非均勻入口流場效應下熔融碳酸鹽燃料電池堆之性能分析

國 立 交 通 大 學

機 械 工 程 學 系

博

士

論

文

非均勻入口流場效應下熔融碳酸鹽燃料

電池堆之性能分析

Effects of Inlet Flow Maldistribution in Stack and Transverse

Direction on the Performance of a Molten Carbonate Fuel Cell

研究生

： 劉旭昉

指導教授 ：陳俊勳教授

曲新生教授

非均勻入口流場效應下熔融碳酸鹽燃料電池堆之性能分析

Effects of Inlet Flow Maldistribution in Stack and Transverse

Direction on the Performance of a Molten Carbonate Fuel Cell

研究生： 劉旭昉 Student:

Syu-Fang

Liu

指導教授：陳俊勳 曲新生 Advisor:

Chiun-Hsun

Chen

Hsin-Sen

Chu

國 立 交 通 大 學

機械工程學系

博 士 論 文

A dissertation

Submitted to Department of Mechanical Engineering

College of Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

in

Mechanical Engineering

July 2007

Hsinchu, Taiwan, Republic of China

非均勻入口流場效應下熔融碳酸鹽燃料電池堆之性能分析

研究生：劉旭昉

指導教授：陳俊勳 曲新生

摘 要

本論文主要在探討，陽極及陰極氣體入口處使用非均勻莫耳流率

( mole flow rate )時，對單體熔融碳酸鹽電池(Molten Carbonate Fuel

Cell)及電池堆 (Molten Carbonate Fuel Cell Stack)的性能影響。以有限

差分法(finite difference method)對質量守恆、能量守恆及化學計量守

恆式等偏微分方程式進行解析。並對部份的結果用套裝軟體 FlexPDE

以有限元素法( finite element method)進行驗證。在分析的方法中，本

文利用莫耳流率的不均勻性設計成八種不同型式(patterns)的入口流

場，然後分析入口流場對單體熔融碳酸鹽電池及電池堆性能的影響。

首先，分析非均勻入口流場對單體熔融碳酸鹽電池性能的影響。

在溫度場及電流密度場方面，以有限差分法進行求解，再利用套裝軟

體 FlexPDE 加以驗證，兩者之間的數據相當吻合。對於單體的融碳酸

鹽電池而言，當氣體入口處的非均勻莫耳流率偏差量為 0.25 時

(d=0.25)，在 G 式樣的電池溫度比均勻流(d=0)的高出 12%，而在 D

式樣的電流密度場要比均勻型式樣的高出 37%，根據結果，在入口處

之非均勻流會對電池的溫度及電池密度分佈範圍產生明顯的影響。

此外，本研究第二部份探討具交叉流氣體供應方式的熔融碳酸鹽

燃料電池在高氣體使用率與非均勻陽極氣流下的性能表現。數學模式

方面採用二維之質量，能量等守恆方程式，而不考慮堆疊方向的性質

變化。由數值計算結果顯示陽極氣體使用率隨入口莫耳流率之減少而

增加。此外，陽極端入口的非均勻流將導致在陽極出口端及陰極入口

端產生不反應區域，進而影響到電池的局部電流密度與性能。

最後，本文第三部份則進一步探討非均勻入口流場對電池堆性能

的影響。電池堆使用 10 個單體熔融碳酸鹽電池所組成，在陽極及陰

極入口處之流場皆為不均勻流場。本文主要是利用近似三維之數值模

擬，來分析一個具有 10 層單體的熔融碳酸塩燃料電池堆之溫度場及

電壓分佈。在陽極及陰極入口處，假設莫耳流率分佈曲線沿電池堆方

向為漸增式及均勻式設計，並將其組成四組不同的入口流場形式

(patterns)。結果顯示在陰極入口處使用非均勻的莫耳流率會明顯的改

變熔融碳酸塩燃料電池堆的溫度場，而且電池堆的入口有最少的莫耳

流率時，那麼在此時的陰極出口處會產生最高的溫度。此外，在沿著

電池堆的陽極入口處，如果具有非均勻的莫耳流率時，會強烈影響電

壓的分佈。沿著電池堆的方向來看，各單體電池的平均溫度變化率約

為百分之二，而平均的電池電壓變化率約為百分之四十。此結果與作

者先前所探討的非均勻流對單體電池的變化率要有比較明顯的不同。

Effects of Inlet Flow Maldistribution in Stack and Transverse

Direction on the Performance of a Molten Carbonate Fuel Cell

Student: Syu-Fang Liu

Advisor:

Chiun-Hsun Chen

Hsin-Sen

Chu

Abstract

This study investigates the temperature and current density distributions in a molten carbonate fuel cell unit and stack when the inlet flows of the anode gas and the cathode gas are mal-distributed. Furthermore, this study extends the research to the temperature and current density distributions in a molten carbonate fuel cell when there is higher utilization of anode gas. In the analysis of a unit cell, the two-dimensional simultaneous partial differential equations of mass, energy and electrochemistry are solved numerically. The numerical method is reliable through the accuracy comparison between this FORTRAN program and a software package. The results indicate that the maldistribution of anode and cathode gases dominates the current density field and the cell temperature field, respectively. Moreover, the non-uniform inlet flow slightly affects the mean temperature and mean current density, but worsens the distribution of temperature and current density for most maldistribution patterns. According to the results, the variations of the cell temperature in Pattern G and the current density in Pattern D are 12% and 37% greater than those in the uniform pattern when the deviation of the non-uniform profile is 0.25. Consequently, the effect of non-uniform inlet flow in the transverse direction on the temperature and current density distribution on the cell plane is evident, and cannot be neglected. In the analysis of a MCFC stack, this study considers that the MCFC is composed by ten stacks, and the molar flow rate in each

stack is different because of the inlet distributor. This study employs the procedure of calculation in a MCFC unit to calculate the results of each stack, and then averages the temperatures of up separator and down separator, which connect together between stacks. The FORTRAN program iterates the whole procedure to get the quasi three- dimensional temperature and current density distributions until the relative errors of average temperature of separators satisfy the converge criterion. The primary results show that the effect of non-uniform in the stacking direction is more apparent than that of non-uniform in the transverse direction on the thermal and electrical performance of a MCFC.

Then, the second part of this dissertation, the electric performance of a planar MCFC unit with cross-flow configuration when there is higher gas utilization in anode and cathode is investigated in the final part of this dissertation. A two-dimensional model, considering the conservation equations of mass, energy and electro-chemistry is applied. The results show that the anode gas utilization increases with a decrease in the molar flow rate, and the average current density decreases when the molar flow rate drops. In addition, non-uniform inlet profile of the anode gas will induce a happening of non-reaction area in the corner of the anode gas exit and the cathode gas inlet. This non-reaction area deteriorates the average current density and then reduces the electrical performance up to 4% when the anode gas molar flow rate is 0.01242 mol s and anode gas utilization is 73%.

