZnO mixture

powder was used as the anode, and NaOH or KOH was used as electrolyte. These improvements

increased the capacity of the battery, making it suitable for continuous discharge at relative high

current, preventing large temperature variation while the battery discharged. In the meantime, the

battery could be reserved for a long time without leakage.

2

Due to environmental concern, the further development of Zn-MnO2 concentrated on reducing

the use of amalgam. Around the 1990s, no-amalgam alkaline Zn-MnO2 cell entered the battery

market.

Now researchers are focused on the development of secondary Zn-MnO2 batteries. Due to the

low cost, high energy density, and environmental friendliness, Zn-MnO2 is a good candidate for

grid-scale energy storage. After 30 years development, researchers have found various methods for

improving the rechargeability of alkaline Zn-MnO2 batteries.

In the United States, the alkaline zinc-manganese dioxide (Zn-MnO2) has generally replaced

the Leclanche cell. Comparing to the modern alkaline cells, Leclanche cells have the following

disadvantages:

• Leclanche cells are not suitable for high-rate continuous discharge.

• The capacity of Leclanche cells is much lower than modern alkaline cells.

• The leakage phenomenon is common in Leclanche cells.

However, due to the lower cost and suitability for low-current electric appliance, Leclanche

cells are still widely used in most developing countries like China and Brazil.

Modern Alkaline cells can be made in different size such as AA, AAA, 4/5AA, B, C, and D

according to different capacity demands. These cells are suited to extensive applications, such as

camera power, toys, and TV sets etc.

1.2 Structure of primary Zn-MnO2 cylindrical cells

Figure 1 Cross Section of a Duracell AA battery [25]

Figure 1 shows the cross-section of a Duracell AA battery. It consists of five parts: inner current

collector (pin), anode, separators, cathode, and outer current collector (can). As shown in Figure 1,

the porous anode and cathode make up the largest parts of the battery. Both electrodes are made of

particles, with electrolyte filling the void areas of the electrode. Electric power of the Zn-MnO2

3

battery comes from the electrochemical reactions of the cathode and anode active materials. A

porous separator keeps the cathode and anode from touching each other. The function of the two

current collector is to build a connection between the battery the outer electrical circuit.

1.3 Cathode of the Zn-MnO2 system

The active material of the cathode is manganese dioxide. There are three kinds of manganese

dioxide in chemistry world: electrolytic MnO2 (EMD) which is made using electrochemical

methods, chemical MnO2 (CMD) made my chemical methods, and natural MnO2 (NMD). Many

different polymorphs of MnO2 exist, including pyrolusite (β-MnO2), ramsdellite (R-MnO2),

hollandite (α-MnO2), intergrowth (γ-MnO2), spinel (λ-MnO2),and layered (δ-MnO2), which are

often applied for energy storage and other applications.[2] The reason why manganese dioxide is a

widely used cathode material is its high electrochemical reactivity, its low cost, and its non-toxic

nature.

Although manganese dioxide has superior electrochemical reactivity, the conductivity of

manganese dioxide is fairly low. Thus in battery systems, carbon or acetylene black is added to the

cathode to increase the conductivity. [3]

The reaction mechanism of manganese dioxide cathode contains two steps. The first step is the

proton insertion reaction: [4][5]

MnO2+H2O+ e-→MnOOH+ OH-

which is the main reaction of the cathode in primary batteries. The second step is the further

reduction of MnOOH, which happens at low potential.

MnOOH+H++e-→Mn(OH)2

The second mechanism occurs only in certain circumstances.

ThusMnO2 discharge begins via a proton insertion reaction. The second step is a conversion

reaction, and Mn(OH)2 can be reversibly recharged back to MnO2, and its formation is not a cause

of final failure of the cell. However, two spinel materials, hausmannite (Mn3O4) and hetaerolite

(ZnMn2O4) also form during discharge, and these are considered as electrochemically inactive

species, leading to failure. The formation of spinel structures Mn3O4 and ZnMn2O4 is the true reason

alkaline batteries are not rechargeable.

4

Gallaway et al [6] proposed the mechanism for formation of hasmannite and hetaerolite. The

reaction mechanism is listed below.

2α-MnOOH+Zn(OH)42-→ZnMn2O4+2OH-+2 H2O

2α-MnOOH+Mn(OH)42-→Mn3O4+2OH-+2 H2O

The dissolution of 𝑀𝑛%& produces 𝑀𝑛(𝑂𝐻))*+ , which reacts with 𝛼-MnOOH leading to

irreversible crystal change of the cathode. This was shown to occur after 0.79 proton equivalents

were inserted into the cathode active material 𝛾-MnO2. After this precisely defined point, 𝛼 -

MnOOH formed. The 𝛼-MnOOH material was found to be the precursor to hausmannite (Mn3O4)

and hetaerolite (ZnMn2O4), which quickly formed after 𝛼 -MnOOH. This showed that the

proximate cause of cathode non-rechargability was the insertion of 0.79 proton equivalents. After

this point, a series of chemical transformations led to the unrechargable spinel structures.

1.4 Anode of the Zn-MnO2 system

The active material in the anode is Zn. Low equilibrium potential and high overpotential for

the hydrogen evolution reaction on Zn metal make it a good anode for high energy density batteries.

In addition, Zn anode is cheap and environmentally friendly.

The Zn anode reaction is governed by a dissolution-precipitation mechanism, which is shown

below

Zn+ 4OH- →Zn(OH)42-+2e-

Zn(OH)42-→ZnO+2OH-+H2O

While the battery discharges, the Zn first dissolves in KOH electrolyte, producing zincate ions

(Zn(OH)42-). Then the zincate ions further precipitate as solids the form ZnO. [7]. The utilization of

Zn in a commercial battery is reduced because a passive film of ZnO forms on the surface of Zn

particles, preventing Zn metal beneath the film from reacting. The discharge rate affects the spatial

distribution of ZnO in the anode. At low discharge rate, ZnO forms uniformly in the anode region.

At high discharge rate, ZnO concentrates on the region near the separator, which reduces battery

cell life. [8]

5

Chapter 2: Porous electrode theory and its application to MnO2

cathodes

2.1 Porous electrode theory

A simple electrode is composed by a solid metal electrode and electrolyte in contact. The

electrochemical reaction in this system is restricted to the surface of the electrode. However, in an

actual battery, a porous electrode is used. Being porous means that active electrode materials are

different distances from the other electrode. A porous electrode provides a large reaction region due

to its high surface area. Porous electrode theory was developed in the 1950s and 1960s to allow

modeling of these kinds of electrochemical devices. In 1960, Euler and Nonnenmacher performed

research on current distributions in porous electrodes assuming a constant concentration and

linearized kinetics. In 1962, Newman and Tobias amended Euler and Nonnenmacher’s model using

Tafel kinetics. In 1975, Newman published a paper entitled ‘Porous electrode theory with battery

applications’ and summarized the basic methodology of modelling porous electrodes, which is

termed the Newman method at present [9,10] Since then, many battery models have been developed

based on porous electrode theory.

2.2 Macroscopic description of porous electrode

It’s impossible to catch all geometrical details when one conducts a complex battery model.

Thus, to perform theoretical analysis of complex charge and discharge behavior, it is essential to

establish a model capturing essential features of the cell without considering all geometric details.

Also, the model should be depicted by parameters which can be easy to measure by some simple

experiments. For example, a porous electrode with arbitrary and random structures can be

characterized by porosity, tortuosity, average surface area per unit volume, average conductivity of

solid matrix, etc. An appropriate model identifying the governing processes inside the battery should

involve averages of variables over a small region with respect to the whole cell but large enough to

contain pore structure. Because the model is being developed to describe the charge and discharge

behavior through a whole cell, the double-layer charging is ignored.[10]

6

2.3 A simple porous electrode model

When a battery is in use, the battery is a part of an electrical circuit and current must travel

across the cell. Inside a battery, the current is transported from anode to cathode (from pin to can in

Figure 1). Pin and can are current collectors inside the battery system. The current collectors serve

as a bridge from the battery to the external current. If a porous electrode is designed well, there are

many conduction pathways through both the particle and electrolyte phases. Fundamentally, current

is the movement of charge in either phase. In solid species, charges move as negative electrons; in

the electrolyte, charges move as ions, which may be positive or negative.

Figure 2 Simple schematic diagram of porous electrode [25]

In Figure 2, the complex electrode structure is assumed to have a single pore, which runs from

separator to current collector. The direction across the electrode from separator to current collector

is the x-direction. In this simple porous electrode model, the current density and potential of the

solid active material and electrolyte is assumed only as a function x.

The separator will not allow electrons to pass through, and thus all current is ionic current at

the separator. The current collector only allows electrons to pass, thus all the current at the current

collector is electronic. The total current through the separator and current collector must be the same

while current is passing. In the region between separator and current collector, the transformation

7

between ionic and electronic current takes place due to the electrochemical reaction at the solid-

electrolyte interface.

Discharge behaviors governed by different kinetics (linear kinetics and Tafel kinetics) are

investigated in this simple porous electrode model. Linear kinetics and Tafel kinetics are two special

cases of the Butler-Volmer equation.

The Butler-Volmer equation is one of the basic electrochemical kinetic relationships, which

shows the relation between current density and overpotential.

𝑖 = 𝑖1 (exp 5𝛼6𝑧𝐹𝜂𝑅𝑇 < − 𝑒𝑥𝑝 5−

𝛼A𝑧𝐹𝜂𝑅𝑇 <) (2-1)

Equation 2-1 is Butler-Volmer equation, 𝑖B is the current density of solid phase and 𝑖* is the

current density of electrolyte. 𝜂 is the overpotential which is equal to the potential difference

between potential of solid phase 𝜙B and potential of electrolyte 𝜙*. 𝑖1 is the exchange current

density. 𝛼6 is the anodic transfer coefficient, 𝛼A is the cathodic transfer coefficient, R is the gas

constant, F is the Faraday constant, z is the number of electrons participating in the electrochemical

reaction, and T is the temperature.

When the overpotential is low, a Maclaurin expression is applied to simplify the Butler-Volmer

equation. The results of simplification are shown below:

𝑖 = 𝑖1𝑧𝐹𝑅𝑇 𝜂

(2-2)

This is the expression of linear kinetics, which is only valid in low overpotential region.

Another limiting case for Butler-Volmer equation is Tafel equation, which is also referred as

Tafel kinetics. There are two cases of Tafel kinetics:

When 𝜂 ≪ 0,

𝑖 = −𝑖1𝑒𝑥𝑝 5−𝛼A𝑧𝐹𝜂𝑅𝑇 < (2-3)

When 𝜂 ≫ 0,

𝑖 = 𝑖1𝑒𝑥𝑝 5−𝛼6𝑧𝐹𝜂𝑅𝑇 < (2-4)

Tafel kinetics is valid at high overpotential regions, when the electrochemical reaction is

dominated by either the anodic or cathodic reaction.

8

2.3.1 Assumptions and basic equations of the model

To solve this simple model, the following assumptions are made:[9]

• It assumed the system is at a steady-state while discharging.

• The concentration effect on potential and current density is ignored.

• The basic equations for this model are shown below:

𝑑𝑖B𝑑𝑥 +

𝑑𝑖*𝑑𝑥 = 0@𝑥 = 𝐿, 𝑖B = 𝐼 (2-5)

𝑑𝑖*𝑑𝑥 = 𝑎𝑖1

𝑛𝐹𝑅𝑇

(𝜙B − 𝜙*)𝑜𝑟𝑑𝑖*𝑑𝑥 = −𝑎𝑖1 exp P

−𝛼A𝑛𝐹𝑅𝑇

(𝜙B − 𝜙*)Q @𝑥 = 0, 𝑖* = 𝐼 (2-6)

𝑖B = −𝜎𝑑𝜙B𝑑𝑥 @𝑥 = 0,𝜙B = 0 (2-7)

𝑖* = −𝜅𝑑𝜙*𝑑𝑥 @𝑥 = 𝐿, 𝑖* = 0 (2-8)

In equation 2-5, 𝑖B is the current density of solid phase and 𝑖* is the current density of

electrolyte. Equation 2-5 is a statement of charge conservation, stating that disappearance of 𝑖*

must be matched by appearance of 𝑖B and vice versa. The boundary condition states all current is

electronic at the current collector. Equation 2-6 states the rate of electrochemical reactions. The two

examples shown are linear kinetics and Tafel kinetics. The boundary condition states that all the

current at the separator is ionic current. Equation 2-7 is Ohm’s law in the solid. The boundary

condition arbitrarily sets the solid potential to zero at the separator. Equation 2-8 describes Ohm’s

law in the liquid. The boundary condition states the liquid potential gradient is zero at the current

collector.

The constant values used to solve the model below are shown in Table 1.[11]

9

Table 1: Value used to solve the simple porous electrode model [11]

Name of parameter Value and units

𝑎 specific area of electrode

per unit volume

23300 cm-1

𝑖1 exchange current density 2×10-7 A/cm2

𝐼 discharge current density 0.1A/cm2

𝐿 length of the electrode 1cm

𝜅 electrolyte conductivity 0.06S/cm

𝜎 electrode conductivity 20S/cm

2.3.2 The results of the model

Python is applied to solve this simple model. Scipy.integrate.solve_bvp in python scipy

package is used to solve boundary problems here. Compared to the BAND method, solve_bvp

simplifies the solution technique.

Figure 3 Current distribution through the electrode

The dot line in Figure 3, shows the current-density distribution when electrode discharge

behavior is governed by linear kinetics. The solid line shows the current-density distribution when

discharge behavior is governed by Tafel kinetics. As shown in Figure 3, the Tafel kinetic expression

predicts that the current transformation concentrates at the region near the separator, and linear

10

kinetic expression shows a more even current distribution through the electrode.

Figure 4 Potential distribution across the electrode

The Tafel kinetic expression predicts that current exchange from liquid to solid phases (ionic

current to electronic current) will be concentrated near x = 0, the location closest to the counter

electrode. There 𝜙B − 𝜙* will be near 300 mV, falling away further into the electrode. In contrast,

the linear kinetic expression predicts a more even exchange of current through the electrode, and

much higher values of 𝜙B − 𝜙* . This is because the linear expression is a softer function of

overpotential than the exponential one. The Tafel approximation is clearly the more appropriate in

this case because 𝜙B − 𝜙* ≫ 𝑅𝑇/𝐹, which is 25.7 mV at room temperature.

Figure 5 and Figure 6 show the current distribution and potential variation across the electrode

when the discharge current density is equal to 0.01A/cm2 (which is pretty low and in a low

overpotential region). In these two figures, the results show that linear kinetics expression and Tafel

kinetics expression predict the same current distribution and potential variation through the cell.

11

Figure 5 Current distribution across the electrode when I = 0.01 A/cm2

Figure 6: Voltage distribution when I=0.01 A/cm2

2.4 A simple model in cylindrical coordinates

The same model is then developed in cylindrical coordinates. The cylindrical cell is one of the

most popular battery configurations in the battery market. Cylindrical cell has several advantages:

1) Easy to manufacture and mechanically stable. It is not easy to rupture when the internal

pressure of battery increases because the tubular cylinder endures high internal pressure.

2) Cylindrical cells have a pressure relief system.

12

The simplest design of pressure-relief system is a membrane, which ruptures under high

pressure. Unwanted phenomena will appear after the membrane breaks such as leakage and dry-out.

Due to these advantages, cylindrical cell has remained popular for a long time.

A secondary current distribution model is presented here to examine the reaction distribution

in annular electrodes, like those found in cylindrical cells. Current distribution is divided into

primary and secondary current distribution. To examine the primary current distribution, surface

overpotentials are neglected, and the electrode is taken to be an equipotential surface. Primary

current distribution is highly nonuniform. To examine secondary current distribution, concentration

overpotential is neglected, which ignores electrolyte concentration effects. The secondary current

distribution is generally more uniform than the primary current distribution. However, due to the

electrochemical reactions occurring while a battery discharges, the concentration variation across

the battery increases, and thus the analysis mentioned here is only appropriate for the initial

discharge stage before concentration gradients have developed.