Finally, in the third part of this dissertation, the effects of the non-uniform inlet flow on the MCFC stack are investigated. We develop a quasi-three dimensional numerical method for analyzing three-dimensional temperature and cell voltage distribution in a ten-layer molten carbonate fuel cell. The authors consider the non-uniform profile as progressively increasing along the stacking direction, and

assign it to the anode gas inlet or cathode gas inlet to form four kinds of patterns. Results indicate that the non-uniform molar flow rate of cathode gas obviously changes the temperature field of a molten carbonate fuel cell stack, and highest cell temperature occurs at the cathode gas exit in the layer with the lowest molar flow rate. Moreover, non-uniform anode gas in the stacking direction strongly affects cell voltage distribution in the molten carbonate fuel cell stack. The variation of average cell temperature and cell voltage among different layers along the stacking direction are 2% and 40%, apparently larger than the variation rate due to non-uniformity in the transverse direction in previous chapter.

誌 謝

首先誠摯的感謝指導教授U曲新生U博士，老師悉心的教導使我得以一窺融熔碳 酸鹽燃料電池領域的深奧，不時的討論並指點我正確的方向，使我在這些年中獲 益匪淺。老師對學問的嚴謹更是我輩學習的典範。同時亦得感謝成功大學機械系 的U陳朝光U教授、清華大學動機系U洪哲文U教授、華梵大學機電系U顏維謀U教授，以及 本校U盧定昶U教授、U陳俊勳U教授和U楊文美U教授對我研究上的建議與指正，使得本論 文能夠更完整而嚴謹。 特別感謝同僚U袁平U博士及U王士賓U博士在這段期間，於學理上的協助和討論及 工作上的支援，在此致上最誠摯的謝意。 在漫長的學習過程中，實驗室裡共同的生活點滴，學術上的討論、言不及義 的閒扯、讓人又愛又怕的宵夜、趕作業的革命情感，感謝學長、同學、學弟妹的 共同砥礪，你們的陪伴讓研究生活變得絢麗多彩。感謝實驗室的U森溥U學長，同窗 U 致廣U，學弟U志文U、U世國、U U木勝U、U時明U、U家輝U、U章裕U、U謙成U、銘恩U 、U U清益、U U榮祥U、 U 式堯U及U祥哲U、學妹U淑惠U，你們的幫忙讓我銘感在心。 感謝父、母親的養育之恩與妹妹及兩個弟弟的關心及鼓勵，使我無後顧之憂 地順利完成學業。 妻子U明妃U在背後的默默支持更是我前進的動力，沒有U明妃U的體諒、包容，相 信是無法順利完成這篇論文。 最後，謹以此文獻給我摯愛的雙親及愛妻U明妃U。

Table of Contents

U

摘 要

U

... i

U

Abstract

U

... iii

U

誌 謝

U

... vi

U

Table of Contents

U

... vii

U

List of Tables

U

... ix

U

List of Figures

U

...x

U

Nomenclature

U

... xii

U

1.

U U

Introduction

U

...1

U

1.1.U UBackground & MotivationU... 1

U

1.2.U UTypes of Fuel CellsU... 2

U

1.3.U UBrief Overview of MCFCU... 2

U

1.4.U ULiterature SurveyU... 10

U

1.5.U UObjectives of Present StudiesU... 19

U

2.

U U

Effect of Inlet Flow Maldistribution on the Thermal and Electrical

Performance of an MCFC Unit

U

...25

U

2.1.U UPhysical Model DescriptionU... 25

U 2.2.U UBasic AssumptionU... 25 U 2.3.U UGoverning EquationsU... 26 U 2.3.1.U UReaction EquationsU... 26 U

2.3.2.U UMass Conservation EquationsU... 27

U

2.3.3.U UEnergy Conservation EquationsU... 27

U

2.3.4.U UNernst Voltage and PolarizationsU... 29

U

2.4.U UMethod of SolutionU... 30

U

2.5.U UResults and DiscussionU... 35

U

2.6.U UConcluding RemarksU... 43

U

Flow and High Anode Gas Utilization

U

...67

U

3.1.U UPhysical Model DescriptionU... 68

U

3.2.U UMethod of SolutionU... 73

U

3.3.U UResults and DiscussionU... 74

U

3.4.U UConclusionsU... 77

U

4.

U U

Effect of Inlet Flow Maldistribution on the Thermal and Electrical

Performance of an MCFC Stack

U

...86

U

4.1.U UPhysical Model DescriptionU... 86

U 4.2.U UBasic AssumptionsU... 87 U 4.3.U UGoverning EquationsU... 88 U 4.3.1.U UReaction EquationsU... 88 U

4.3.2.U UMass Conservation EquationsU... 88

U

4.3.3.U UEnergy Conservation EquationsU... 89

U

4.3.4.U UNernst Voltage and PolarizationsU... 90

U

4.4.U UMethod of SolutionU... 92

U

4.5.U UResults and DiscussionU... 95

U

4.6.U UConcluding RemarkU... 100

U

5.

U U

Conclusions and Future Perspectives

U

...110

U 5.1.U UConcluding RemarksU...110 U 5.2.U UFuture PerspectivesU...111 U

References

U

...113

U

Appendix A

U

...122

U

Publication List

U

...125

U

Author

U

...126

U

Publication List

U

...127

List of Tables

U

Table 1.1U UThe types of fuel cellsU... 22

U

Table 2.1U UExpressions of energy source terms in Eq.X(2.5)X to Eq.X(2.11)XU... 46

U

Table 2.2U UParameters and conditions in this studyU... 47

U

Table 2.3U URelative variation of cell temperature and current density at different

non-uniform inlet flow patterns related to at uniform inlet flow patternU

... 48

U

Table 3.1U UAverage current density and anode gas utilization at different inlet molar

flow rate and patternsU... 79

U

Table 4.1U UExpressions of energy source terms in energy conservation equations

X

(4.5)X to X(4.8)XU... 102

U

List of Figures

U

Fig 1.1.U UOperating principle of a MCFCU... 23

U

Fig 1.2.U UComponent diagram of a unit cellU... 24

U

Fig 2.1.U USchematic diagram of a molten carbonate fuel cell unit in crossflowU 49

U

Fig 2.2.U UPatterns of non-uniform inlet flow profile in Chapter 2U... 50

U

Fig 2.3.U UCalculated node arrangement in this studyU... 51

U

Fig 2.4.U UAnode gas temperature at the central point versus grid numbers in

numerical program.U... 52

U

Fig 2.5.U UCell temperature at the central point versus grid numbers in numerical

program.U... 53

U

Fig 2.6.U UCathode gas temperature at the central point versus grid numbers in

numerical program.U... 54

U

Fig 2.7.U USeparator temperature at the central point versus grid numbers in

numerical program.U... 55

U

Fig 2.8.U UCurrent density at the central point versus grid numbers in numerical

program.U... 56

U

Fig 2.9.U UCell temperature distribution calculated by the numerical method in

Chapter 2 and FlexPDE software with uniform inlet flow rateU... 57

U

Fig 2.10.U UCurrent density distribution calculated by the numerical method in

Chapter 2 and FlexPDE software with uniform inlet flow rateU... 58

U

Fig 2.11.U UTotal Resistance distribution on the cell plane with uniform inlet flowU59

U

Fig 2.12.U UCell temperature distribution of eight non-uniform patterns with

deviation of 0.5U... 60

U

Fig 2.13.U UTemperature difference related to uniform pattern on the cell planeU.. 61

U

Fig 2.14.U UCurrent density distribution of eight non-uniform patterns with deviation

of 0.5U... 62

U

U

Fig 2.16.U URelative variation of UT_cU and UI_rU at different non-uniform inlet flow

patterns related to at uniform inlet flow patterU... 66

U

Fig 3.1.U USchematic diagram of a unit of molten carbonate anode gas cell in

cross-flow.U... 80

U

Fig 3.2.U UArrangements of non-uniform inlet flow patterns in this chapter.U... 81

U

Fig 3.3.U UCurrent density distribution in Pattern A and B when U f n U=0.0621 and U a n U=0.1841 mol/s.U... 82 U

Fig 3.4.U UCurrent density distribution in Pattern A and B when Un_f U=0.01242 and

U

a

n U=0.0526 mol/sU... 83

U

Fig 3.5.U UCurrent density distribution in Pattern A and B when Un_fU=0.00621 and Un_aU=0.0263 mol/s.U... 84

U

Fig 3.6.U URelative change of average current density in non-uniform pattern.U.. 85

U

Fig 4.1.U USchematic diagram of a molten carbonate fuel cell unit in crossflow.U104

U

Fig 4.2.U UVarious inlet flow patterns in an MCFC stack.U... 105

U

Fig 4.3.U UTemperature distribution in bottom, middle, and top layers of an MCFC

stack.U... 106

U

Fig 4.4.U UTemperature distribution on exit face and top face of an MCFC stack.U

107

U

Fig 4.5.U UAverage temperature of each layer in an MCFC stack .U... 108

U

Nomenclature

a area of heat transfer area per base area (m2⋅m−2)

p

c specific heat capacity (J mol⋅ −1⋅K−1)

d deviation of inlet flow profile, as shown in Fig. 2 E Nernst voltage (V)

0

E reversible open circuit voltage (V) F Faraday’s constant (96485 As mol-1) h heat transfer coefficient ( 2 1

W m⋅ − ⋅K− )

i current density (A m⋅ −2)

0

i exchange current density (A m⋅ −2)

i average current density in the x-y plane (A m⋅ −2)

i

Δ current density variation in the x-y plane (A m⋅ −2) k conductivity ( 1 1

W m⋅ − ⋅K− )

L length in the x or y direction, indicated by subscript x or y (m)

n molar flow rate of anode gas or cathode gas per unit width, indicated by subscript ag or cg (mol m⋅ −1⋅s−1)

N molar flow rate of anode gas or cathode gas, indicated by a subscript ag or cg (mol s⋅ −1)

e

n number of electrons transferred in reactions of anode and cathode P pressure (Pa)

q heat generation due to chemical reaction (W m⋅ 2) R universal gas constant (8.314 J mol⋅ −1⋅K−1) Rtot total cell resistance (

2

m

T temperature (K)

T _{average temperature in the x-y plane (K)}

ΔT temperature variation in the x-y plane (K) V cell voltage (V)

x x direction

X molar fraction indicated by subscript k y y direction

Greek symbols

δ thickness (m)

ν stoichiometric coefficient of k-component indicated by subscript k

Subscripts

ag anode gas

ag-c interface between anode gas and cell ag-s interface between anode gas and separator c cell

c-s interface between cell and separator cg cathode gas

cg-c interface between cathode gas and cell cg-s interface between cathode gas and separator k components in anode gas or cathode gas s separator

x x direction y y direction

1.

0B

Introduction

1.1.

5B

Background & Motivation

A fuel cell is an electrochemical device for transforming chemical energy into electricity directly. Fuel cells are different from traditional thermal engines which transform chemical energy to mechanical energy via combustion, and then to electricity by power generators. As a result, fuel cells have higher transfer efficiency than thermal engines because they are not restricted by Carnot cycle efficiency. The maximal theoretical thermodynamic efficiency, ε_th, of a fuel cell is the ratio of Gibb’s free energy to the standard enthalpy of formation:

ε = Δ

Δ

th

G

H (1.1)

The theoretical efficiency of fuel cells is about 80%, but in practice, it is determined by the cell voltage, V_cell:

ε = − Δ cell real zFV H (1.2)

The practical electrical efficiencies of fuel cells are about 40-60%, depending on the type of fuel cell. Losses that limit cell voltage include ohmic losses, kinetic loss, and mass transfer limitations in the system. Increasing energy demands and global environment preservation concerns have increased the necessity of developing energy systems with high energy conversion efficiency and very low environmental pollution.

1.2.

6B

Types of Fuel Cells

Researchers have focused their attention on four kinds of fuel cells that have potential application in various industries. These fuel cells include the proton exchange membrane fuel cell (PEMFC), phosphoric acid fuel cell (PAFC), molten carbonate fuel cell (MCFC), and solid oxide fuel cell (SOFC), which are classified by electrolytes, operation temperature, etc. Table 1-1 lists these fuel cells.

1.3.

7B

Brief Overview of MCFC

In the 1930s, Emil Baur and H. Preis experimented with high-temperature, solid oxide electrolytes in Switzerland. They encountered problems with electrical conductivity and unwanted chemical reactions between the electrolytes and various gases (including carbon monoxide). The following decade, O. K. Davtyan of Russia explored this area further, but met with little success. By the late 1950s, Dutch scientists G. H. J. Broers and J. A. A. Ketelaar began building on this previous work. They determined that limitations on solid oxides at that time made short-term progress unlikely. Instead, they focused on electrolytes of fused (molten) carbonate salts. By 1960, they reported making a fuel cell that ran for six months using an electrolyte "mixture of lithium-, sodium- and/or potassium carbonate, impregnated in a porous sintered disk of magnesium oxide." However, they found that the molten electrolyte was slowly lost, partly through reactions with gasket materials. At approximately the same time, Francis T. Bacon was developing a molten cell using two-layer electrodes

on either side of a "free molten" electrolyte. At least two groups were working with semisolid or "paste" electrolytes and most MCFC research groups were investigating "diffusion" electrodes rather than solid electrodes. In the mid-1960s, the U.S. Army's Mobility Equipment Research and Development Center (MERDC) at Fort Belvoir tested several molten carbonate cells made by Texas Instruments. These cells ranged in size from 100 watts to 1,000 watts output and were designed to run on "combat gasoline" using an external reformer to extract hydrogen. In particular, the Army wanted to use fuels already available rather than a special fuel that might be difficult to supply to field units.