The extent of polarization from kinetic and ohmic limitations is assessed in this simplified

model. The effects of electrode porosity and curvature effects on electrode performance are also

considered here.

2.4.1 Assumption and basic equation

To solve this simple model, the following assumptions is made:[12]

• The system maintains steady state while discharging

• The concentration effect on potential and current density is ignored.

• This model is used to analyze the MnO2 cathode in a Duracell AA battery.

The basic equations for this model are shown below:

𝜕𝜙B𝜕𝑟 = −

𝑖B𝜎 @𝑟 = 𝑟 ,𝜙B = 0 (2-9)

𝜕𝜙*𝜕𝑟 = −

𝑖*𝜅 @𝑟 = 𝑟_, 𝑖* = 0 (2-10)

𝜕𝑖*𝜕𝑟 = −

𝑖*𝑟 +

𝑎𝑖1(𝛼6 + 𝛼A)𝐹𝑅𝑇

(𝜙B − 𝜙*)@𝑟 = 𝑟 , 𝑖* =𝐼

2𝜋𝑟 𝐿 (2-11)

1𝑟𝜕(𝑟𝑖B)𝜕𝑟 +

1𝑟𝜕(𝑟𝑖*)𝜕𝑟 = 0@𝑟 = 𝑟_, 𝑖B =

𝐼2𝜋𝑟_𝐿

(2-12)

13

𝑗 =𝑎𝑖1(𝛼6 + 𝛼A)𝐹

𝑅𝑇(𝜙B − 𝜙*)

(2-13)

Equations 2-9 and 2-10 state the conduction mechanisms in the cathode. In the first two

equations, 𝑖B is the current density of solid matrix, 𝑖* is the current density of electrolyte, 𝑟 is

the radius at the cathode-separator interface, and 𝑟_ is the cathode-current collector interface.

Equation 2-11 describes the electrochemical reaction kinetics of the cathode active material, where

linear kinetics have been assumed. Equation 2-12 describes the charge balance across the cathode,

which states that 𝑖* that disappearance must be matched by appearance of 𝑖B and vice versa. The

boundary condition of equation 2-9 arbitrary sets the potential of the solid matrix equal to zero at

the cathode-separator interface. The boundary condition of equation 2-10 states that there is no ionic

current in current collector. The boundary condition of equation 2-11 states that all current in the

separator is ionic. Equation 2-13 represents the transfer current, which is equivalent to the reaction

distribution.

2.4.2 Dimensionless parameters

In order to normalize the system analysis, several dimensionless parameters are introduced[12].

𝑗∗ =𝑗𝐼

𝜋(𝑟_* − 𝑟*)𝐿

(2-14)

𝛿 =(𝑟_ − 𝑟 )*𝑎𝑖1(𝛼6 + 𝛼A)𝐹

𝑅𝑇 P51𝜎<+ 5

1𝜅<Q (2-15)

𝜉 =𝑟 − 𝑟𝑟_ − 𝑟

(2-16)

𝛾 = 𝛿 e𝜅

𝜅 + 𝜎f (2-17)

𝜔 = 1 −𝑟𝑟_ (2-18)

𝑗∗ is the dimensionless transfer current density. 𝛿 measures the relative importance between

ohmic effects and charge-transfer effects, which quantifies the polarization effects of electrode.

Large values of 𝛿 imply that ohmic limitations control the system, which shows more nonuniform

current distribution. Small 𝛿 means that charge-transfer effects control the system behavior, which

shows more uniform current distribution. 𝜉 is dimensionless radius. 𝛾 measures relative

difference between solid conductivity𝜎 and electrolyte conductivity 𝜅. 𝜔 is the curvature of the

electrode. For cylindrical cells, the curvature is a factor which influences the current distribution.

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2.4.3 Constants used to solve the model

The constants value using to solve the model are shown in Table 2: [11]

Table 2: Values used to solve the second current distribution model [11]

Parameter Name Value & Unit Electrode area per volume, 𝒂

23300 cm-1

Exchange current density, 𝒊𝟎

2.0×10-7A/cm2

Battery discharge current, 𝑰

0.1A

Length, 𝑳 4cm Bulk electrolyte conductivity, 𝜿r

0.6S/cm

Electrode conductivity, 𝝈

20S/cm

Outer radius of cathode, 𝒓𝒐

0.66cm

Inner radius of cathode, 𝒓𝒊

0.44cm

Porosity of cathode, 𝝐

0.02,0.04,0.07,0.2,and0.5

Transfer coefficient of anode, 𝜶𝒂

0.5

Transfer coefficient of anode, 𝜶𝒄

0.5

15

2.4.4 Results of the model

Figure 7: The effect of 𝜹 on reaction distribution with 𝝎 = 𝟎. 𝟑 and 𝜸 = 𝟎

Figure 7 shows the effect of electrode porosity on reaction distribution when the conductivity

of the solid is far more than the conductivity of electrolyte, as is normal for practical batteries. The

electrolyte concentration and the porosity of the cathode affect the polarization parameter. In this

part, electrode material with different porosity is applied to investigate the effect of porosity on

electrode discharge performance. The conductivity of electrolyte phase 𝜅 is the effective

conductivity, which is defined as 𝜅 = 𝜅r𝜖B.� where 𝜅r is the conductivity of bulk electrolyte.

Here, we take the conductivity of the solid as a constant 20 S/cm. The polarization parameter 𝛿 is

calculated via equation 2-15. Results show that when porosity 𝜖 increases, 𝛿 decreases. As shown

in the Figure 7, the smaller 𝛿 the more uniform distribution inside the battery. When𝛿 increases,

more reaction happen in the region near separator.

16

Figure 8: The effect of 𝜹 on reaction distribution with 𝝎 = 𝟎. 𝟑 and 𝜸 = 𝟎. 𝟓𝜹

The conductivity of MnO2 is relatively low. Usually, carbon is added to the cathode to improve

its conductivity. Thus, it is important for us to investigate the effect of the dimensionless parameter

γ to optimize the cathode blend. Figure 8 shows the effect of 𝛿 on reaction distribution when the

conductivity of electrolyte and solid are comparable. When the conductivity of the solid matrix and

the active material are comparable, the reaction zone in the electrode tends to shift toward

cathode/separator interface and cathode-current collector interface to minimize the ohmic potential

drop of the system. When 𝛿 is large, the reaction distribution will be more non-uniform compared

to the situation when 𝛿 is small.

Figure 9: The effect of 𝜸 on overpotential when 𝝎 = 𝟎. 𝟑

Overpotential distribution with respect to 𝛾 is shown in Figure 9. When𝜅 is close to𝜎, the

17

current will split between the electrode/separator and the electrode/current collector interface to

minimize the ohmic potential drop in the system. The reaction zone will concentrate on the region

near cathode-separator interface and cathode-current collector interface, which corresponds the blue

line in Figure 9, showing higher overpotential near the cathode-separator interface and cathode-

current collector interface. On the other hand, when𝜅 is far smaller than𝜎, like the purple line in

Figure 9, the reaction zone concentrates at the cathode-separator interface, corresponding to higher

overpotential near cathode-separator interface.

2.5 Method used for numerical solution

Scipy.intergrate.solve_bvp is applied to get numerical solution in this part. Scipy.solve_bvp is

designed to solve a boundary-value problem for a system of ODEs. This package is built based on

a finite difference method. The format to use this package is shown below:

Figure 10: Format using solve_bvp [13]

Using this package to solve a boundary value problem, a function to describe the ODE system

should be set as the first step. That is the first parameter in the parenthetical. In this function, the

right-hand side of the system should be included. The calling signature of the function is fun(x,y)

or fun(x,y,p) if unknown parameters are included.

As the second step, a function describing the boundary conditions of the system should be set.

That is the second parameter, bc in the format. This function evaluates residuals of the boundary

conditions. The calling signature of bc function is bc(ya,yb). In this calling signature, ya is the left

boundary of the system and yb is the right boundary of the system.

As the third step, initial mesh for the boundary problem should be set. The numerical solution

for the system is based on this initial mesh. It must be an increasing sequence of real numbers with

x[0] = a, x[-1] = b.

As the fourth step, the initial guesses for the function values at the mesh nodes should be

18

provided, with the i-th column corresponding to x[i]. Sometimes, the initial guess of the function

will affect the final solution of the system. If a bad initial guess is provided, errors can be returned,

such as ‘Singular Jacobian encountered when solving the system’ or ‘The maximum number of

mesh nodes is exceeded.'

A solution method for coupled, partial differential equations for electrochemical systems was

developed by Newman, and is referred to in the literature as the BAND method[9]. BAND can also

be used to solve the system describing a Zn-MnO2 battery. Compared to the solve_bvp package,

BAND is like the basic algorithm used to execute solve_bvp package. Solve_bvp is the easier way

to solve boundary problem than the BAND method. The detailed introduction of the BAND method

will be provided in Chapter 3.

2.6 Conclusion

In this part, two simple porous electrode models were provided, which described the initial

discharge behavior of porous electrodes. The difference between Tafel kinetics and linear kinetics

was shown in part 2.3. Tafel kinetics resulted in a more nonuniform reaction distribution than linear

kinetics. In both cases, more reaction happened near the cathode-separator interface. At higher

overpotential, Tafel kinetics is closest to describing real initial discharge behavior.

In part 2.4, the discharge performance within annular porous electrode was investigated. 𝛿 is

an important parameter measuring the polarization effects within the porous electrode. When 𝛿 is

small, reaction distribution across the electrode is more uniform and ohmic limitation controls the

system. When 𝛿 is large, reaction distribution across the electrode is nonuniform, and charge-

transfer effects control the system. The effect of 𝛾 on reaction distribution was also investigated.

When the conductivity of the electrolyte and the conductivity of the solid matrix is comparable, the

transefer current density will split between electrode-separator and electrode-current collector

interfaces, and thus the reaction zone tends to shift towards these interfaces.

19

Chapter 3 Simplified model describing discharging behavior of

cathode

3.1 Introduction

In Chapter 2, secondary reaction distribution was provided to describe the discharge behavior

of a porous MnO2 electrode, which neglected the concentration variation while discharging.

However, as discharge continues, the effect of concentration variation influences the discharge

behavior of the cell. Here, a detailed porous electrode model including changes in concentration and

porosity is provided mainly to describe discharge behavior of the cathode.

The charge and discharge behavior inside a battery is governed by kinetics, transport, and

thermodynamic processes. Building a battery model including all these processes is relative

complicated task.

Chen and Cheh [11] made some progress based on the work of a secondary distribution reaction

model, including concentration, porosity, and volume average velocity effect on cathode region,

using a simplified mixed-reaction model to describe the discharge behavior of anode. Then, Chen

and Cheh [14] made further improvement to this model, including a dissolution-precipitation

reaction mechanism to describe anode discharge behavior. The zincate ions were assumed only

existed in anode region. Then Poldlha and Cheh [15] made further improvement, including the

effects of zincate ion transport throughout the entire cell during discharge. The temperature variation

while discharging was also investigated in this model. It is found that the temperature effects on

alkaline Zn-MnO2 is so small and can be neglected. Farrel et al [16] made further improvements of

alkaline battery model. These authors analyzed alkaline battery discharge behavior based on three

different length scales, submicroscopic, microscopic, and macroscopic, which describes alkaline

battery behavior more precisely. It is our group's goal to add charging behavior to the Zn-MnO2

alkaline model, and this work is part of that effort.

In this part, a simplified model is developed in order to analyze the discharge behavior of

alkaline battery.

20

3.2 Electrode Reactions

The electrochemical reactions of both electrodes are shown in the sections below.

3.2.1 Cathode Reactions

Research shows that in simplified form, the MnO2 discharge reaction can be considered as a

two-step electrochemical reaction. The first step is the proton insertion reaction of MnO2, like

reaction (3-1) show below:

MnO* +x�H*O + x�e+ → MnO*+��(OH)�� + OH+ (3-1)

This reaction contributes most discharge capacity of MnO2, which is reversible before 𝑥�

reaches to 0.79, as shown by Gallaway et al. [6] The second reaction step is further reduction of

MnOOH to Mn(OH)2. The second step reaction, which appears when the battery operating voltage

is low, is shown below:

MnOOH+H& + e+ → Mn(OH)* (3-2)

In the model mentioned in part 3, only reaction 3-1 is being considered because reaction 3-1 is

the primary cathode reaction and contributes most usable capacity in the battery.

3.2.2 Anode Reactions

The anodic reaction mechanism is relatively complex. In order to simplify the anode reaction

mechanism, a mixed reaction model is applied. It assumes that two extreme reactions happen in the

anode.

Case1:Zn +2OH+ → ZnO+ H*O + 2e+

Case2:Zn +4OH+ → Zn(OH))*+ + 2e+

In Case 1, it is assumed that all active material zinc reacts with hydroxide ions to form solid

ZnO. ZnO is one stable species which is assumed to have no further dissolution after being produced.

For Case 2, it is assumed that all active material Zn reacts with hydroxide ions producing zincate

ions. In this case, the supersaturation of Zn is considered to have an indefinite limit. Thus, ZnO does

not form by precipitation. Experimental results show that both of these reactions happen in the anode.

[11] In this part, a parameter 𝜃 is defined as the ratio of Case 2 extent to the total extent of Case 1

and Case 2.

21

3.3 Assumptions adopted in the model

In order to build a macrohomogenous model to describe alkaline battery discharge behavior,

the following assumptions are made in order to simplify the calculation.

1) Double-layer charging is ignored because the double layer is extremely thin compared to the

pore wall.

2) Only the reduction of MnO2 to MnOOH is considered.

3) The cathode is made up of several spherical manganese dioxide particles.

4) The manganese dioxide particles are exhausted when the extent of reaction 𝑥𝑟 is equal to 1.

5) The cell is considered to be isothermal while discharging.

6) Since the polarization of zinc electrode is low, the anode overpotential is neglected.

7) The mixed-reaction model is applied to describe the anode discharge mechanism.

a) The zinc electrode is considered as reversible and highly non-polarizable, i.e. the reaction

distribution across the anode is uniform.

b) The electrolyte solution is assumed to only contain only three species: K+, OH-, and H2O

3.4 Method used to solve the model

Mathematical modelling of transport processes results in a series of simultaneous partial

differential equations. These partial differential equations are too difficult to derive an analytical

solution by hand. Computational software is an option for us to get the numerical solutions. In this

thesis, Newman's BAND method [17] is used, with the solution computed using the Python

environment.

3.4.1 Finite-difference method

The finite difference method is the algorithm used to get the numerical solution of the partial

differential equations for this battery model. In mathematics, finite difference methods are used to

get the numerical solutions for differential equations by approximating them with difference

equations.

To use a finite difference method to approximate the solution, discretizing the problem domain

is the first step. The second step is using finite difference formulas to calculate the derivatives. There

22

are three kinds differences approximation commonly used: forward, backward, and central

difference. Forward and backward difference are used to calculate the derivative of problem domain

boundary, and central difference is used to approximate the derivatives. These formula are shown

below:

𝑑*𝑐�𝑑𝑥* =

𝑐��𝑥� + ℎ� + 𝑐��𝑥� − ℎ� − 2𝑐��𝑥��ℎ + 𝑂(ℎ*) (3-3)

𝑑𝑐�𝑑𝑥 =

𝑐��𝑥� + ℎ� + 𝑐��𝑥� − ℎ�ℎ + 𝑂(ℎ*) (3-4)

In these two equations, c is general term for variables, k is the number of variables, h is the

mesh distance, and xj represents one certain point of the mesh.