Molten Carbonate fuel cells (MCFCs) contain a liquid solution of lithium, sodium and/or potassium carbonates, soaked in a matrix for an electrolyte. They promise high fuel-to-electricity efficiencies, about 60% normally or 85% with cogeneration, and operate at approximately 1,200 ℉ (650 ℃). This high operating temperature is necessary to achieve sufficient electrolyte conductivity. Because of this high temperature, noble metal catalysts are not required for the fuel cell's electrochemical oxidation and reduction processes. To date, MCFCs have been operated with hydrogen, carbon monoxide, natural gas, propane, landfill gas, marine diesel, and simulated coal gasification products. MCFCs from 10 kW to 2 MW have been tested on a variety of fuels, and are primarily targeted toward electric utility applications. Carbonate fuel cells for stationary applications have been successfully

demonstrated in Japan and Italy. Their high operating temperatures create a big advantage because this allows higher efficiency and the flexibility to use more types of fuels and inexpensive catalysts. This is because reactions involving the breaking of carbon bonds in larger hydrocarbon fuels occur much faster at higher temperatures. A disadvantage to these phenomena, however, is that high temperatures enhance the corrosion and breakdown of cell components. The higher working temperature of these fuel cells, between 650~1000℃, and the heat transfer caused by conduction and convection, generates radiation heat transfer. The mechanism in this case is electromagnetic radiation propagated because of temperature difference, called thermal radiation.[1]

Davtyan was the first to realize the necessity of “support” for the electrolyte, i.e. a matrix which holds the electrolyte in place and prevents direct combination of reacting gases. In 1964, Broers reported on LiAlO , which was chemically stable and ₂ gave much performance. Broers was also the first to introduce porous nickel as the anode material. Clauss and Genin showed that porous nickel oxide, oxidized in situ,

provides stable performance for the cathode. An MCFC uses a salt mixture of alkali carbonates as the electrolyte. This mixture provides mass and charge transfer from the cathode to the anode via carbonate ions. The electrolyte in modern applications is a mixture of lithium carbonate and potassium carbonate. Mixtures of lithium carbonate and sodium carbonate and carbonates of alkaline-earth metals are also in use. The typical operating temperature of a MCFC is about 650°C. At that high operating

temperature, the carbonate mixture is in a molten state and becomes a good ionic conductor. The molten electrolyte is contained in a porous electrolyte matrix of

2

LiAlO , which is an electrically insulating and chemically inert ceramic. Thus, royal metals are not required to act as catalysts, reducing the material cost of the MCFC. A molten carbonate fuel cell has many features as follows.

1. The electrolyte material is a eutectic mixture of lithium carbonate and potassium carbonate. It is in liquid phase at temperatures higher than 500° C.

2. An MCFC exhibits an internal reforming ability because of its high operating temperature. Therefore, it does not require pure hydrogen as fuel, but can use hydrocarbons such as natural gas and coal gas etc., Moreover, a MCFC produces 40% lower carbon dioxide emission than a thermal power plant.

3. The waste heat of reacting gases emitted by the MCFC can be utilized to generate electric power through gas turbines.

4. An MCFC can be used as a device for separating and concentrating carbon dioxide because the anode gas has the ability to concentrate carbon dioxide[2]

(1). Cell Sealing

In addition to the cathode, anode, and electrolyte, each cell structure also contains an electrolyte matrix that holds the liquid electrolyte in place. This matrix structure is composed of a mixture of ceramic powder (usually lithium aluminates, LiAlO ) and carbonate electrolyte. The mixture is ₂

semisolid (paste-like) and the molten carbonate electrolyte is immobilized by the capillary force. The resulting matrix structure is stiff and impermeable to the reactant gases, but also deformable. The plasticity of the matrix provides a gastight seal around the periphery of the cell. Gas sealing is a major challenge in high-temperature fuel cells. This edge sealing technique is often called a wet seal. The wet seal concept is very similar to the sealing technique used in PEM fuel cells in that both techniques use the electrolyte itself as the sealing material to provide gas-tight sealing. This works because the electrolyte itself is gas impermeable, and is compatible with the rest of the cell components. In the molten carbonate fuel cell, however, wet sealing the cell is the only feasible sealing technique when the cell housing is made of metals. This is because the carbonate electrolyte is very corrosive and very few materials can remain stable under MCFC operating conditions. Although high-density alumina and other dense ceramics are suitable sealing materials, they cannot withstand thermal cycling.

(2). Current Collectors

Current collectors enhance the rate of electric current collection and reduce ohmic losses. They are usually made of stainless steel or nickel metal screens and are located between the electrodes and the cell housing for good electrical contact between both components. The cell housing is made of metal shells with flow distribution channels built on its inside surface for proper distribution of the gas supply to the respective electrode.

(3). Electrolyte Management

Another unique feature of the molten carbonate fuel cell structure is its unique method of electrolyte management. PAFC and PEMFC electrolyte management uses hydrophobic materials such as PTFE. The dispersed PTFE in the porous electrodes acts as a binder for the integrity of the electrode structure and as a wet-proofing agent for the establishment of a stable gas-liquid interface. However, this method cannot be used for MCFCs because similar de-wetting materials do not exist in molten carbonate under oxidizing conditions. Hence, capillary equilibrium is used to control electrolyte distribution in the porous electrodes, and stable electrolyte/gas interfaces in MCFC porous electrodes (the so-called three-phase zone).

available, and has the potential to replace conventional thermal power plants. The principal reason for this is that MCFC power plants have a higher energy conversion efficiency and are able to use both LNG and coal gas as fuel. Furthermore, an MCFC power plant can be applied in the electric power industry as a dispersed power source or a central power source fueled by LNG or coal. Many manufacturers and organizations have developed conceptual designs of MCFC power plants. The efficiency in most of these designs is 45%~70% (LHV, low heat value). MCFC power plants therefore carry great promise as primary power plants in the future, especially for a decentralized power supply.

Research and development on MCFCs is conducted primarily in the USA, Japan, and Europe. The USA led MCFC technology initially, but Japan and several European countries, which started their own R&D programs in the 1980s, have greatly increased their activities. The goal of the development programs in all of these countries is to develop and commercialize simple, low-cost power plants that can compete favorably with conventional thermal power plants. Many R&D programs have now reached the commercial stage, where prototype stacks and plants are being constructed and tested.

The principle of an MCFC is that, at high operating temperatures, carbonate ions migrates in a molten electrolyte. This carbonate ion produced from carbon dioxide and oxygen in the cathode passes through the electrolyte, and reacts with the hydrogen in anode. At the same time, this reaction in the anode produces carbon

dioxide, vapor, and electrons. The electrons are conducted to the external load circuit through the anode electrode, and back to the cathode through the cathode electrode. Figure 1.1 shows the principle of electric power generation in an MCFC, and Fig. 1.2 shows the basic components of a fuel cell. The key materials in an MCFC are anode electrode, cathode electrode, electrolyte, and bipolar plate. The chemical reaction equations in the anode and cathode of an MCFC are as follows.

In the cathode, 2 2 2 3 1 2 2O CO e CO − − + + → (1.3) In the anode, 2 2 3 2 2 2 H +CO− →CO +H O+ e− (1.4)

The total reaction is

2 2 2 2 2

1

2O +H +CO →CO +H O (1.5)

In equation (1.1) and (1.2), the carbon dioxide is the product and reactant in both the anode and the cathode. The overall reaction in MCFCs is similar to other fuel cells, but CO₂ is produced at the anode and consumed at the cathode. This implies that a CO₂ recycling system is needed to supply CO₂ from the anode chamber to the cathode chamber in a power plant. When carbon dioxide produced in the anode is transferred to the cathode as the reactant, it creates a closed cycle and reduces overall carbon dioxide emissions.

Because of their many advantages such as low pollution, low noise, high efficiency, wide application, etc., MCFCs can assist or even replace thermal electric generators in the future. Therefore, this study investigates the thermal and electrical performance of a MCFC based on its potential development in electric power generation [4, 5].

1.4.

8B

Literature Survey

The technology of the molten carbonate fuel cell (MCFC) has received much attention in the last two decades, and is now at the stage of being scaled-up for commercialization. Since the MCFC operates at a high temperature of around 650℃, the prediction of the temperature distribution is important to avoid hot spots in cells. Hot spots of extra high temperature cause electrolytic loss by corrosion and reduce the lifetime of the fuel cells. Moreover, the variation of the temperature influences the local current density, and changes the electrical performance of the MCFC. Therefore, many researchers have investigated the thermal and electrical performance of molten carbonated fuel cells.