3.4.2 Linearizing non-linearized equations

BAND algorithm is applied to get numerical solution of the mode. To apply BAND algorithm,

all the non-linear equations in the model should be linearized. Often, coupled non-linear differential

equations need to be solved after building a model. To linearize the non-linear terms in the system,

the first thing to do is to adopt a form of each variable. The new form is the sum of an initial guess

and a small variation of the variable. For example, if we meet the term like 𝑎𝑏, the way to adopt

the form of 𝑎 and 𝑏 is shown below:

𝑏 = 𝑏1 +∆𝑏 (3-5)

𝑎 = 𝑎1 +∆𝑎 (3-6)

Then, we calculate the product of new a and b. The result is shown below:

𝑎𝑏 =𝑎1𝑏1 + 𝑎1∆𝑏 +𝑏1∆𝑎 + ∆𝑎∆𝑏 (3-7)

The last term ∆𝑎∆𝑏 can be ignored, because ∆𝑎 and ∆𝑏 are small variations of the

variables, and the product of these two terms is quite small. For expressions such as exp, ln, etc. the

way we = linearize is by a Taylor series, which is shown below:

𝑓(𝑥) = 𝑓(𝑥1) + 𝑓�(𝑥1)∆𝑥 (3-8)

3.4.3 BAND Method

The BAND method is based on finite difference method. It was first developed by Newman in

1962. [9] The BAND method to describe the partial differential equation system is shown below:

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�𝑎�,^𝑑*𝑐�,^𝑑𝑥* +�𝑏�,^

𝑑𝑐�,^𝑑𝑥 +�𝑑�,^𝑐�,^ = 𝑔 (3-9)

In this formula, k means the number of dependent variables in the system and i represents the

equation number of the partial differential system. 𝑎�,^,𝑏�,^ ,𝑔 and 𝑑�,^ are the coefficients. A

different formula is used to describe the boundary of the system.

�𝑝�,^𝑑𝑐�,^𝑑𝑥 +�𝑒�,^𝑐�,^ = 𝑓 (3-10)

Equation (3-10) should be inserted back into the first and last element of (3-9). To apply the BAND

sub-rountine, each nonlinear equation should be linearized and initial guesses for each parameter

should be provided, then the finite difference method is applied to solve the system.

3.4 Model formulation

The theoretical model in this part is built based on the macrohomogeneous theory of porous

electrodes, disregarding the actual geometric detail of the pores. The electrode is assumed to be

made up of two parts: electrolytic solution and solid material.

3.4.1 Governing equations in the cathode

Figure 11: Duracell AA battery inner structure

The model developed in this part aims to describe the discharge behavior of alkaline battery at

low discharge rate. Figure 11 shows the inner structure of Duracell AA battery. A Duracell AA

24

battery consists of pin, anode, separator, cathode, and current collector. All parameters in this part

are assumed to be the function of radius and time. The pin is located in the middle of battery and

the center of pin is 𝑟 = 0 cm. The anode region is from 𝑟 = 0.10 cm to 𝑟 = 0.40 cm, the

separator region is from 𝑟 = 0.40 cm to 𝑟 = 0.42 cm, and the cathode region is from 𝑟 = 0.42

cm to 𝑟 = 0.69 cm.

In the whole system, the KOH electrolyte is assumed to include only three species, potassium

cation, hydroxide anion, and water. To build a complex model, a control volume should be chosen

first. Mass conservation of each species within the control volume shows that the accumulation of

each species is equal to the sum of the net influx of that species into the control volume and the

production of that species due to electrochemical or chemical reaction. The governing equation of

mass balance for each species is shown below:

𝜕(𝜀𝑐^)𝜕𝑡 = −∇ ∙ 𝑁^ +𝑅^ (3-11)

In this equation, 𝑐^ means the concentration of species i in the pore solution, 𝜀 is the porosity

of each electrode which means the space occupied by electrolyte. 𝑁^ is the superficial ionic flux

across the cross-sectional area of the porous electrode, and 𝑅^ is the production rate of species i

via electrochemical reaction. From Faraday’s law, 𝑅^ can be expressed by the following equation:

𝑅^ = 𝑎𝑗^¤ = −𝑠^𝑛𝐹 𝑎𝑖¤ = −

𝑠^𝑛𝐹 𝑗 (3-12)

In this equation, 𝑗^¤ is the pore wall flux of species i from the solid phase to the solution, 𝑖¤

is the normal current density at the pore wall interface, 𝑗 is the transfer current density per unit

volume, n is the number of electrons transferred, and 𝑠^ is the stoichiometric coefficient of species

i in the electrochemical reaction.

The movement of charged species inside the battery can be divided into three parts: diffusion,

convection, and migration. Diffusion appears due to concentration variation across the cell.

Convection means that the charged species are moving with bulk electrolyte. Migration describes

the movement of charged species under the effect of an electric field. The flux of species i is

governed by the Nernest-Planck equation, describing the motion of charged species inside battery.

The Nernest-Planck equation is shown below.

𝑁^ = −𝜖𝐷6∇𝑐^ +𝑖*𝑡^∎

𝑧^𝐹+𝑐^𝑣∎ (3-13)

The first part of this equation −𝜖𝐷6∇𝑐^ is the diffusion part, where 𝜖 is porosity of the

25

cathode, 𝐷𝑎 is the effective diffusion coefficient of electrolyte, and ∇𝑐^ is the concentration

gradient of species i across the cathode. The second term of this equation ^©ª«∎

¬« is the migration part

inside the battery. In this term, 𝑖* is the ionic current density, 𝑡^∎ is the transference number of

species i (which is assumed to be a constant while battery discharging), 𝑧^ is the charge number,

and F is Faraday constant. The last part of this equation is the convection part. In this part, 𝑐^ is the

instantaneous concentration of species i.

For a binary electrolyte, electrolyte concentration can be expressed by

𝑐6 =𝑐B𝑣B6

= 𝑐*𝑣*6

(3-14)

Then inserting equation 3-13, 3-14, and 3-12 into 3-11, the first equation which describes the

mass balance of electrolyte is derived.

𝜕(𝜖𝑐6)𝜕𝑡 +𝑣∎ ∙ ∇𝑐6 +𝑐6∇ ∙ 𝑣∎ = ∇ ∙ �𝜖B.�𝐷6,r∇𝑐6� − P

1 − 𝑡*∎

𝐹 Q∇ ∙ 𝑖* (3-15)

In equation (3-15), 𝑐6 is the concentration of electrolyte, 𝐷6,r is the diffusion coefficient of

bulk electrolyte. The effective diffusion coefficient 𝐷6 of electrolyte is equal to 𝐷6 = 𝐷6,r𝜖B.�.

The mass balance of water species is described below,

𝜕(𝜖𝑐1)𝜕𝑡 +∇ ∙ (𝑐1𝑣∎) =∇ ∙ �𝜖B.�𝐷6,r∇𝑐1� −

𝑠1𝑛𝐹 𝑗

(3-16)

which is derived from 3-12 and 3-13. Because water is neutral the migration term in equation 3-13

is neglected.

Multiplying 3-15 with 𝑉6+ (molar volume of electrolyte) and 3-16 with 𝑉1+ (molar volume

of water), using the thermodynamic relationship 𝑐6𝑉6+ +𝑐1𝑉1+ = 1, the continuity equation inside

the battery is shown below:

𝜕𝜖𝜕𝑡 +∇ ∙ 𝑣

∎ = 5𝑉1+

𝐹 + (𝑡*∎ − 1)𝑉6+

𝐹 <∇ ∙ 𝑖* (3-17)

The conduction mechanism inside a battery is governed by Ohm’s law. The transfer between

ionic current to electronic current happen at the active material-electrolyte interface. Charge

conservation law requires the current density inside the battery is zero, which is expressed by

∇ ∙ 𝑖B + ∇ ∙ 𝑖* = 0 (3-18)

With Gauss's theorem, equation 3-18 can be further integrated into equation 3-19 in cylindrical

coordinates.

26

𝑖B + 𝑖* = 𝐼

2𝜋𝑟𝐿 (3-19)

The conduction mechanism in solid phase can be shown by the following equation

𝑖B = −𝜎∇𝜙B (3-20)

The conduction mechanism in electrolyte is expressed by:

𝛻𝜙* = −𝑖*𝜅 +

2𝑅𝑇𝐹 51 − 𝑡*∎ +

𝐶6𝐶1<𝛻𝑙𝑛𝛼6 (3-21)

In the electrolyte, the potential of electrolyte is a function of chemical potential and ionic

current density. By using equation 3-19, 3-20, and 3-21, the overpotential of the cathode can be

described by

∇𝜂 = 𝑖* 51

𝜅r𝜖B.�+1𝜎< −

𝐼2𝜋𝑟𝐿𝜎 +

2𝑅𝑇𝐹 51 − 𝑡*∎ +

𝑐6𝑐1< ∇𝑙𝑛𝑎6 (3-22)

Figure 12 Particle model of MnO2

Figure 12 shows a core-shell model of MnO2 particles during discharge. At the initial stage of

discharge, the outer region of MnO2 particles reacts with protons near the surface of particles. A

shell of 𝛼-MnOOH forms at the surface of MnO2. Due to the formation of the 𝛼-MnOOH shell at

the surface of MnO2, the protons need to pass through an 𝛼-MnOOH layer to the inner MnO2 core

for further reaction. Because of the molar volume difference between the 𝛼-MnOOH and MnO2,

manganese dioxide particles swell during discharge. The swelling of active particles leads to a

porosity decrease. With Faraday’s law, the overall solid balance is shown by the following

27

expression:

𝜕𝜖𝜕𝑡 = 5

𝑉²+

𝐹 −𝑉³+

𝐹 <∇ ∙ 𝑖* (3-23)

In equation (3-23), 𝑉²+ is the partial molar volume of α-MnOOH and 𝑉³+ is the partial molar

volume of 𝛾-MnO2. The change in radius of the particles caused by this swelling is described by a

mass balance of each EMD particle, as shown below:

𝜕𝑟_%

𝜕𝑡 = −34𝜋𝑁𝐹 �𝑉²

+ − 𝑉³+�∇ ∙ 𝑖* (3-24)

The electrochemical reaction rate of the battery is given by Butler-Volmer equation.

∇ ∙ 𝑖* =exp e𝑎6𝐹𝜂 𝑅𝑇µ f − exp e−𝑎A𝐹𝜂 𝑅𝑇µ f

1𝑎𝑖_

+−exp e−𝑎A𝐹𝜂 𝑅𝑇µ f

(∇ ∙ 𝑖*)¶^·

(3-25)

The proton-insertion reaction is diffusion-limited by the transport of protons through the

particle shell. Thus the limiting transfer current is given by the term (∇ ∙ 𝑖*)¶^· . The limiting

transfer current is given by the following expression for diffusion in spherical coordinates:

(∇ ∙ 𝑖*)¶^· =4𝜋𝐹𝑁𝐷¸𝑐61𝑟_− 1𝑟

(3-26)

Here, 𝐷¸ is the proton diffusion coefficient inside MnO2 particles, 𝑟_ is the outer radius of

cathode particles, 𝑟 is the inner, unreacted core of MnO2, N is the total number of cathode particles

in cathode region, and 𝑐6 is the concentration of protons at the outer radius of cathode particles,

which is assumed to be equal to the concentration of hydroxide ions. The assumption is rational

because the variation of hydroxide ion at the outer radius of cathode particles is small.

In the cathode, the reaction extent of MnO2 is defined as the extent of MnO2 exhausts through

cathode reaction.

MnO* + x�H*O + x�e+ → x�MnO*+��(OH)�� + x�OH+ (3-27)

where the extent of reaction 𝑥� is given by:

𝑥� = �𝑟_,^% − 𝑟_%�(𝑉³+ − 𝑉²+)

𝑉³+

𝑟_,^% (3-28)

In equation 3-17, 𝑟_,^ is the initial outer radius of cathode particles, which is assumed equal

to 20 𝜇𝑚 according to relative literature. 𝑟_ is the outer radius of cathode particles while the

battery is discharging.

Depth of discharge (DOD) is the discharged capacity relative to the total capacity of the battery

28

𝑄, which is evaluated by equation 3-29 for constant current or galvanostatic discharge.

DOD = ItQ (3-29)

In equation 3-29, I is discharge current, t is discharge time, and Q is total capacity of battery.

3.4.2 Governing equations in the anode

Because the thickness of the separator is small compared to the thickness of cathode and anode,

the effect of separator is neglected in my thesis. The anode in this model is assumed as a reversible,

nonpolarizable electrode with uniform current distribution. A mixed-reaction model is adopted to

analyze anode reaction. The main expressions of anode are shown below:

𝜕(𝜖6𝑐*𝑉6)𝜕𝑡 =−2𝜋𝑟6𝐿𝑁*(𝑟 = 𝑟6) − (2 − 𝜃)

𝐼𝐹

(3-30)

According to the Chen’s paper [11], 𝜃 =0.8 was appropriate for AA batteries. In this

expression,𝑟6 is the anode-separator boundary, N2 is the flux of hydroxide ions into anode, 𝜖6 is

anode porosity, and 𝑉6 is the anode volume. The porosity variation in the anode is shown by the

following expression:

𝜕𝜖6𝜕𝑡 =

𝑉À¤+

2𝐹𝑉6𝐼 − 𝜃

𝑉À¤Á+

2𝐹𝑉6𝐼 (3-31)

In equation 3-31, 𝑉À¤+ is the partial molar volume of Zn and 𝑉À¤Á+ is the partial molar volume

of ZnO.

3.4.3 Boundary conditions of the system

At the cathode-current collector, the boundary conditions are given by

𝑖* = 0 (3-32)

𝑁6 = 0 (3-33)

At current collector, all currents are electronic. And no hydroxide flux enters current collector.

At cathode-anode interface:

𝑖* =𝐼

2𝜋𝑟𝐿 (3-34)

𝑐6, 𝑣∎ = 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 (3-35)

29

3.4.4 Physical properties of KOH electrolyte

Electrolyte properties used in this model is shown in the table below.[11] Density of electrolyte

was given by equation 3-36,

𝜌 = 𝐺1 +𝐺B𝑀 + 𝐺*𝑀* + 𝐺%𝑀% + 𝐺)𝑀) + 𝐺�𝑀� + 𝐺Å𝑀Å (3-36)

In this equation, the molarity of electrolyte 𝑀 = 𝑐6 × 10% with 𝑐6 in 𝑚𝑜𝑙/𝑐𝑚% and the

constants in the equation are shown in Table 3.

Table 3: Constant value using to evaluate density of electrolyte

𝐺1 0.9966872 𝐺� 1.3081735 × 10-6

𝐺B 5.0065284 × 10-2 𝐺Å -3.012443 × 10-8

𝐺* -2.006849 ×10-3

𝐺% 2.2577517 × 10-4

𝐺) -2.276481 × 10-5

The bulk conductivity of electrolyte was given by equation 3-37:

𝑘∞ =𝐸1 + 𝐸B𝑊 + 𝐸*𝑊* + 𝐸%𝑊% + 𝐸)𝑊) (3-37)

The constants of this equation are shown in Table 4.

Table 4: Constant value using to evaluate electrolyte conductivity

𝐸1 8.782 × 10-3 𝐸% -3.026 × 10-5

𝐸B 3.720 × 10-2 𝐸) 3.346 × 10-7

𝐸* 8.109 × 10-5

The diffusion coefficient of the electrolyte was given by Equation 3-38.

𝐷6 = exp É𝐹B + 𝐹*𝐶61.�

(1 + 𝑐61.�)+ 𝐹%𝑐6 + 𝐹)𝑐6B.� + 𝐹�𝑐6*Ê (3-38)


30

Table 5: Constant value using to evaluate diffusion coefficients of bulk electrolyte

𝐹B -1.0464 × 101 𝐹) -9.1543 × 10-2

𝐹* -4.1100 × 10-1 𝐹� 5.9489 × 10-3

𝐹% 2.9182 × 10-1

Molality of the electrolyte was given in Equation 3-39.

𝑚 =𝐻1 + 𝐻B𝑀 +𝐻*𝑀* + 𝐻%𝑀% + 𝐻)𝑀) + 𝐻�𝑀� + 𝐻Å𝑀Å (3-39)


Table 6: Constant value using to evaluate electrolyte molality

𝐻1 4.0074878 × 10-6 𝐻) -1.478333 × 10-5

𝐻B 1.002374947 𝐻� 2.3063735× 10-7

𝐻* 7.4141713× 10-3 𝐻Å 3.562232× 10-8

𝐻% 1.3054688× 10-3

The molality activity coefficient was given by Equation 3-40.

𝛾& = exp É−𝐾B𝑚1.�

1 + 𝐾*𝑚1.� + 𝐾%𝑚 + 𝐾)𝑚* + 𝐾�𝑚% + 𝐾Å𝑚)Ê (3-40)


Table 7: Constant value using to evaluate electrolyte molality activity coefficients

𝐾B 1.1762 𝐾� -2.9934 × 10-4

𝐾* 1.15 𝐾Å 3,9144× 10-7

𝐾% 0.2302

𝐾) 6.0489× 10-3

The activity of the electrolyte was given by equation 3-41.

31

𝑎6 = 𝑐6 Ì𝑚𝜌1𝑀 Í𝛾& (3-41)

where ρ0=0.997 g/cm3 is the solvent density

3.4.5 Constants using in the model

The constants using to solve to solve this model is related to the parameters of commercial AA

batteries, given in Table 8. [11]

Table 8: Main parameters value used to solve the model

Parameter Name Value & Unit Parameter Name Value & Unit Transference number of hydroxide ions, 𝒕𝟐∎

0.78 Molecular weight of MnO2, 𝑀Ð¤Á*

84.94 g/mol


2.0×10-7 A/cm2

Molar volume of water 𝑉Ñ1

18.07 cm3/mol

Battery discharge rate

C20

,C30

Molar volume of water 𝐾𝑂𝐻,𝑉Ñ6

17.8 cm3/mol

Length, 𝑳 4 cm Average MnO2 particle size 𝒓𝒐,𝒊

0.002 cm

Molar volume of MnOOH, 𝑉Ñ²

20.45 cm3/mol Proton diffusion coefficient in solid 𝑫𝑯

6 × 10-10 cm2/s


20 S/cm

Faraday constant 𝑭

96485 C/mol


0.69 cm Gas constant 𝑹 8.314 J/mol·K


0.42 cm Weight of MnO2 particles, 𝑾

10.35 g

Initial Porosity of cathode, 𝝐𝒄𝟎

0.26 Radius of the pin 𝒓𝒑𝒊𝒏

0 cm


0.5 Initial Porosity of anode, 𝝐𝒂𝟎

0.85


0.5 Molar volume of Zn, 𝑉ÑÀ¤

9.15 cm3/mol

Molar volume of MnOOH, 𝑉Ñ³

17.29 cm3/mol Molar volume of Zn, 𝑉ÑÀ¤Á

14.51 cm3/mol

Initial electrolyte concentration, CaO

9 M Capacity of the cell

3160 mA·h

32

3.5 Results of the model

The model in this part is being developed in order to investigate Zn-MnO2 alkaline battery

discharge behavior at low discharge rates. When the discharge rate is high, the effect of zincate ion

become more important. [15] The initial electrolyte concentration is 9 M and the battery discharge

at C/30 [11]. The numerical solution of the model is obtained using the BAND method mentioned

in Python.

𝐸_ÑÙ�6ª^¤Ú =𝐸1 +2𝑅𝑇𝐹 ln 5

1 − 𝑓𝑓 < +∆𝜂A6ªÜ_Ý^A (3-42)

Equation 3-42 evaluates the operating voltage of the battery while discharging. 𝐸1 is the open

circuit voltage when half of the cathode capacity is discharged. 𝑓 is the extent of discharge which

is the ratio of capacity lost while discharging to the total capacity of the cell, which is equal to Þ∗ªß

(I-discharge current, t-discharge time, and Q-total capacity of the battery). ∆𝜂A6ªÜ_Ý^A is the

cathode overpotential, which is evaluated through equation 3-22.

Figure 13: Simulated battery discharge curve when the discharge rate is C/30

Figure 13 is the discharge curve of the battery when the discharge rate is equal to C/30 (0.1 A,

with battery capacity 3160 mA·h). The potential has an obvious drop at the beginning of discharge.

Then, as discharging continues, the potential drops more slowly until the active material (MnO2)

becomes exhausted. The obviously potential drop at the beginning of discharge is because of the

33

nonuniform secondary current distribution under charge-transfer and ohmic control. Further,

smaller potential drop appears, which is mainly due to the formation of MnOOH on the surface of

MnO2. As discharge continues, hydroxide ions are further consumed in the anode. The hydroxide

ions are being transported to the anode due to diffusion, migration, and convection in the battery.

At later discharge stage, due to the exhaustion of hydroxide ions, the potential drops quickly, finally

leading to cell failure. [11] Five points are selected from the discharge curve to analyze in detail

with respect to their parameter variation.

Figure 14: Reaction distribution across the cathode for 0.1 A discharge

Figure 14 shows the reaction distribution in the cathode. The reaction distribution is higher at

the separator than at the current collector in the initial stage of discharge. As discharge continues,

the reaction rate near cathode-separator interface increases further. Once MnO2 particles near

cathode-separator (r = 0.42 cm) lose reaction activity, the reaction rate falls off to zero. After the

reaction rate near the cathode-separator interface drops to zero, the active material (MnO2 particles)

in other regions of cathode experience an increase in reaction rate because the total current is

constant. Thus, the reaction rate near cathode-current collector interface (r = 0.69 cm) increases.

34

Figure 15: Porosity variation across the cathode

Figure 15 shows the porosity variation across the cathode. Porosity variation while discharging

is due to the molar volume difference between MnOOH and MnO2. While the battery discharges,

MnO2 is reduced to MnOOH and the particles in cathode region increase their radius. The variation

of porosity is in agreement with reaction distribution across the cathode. In Figure 15, the reaction

rate is highest in the region near cathode-separator (interface). Thus, an obvious bend appears at

cathode-separator interface. At late discharge, the porosity near the separator (x = 0.42 cm) remains

the same because the active material in this area has lost reaction activity.

Figure 16: Concentration variation across the cathode

35

The electrolyte concentration variation across the cathode is due to the combined effects of

diffusion, migration, and convection of hydroxide ions, as well as the fact that OH- is generated by

the cathode reaction and consumed by the anode reaction. Figure 16 shows the concentration

profiles across the cathode. Near the cathode-separator interface (x = 0.42 cm), the concentration

decreases because the exhaustion rate of hydroxide ion transport into the anode surpasses its

replenishment rate from the cathode reaction. Near cathode-current collector interface (r = 0.69 cm),

the electrolyte concentration increases because the generation rate of hydroxide is higher than

exhaustion rate by transport to the anode. As discharge continues, electrolyte concentration near

cathode-separator interface (r = 0.42 cm) continuously declines. In the later discharge stage,

electrolyte concentration near the cathode-current collector interface (r = 0.69 cm) increases because

the reaction rate in this region increases.

Figure 17: Overpotential variation across the cathode

Figure 17 shows the overpotential variation across the cathode. Overpotential is the driving

force of electrochemical reaction. Like the analysis in Figure 14, more reaction happens near

cathode-separator interface (x = 0.42 cm), and thus the overpotential near the separator is more

negative than other parts in the cathode. As discharge continues, the overpotential across the cathode

decreases.

36

3.5.1 The effect of electrolyte concentration on battery discharge behaviors

In this part, the electrolyte concentration is adjusted to 7 M in order to investigate the effect of

electrolyte concentration on battery discharge behavior. The reaction distribution and porosity

variation across the cathode is compared here.

Figure 18: The effect of initial electrolyte concentration on reaction distribution

Figure 18 shows the effect of initial electrolyte concentration on reaction distribution across

the cathode. When the initial electrolyte concentration is higher, the reaction rate near cathode-

separator interface (r = 0.42 cm) is higher. That is because higher initial electrolyte concentration

promotes the anode reaction to a certain degree, which increases the OH- exhaustion rate near

cathode-separator interface. Near the cathode-current collector interface (r = 0.69 cm), the

hydroxide ions concentration is higher when the initial electrolyte concentration is higher.

Hydroxide ions is a product of the cathode reaction. According to Le Chatelier’s principle, higher

hydroxide ions concentration will decrease the cathode reaction rate. Thus, the reaction rate near

cathode-current collector interface is lower near cathode-current collector interface (r = 0.69 cm).

37

Figure 19: The effect of initial electrolyte concentration on porosity variation

Figure 19 shows the effect of initial electrolyte concentration on porosity variation, which is

consistent with reaction rate distribution. Lower porosity means higher reaction rate in a region.

Porosity is lower near cathode-separator interface (r = 0.42 cm), and the porosity near the cathode-

current collector interface is higher when the initial electrolyte concentration is 9M.

3.5.2 Discharge behaviors at different C-rates

A C-rate is a measure of the rate at which a battery is discharged relative to its maximum

capacity. A 1C rate means that the discharge current will discharge the entire battery in 1 hour. For

a battery with a capacity of 500 Ah, this equates to a discharge current of 500A. The total capacity

of AA battery is 3.16 Ah [6], C/20 means the battery discharges at 0.153 A and C/30 means the

battery discharges at 0.1 A.

The effect of C-rate on battery discharge behavior is being investigated in this part. The

discharge behavior of an alkaline battery at C/20 discharge rate is compared with that at C/30. The

results are shown below.

38

Figure 20 Simulated battery discharge curves at different discharge rates (Orange line: C/20

0.153 A, Blue line: C/30 0.1 A)

In Figure 20, the orange line is the simulated battery discharge curve when discharge rate is

C/20 and the blue line is that at C/30. The potential drop within the cell is due to the exhaustion of

active material (MnO2). Higher discharge current leads to a more nonuniform current distribution

across the cathode, which gives rise to lower overpotential at cathode-separator interface. Thus, the

potential drop at C/20 (orange line) is faster than the potential drop at C/30 (blue line). In addition,

higher discharge increases cathode reaction rate, which means the exhaustion rate of MnO2 active

species is higher. Higher MnO2 active species exhaustion rate leads to a faster overpotential drop

across the cathode, which results in shorter cell life. Five point from each discharge curve are picked

to analyze cathode parameter variation. Points in the figure which have the same color represents

the cell discharge at the same DOD.

39

Figure 21 Electrolyte concentration profile in cathode region at different C-rates

Electrolyte concentration profiles across the cathode are shown in Figure 21. The concentration

profile of the same DOD is comparing here. Higher C-rate (C/20) leads to a larger concentration

gradient across the cathode. Higher discharge current increases anode and cathode reaction rates in

the cell. At the cathode-separator interface (x = 0.42 cm) the exhaustion rate of hydroxide ions is

higher at C/20 discharge rate. Thus, the electrolyte concentration at the cathode-separator interface

(x = 0.42 cm) at C/20 discharge rate is lower than it at C/30 discharge rates. At the cathode-current

collector interface (x = 0.69 cm), the exhaustion rate of hydroxide ions is lower than production rate.

The electrolyte concentration at the cathode-current collector interface (x = 0.69 cm) is larger than

it at the cathode-separator (x = 0.42 cm). In addition, owing to higher cathode reaction rate, the

production rate of hydroxide ions is larger at C/20 discharge rate. So, at cathode-current collector

interface (x = 0.69 cm), the electrolyte concentration is higher at C/20 rates.

40

Figure 22 Overpotential profile in cathode region at different C-rates

Overpotential profiles at different C-rates is shown in Figure 22. According to equation 3-22,

the overpotential of cathode is related to discharge current, ionic current density, and electrolyte

concentration. Higher discharge current results in lower overpotential in the cathode. The

overpotential at C/20 discharge rates is lower than it at C/30 discharge rate. In addition, the

electrolyte concentration affects overpotential. Higher concentration leads to higher overpotential.

At the cathode-current collector interface, the electrolyte concentration increases first and then

decreases. Thus, in Figure 22, at the cathode-current collector interface, the overpotential increases

first and then drops.

Figure 23 Reaction distribution profile in cathode region at different C-rates

41

Figure 23 shows reaction distribution profiles at different C rates. Higher discharge rate

increases cathode reaction rate. Thus, in Figure 23, we can see the dotted line is higher than the

solid line. The reaction rate increases first and then decreases at the cathode-separator interface (x

= 0.42 cm). From a kinetics viewpoint, at the beginning of discharge, the exhaustion of hydroxide

ions causes cathode reaction rate increase. After several hours of discharge time, the cathode

reaction reaches an equilibrium again and get its highest reaction rate. Then, the reaction rate falls

down because some active material MnO2 is exhausted.

3.6 Conclusions

A simplified model is developed and the BAND algorithm is used to get numerical solutions

of the model. The model is mainly used to predict discharge behavior of Duracell AA battery

cathodes at constant discharge current. The reaction rate near the cathode-separator is always the

largest in the cathode region. The porosity profile is consistent with the reaction distribution profile.

The effect of initial electrolyte concentration is limited on battery discharge behavior. Higher

electrolyte concentrations will affect anode reaction rate, which will increase the exhaustion rate of

hydroxide ions near the cathode-separator interface (r = 0.42 cm). Near the cathode-current interface

(r = 0.69 cm), according the Le Chatelier's principle due to higher electrolyte concentration, the

reaction rate is lower when the initial electrolyte concentration is high.

Higher discharge current affects the reactions distribution across the cathode. Near the cathode-

current collector interface (x = 0.69 cm), the electrolyte concentration is higher at high C-rate

discharging. Near the cathode-separator interface (x = 0.42 cm), the electrolyte concentration is

lower at high C-rate discharge. High discharge current also leads to a large overpotential drop across

the cathode, which shortens the cell life.

42

Chapter 4 Simplified model describing prismatic Zn-MnO2 battery

cathode discharge behavior

4.1 Introduction

With increasing concern for the environment, the demand for alternative energy resources is

rising. Secondary batteries are seen as an excellent option for energy storage. In recent years,

multivalent ions battery has been developed, such as Mg-ion and Al-ion batteries, and Zn-based

batteries. They receive attention among modern researchers because of the qualities of the basis

metals, natural abundance, high energy density, and low standard redox potential. However, the

reaction mechanisms of these modern secondary batteries needs to be understood well. For example,

the reason why the Mg-ion battery is not widely used is because of low diffusion rate of the Mg ion

within the cathode host material. [18]

Due to low cost, higher energy density, safety, and environmental friendliness, alkaline Zn-

MnO2 batteries are good candidates for grid-scale energy storage. However, limited cycle life is a

major issue which prevents the deployment of rechargeable Zn-MnO2 batteries.

The formation of irreversible materials within the battery cathodes via electrochemical and

chemical reactions during discharge is the chief phenomenon leading to limited cycle life. While

the system is discharging, the proton insertion reaction occurs in the MnO2 cathode, followed by a

dissolution-precipitation mechanism forming lower valent manganese compounds such as Mn(OH)2,

Mn2O3 and Mn3O4. Among these lower valent manganese compounds Mn(OH)2 is usually

considered rechargeable, as it can be oxidized to δ −MnO*. But Mn2O3 and Mn3O4 (hausmannite)

are electrochemical inactive species, and their formation leads to the failure of the battery. Another

inactive species ZnMn2O4 (hetaerolite) is similarly nonrechargeable, and forms in the presence of

zincate ions that migrate to the cathode from the anode. [19]

In order to improve the rechargeability of Zn-MnO2 batteries, researchers have applied several

methods. First, additives like CeO2, BaO, and CaO are mixed with MnO2 to keep rechargeability.