In the analysis of an MCFC, the major researches focus on the temperature and current density field in a unit, a stack, or transient state. For a MCFC unit analysis, Wilemski and Wolf [6] used a numerical method to analyze a two-dimensional heat and mass transfer in a large MCFC unit with considering different cell operating conditions and design parameters. Kobayashi et al. [7] used a numerical method to

solve the steady-state temperature distribution of fuel cells with reaction areas of

900cm and 3600 2 cm . They compared experimental data with numerical results, 2

and reasonable agreement between them was reached. Lee et al. [8] calculated the temperature distribution, hydrogen conversion, and current density distribution of a unit molten carbonate fuel cell using constant voltage and constant current density methods. The results indicated that cell performance calculated by the constant voltage method fits better the experimental data than that calculated by the constant current density method.

The analysis on the thermal and electrical performance of a MCFC stack recently grows up, because a MCFC stack had applied in industry. Yoshiba et al. [9] developed a three-dimensional numerical model to analysis the cell voltage, temperature, and current profile in molten carbonate fuel cell stacks. They compared the effects of flow patterns such as co-flow, counter-flow, and cross-flow, and found that the net output power was highest in co-flow geometry. Later, Yoshiba et al. [10] investigated the temperature and performance of molten carbonate fuel cell stacks with co-flow configuration by applying a numerical model. Their results indicated that the increase in the partial internal resistance and an insufficiency of supplied fuel gas to the cell could induce differences in cell voltage. He and Chen [11] investigated the three-dimensional temperature distribution, the pressure, the gas concentration, and the current density of a molten carbonate fuel cell of five stacks with three manifolds, using CFD software. The results showed that the maximum temperature

locates at different positions under co-flow, counter-flow, and cross-flow configurations. The maximum temperature difference among the flow configurations is 10-20 ℃ . Recently, Ma et al. [12] developed a practical computational model for a MCFC stack. This model included three dimensional fluid flow, heat and mass transfer, gas-phase and surface chemistry, electrochemistry and structural mechanics, and this model was validated by comparing experimental data. Moreover, the materials and design of an MCFC stack are reviewed by Mugikura et al [13].

In the transient analysis of a MCFC, Lukas et al [14] developed a nonlinear mathematical model of an internal reforming MCFC stack for control system applications. This model can be used to provide realistic evaluations of the responses to varying load demands on the fuel cell stack and to define transient limitations and control requirements. Koh et al. [15] used a software package to predict the dynamic pressure and temperature distribution of gas in a co-flow molten carbonate fuel cell stack based on an assumption of uniform current density. The results indicated that the predicted axial velocity profile precisely reflects the mass change in MCFC, by showing a drop in the volumetric flow in the cathode and an increase in the anode. Later, Koh et al. [16] used computational fluid dynamics code to predict the temperature distribution of a co-flow MCFC stack considering the effects of radiation and variable gas properties. The results showed that the thermal radiation only weakly affects the calculation of the temperature field using the model,

and most of the gas properties can be treated constant, except for the specific heat capacity of the anode gas. He and Chen [17] extended their simulation to investigate the transient behavior of an MCFC stack with the cross-flow configuration; the results showed that the current density profile changes rapidly in the beginning and slowly in the following stage, and the temperature response is slow when the MCFC was under a step voltage change. Xu et al [18] developed a voltage drop and recovery analysis method to estimate the different contributions to the transient behavior of a MCFC. Their results showed that the model predictions were in reasonable agreement with the experiment data, and it is an efficient tool to analyze the transient characteristics of a MCFC. Heidebrecht and Sundmacher [19] used a general notation in dimensionless form to analyze the transient state of a single counter-flow MCFC with considering the concentration, temperature, and potential field of the gas and the solid phases. This general notation of calculation can easily be extended to describe cross-flow 2D unit and 3D stacks. Lee et al. [20] used a numerical method to analyze the beginning of the operation of a MCFC unit, and investigated the effects of the molar flow rates of gases and the utilization of fuel gas. Their results showed that the time required to approach a steady-state decreases with an increase in the inlet gas-flow rates or the hydrogen utilization.

The electrodes and electrolyte phenomena are important to the overall performance in a MCFC unit or stack, so there are many literatures investigating the analysis on the anode, cathode, and electrolyte. Vallet and Braunstein [21] modified

steady-state equations for composition gradients in battery analogs with binary mixtures of molten salts as electrolytes to apply to a MCFC, and used a numerical method to solve the diffusion-migration equation to predict the development with time of the concentration gradient. Wilemski [22] used individual porous electrode models for calculating the local cell overpotential and current density in a MCFC, and their results had compared with experiment data. Kunz et al [23] developed a cathode model of a MCFC, which was a function of cathode electrolyte content including the effective agglomerate diameter, porosity, tortuosity, and number based on knowledge of the electrode’s pore spectrum. Lee et al. [24] presented the experimental characterization of a MCFC unit with transient response analysis methods such as electrochemical impedance spectroscopy and current interrupt method. They found that the cathode over-potential was controlled by mixed diffusion of oxygen and carbon dioxide. Prins-Jansen et al [25, 26] considered the cathode was constructed by an easiest-to-handle shape of semi-infinite slabs, and used the agglomerate model for porous electrodes in MCFC. Using analytical mathematical tools, this model can give the optimal electrode thickness and agglomerate size based on general problem properties and analytic solutions for special cases. Fehribach et al [27] derived an electrochemical-potential model for the peroxide mechanism describing the electrochemistry of a MCFC cathode. This model made clear the connection to the underlying reaction stoichiometry, and requiring the fewest equations consistent with that stoichiometry. Their results

showed that the mean current density associated with a small portion of electrode may be increased by as much as a factor of five, and on this scale the current density is most sensitive to the electrolyte diffusivity. Bergman et al [4] investigated two cathode materials to elucidate the impact of the cathode material on the formed corrosion layer by polarization measurements and electrochemical impedance spectroscopy. The results indicated that the contact resistance between the cathode and the current collector contributed with a large value to the total cathode polarization. Morita et al [28] estimated the potential of Li/Na carbonate as the MCFC electrolyte by investigating the dependence of the cell performance on the operating conditions and the behavior during long-term performance in several bench-scale cell operations. Arato et al. [29] investigated the limitation on the performance of molten carbonate fuel cells due to gas diffusion phenomena in the porous electrodes when high reactant utilization factors were used. The expression of voltage decay depends on concentration polarization due to hydrogen and carbon dioxide, while oxygen diffusion effects have been considered to be negligible. Furthermore, the limiting diffusion conditions must also be correctly evaluated for the local temperature and pressure drops. For over-potential from the anode gas to the cathode gas, Bosio et al. [30] presented a model and experimental investigation of electrochemical reactors in the molten carbonate fuel cell. Additionally, they used their formula for total cell resistance, tested with experimental data, to analyze the temperature distribution and current density distribution for a single cell and stacks,

using FORTRAN program. Their numerical results agree with the experimental results, and showed that the thermodynamics fails to predict the open circuit voltage because of the effects of gas crossover phenomena at the cell level. Although thermodynamic equilibrium should be established under open circuit conditions in principle, short circuit electrical currents circulate within the cell, and the consequent voltage loss is responsible for irreversibility.