These additives have various function. Some of them reduce the access of Zn-ion to the MnO2, some

of them hinder the formation of non-rechargeable products such as Mn3O4, and some of them

maintain the crystal structure of MnO2. [20,21,22] It has also been reported that LiOH can replace

43

some NaOH or KOH electrolyte in order to keep the rechargeability of Zn-MnO2 batteries. Due to

the change of electrolyte, the reaction mechanism in the cathode is altered, and LixMnO2 is produced

instead of MnOOH. Because no MnOOH is produced, ZnMn2O4 will not form, which significantly

improves rechargeability. [19] New separators have also been reported in order to prevent zincate

ions being transported to cathode region. Duay et al. focused on one new separator material [23],

sodium super ion conducting ceramic (NaSICON). NaSICON is a cation conductor rather than an

anion conductor, eliminating zincate ions transport through the separator. Thus it is another method

preventing the formation of ZnMn2O4.

In addition, it has been shown that Zn-MnO2 batteries can keep rechargeability when the DOD

is low, up to 4000 cycles at 10% DOD. The reversibility improves in this situation because the main

crystal structure of MnO2 is maintained while the battery cycles.[24] Gallaway et al. showed that

the formation of the spinel structures hetaerolite (ZnMn2O4) and hasumannite (Mn3O4) compromise

the rechargeability of alkaline Zn-MnO2 battery due to their low reactivity. The formation of these

spinel structures appears after insertion of 0.79 proton equivalents per MnO2 atom [6]. Engineers

want to avoid the formation of these deleterious materials. The simplified model mentioned in this

part can tell engineers when and where formation of these occurs in the cathode during a single,

initial discharge of a battery. This information establishes the C rate, DOD, and cathode thickness

combinations that are initially acceptable for a limited-DOD rechargeable Zn-MnO2 battery. This

parameter space can then be searched once charging capability is added to the model.

4.2 Prismatic Zn-MnO2 battery model

The same model in Chapter 3 is developed in Cartesian coordinates to investigate the discharge

behaviors of prismatic Zn-MnO2 cells like those developed by Ingale, et al [24]. Because the

prismatic cell has low production costs, easy scale-up, more uniform current distribution and active

material utilization, the prismatic Zn-MnO2 cell is more appropriate for grid-scale energy storage.

44

Figure 24 Prismatic Zn-MnO2 battery

Figure 24 shows the schematic of a prismatic Zn-MnO2 battery, in which each part of the

battery is produced as a plate. The surface area of the electrodes is 𝐻 ×𝑊, which is different from

the surface area of a cylindrical cell. Thus, the charge balance in this case is:

𝑖B + 𝑖* = 𝐼𝐻𝑊 (4-1)

The assumptions in prismatic Zn-MnO2 battery are the same with the assumptions in Chapter

3.2. The governing equations of anode and boundary conditions is also the same as in Chapter 3.

The governing equations of the prismatic Zn-MnO2 model are shown below:

∂η∂x = 𝑖* 5

1𝜅r𝜖B.�

+1𝜎< −

𝐼𝐻𝑊𝜎 +

2𝑅𝑇𝐹 51 − 𝑡*∎ +

𝑐6𝑐1<∂𝑎6𝑎6 ∂x

(4-2)

𝜕𝑖*𝜕𝑥 =

exp e𝑎6𝐹𝜂 𝑅𝑇µ f − exp e−𝑎A𝐹𝜂 𝑅𝑇µ f

1𝑎𝑖_

+−exp e−𝑎A𝐹𝜂 𝑅𝑇µ f

(∇ ∙ 𝑖*)¶^·

(4-3)

𝜕(𝜖𝑐6)𝜕𝑡 +𝑣∎ ∙

∂𝑐6∂x +𝑐6

∂𝑣∎

∂x = ∇ ∙ P𝐷6,r1.5𝜖1.�∂𝑐6∂x

∂ϵ∂x + 𝐷6,r𝜖

1.5 ∂*𝑐6∂x* Q

−P1 − 𝑡*∎

𝐹 Q𝜕𝑖*𝜕𝑥

(4-4)

45

𝜕𝜖𝜕𝑡 +

∂𝑣∎

∂x = 5𝑉1+

𝐹 + (𝑡*∎ − 1)𝑉6+

𝐹 <𝜕𝑖*𝜕𝑥

(4-5)

𝜕𝜖𝜕𝑡 = 5

𝑉²+

𝐹 −𝑉³+

𝐹 <𝜕𝑖*𝜕𝑥

(4-6)

𝜕𝑟_%

𝜕𝑡 =−34𝜋𝑁𝐹 �𝑉²

+ − 𝑉³+�𝜕𝑖*𝜕𝑥

(4-7)

The parameters in these six equations have the same meanings with the parameters in part 3.

The capacity of the cell is calculated according to the capacity of the cathode, meaning the anode is

always higher in capacity. The theoretical capacity of MnO2 is 0.308 Ah/g. Thus the capacity of the

cell is evaluated by the following expression:

𝑄 = 0.308 ×𝑚A6ªÜ (4-8)

Here in equation 4-8, 𝑄 is the capacity of the cell in Ah and 𝑚A6ªÜ is the total mass of the

cathode.

The anode of prismatic Zn-MnO2 consists of three parts, Zn, ZnO, and colloidal Teflon. The

mass fraction of each anode part is shown in Table 9.

Table 9 Anode information of Prismatic Zn-MnO2

0.85 Zn mass fraction

0.10 ZnO mass fraction

0.05 colloidal Teflon mass fraction

The cathode of prismatic Zn-MnO2 cells is made up of three parts, 𝛾-MnO2, graphite, and

colloidal Teflon. The composition of cathodes is given in Table 10.

Table 10 Cathode information of Prismatic Zn-MnO2

0.65 γ-MnO2 mass fraction

0.30 KS44 graphite mass fraction

0.05 colloidal Teflon mass fraction

Mixture density can be calculated by equation 4-9 where xi is the mass fraction of the ith

component and ρi is the density of the ith component.

46

𝜌·^åªæ�Ù = ç�𝑥^𝜌^^

è+B

(4-

9)

MnO2 particles are assumed to be spherical and 50 µm in radius. Zn particles are 75 µm in

diameter. The manufacturing process is known to result in the measured porosities given here.

The initial porosity of anode and cathode is shown in Table 11.

Table 11 Initial porosity of each cell

εanode 0.266

εcathode 0.274

The thickness of each electrode is evaluated according to the porosity of each cell which is

given by the following equations.

𝑉é_¶^Ý =𝑚Ù¶ÙAª�_ÝÙ

𝜌Ù¶ÙAª�_ÝÙ (4-10)

𝑉Ù¶ÙAª�_ÝÙ =𝑉é_¶^Ý

𝜖Ù¶ÙAª�_ÝÙ (4-11)

𝐿Ù¶ÙAª�_ÝÙ =𝑉Ù¶ÙAª�_ÝÙ𝐻𝑊

(4-12)

In equation 4-10, 𝑉é_¶^Ý is the volume of active species of each electrode, 𝑚Ù¶ÙAª�_ÝÙ is the

total mass of electrode, and 𝜌Ù¶ÙAª�_ÝÙ is the mixture density of each electrode. In equation 4-11,

𝑉𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑑𝑒 is the total volume of electrode (including the void area of electrode) and 𝜖𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑑𝑒 is

the porosity of electrode. In equation 4-12, 𝐿Ù¶ÙAª�_ÝÙ is the thickness of each electrode and H, W

are the width and length of electrode plate separately.

The reaction extent 𝑦 is the localized DOD of the cell, x is the DOD of the whole cell .

MnO2 + yH2O + ye- → MnO2-y(OH)y + yOH-

The local DOD y is evaluated by equation 4-12, which is shown below:

𝑦 =�𝑟_,^% −𝑟_%�(𝑉³+ − 𝑉²+)

𝑉³+

𝑟_,^% (4-13)

In equation 4-13, 𝑟_ is the outer radius of MnO2 particles while discharging, 𝑟_,^ is the initial

outer radius of MnO2 particles.

Dimensionless length is applied in this part, calculated in equation 4-14.

𝐿Ý =𝐿

𝐿Ù¶ÙAª�_ÝÙ (4-14)

47

L in equation 4-14 is the position within the electrode.

4.3 Results of the model

The prismatic model aims to analyze the discharge behavior of prismatic Zn-MnO2 batteries.

[6,11] Constant values ued to solve the model are given in Table 12.

Table 12 Constant value use to solve the model [14]

Parameter Name Value & Unit Parameter Name Value & Unit Transference number of hydroxide ions, 𝒕𝟐∎

0.78 Molecular weight of MnO2, 𝑴𝑴𝒏𝑶𝟐

86.94 g/mol


2.0×10-7 A/cm2

Molar volume of water 𝑉1+

18.07 cm3/mol

Battery discharge current, 𝑰

According to discharge rate

Molar volume of water 𝐾𝑂𝐻,𝑉6+

17.8 cm3/mol

Length, 𝑳 4 cm Average MnO2 particle size 𝒓𝒐,𝒊

0.005 cm

Molar volume of MnOOH, 𝑽𝜶+

20.45 cm3/mol Proton diffusion coefficient in solid 𝑫𝑯

6 × 10-10 cm2/s


20 S/cm

Faraday constant 𝑭 96485 C/mol


0.69 cm Gas constant 𝑹 8.314 J/mol·K


0.42 cm Weight of MnO2 particles, 𝑾

10.35 g

Initial Porosity of cathode, 𝝐𝒄𝟎

0.26 Radius of the pin 𝒓𝒑𝒊𝒏 0 cm


0.5 Initial Porosity of anode, 𝝐𝒂𝟎

0.85


0.5 Molar volume of Zn, 𝑉ÑÀ¤

9.15 cm3/mol

Molar volume of MnOOH, 𝑽𝜸+

17.29 cm3/mol Molar volume of Zn, 𝑉ÑÀ¤Á

14.51 cm3/mol

Initial electrolyte concentration, 𝒄𝒂

9 M Capacity of the cathode

308 mA·h/g

Density of active material zinc 𝝆𝒁𝒏

7.14 g/cm3 Density of active material zinc oxide 𝝆𝒁𝒏𝑶

5.61 g/cm3

48

Density of active material manganese dioxide 𝝆𝜸+𝑴𝒏𝑶𝟐

5.026 g/cm3 Density of graphite 𝝆𝑲𝑺𝟒𝟒

2.26 g/cm3

Density of colloidal Telflon 𝝆𝑻𝒆𝒇𝒍𝒐𝒏

1 g/cm3

4.3.1 The effect of anode mass on cathode discharge behavior

In this part, effect of anode mass on discharge behavior is evaluated. Cathode mass is held

constant at 150 g as well as discharge rate. Three experimental conditions are given in the Table 13,

amounting to an increase in anode mass. The discharge rate is 7.6 mA/g-MnO2. And all cells in this

part are discharged to 15% DOD.

Table 13 Three conditions investigating the effect of anode mass on modeled discharge

behavior

Cathode total mass

Anode total mass

Discharge current

A 150 g 10 g 0.75 A

B 150 g 50 g 0.75 A

C 150 g 100 g 0.75 A

Figure 25 Effect of anode mass on electrolyte concentration across the cathode (DOD = 15%)

49

The effect of anode mass on electrolyte concentration variation across the cathode is shown in

Figure 25. There is slight effect on electrolyte concentration at cathode-current collector interface.

The blue line is higher than orange line and green line in Figure 25. The exhaustion rate of hydroxide

ions when the anode mass is 100 g is the largest among three cells. Difference of hydroxide ions

exhaustion rate leads to slight electrolyte concentration difference at the cathode-current collector

interface.

Figure 26 Effect of anode mass on proton insertion reaction extent

across the cathode (DOD = 15%)

Figure 27 Effect of anode mass on overpotential across the cathode (DOD = 15%)

50

Figure 28 Effect of anode mass on porosity across the cathode (DOD = 15%)

The effect of anode mass on cathode reaction extent, overpotential, and cathode porosity is

shown in Figures 26, 27, and 28. The difference in these parameters is too small to differentiate in

the three figures. The conclusion is that anode mass is not critical in the analysis that follows.

4.3.2 I-L figure to judge the rechargeability of MnO2 cathodes

Gallaway et al. stated that the proton insertion reaction is reversible until y reaches a value of

0.79. When y is higher than 0.79, spinel structure and hetaerolite forms quickly in the cathode,

which destroys cell rechargeability. [6] Thus this value of y is a good metric to judge MnO2

rechargeability. The model in part 4.2 can tell us when and where this condition is expected in MnO2

cathodes as a function of DOD, cathode thickness L, and C rate.

The model developed in part 4.2 is appropriate to analyze Zn-MnO2 cathode discharge

behavior at low discharge rate (around 10 mA/g-MnO2). When the discharge rate is higher than 10

mA/g-MnO2, the effect of zincate ions in the cell become more obvious which will increase the

conductivity of the cell. [14] Thus, in this part, low discharge rate (1mA/(g-MnO2) – 10 mA/(g-

MnO2)) applied. The discharge behavior of prismatic Zn-MnO2 cathodes with different thickness,

DOD, and discharge rate (1mA/(g-MnO2) – 10 mA/(g-MnO2)) is investigated. The scope of cathode

thickness Lc is 0.2 cm – 2.0 cm. In this part, the scope of cathode thickness is divided into three part:

thin-electrode region (0.2 – 0.8 cm), mid-electrode region (0.8 – 1.4 cm), and thick-electrode region

51

(1.4 – 2 cm).

A map showing the relationship of the three variables is shown in Figures 29-44 for different

DOD values. Partially/totally non-rechargeable regions are marked as light-grey areas, and are

defined as conditions where the localized DOD in some or all parts of cathode is higher than 0.79.

Blank regions mean all localized DOD y values are lower than 0.79 across the cathode.

Rechargeable line is the boundary differentiating the partially/totally non-rechargeable region and

the blank region. Each point on the rechargeable line has the same characteristics as the blank area.

Figure 29 I – Lc figure when DOD = 70%

Figure 29 is I-Lc figure at 70% DOD which means all cathodes (thickness scope 0.2 cm – 2cm)

were discharge to 70% DOD. The partially/totally non-rechargeable region takes most parts of the

figure, which means the formation of spinel structures appear in most cathodes (with different

thickness) shown in the figure. The blank region was located in the thinner electrode region. If

someone wants to make use of more cathode capacity in the meantime keep the rechargeability of

cathode, thinner electrode is better choice.

52


Figure 30 shows the I-L map when DOD is equal to 60%. When the DOD decreases, less

capacity can be used while discharging. The partially/totally non-rechargeable region was reduced

and the blank region expanded. More cathode thickness was available to keep rechargeability at a

constant discharge rate to 60 % DOD.


Figure 31 shows the I-L map when DOD is 50%. The partially/totally non-rechargeable region

further reduced and shifted towards upper right region of the figure. The healthy region (blank

region) increased. Comparing to figure 30, the cathode in the mid-thickness region can keep

53

rechargeability at low discharge rate. In summary, when the DOD of cathode is high (50% DOD,

60%DOD, and 70 %DOD), although more capacity in the battery is available, only cathode with

small thickness (thin-electrode region and some part of mid-electrode region) can keep reversibility.


Figure 32 shows the I – L map when the DOD of the cell at 40 % DOD. Compared to figure

31, the healthy region increased. Cathodes in the thick electrode region can maintain rechargeability

at low discharge rate.