For a power plant, there are also many researches analyze its overall efficient by using a simple or rapid calculation for a MCFC. Mangold and Sheng [31]applied a reduced nonlinear model to solve a planar molten carbonate fuel cell with cross-flow. Since the reduced model was of the lower order than the original model, it markedly reduced the computational time. Therefore, this model was suitable for application in predicting the behavior of a control system in a power plant. He [32]presented a simulation model for investigating the dynamic performance of MCFC power-generation systems. This simulation model consists of nine types of component models, which are fuel cell, external reformer, steam generator, water separator, rotation equipment, heat exchanger, DC/AC invertor, pipeline and control valve. Later, He [33] extended his analysis to a MCFC power generation system including twelve types of component models. De Simon et al [34] simulated a global MCFC power plant in steady state. This simulation can conduct a sensitivity analysis with the preliminary input specification, and find the process parameters whose change improves the global efficiency. Yoshiba et al [35] calculated the

materials and heat balance of integrated coal gasification and MCFC combined system with considering the electricity generating performance of the practical cell. The results showed that the net thermal efficiency of the anode gas recycling system has a peak for carbon dioxide partial pressure where the net thermal efficiency of the anode heat exchange system increases as the carbon dioxide partial pressure of the cathode gas decreases. Recently, Baranak et al [36] developed a MCFC model for a unit analysis with considering several performance model equations separately for anode and cathode, and then they applied this model into a process simulation software to simulate a power system.

In a MCFC, there are simultaneous reactions in anode side, which are chemical reaction in anode, reforming reaction, and water-gas shift reaction. Most MCFC use internal reformer because of its simplicity in structure. Park et al [37] investigated the effects of the reformer in an internal-reforming MCFC on the temperature distributions, conversion of methane, and compositions of gases by a numerical method. Their results indicated that the methane-reforming reaction and the water-gas shift reaction occur simultaneously and the conversion of methane to hydrogen reached 99%, and the endothermic-reforming reaction contributes to a uniform temperature distribution. Seo et al [38] analyzed the performance and operation results of an external-reformer that supplied synthesis gases to a 100kW class MCFC. In order to maintain the outlet temperature of the reforming reactor over 580°C, it is necessary to heat the reformed gases at the convection zone of

combustion gases. Kim et al [39] discussed the effects of water-gas shift reaction on the temperature distribution, voltage distribution, conversion, and performance in a MCFC unit. Their results indicated that the voltage calculated without the shift reaction would be higher than the real value, and the effect of the shift reaction on the voltage distribution and cell performance is quite small.

Bosio et al [40] reported the development of molten carbonate fuel cell technology at Ansaldo Ricerche, from small-scale single cell up to stacks of several KW capacities, for industrial applications. Although the report showed that MCFC technology had been successfully tested on stacks in the kW power class, the control of the start-up phase, electrolyte migration through the manifolds and gas feed distribution have not yet been to be solved. Notably, the gas feed distribution in [40] identified the variation of mole flow rate in different stacks. Hence, the stack nearest the anode gas inlet duct has largest flow rate and the farthest one has the lowest. The cross-sectional geometry of a fuel cell is similar to that of a heat exchanger, whose inlet distributor is responsible for a non-uniform flow distribution in the frontal area. Therefore, the maldistribution of the inlet flow rate on the frontal area is realistic and it must affect the performance of fuel cells. In the research of a heat exchanger, Chiou [41] first investigated the thermal performance deterioration in a cross-flow heat exchanger due to the flow non-uniformity. Later, Yuan [42, 43] analyzed the thermal performance and exergy of a three-fluid cross-flow heat exchanger with considering a non-uniform inlet flow. The results showed that most non-uniform

will drop the performance of a heat exchanger, but some of the non-uniform profiles in a three-fluid cross-flow heat exchanger may promote the performance.

Hirata and Hori [44] adopted a numerical method to examine the relationships among the gas flow uniformities in the planar direction, the gas flow uniformity in the stacking direction, and the cell performance in a co-flow MCFC. Their results showed that the gas flow uniformity in the stacking direction is about two to ten times that in the planar direction. Later, Hirata et al [45] investigated the relationship between the gas channel height, the gas flow characteristics, and the gas diffusion characteristics in a plate heat-exchanger type MCFC stack. They used numerical method to evaluate the effect of the gas channel height on the uniformity and pressure loss of the gas flow. Recently, Okada et al [46] presented an investigation of the gas distribution in a large-scale stack with internal reforming MCFC stack. They proposed a large-scale stack divided into four blocks from the point of view of the gas flow scheme in order to achieve more uniform supply gas to each cell. The results showed that the flow variation among the four blocks is less than 1.5%, and it can improve the prospects for a MCFC stack.

1.5.

9B

Objectives of Present Studies

The above literature review shows that there is still much necessity to research the complex temperature and current density distributions in the MCFCs due to the mal-distributed inlet flow rate. Moreover, the relationships between the mole

fraction of each species, concentrations, over-potential, temperature, and current density can be more clearly by our investigation when the MCFC is under cross-flow configuration. In this study, the model of an MCFC with inlet flow mal-distribution is developed and numerically solved to enhance the understanding of the underlying characteristics of non-uniform effects, as well as various factors that dominate the temperature and current density distribution of the MCFCs.

The scopes of this investigation mainly focus on three parts. First, we study the effect of non-uniform inlet flow in the transverse direction on the temperature and current density of an MCFC unit. In this study, the distribution of inlet flow rate in the transverse direction includes the uniform, increasing, and decreasing profiles, and these profiles in anode and cathode assemble to eight patterns. Furthermore, this study examines the temperature and current density distributions in the eight patterns, and compares them to the results in uniform inlet flow. The second part investigates the effects of inlet flow mal-distribution in the stack direction on the thermal and electrical performance of a ten-stack MCFC. This study uses the numerical procedure for a MCFC unit to calculate the quasi-three dimensional temperature and current density fields in the MCFC stack by averaging the temperatures of top and down separators, which connect together between stacks. Scenarios of non-uniform inlet flow of either anode side or cathode side have been calculated and discussed in this study. The electric performance of a planar MCFC unit with cross-flow configuration when there is higher gas utilization in anode and cathode investigate in

the final part of this dissertation. A phenomenon of the existence of non-reaction zone arising from the non-uniform inlet pattern is demonstrated and its influence on cell performance is explored.

Table 1.1 The types of fuel cells Name Electrolyte Operating Temperature (℃) Charge carrier Catalyst Prime cell components Alkaline

Fuel Cell Liquid KOH 65-220

−

OH Pt Carbon based Phosphoric Acid

Fuel Cell Liquid H PO3 4 200 H+ Pt

Carbon based Polymer Electrolyte Fuel Cell Ion Exchange membranes 80 + H Pt Graphite based Molten Carbonate

Fuel Cell CO3melt 650

2 3 −

CO Ni Stainless steel based Solid Oxide Fuel

Cell Ceramic 600-1000

2−

Fig 1.2. Component diagram of a unit cell

Separator (bipolar) plate

Separator (bipolar) plate

Anode Current Collector

(stainless steel)

Cathode Current Collector

(stainless steel)

Anode

(porous Ni+2-10% Cr or Al)

Cathode

(porous lithiated NiO)

2.

1B

Effect of Inlet Flow Maldistribution on the Thermal

and Electrical Performance of an MCFC Unit

2.1.

10B

Physical Model Description

This study investigates a unit of a molten carbonate fuel cell with an area of 0.6m ×0.6m when the anode gas and the cathode gas flow in the cross-flow configuration, in which the anode gas flows in the x direction, and the cathode gas flows in the y direction, as shown in Figure 2.1. The ribs of separator cause the anode gas and the cathode gas to flow without mixing, therefore each flow is considered to composed of many flow tubes that are parallel to each other. As mentioned in Chapter 1, the position of the manifold may induce different mal-distributions of the flow rate in the inlet ports. This study considers eight patterns of inlet flow scenarios from three kinds of profiles: uniform, progressively decreasing, and progressively increasing, as shown in Figure 2.2. The term d represents the unilateral deviation of the non-uniform profile, which is the ratio of the variation of flow rate in one side to the mean flow rate.

2.2.

11B

Basic Assumption

The formulations of the governing equations are based on the following assumptions.

6. The gas flow in the anode channel and the cathode channel are treated as a plug-flow.

7. The inlet temperature and molar fractions of species in the anode gas and the cathode gas are constant and uniform.

8. The thermal properties of the anode gas, the cathode gas, the cell and the separator are constant, except for the specific heat capacities of the anode gas and the cathode gas.

9. The boundaries of the cell and separator are adiabatic. 10. The properties variations in the z direction are negligible.

11. The cross-sectional geometry of separator is unchanged throughout the x-y plane.

12. The water-shift reaction in the anode gas is negligible. 13. The cell voltage is uniform over the cell plane.

2.3.

12B

Governing Equations

2.3.1.

28B

Reaction Equations

This study considers a molten carbonate fuel cell unit with external reforming and the reforming reaction in the anode gas is neglected. The cathode gas is the air obtained from the atmosphere, and the electricity is generated in the cell through the

electrochemical reaction in the anode and the cathode. Meanwhile, the electrolyte is a porous matrix that contains migrating ions of molten carbonate. The reactions in the anode and the cathode are as follows.

2 2 3 2 2 2 H +CO − →H O CO+ + e− (2.1) − −_→ + + 2 3 2 2 2CO 4e 2CO O (2.2)

2.3.2.

29B

Mass Conservation Equations

The relationship between the current density and gas molar flux for species at the electrodes surface is described from Faraday’s law. Therefore, the mass balances of the anode and the cathode gases are as follows.

, 1 ag k y e dn _i L dx = ±n F (2.3) F n i dy dn L _e k cg x ± = , 1 (2.4)

where n is the mole flow rate of the k-component, and n is the number of e

electrons transferred in the reactions of the anode and the cathode. The plus/minus symbol represents an increase/decrease in each species’ mole flow rate caused by the electrochemical reaction in the anode and the cathode. It is positive for reactants, and negative for products in the anode and the cathode.

2.3.3.

30B

Energy Conservation Equations

and the separator. For the gas in the anode flow channel,

(

nagkcpkTag

)

qconvs ag qconvc ag qmassc ag

dx d − − − + + =

∑

, ,  ,  ,  , (2.5) (0, ) 858 ag T y = K (2.6)

For the gas in the cathode flow channel,

(

ncgkcpkTcg

)

qconvs cg qconvc cg qmasscg c

dy d − − − + − =

∑

, ,  ,  ,  , (2.7) ( , 0) 867 cg T x = K (2.8)

For the cell,

( )

2 2

( )

2 2 ,

, , , 0

c c

cont conv c ag

c c

conv c cg mass cg c mass c ag reac

T T k k q q x y q q q q δ δ ₋ − − − ∂ ₊ ∂ _{− −} ∂ ∂ − + − + =       (2.9) 0 0 0 = ∂ ∂ = ∂ ∂ = ∂ ∂ = ∂ ∂ = = = = x y L_y c y c L x c x c y T y T x T x T _(2.10)

For the separator,

( )

( )

_{2 , ,} 0 2 2 2 = − − + ∂ ∂ + ∂ ∂ − −ag convs cg s conv cont s s s s q q q y T k x T kδ δ    (2.11) 0 0 0 = ∂ ∂ = ∂ ∂ = ∂ ∂ = ∂ ∂ = = = = x y L_y s y s L x s x s y T y T x T x T (2.12)

where the heat transfer rate terms are described in Table 2.1. In Eqs X(2.3)X to

X

(2.7)X, this study considers the molar flow rate of the anode gas and cathode gas are a

profile in the inlet. The inlet conditions are described as following.

( )

_⎟⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + = y d L d L N y n y y ag ag 1 2 , 0 (2.13)

( )

2 , 0 = ⎛_⎜ + −1 ⎞_⎟ ⎝ ⎠ cg cg x x N d n x x d L L (2.14)

where N and _ag N are the total molar flow rate of the anode gas and the _cg

cathode gas from inlet ducts, and the deviation of d may be negative, zero, and positive, representing the progressively decreasing profile, uniform profile, and progressively increasing profile, respectively. In Table 2.1, k_{c s−} is the thermal conductivity due to the contact resistance between the cell and the separator in the z direction. The thermal conductivity, k_{c s−} , is set to 1.0 W m K⋅ −1 −1 .

2.3.4.

31B

Nernst Voltage and Polarizations

The Nernst voltage is calculated using the Nernst equation, as follows.

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ag CO O H cg CO O H P P P P P F RT E E , , 5 . 0 0 2 2 2 2 2 ln 2 (2.15) T . . E₀ =12723−27654×10−4 (2.16)

Meanwhile, E is the reversible voltage under standard conditions. According ₀

to the results in, this study uses the total cell resistance, including that due to cathode polarization, the electrolyte tile contribution, and the Ohmic resistance of the contacts. Note that this total cell resistance did not include the concentration polarization,

Because we assumed the diffusion is non-limiting in the electrode. The cell voltage is the Nernst voltage minus the over-potentials, as follows.

= − tot V E iR (2.17) T F ir i i T B tot c D e p Ae R i / / ⋅ + + =

∏

β (2.18)

where the parameters are βO2 =_{0.67, A}=_{1.38 10}× −7Ω_{m Pa}2 0.67_{, B=11400K,}

4 2

0.348 10

ir

c = × − Ωm , D=4.8 10× −8Ωm2 and F=6596K .

The above simultaneous equations of the MCFC contain seven unknown variables, which are mole flow rate of each species (n_ag,k and n_cg,k), anode gas temperature (T_ag), cathode gas temperature (T_cg), cell temperature (T_c), separator temperature (T_s), current density (i), and cell voltage (V). The mass equations are used to determine mole flow rate of each species, and energy equations are used to determine the temperatures. Nevertheless, in Eq. X(2.17)X, both current density and

cell voltage variables must be evaluated. Therefore, this study assumes that the cell voltage is uniform over the reaction area of the cell and then calculates the current density using Eq. X(2.17)X.

2.4.