Figure 33 shows the I-L map when the DOD of the cathode is equal to 30%. The

54

partially/totally non-rechargeable region reduced further and shifted toward the right part of the

figure (high discharge current, thick electrode). Cathodes in thin-electrode region and mid-electrode

region could keep heathy at all discharge rate scope (1mA/(g-MnO2) – 10 mA/(g-MnO2)) to 30 %

DOD and cathodes in the thick-electrode region could maintain rechargeability too.


Figure 34 shows the I-L map when the DOD of the cathode was equal to 20 %. The

partially/totally non-rechargeable region decreased further. More non-uniform reaction distribution

appears in thick cathode at the same discharge rates. For this reason, even though the depth of

discharge is low, high local DOD in the active material still occurred near the cathode-separator

interface. The blank region further increased.

Limiting the DOD of MnO2 cathodes can extend cell life because it limits the structure change

and maintains reversibility of the proton insertion reaction in the cathode. When the DOD decreases,

the blank region (healthy region) increases. In conclusion, in order to keep rechargeability of Zn-

MnO2 cell, engineers should choose appropriate thickness and discharge rate in the blank region. If

you want to make higher utilization of active material (which means higher DOD), thinner electrode

and lower discharge rate would be a better discharge condition for the cell to maintain

rechargeability. When DOD is lower, a cathode with large thickness (up to 1.7 cm) can keep

rechargeability. This is an important optimization problem, as cost of the battery is higher with

55

thinner electrodes, as the current collectors are the chief cost in this cell.

4.4 Conclusion

Rechargeable Zn-MnO2 batteries are good candidates for grid-scale energy storage due to the

low cost, environmental friendliness, and high energy density. Prismatic Zn-MnO2 batteries are used

here because they have low fabrication cost and ease of scale-up. The effect of anode mass and the

relationship between discharge current rate (I) and cathode thickness (Lc) at different depth of

discharge is investigated with regard to the impact on rechargeability after the first discharge. Large

anode mass can promote cathode reaction rate, but this effect of the anode is limited.

Relative low discharge current rates (1mA/(g-MnO2) – 10 mA/(g-MnO2)) are applied to

investigate the discharge behaviors of prismatic Zn-MnO2. From I-Lc figures at different DOD

values, the following conclusion can be made: In order to keep rechargeability of the cell, if desired

DOD is high, relative thinner electrode should be chosen. Limiting DOD of the MnO2 cathode

extends cycle life of Zn-MnO2 to a certain degree, which corresponds to large blank region of the

I-Lc figure at low DOD. When the DOD decreases, this blank region increases, and more cathode

thickness and discharge current are available. In order to prevent formation of hetaerolite (ZnMn2O4)

and hausmannite (Mn3O4) and maintain rechargeability of Zn-MnO2 batteries, the I-Lc figures at

different DOD are a good guide as to what rechargeable Zn-MnO2 battery designs are applicable

for grid – scale energy storage.

56

Chapter 5 Conclusion & future work

Simplified models (with the anode treated as uniform) were used to analyze the discharge

behavior of cylindrical and prismatic alkaline Zn-MnO2 batteries. The model in Chapter 3 mainly

analyzed the discharge behavior of cylindrical Zn-MnO2 battery cathodes. The model in Chapter 4

investigated the discharge behavior of prismatic Zn-MnO2 battery cathodes for grid-scale energy

storage. Conclusions drawn from the study are shown below:

• The secondary current distribution of cylindrical porous electrode is found to be governed by

two polarization parameters 𝛿and 𝛾 . When 𝛿 is small, reaction distribution across the

electrode is more uniform and ohmic limitation control the system. When 𝛿 is large, reaction

distribution across the electrode is nonuniform, charge-transfer effects control the system. For

small values of 𝛾, the current will split between the electrode-separator interface and the

electrode-current collector interface. In this case the reaction zone tends to shift towards the

cathode-separator and cathode-current collector interfaces.

• A mixed-reaction model is used to analyze the discharge behavior of cylindrical Zn-MnO2

battery cathodes at relative low discharge current. Near the cathode-separator interface (r =

0.42 cm), the reaction rate is the largest across the cathode. The porosity profile is consistent

with reaction distribution profile. As discharge proceeds, cathodic overpotential decreases with

the accumulation of MnOOH in cathode region. The effect of electrolyte concentration and

discharge current is also investigated. Higher electrolyte concentration will increase anode

reaction rate, and the reaction rate near cathode-separator interface (r = 0.42 cm) is higher when

the initial electrolyte concentration is high. Higher discharge current will increase the reaction

rate across the battery, which leads to a more nonuniform reaction distribution across the

cathode.

• The mixed reaction model in part 3 is adjusted to analyze the initial discharge behavior of

prismatic rechargeable Zn-MnO2 batteries. The effect of anode mass and the relationship of

discharge rate (I) and cathode thickness (Lc) at different DOD is investigated. Large anode

mass will promote the reaction rate of cathode, but the effect is limited. A series of I-Lc figures

at different DOD is a good guide to designing rechargeable Zn-MnO2 batteries to remain

rechargeable after the initial discharge. In order to maintain the reversibility of cathode reaction,

57

at higher DOD thinner electrodes should be chosen. When the DOD decreases, the blank area

(healthy area) increases in size, as more cathode thickness and discharge rate are available to

maintain the rechargeability of Zn-MnO2 batteries.

In the future, the following work can be conducted to make further development of the model

Future work for the model includes:

• Inclusion of the second step cathode reaction

• Inclusion of rechargeability

• Applying dissolution-precipitation mechanism to describe anode reaction and considering

anode polarization

58

Reference

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7 (2003): 118-121

[2] Juran.T.R, Young.J, Smeu.M. “Density functional theory modeling of MnO2 polymorphs as

cathodes for multivalent ion batteries.” J.Phys.Chem.C, 122 (2018): 8788-8795

[3] Jenn-Shing C. Mathematical Modelling of cylindrical alkaline cells (Doctorial dissertation)

Retrieved from https://search-proquest-com.ezproxy.neu.edu/docview/303695814?pq-

origsite=primo

[4] Kozawa.A, Powers.R.A. “The manganese dioxide electrode in alkaline electrolyte; The electron-

proton mechanism for the discharge process from MnO2 to MnO1.5.” J.Electrochem Soc. 113

(1966): 870-878

[5] Kozawa.A, Powers.R.A. “Cathodic polarization of the manganese dioxide electrode in alkaline

electrolyte.” J.Electrochem Soc.115 (1968): 122-126

[6] Gallaway.J.W, Hertzberg.B.J et al. “Operando identification of the point of [Mn2]O4 spinel

formation during 𝛾-MnO2 discharge within batteries.” Journal of Power Sources 321 (2016): 135-

142

[7] Minakshi.M, Appadon.D, Martin.D.E. “The anodic behavior of planar and porous zinc

electrodes in alkaline electrolyte.” Electrochemical and Solid-State Letters, 13.7 (2010): A77-A80

[8] Horn.Q.C, Shao-Horn.Y. “Morphology and spatial distribution of ZnO formed in discharged

alkaline Zn/MnO2 AA cells.” Journal of the Electrochemical Society 150.3 (2003): A652-A658

[9] Newman.J.S, Tobias.C.W. “Theoretical analysis of current distribution in porous electrodes.”

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[10] Newman.J, Tiedemann.W. “Porous-electrode theory with battery applications.” Alche Journal

21.1 (1975): 25-38

[11] Chen.J.S, Cheh.H.Y. “Modeling of cylindrical alkaline cells Ⅲ.” J.Electrochem.Soc. 140.5

(1993): 1205-1213

[12] Mak.C.Y, Cheh.H.Y. “Modeling of cylindrical alkaline cells Ⅱ.” J.Electrochem.Soc. 138.6

(1991): 1611-1615

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[13] Kierzenka.J, Shampine.L.F. “A bvp solver based on residual control and the Matlab PSE.”

ACM.Trans.Math.Software. 27.3 (2001): 299-316

[14] Chen.J.S, Cheh.H.Y. “Modeling of cylindrical alkaline cells Ⅳ.” J.Electrochem.Soc. 140.5

(1993): 1213-1216

[15] Podlaha.E.J, Cheh.H.Y. “Modeling of cyclindrical alkaline cells Ⅴ.” J.Electrochem.Soc.

141.1 (1994): 15-26

[16] Farrell.T.J, Please.C.P et al. “Primary alkaline battery cathodes.” J.Electrochem.Soc. 147.11

(2000): 4034-4044

[17] Newman J, Alyea K E T. “ Numerical solution of coupled, ordinary differential equations.”

Electrochemical Systems. Hoboken,NJ: John Wiley & Sons, Inc, 2004. 538-555

[18] Kim.S.U, Perdue.B, Apblett.C.A et al. “Understanding performance limitations to enable high

performance magnesium-ion batteries.” J.Electrochem.Soc. 163.8 (2016) : A1535-A1542

[19] Hertzberg.B.J, Huang.A et al. “Effect of multiple cation electrolyte mixtures on rechargeable

Zn-MnO2 battery.” Chem.Mater. 28 (2016): 4536-4545

[20] Minakshi.M. “Alkaline-earth oxide modified MnO2 cathode: enhanced performance in an

aqueous rechargeable battery.” Ind.Eng.Chem.Res 50 (2011): 8792-8795

[21] Minakshi.M. “Characterization of alkaline-earth oxide additions to the MnO2 cathode in an

aqueous secondary battery.” Journal of Alloys and Compounds 509 (2011): 5974-5980

[22] Minakshi.M, Singh.P. “Synergistic effect of additives on electrochemical properties of MnO2

cathode in aqueous rechargeable batteries.” J Solid State Electrochem 16 (2012): 1487-1492

[23] Duay.J, Kelly.M et al. “Evaluation of a ceramic separator for use in rechargeable alkaline Zn-

MnO2 batteries.” Journal of Power Sources. 395 (2018): 430-438

[24] Ingale.N.D, Gallaway.J.W et al. “Rechargeability and economic aspects of alkaline Zn-MnO2

cells for electrical storage and load leveling.” Journal of Power Sources 276 (2015): 7-18

[25] Gallaway.J.W. “Modeling porous electrodes: Part 1.” Joshua Gallaway-Electrochemist.1 Nov.

2014

60

Appendix

Basic code to solve the model.

import numpy as np

from scipy.sparse import coo_matrix

from scikits.umfpack import spsolve

import matplotlib.pyplot as plt

N=2 #initial problem rectilinear

Ncath = 5 #standard cathode problem

NJ=100

itmax=500

tol=1e-6

# ====== Discharge info

I = 0.12285 # A (amps) battery discharge current

# ====== General cathode info

ep = 0.274 # initial porosity of cathode

sigma = 20.0 # S/cm graphite conductivity

H = 6.0*2.54 # cm battery inside height

W = 3.0*2.54 # cm electrode plate width

t2 = 0.78 # transferrence number wrt vol av velocity of OH-

V_alpha = 20.45 # cm3/mol molar vol MnOOH

V_gamma = 17.29 # cm3/mol molar vol MnO2

MWmn = 86.94 # g/mol MW of MnO2

dmn = MWmn/V_gamma # g/cm3 density of MnO2

V_0 = 18.07 # cm3/mol molar vol H2O

V_a = 17.8 # cm3/mol molar vol KOH

initKOH = 0.009 # mol/cm3 initial KOH conc

61

ro_Teflon = 1 # g/cm3 density of colloidal Teflon

ro_MnO2 = 5.026 # g/cm3 density of manganese dioxide

ro_KS44 = 2.26 # g/cm3 density of ks44

x_Teflon = 0.05 # mass fraction (Teflon)

x_MnO2 = 0.65 # mass fraction(MnO2)

x_KS44 = 0.30 # mass fraction(ks44)

ro_mix = ((x_Teflon/ro_Teflon)+(x_MnO2/ro_MnO2)+(x_KS44/ro_KS44))**(-1) #density

mixture

L_c = 0.7

V_c = L_c*H*W

Vc = V_c*(1-ep)

m_cathode = Vc*ro_mix

# ====== General anode info

epa = 0.266 # initial porosity of anode (Chen 0.75)

V_Zn = 9.15 # cm3/mol molar vol Zn

V_ZnO = 14.51 # cm3/mol molar vol ZnO

theta = 0.8 # mixed reaction parameter

ro_Zn = 7.14 # g/cm3 density of zinc

ro_ZnO = 5.61 # g/cm3 density of zinc oxide

x_Zn = 0.85 # mass fraction Zn

x_ZnO = 0.10 # mass fraction ZnO

roa_mix = ((x_Teflon/ro_Teflon)+(x_Zn/ro_Zn)+(x_ZnO/ro_ZnO))**(-1) #density of

mixtured anode

m_anode = 430 # g mass of anode

Va = m_anode/roa_mix # cm3 anode volume

V_a = Va/(1-epa) # thickness of anode

L_a = V_a/H/W

62

# ====== MnO2 particle info

r_ri = 0.005 # cm init MnO2 av particle radius

W1 = 0.65*m_cathode # g total mass of MnO2 in cathode

MW = 86.9368 # g/mol MW of MnO2

DH = 6e-10 # cm2/s proton diffusion coeff in solid

Vpart_i = (4.0/3)*3.14*r_ri**3 # cm3 init volume of MnO2 particle

Vc = H*W*L_c

NN = W1/Vc/Vpart_i/dmn # cm-3 number of MnO2 particles per cath vol

concMn = W/MW/Vc # mol/cm3 total conc of MnO2 in cathode

romax = (r_ri**3*V_alpha/V_gamma)**(1./3.) # cm max particle radius

Asep = H*W

# ====== Kinetic info

io = 0.0000002 # A/cm2 exch current density

ac = 0.5 # cathodic symmetry parameter

F = 96500.0 # C/mol faraday's number

R = 8.314 # J/mol-K gas constant

T = 298.0 # K temp

ba = (1.0-ac)*F/R/T # 1/V anodic kinetic constant

bc = ac*F/R/T # 1/V cathodic kinetic constantm2 separator area

h=L_c/float(NJ-1)

# region for the problem

rr = np.linspace(0,L_c,NJ)

# ===========

# INITIAL GUESSES

63

# ===========

def initguess():

cold = np.ones([N,NJ])

cold[0,:]=-0.05

cold[1,:]=0.1

return cold

# INTERPOLATE

# ===========

def interpolate( c, h):

( N, NJ )=c.shape

cE=np.concatenate( ( c[:,1:NJ], np.zeros((N,1)) ), axis=1 )

cW=np.concatenate( ( np.zeros((N,1)), c[:,0:NJ-1] ), axis=1 )

dcdx=(cE-cW)/2.0/h;

dcdx[:, 0 ] = ( - 3.0*c[:, 0 ] + 4.0*c[:,1 ] - c[:,2 ] ) /2.0 /h;

dcdx[:,NJ-1] = ( 3.0*c[:,NJ-1] - 4.0*c[:,NJ-2] + c[:,NJ-3] ) /2.0 /h;

d2cdx2= (cE + cW - 2*c)/h**2;

d2cdx2[:,0] = np.zeros((1,N));

d2cdx2[:,NJ-1] = np.zeros((1,N));

return dcdx , d2cdx2

# FILLMAT

# ===========

64

def fillmat1(cold,dcdx,d2cdx2):

#________first column refers to equation

#________second column refers to position

#________third column refers to species

( N, NJ )=cold.shape

sma=np.zeros((N,NJ,N))

smb=np.zeros((N,NJ,N))

smd=np.zeros((N,NJ,N))

smg=np.zeros((N,NJ))

# Calculate varying physical params

kappa = 0.45 # S/cm electrolyte conductivity

# N particles per volume will not change. However particle radius will

# This will affect area per volume a

r_r = r_ri

Apart = 4.0*3.14*r_r**2 # cm2 area of MnO2 particle

a = NN*Apart # cm-1 area per volume MnO2 in cathode

aio = a*io # A/cm3 exchange transfer current

# lim will initially be very large because ro and ri are equal

# We begin by setting it to a high number

ep = 0.4

# Cathode Model is entered HERE

65

# Initial, steady-state problem:

smb[0,:,0] = 1.0

smd[0,:,1] = -1.0/kappa/ep**1.5 - 1.0/sigma

smg[0,:] = -dcdx[0,:] + (1.0/kappa/ep**1.5 + 1.0/sigma)*cold[1,:] - \

I/H/W/sigma

smd[1,:,0] = aio*(-ba*np.exp(ba*cold[0,:]) - bc*np.exp(-bc*cold[0,:]))

smb[1,:,1] = 1.0

smg[1,:] = -dcdx[1,:] + \

aio*np.exp(ba*cold[0,:]) - aio*np.exp(-bc*cold[0,:])

#__________________________________________________ Boundary-Condition 1

smp = np.zeros([N,N])

sme = np.zeros([N,N])

smf = np.zeros([N,1])

sme[1,1] = 1.0

smf[1] = I/H/W - cold[1,0]

# Non-B.C.

smp[0,0] = 1.0

sme[0,1] = -1.0/kappa/ep**1.5 - 1.0/sigma

smf[0] = -dcdx[0,0] + (1.0/kappa/ep**1.5 + 1.0/sigma)*cold[1,0] - \

I/H/W/sigma

66

# Insert (sme smp smf) into (smb smd smg)

smb[:,0,:] = smp[:,:]

smd[:,0,:] = sme[:,:]

smg[:,0] = np.transpose(smf)

#_______________________________________________ Boundary-Condition 2




# B.C.

sme[1,1] = 1.0

smf[1] = - cold[1,NJ-1]

# Non-B.C.

smp[0,0] = 1.0

sme[0,1] = -1.0/kappa/ep**1.5 - 1.0/sigma

smf[0] = -dcdx[0,NJ-1] + (1/kappa/ep**1.5 + 1/sigma)*cold[1,NJ-1] - \

I/H/W/sigma


smb[:,NJ-1,:] = smp[:,:]

smd[:,NJ-1,:] = sme[:,:]

smg[:,NJ-1] = np.transpose(smf)

67

return sma, smb, smd, smg

def ABDGXY(sma, smb, smd, smg):

( N, NJ )=smg.shape

sma = np.transpose(sma, (0, 2, 1))

smb = np.transpose(smb, (0, 2, 1))

smd = np.transpose(smd, (0, 2, 1))

A = sma-h/2.0*smb

B = -2.0*sma+h**2*smd

D = sma+h/2.0*smb

G = h**2*smg

# Old version

B[:,:,0] = h*smd[:,:,0]-1.5*smb[:,:,0]

D[:,:,0] = 2.0*smb[:,:,0]

G[:,0]=h*smg[:,0]

X = -0.5*smb[:,:,0]

# Old version

A[:,:,NJ-1]=-2.0*smb[:,:,NJ-1]

B[:,:,NJ-1]=h*smd[:,:,NJ-1]+1.5*smb[:,:,NJ-1]

G[:,NJ-1]=h*smg[:,NJ-1]

Y=0.5*smb[:,:,NJ-1]

ABD = np.concatenate((A, B, D), axis=1)

68

BC1 = np.concatenate((B[:,:,0] , D[:,:,0] , X), axis=1)

BC2 = np.concatenate((Y , A[:,:,NJ-1] , B[:,:,NJ-1]), axis=1)

ABD[:,:,0] = BC1

ABD[:,:,NJ-1] = BC2

return ABD, G

def band(ABD, G):

BMrow = np.reshape(np.arange(1,N*NJ+1), (NJ,N))

BMrow = BMrow[:, :, np.newaxis]

BMrow = np.transpose(BMrow, (1, 2, 0))

BMrow = BMrow[:,[0 for i in range(3*N)],:]

a = np.arange(1,3*N+1)

a = a[np.newaxis,:]

a = np.repeat(a,N,0)

a = a[:,:,np.newaxis]

a = np.repeat(a,NJ,2)

b = np.arange(0,(N)*(NJ-3)+N,N)

b = np.hstack((b[0], b, b[len(b)-1]))

b = b[np.newaxis,np.newaxis,:]

b = np.repeat(b,N,0)

b = np.repeat(b,3*N,1)

BMcol = a + b

BMcol = BMcol - 1

BMrow = BMrow - 1

BMrow = np.ravel(BMrow)

BMcol = np.ravel(BMcol)

69

ABD = np.ravel(ABD)

BigMat = coo_matrix((ABD, (BMrow, BMcol)), shape=(N*NJ, N*NJ)).tocsc()

BigG = np.transpose(G)

BigG = np.ravel(BigG)

delc = spsolve(BigMat, BigG)

delc = delc.reshape((NJ, N))

delc = np.transpose(delc)

return delc

def bound_val1(cold1):

for iter in range(1,itmax):

( dcdx , d2cdx2 )=interpolate( cold1,h)

(sma,smb,smd,smg)=fillmat1(cold1,dcdx,d2cdx2)

(ABD,G)=ABDGXY(sma,smb,smd,smg)

delc= band(ABD,G)

error=np.amax(np.absolute(delc))

print ("iter, error = %i, %g" % (iter,error))

cold1=cold1+delc

if error < tol:

return cold1

print ('The program did not converge!!')

70

#disp([h*[1:NJ]',cold',delc'])

return cold

cold = initguess()

#initial problem solution(which includes overpotential, and ionic current variable)

Sol_init=bound_val1(cold)

#Including time derivative

#1 Time discretization infomation

tstep= 20 #

total_time= 216.6 #h discharge time

totalsteps=int(total_time*3600/tstep)

# Transfer Current

# ===========

def transfer(jcold):

# Calculates the tranfer current j

# Uses the 2nd variable in cold, which is i2

jdcdx, jd2cdx2 = interpolate(jcold,h)

trans = jdcdx[1,:]

if abs(trans[1]) <= 1e-3:

trans[0] = trans[1]

trans[NJ-1] = trans[NJ-2]

return trans

# DERIV

# ===========

71

def deriv(c):

# Derivative of a 1D array

cE = np.zeros_like(c)

cW = np.zeros_like(c)

cE[0:NJ-1] = c[1:NJ]

cW[1:NJ] = c[0:NJ-1]

dd = (cE-cW)/2.0/h

dd[0] = (-3.0*c[0] + 4.0*c[1] - c[2]) /2.0 /h

dd[NJ-1] = (3.0*c[NJ-1] - 4.0*c[NJ-2] + c[NJ-3]) /2.0 /h

return dd

# Electrolyte Properties

# ===========

def electrolyte(concold):

M = concold[2,:]*1000 # mol/L molarity

p = (0.99668742) + \

(5.0065284e-2)*M + \

(-2.006849e-3)*M**2 + \

(2.2577517e-4)*M**3 + \

(-2.276481e-5)*M**4 + \

(1.3081735e-6)*M**5 + \

(-3.012443e-8)*M**6 # g/cm3 density

molal = (4.0074878e-6) + \

(1.002374947)*M + \

(7.4141713e-3)*M**2 + \

(1.3054688e-3)*M**3 + \

(-1.478333e-5)*M**4 + \

(2.3063735e-7)*M**5 + \

72

(3.562232e-8)*M**6 # mols/kg-H2O molality

WP = 5610.8*concold[2,:]/p[:] # weight % KOH

kappainf = (8.782e-3) + \

(3.720e-2)*WP + \

(8.109e-5)*WP**2 + \

(-3.026e-5)*WP**3 + \

(3.346e-7)*WP**4 # S/cm conductivity

Dainf = np.exp((-1.0464e1) + \

(-4.1100e-1)*concold[2,:]**0.5/(1+concold[2,:]**0.5) + \

(2.9182e-1)*concold[2,:] + \

(-9.1543e-2)*concold[2,:]**1.5 + \

(5.9489e-3)*concold[2,:]**2) # cm2/s KOH diff coeff

gam = np.exp(-(1.1762*molal**0.5)/(1+1.15*molal**0.5) + \

(0.2302)*molal + \

(6.0489e-3)*molal**2 + \

(-2.9934e-4)*molal**3 + \

(3.9144e-7)*molal**4) # KOH molality activity coeff

faa = 0.997/(p-56.1056*concold[2,:])*gam # KOH activity coeff

cwater = (1.0 - concold[2,:]*V_a)/V_0 # mol/cm3 water conc

return kappainf,Dainf,cwater,faa,WP

# MnO2 Particle Properties

# ===========

def particle(part,jp,tstep):

# Evolves the MnO2 particle properties

# 0 is particle radius

# 1 is inner MnO2 core

# 2 is lim (proton diffusion across the shell)

73

# 3 is aio

# 4 is X in MnOx

newpart = np.zeros_like(part)

ro3 = (-3.0/4.0/3.14/NN/F*(V_alpha-V_gamma)*jp)*tstep + part[0,:]**3

ro = ro3**(1./3.)

ro[ro > romax] = romax

newpart[0,:] = ro

ri3 = r_ri**3*(newpart[0,:]**3/r_ri**3-V_alpha/V_gamma)/(1.0-V_alpha/V_gamma)

ri3[ri3 <= 0] = 0

ri = ri3**(1./3.)

ri[ri <= 0] = 1.0e-30

newpart[1,:] = ri

newpart[2,:] = 4.0*3.14*F*NN*DH/(1.0/newpart[0,:]-1.0/newpart[1,:])

Apart = 4.0*3.14*newpart[0,:]**2 # cm2 area of MnO2 particle

a = NN*Apart # cm-1 area per volume MnO2 in cathode

newpart[3,:] = a*io # A/cm3 exchange transfer current

newpart[4,:] =((r_ri**3-newpart[0,:]**3)/(V_gamma-V_alpha))*(V_gamma/r_ri**3)

return newpart

# Anode

# ===========

def anode(anprop,tstep):

# Evolves the anode properties

# 0 = anode porosity

# 1 = convection velocity from anode

# 2 = anode OH- conc

# 3 = OH- conc at cathode interface (unchanged here)

# 4 = overpotential at cathode interface (unchanged here)

74

# 5 = OH- flux to anode (unchanged here)

# 6 = OH- flux from cathode (unchanged here)

newanprop = np.zeros_like(anprop)

newanprop[0] = I*(V_Zn - theta*V_ZnO)/2.0/F/V_a*tstep + anprop[0]

newanprop[1] = -(newanprop[0]-anprop[0])*V_a/Asep/tstep

newanprop[2] = ((-Asep*anprop[6] - theta*I/F - (1-theta)*2.0*I/F)*tstep/V_a +

anprop[2]*anprop[0])/newanprop[0]

if newanprop[2] <= 0.0:

newanprop[2] = 0.0

newanprop[3] = anprop[3]





return newanprop

# Fluxes

# ===========

def fluxes(somecold):

# Calcualtes the OH- flux at each side of something

kappa,Da,cw,fa,WKOH = electrolyte(somecold)

dcdr, d2cdr2 = interpolate(somecold,h)

cE=np.concatenate((somecold[:,1:NJ], np.zeros((N,1)) ), axis=1 )

cW=np.concatenate(( np.zeros((N,1)), somecold[:,0:NJ-1] ), axis=1 )

dcdr=(cE-cW)/2.0/h

75

dcdr[:,0] = (-3.0*somecold[:,0] + 4.0*somecold[:,1] - somecold[:,2] ) /2.0 /h

dcdr[:,NJ-1] = (3.0*somecold[:,NJ-1] - 4.0*somecold[:,NJ-2] + somecold[:,NJ-3] )

/2.0 /h

left_flux = -somecold[4,0]**1.5*Da[0]*dcdr[2,0] - somecold[1,0]*t2/F +

somecold[2,0]*somecold[3,0]

right_flux = -somecold[4,NJ-1]**1.5*Da[NJ-1]*dcdr[2,NJ-1] - somecold[1,NJ-1]*t2/F +

somecold[2,NJ-1]*somecold[3,NJ-1]

return left_flux, right_flux

def fillmat(cold,dcdx,d2cdx2,cprev,tstep,part,anprop):

( N, NJ )=cold.shape

sma=np.zeros((N,NJ,N))

smb=np.zeros((N,NJ,N))

smd=np.zeros((N,NJ,N))

smg=np.zeros((N,NJ))

kappa,Da,cw,fa,WKOH = electrolyte(cold)

dfa = deriv(fa)

OHs = anprop[2]

velo = anprop[1]

lim = part[2,:]

aio = part[3,:]

smb[0,:,0]= -cold[4,:]**1.5*cold[2,:]

smd[0,:,1]= 1.0/kappa*cold[2,:] + \

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1.0/sigma*cold[2,:]*cold[4,:]**1.5

smb[0,:,2]= 2.0*R*T/F*(1.0-t2)*cold[4,:]**1.5 + \

2.0*R*T/F/cw*cold[2,:]*cold[4,:]**1.5

smd[0,:,2]= 1.0/kappa*cold[1,:] + \

1.0/sigma*cold[4,:]**1.5*cold[1,:] - \

I/H/W/sigma*cold[4,:]**1.5 - \

dcdx[0,:]*cold[4,:]**1.5 + \

2.0*R*T/F/cw*dcdx[2,:]*cold[4,:]**1.5 + \

2.0*R*T/F*(1.0-t2)*dfa/fa*cold[4,:]**1.5 + \

4.0*R*T/F/cw*dfa/fa*cold[4,:]**1.5*cold[2,:]

smd[0,:,4]= 1.0/sigma*cold[2,:]*cold[1,:]*1.5*cold[4,:]**0.5 - \

I/H/W/sigma*cold[2,:]*1.5*cold[4,:]**0.5 - \

dcdx[0,:]*cold[2,:]*1.5*cold[4,:]**0.5 + \

2.0*R*T/F*(1-t2)*1.5*dcdx[2,:]*cold[4,:]**0.5 + \

2.0*R*T/F/cw*dcdx[2,:]*1.5*cold[2,:]*cold[4,:]**0.5 + \

2.0*R*T/F*(1-t2)*dfa/fa*1.5*cold[2,:]*cold[4,:]**0.5 + \

2.0*R*T/F/cw*dfa/fa*cold[2,:]**2*1.5*cold[4,:]**0.5

smg[0,:]= + cold[4,:]**1.5*cold[2,:]*dcdx[0,:] - \

1.0/kappa*cold[1,:]*cold[2,:] - \

1.0/sigma*cold[2,:]*cold[4,:]**1.5*cold[1,:] + \

I/H/W/sigma*cold[2,:]*cold[4,:]**1.5 - \

2.0*R*T/F*(1-t2)*dcdx[2,:]*cold[4,:]**1.5 - \

2.0*R*T/F/cw*dcdx[2,:]*cold[2,:]*cold[4,:]**1.5 - \

2.0*R*T/F*(1-t2)*dfa/fa*cold[2,:]*cold[4,:]**1.5 - \

2.0*R*T/F/cw*dfa/fa*cold[2,:]**2*cold[4,:]**1.5

smd[1,:,0] = -lim*ba*np.exp(ba*cold[0,:])*cold[2,:] - \

lim*bc*np.exp(-bc*cold[0,:])*cold[2,:] + \

1.0*bc*np.exp(-bc*cold[0,:])*dcdx[1,:] + \

ba*dcdx[1,:]*np.exp(ba*cold[0,:])

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smb[1,:,1] = 1.0/aio*lim*cold[2,:] - \

1.0*np.exp(-bc*cold[0,:]) + \

np.exp(ba*cold[0,:])

smd[1,:,2] = -lim*np.exp(ba*cold[0,:]) + \

lim*np.exp(-bc*cold[0,:]) + \

1.0/aio*lim*dcdx[1,:]

smg[1,:] = lim*cold[2,:]*np.exp(ba*cold[0,:]) - \

lim*cold[2,:]*np.exp(-bc*cold[0,:]) - \

lim*dcdx[1,:]*cold[2,:]/aio - \

dcdx[1,:]*np.exp(ba*cold[0,:]) + \

dcdx[1,:]*np.exp(-bc*cold[0,:])

smb[2,:,1] = (t2-1.0)/F

sma[2,:,2] = Da*cold[4,:]**1.5

smb[2,:,2] = 1.5*Da*cold[4,:]**0.5*dcdx[4,:] - \

cold[3,:]

smd[2,:,2] = - dcdx[3,:] - \

2.0*cold[4,:]/tstep + \

cprev[4,:]/tstep

smb[2,:,3] = -cold[2,:]

smd[2,:,3] = -dcdx[2,:]

smb[2,:,4] = 1.5*Da*cold[4,:]**0.5*dcdx[2,:]

smd[2,:,4] = 1.5*Da*d2cdx2[2,:]*cold[4,:]**0.5 + \

0.75*Da*cold[4,:]**-0.5*dcdx[2,:]*dcdx[4,:] - \

2.0*cold[2,:]/tstep + \

cprev[2,:]/tstep

smg[2,:] = - Da*d2cdx2[2,:]*cold[4,:]**1.5 - \

1.5*Da*cold[4,:]**0.5*dcdx[2,:]*dcdx[4,:] + \

(1-t2)/F*dcdx[1,:]+ \

cold[3,:]*dcdx[2,:] + \

78

cold[2,:]*dcdx[3,:] + \

2.0*cold[4,:]*cold[2,:]/tstep - \

cold[4,:]*cprev[2,:]/tstep - \

cold[2,:]*cprev[4,:]/tstep

smb[3,:,1] = -(V_0/F+(t2-1.0)*V_a/F)

smb[3,:,3] = 1.0

smd[3,:,4] = 1.0/tstep

smg[3,:] = (V_0/F+(t2-1.0)*V_a/F)*dcdx[1,:]- \

dcdx[3,:]- \

cold[4,:]/tstep + \

cprev[4,:]/tstep

smb[4,:,1] = -(V_alpha/F - V_gamma/F)

smd[4,:,4] = 1.0/tstep

smg[4,:] = -1.0/tstep*cold[4,:] + \

1.0/tstep*cprev[4,:] + \

(V_alpha/F - V_gamma/F)*dcdx[1,:]

#Boundary condition




sme[1,1] = 1.0

smf[1] = I/H/W - cold[1,0]

sme[3,3] = 1.0

smf[3] = velo-cold[3,0]

79

sme[2,2] = 1.0

smf[2] = OHs - cold[2,0]

smp[0,0]= -cold[4,0]**1.5*cold[2,0]

sme[0,1]= 1.0/kappa[0]*cold[2,0] + \

1.0/sigma*cold[2,0]*cold[4,0]**1.5

smp[0,2]= 2.0*R*T/F*(1.0-t2)*cold[4,0]**1.5 + \

2.0*R*T/F/cw[0]*cold[2,0]*cold[4,0]**1.5

sme[0,2]= 1.0/kappa[0]*cold[1,0] + \

1.0/sigma*cold[4,0]**1.5*cold[1,0] - \

I/H/W/sigma*cold[4,0]**1.5 - \

dcdx[0,0]*cold[4,0]**1.5 + \

2.0*R*T/F/cw[0]*dcdx[2,0]*cold[4,0]**1.5 + \

2.0*R*T/F*(1.0-t2)*dfa[0]/fa[0]*cold[4,0]**1.5 + \

4.0*R*T/F/cw[0]*dfa[0]/fa[0]*cold[4,0]**1.5*cold[2,0]

sme[0,4]= 1.0/sigma*cold[2,0]*cold[1,0]*1.5*cold[4,0]**0.5 - \

I/H/W/sigma*cold[2,0]*1.5*cold[4,0]**0.5 - \

dcdx[0,0]*cold[2,0]*1.5*cold[4,0]**0.5 + \

2.0*R*T/F*(1-t2)*1.5*dcdx[2,0]*cold[4,0]**0.5 + \

2.0*R*T/F/cw[0]*dcdx[2,0]*1.5*cold[2,0]*cold[4,0]**0.5 + \

2.0*R*T/F*(1-t2)*dfa[0]/fa[0]*1.5*cold[2,0]*cold[4,0]**0.5 + \

2.0*R*T/F/cw[0]*dfa[0]/fa[0]*cold[2,0]**2*1.5*cold[4,0]**0.5

smf[0]= cold[4,0]**1.5*cold[2,0]*dcdx[0,0] - \

1.0/kappa[0]*cold[1,0]*cold[2,0] - \

1.0/sigma*cold[2,0]*cold[4,0]**1.5*cold[1,0] + \

I/H/W/sigma*cold[2,0]*cold[4,0]**1.5 - \

2.0*R*T/F*(1-t2)*dcdx[2,0]*cold[4,0]**1.5 - \

2.0*R*T/F/cw[0]*dcdx[2,0]*cold[2,0]*cold[4,0]**1.5 - \

2.0*R*T/F*(1-t2)*dfa[0]/fa[0]*cold[2,0]*cold[4,0]**1.5 - \

2.0*R*T/F/cw[0]*dfa[0]/fa[0]*cold[2,0]**2*cold[4,0]**1.5

80

smp[4,1] = -(V_alpha/F - V_gamma/F)

sme[4,4] = 1.0/tstep

smf[4] = -1.0/tstep*cold[4,0] + \

1.0/tstep*cprev[4,0] + \

(V_alpha/F - V_gamma/F)*dcdx[1,0]

smb[:,0,:] = smp[:,:]

smd[:,0,:] = sme[:,:]

smg[:,0] = np.transpose(smf)




# B.C.

sme[1,1] = 1.0

smf[1]= -cold[1,NJ-1]

smp[2,2] = 1.0

smf[2] = -dcdx[2,NJ-1]

smp[0,0]= -cold[4,NJ-1]**1.5*cold[2,NJ-1]

sme[0,1]= 1.0/kappa[NJ-1]*cold[2,NJ-1] + \

1.0/sigma*cold[2,NJ-1]*cold[4,NJ-1]**1.5

smp[0,2]= 2.0*R*T/F*(1.0-t2)*cold[4,NJ-1]**1.5 + \

2.0*R*T/F/cw[NJ-1]*cold[2,NJ-1]*cold[4,NJ-1]**1.5

sme[0,2]= 1.0/kappa[NJ-1]*cold[1,NJ-1] + \

1.0/sigma*cold[4,NJ-1]**1.5*cold[1,NJ-1] - \

I/H/W/sigma*cold[4,NJ-1]**1.5 - \

dcdx[0,NJ-1]*cold[4,NJ-1]**1.5 + \

81

2.0*R*T/F/cw[NJ-1]*dcdx[2,NJ-1]*cold[4,NJ-1]**1.5 + \

2.0*R*T/F*(1.0-t2)*dfa[NJ-1]/fa[NJ-1]*cold[4,NJ-1]**1.5 + \

4.0*R*T/F/cw[NJ-1]*dfa[NJ-1]/fa[NJ-1]*cold[4,NJ-1]**1.5*cold[2,NJ-1]

sme[0,4]= 1.0/sigma*cold[2,NJ-1]*cold[1,NJ-1]*1.5*cold[4,NJ-1]**0.5 - \

I/H/W/sigma*cold[2,NJ-1]*1.5*cold[4,NJ-1]**0.5 - \

dcdx[0,NJ-1]*cold[2,NJ-1]*1.5*cold[4,NJ-1]**0.5 + \

2.0*R*T/F*(1-t2)*1.5*dcdx[2,NJ-1]*cold[4,NJ-1]**0.5 + \

2.0*R*T/F/cw[NJ-1]*dcdx[2,NJ-1]*1.5*cold[2,NJ-1]*cold[4,NJ-1]**0.5 + \

2.0*R*T/F*(1-t2)*dfa[NJ-1]/fa[NJ-1]*1.5*cold[2,NJ-1]*cold[4,NJ-1]**0.5 + \

2.0*R*T/F/cw[NJ-1]*dfa[NJ-1]/fa[NJ-1]*cold[2,NJ-1]**2*1.5*cold[4,NJ-1]**0.5

smf[0]= cold[4,NJ-1]**1.5*cold[2,NJ-1]*dcdx[0,NJ-1] - \

1.0/kappa[NJ-1]*cold[1,NJ-1]*cold[2,NJ-1] - \

1.0/sigma*cold[2,NJ-1]*cold[4,NJ-1]**1.5*cold[1,NJ-1] + \

I/H/W/sigma*cold[2,NJ-1]*cold[4,NJ-1]**1.5 - \

2.0*R*T/F*(1-t2)*dcdx[2,NJ-1]*cold[4,NJ-1]**1.5 - \

2.0*R*T/F/cw[NJ-1]*dcdx[2,NJ-1]*cold[2,NJ-1]*cold[4,NJ-1]**1.5 - \

2.0*R*T/F*(1-t2)*dfa[NJ-1]/fa[NJ-1]*cold[2,NJ-1]*cold[4,NJ-1]**1.5 - \

2.0*R*T/F/cw[NJ-1]*dfa[NJ-1]/fa[NJ-1]*cold[2,NJ-1]**2*cold[4,NJ-1]**1.5

smp[3,1] = -(V_0/F+(t2-1.0)*V_a/F)

smp[3,3] = 1.0


smf[3] = (V_0/F+(t2-1.0)*V_a/F)*dcdx[1,NJ-1] - \

dcdx[3,NJ-1] - \

cold[4,NJ-1]/tstep + \

cprev[4,NJ-1]/tstep

smp[4,1] = -(V_alpha/F - V_gamma/F)


smf[4] = -1.0/tstep*cold[4,NJ-1] + \

1.0/tstep*cprev[4,NJ-1] + \

(V_alpha/F - V_gamma/F)*dcdx[1,NJ-1]

82


smb[:,NJ-1,:] = smp[:,:]

smd[:,NJ-1,:] = sme[:,:]

smg[:,NJ-1] = np.transpose(smf)

return sma, smb, smd, smg

N = Ncath

# Transfer initial result to cprevC

cprevC = np.zeros([N,NJ])

cprevC[0:2,:] = Sol_init[:,:]

cprevC[2,:] = initKOH

cprevC[3,:] = 0

cprevC[4,:] = ep

# Expand coldC for time-dependent problem

coldC = np.zeros([N,NJ])

coldC[:] = cprevC

j = transfer(coldC)

# make MnO2 particle array

partC = np.zeros([5,NJ])

partC[0,:] = r_ri

partC[1,:] = r_ri

partC = particle(partC,j,tstep)

83

Lflux, Rflux = fluxes(coldC)

# ====== Solve initial ANODE problem

propA = np.zeros([8])

propA[0] = epa

propA[2] = initKOH

propA[3] = coldC[2,0]


propA[5] = Lflux

propA[6] = Lflux

propA = anode(propA,tstep)

def bound_val(cold,cprev,tstep,part,anprop):

for iter in range(1,itmax):

dcdx, d2cdx2 = interpolate(cold,h)

sma,smb,smd,smg = fillmat(cold,dcdx,d2cdx2,cprev,tstep,part,anprop)

ABD,G = ABDGXY(sma,smb,smd,smg)

delc = band(ABD,G)

error = np.amax(np.absolute(delc))

print ("iter, error = %i, %g" % (iter,error))

cold=cold+delc

84

if error < tol:

return cold

print ('The program did not converge!!')

return cold

#solve subsequent problem

#Values need to draw discharge curve

E_0 = 1.34 #V voltage when half of one electron release

Q = m_cathode*0.65*0.308 #A*h theoritical capacity of duracell AA at I = 0.1

for step in range(1,totalsteps+1):

print

print ('TIME STEP:', step)

print ('cathode')

coldC = bound_val(coldC,cprevC,tstep,partC,propA)

j = transfer(coldC)

partC = particle(partC,j,tstep)

fluxC, junk = fluxes(coldC)



propA[6] = fluxC

propA = anode(propA,tstep)

times= step*tstep/3600

f=I*times/Q

plt.figure(1)

85

plt.plot(times,partC[4,0],'ro')

if step == 2783:

plt.figure(2)

plt.plot(rr,coldC[2,:]*1000,label='DOD=5%')

plt.legend()

plt.figure(3)

plt.plot(rr,coldC[4,:],label='DOD=5%')

plt.legend()

plt.figure(4)


plt.legend()

plt.figure(5)

plt.plot(rr,-j,label='DOD=5%')

plt.legend()

plt.figure(6)

plt.plot(rr,partC[4,:],label='DOD=5%')

plt.legend()

if step == 5567:

plt.figure(2)


plt.legend()

plt.figure(3)


plt.legend()

plt.figure(4)


plt.legend()

plt.figure(5)


86

plt.legend()

plt.figure(6)


plt.legend()

if step == 11135:

plt.figure(2)


plt.legend()

plt.figure(3)


plt.legend()

plt.figure(4)


plt.legend()

plt.figure(5)


plt.legend()

plt.figure(6)


plt.legend()

if step == 16703:

plt.figure(2)


plt.legend()

plt.figure(3)


plt.legend()

plt.figure(4)


plt.legend()

87

plt.figure(5)


plt.legend()

plt.figure(6)


plt.legend()

if step == 22271:

plt.figure(2)


plt.legend()

plt.figure(3)


plt.legend()

plt.figure(4)


plt.legend()

plt.figure(5)


plt.legend()

plt.figure(6)


plt.legend()

if step == 27838:

plt.figure(2)


plt.legend()

plt.figure(3)


88

plt.legend()

plt.figure(4)


plt.legend()

plt.figure(5)


plt.legend()

plt.figure(6)


plt.legend()

if step == 33406:

plt.figure(2)


plt.legend()

plt.figure(3)


plt.legend()

plt.figure(4)


plt.legend()

plt.figure(5)


plt.legend()

plt.figure(6)


plt.legend()

if step == 38974:

plt.figure(2)

89


plt.legend()

plt.figure(3)


plt.legend()

plt.figure(4)


plt.legend()

plt.figure(5)


plt.legend()

plt.figure(6)


plt.legend()

cprevC = coldC

plt.figure(1)

plt.xlabel(r'$times(h)$',size=10)

plt.ylabel(r'$y$',size = 10)

plt.ylim(0.0023,0.0225)

plt.tick_params(axis='both', direction='in', bottom=True, top=True, left=True,

right=True)

plt.savefig('y20.png',dpi=300)

plt.figure(2)

90

plt.xlabel(r'$z(cm)$',size = 10)

plt.ylabel(r'$Ca(M)$',size = 10)


right=True)

plt.savefig('Ca20.png',dpi = 300)

plt.figure(3)


plt.ylabel(r'$porosity$',size = 10)


right=True)

plt.savefig('porosity20.png',dpi = 300)

plt.figure(4)


plt.ylabel(r'$overpotential(V)$',size = 10)


right=True)

plt.savefig('ETA920.png',dpi = 300)

plt.figure(5)


plt.ylabel(r'$-j(A/cm^3)$',size = 10)


right=True)

plt.savefig('j920.png',dpi = 300)

plt.figure(6)

xxx = np.array([0,0.4])

yyy = np.array([0.79,0.79])

91

plt.plot(xxx,yyy,'r--')


plt.ylabel(r'$y$',size = 10)

plt.yticks([0.1,0.15,0.2,0.25,0.3,0.35,0.4,0.45,0.5,0.55,0.6,0.65,0.7,0.75,0.8,0.85,0.

9,0.95,1])


right=True)

plt.savefig('xr920.png',dpi = 300)

print(m_cathode)

Date post:	13-Sep-2020
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MATHEMATICAL MODELING OF THE INITIAL DISCHARGE OF ALKALINE …m... · 2019. 9. 25. · Abstract A...

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