13B

Method of Solution

This study divides the calculation domain of the x-y plane into N×N

ag

T at the inlet and outlet of each subdivision in the x direction of the anode gas flowing. Similarly, the calculating nodes of n_cg,k and T are assigned to the inlet _cg

and outlet of each subdivision in the y direction of the cathode gas flowing. Furthermore, the calculation nodes of i, Tc, and Ts are assigned to the center of the

subdivision. Grid generation and the implicit scheme are adopted to discretize Eqs.

X

(2.3)X to X(2.11)X to finite difference equations, and then employs TDMA (Tri-Diagonal

Matrix Algorithm) to solve simultaneous algebraic equations. This node arrangement avoids the need to apply the upwind method to treat the first-order differential terms, and has been used to calculate the temperature fields of a three-gas cross-flow heat exchanger. [42]

The Eq. X(2.3)X and (2.4) can be discretized to the following based on different

species in anode and cathode side, respectively. For the gas in the anode flow channel,

F x i n n_{H H} (i,j) ) j , i ( ) j , i ( 1 2 ₂ 2 Δ ⋅ − = + , (2.19) F x i n n_{HO H O} (i,j) ) j , i ( ) j , i ( 1 2 ₂ 2 Δ ⋅ + = + , (2.20) F x i n n_{CO CO} (i,j) ) j , i ( ) j , i ( 1 2 ₂ 2 Δ ⋅ + = + (2.21)

For the gas in the cathode flow channel,

F y i n n_{CO CO} (i,j) ) j , i ( ) j , i ( 1 2 ₂ 2 Δ ⋅ − = + (2.22) F y i n n_{O O} (i,j) ) j , i ( ) j , i ( 1 2 ₄ 2 Δ ⋅ − = + (2.23)

The Eqn. X(2.5)X to X(2.11)X for energy conservation equations of the gas in the anode

flow channel, cathode flow channel, cell, and separator can be discretized to the following finite difference equations

For the gas in the anode flow channel,

(

)

(

)

(

)

( 1, ) ( , ) ( , ) ( , ) ( , ) 2 2 2 ( 1, ) ( , ) ( , ) ( , ) ( , ) ( , ) 2 2 2 2 2 + + ⎛_{⎛ ⋅ ⎞} ⎞ ⎜_{⎜ ⎟ + +} ⎟ ₌ ⎜_{⎜ Δ ⎟} ⎟ ⎜_{⎝ ⎠} ⎟ ⎝ ⎠ ⎛_{⎛ ⋅ ⎞} ⎞ ⎜_{⎜ ⎟ − −} ⎟ ⎜_{⎜ Δ ⎟} ⎟ ⎜_{⎝ ⎠} ⎟ ⎝ ⎠ + − + + ⋅ + +

∑

∑

i j i j H CO H O i j i j i j p _{f f cf f sf} f i j p _{f f cf f sf} f i j i j p i j p i j p i j c f cf c f sf s n c _{h a h a} T x n c _{h a h a} T x i c c c T h a T h a T F (2.24)

For the gas in the cathode flow channel,

(

)

(

)

₍

₎

( , 1) ( , ) 2 2 ( , ) ( , ) ( , ) ( , 1) ( , ) ( , ) ( , ) ( , ) 2 2 2 2 2 4 + + ⎛_{⎛ ⋅ ⎞} ⎞ ⎜_{⎜ ⎟ + +} ⎟ ₌ ⎜_{⎜ Δ ⎟} ⎟ ⎜_{⎝ ⎠} ⎟ ⎝ ⎠ ⎛_{⎛ ⋅ ⎞} ⎞ ⎜_{⎜ ⎟ − −} ⎟ _{− + ⋅} ⎜⎜ Δ ⎟ ⎟ ⎜_{⎝ ⎠} ⎟ ⎝ ⎠ + +

∑

∑

i j i j O CO i j i j i j p _{o o co o so} o i j p _{o o co o so i j} o p i j p i j c i j o co c o so s n c _{h a h a} T y n c h a h a i T c c T y F h a T h a T (2.25) For cell,

( )

( )

( )

( )

( )

( )

₍

₎

(

)

( , ) ( , 1) ( , 1) ( 1, ) ( 1, ) ( 1, ) ( . ) ( , 1) ( . ) 2 2 2 2 2 2 2 2 2 2 δ δ δ δ δ δ + − + − + + ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ + + + = + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ _{Δ Δ} ⎟ ⎜ _Δ ⎟ ⎜ _Δ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ +⎜_{⎜ Δ} ⎟_⎟ +⎜_{⎜ Δ} ⎟_⎟ +⎜ + ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ +_⎜ + _⎟− ⎝ ⎠ i j i j i j i j i j i j i j i j i j c c c c f cf o co c c c f cf c c c c f f o co o o k k k k h a h a T T T x y y y k k h a T T T T x x h a T T q (2.26)

For separator,

( )

( )

( )

( )

( )

( )

(

)

(

)

( , ) ( , 1) ( , 1) ( 1, ) ( 1, ) ( 1, ) ( . ) ( , 1) ( . ) 2 2 2 2 2 2 2 2 2 2 δ δ δ δ δ δ + − + − + + ⎛ ⎞ ⎛ ⎞ + + + = ⎜ ⎟ ⎜ ⎟ ⎜ _{Δ Δ} ⎟ ⎜ _Δ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ +⎜_⎜ ⎟_⎟ +⎜_⎜ ⎟_⎟ +⎜_⎜ ⎟_⎟ Δ Δ Δ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ +_⎜ + _{⎟ ⎜}+ + _⎟ ⎝ ⎠ ⎝ ⎠ i j i j i j i j i j i j i j i j i j s s s f sf o so s s s s s s s s f sf o so f f o o k k k h a h a T T x y y k k k T T T y x x h a h a T T T T (2.27)

The calculation proceeds as follows

14. The program guesses a uniform current density distribution and solves the mole flow rate of each species in the anode flow channel and the cathode flow channel using Eqs. X(2.19)X to X(2.23)X

15. The program solves the temperature fields of the gas in the anode channel, the gas in the cathode channel, the cell, and the separator using Eqs. X(2.24)X

to X(2.27)X, respectively.

16. The Nernst voltage and internal total resistance are calculated using Eqs.

X

(2.15)X and X(2.18)X, and then the current density is obtained from the Eq.

X

(2.17)Xby setting the cell voltage to a constant value.

17. The current density is updated to Step 1, and the loop iterated from Step 1 to Step 4 until all relative errors of the mole flow rates, the temperature and the current density satisfy the converge criterion.

cell, and separator at the central point of calculating domain, respectively. In these figures, the number of grid points increases from 100 100× to 1000 1000× for FORTRAN program calculations, and the variations of anode gas temperature, cathode gas temperature, cell temperature, and separator temperature in the central position are stable between 100 100× and 400 400× . However, the variation of temperature becomes unstable when the number of grid points are higher than

500 500× . It downs first and then rise again when the number of grid points are higher than 700 700× . The variations of current density and temperature are similar, as shown in Figure 2-8. Therefore, this study selects the grid number of 400 400×

as the dimension size in the FORTRAN program for calculating the results.

FlexPDE software is adopted to solve the Eqs. X(2.3)X to X(2.18)X, to verify the

accuracy of calculation, because FlexPDE is a flexible solver of partial differential equations by using the finite element algorithm.

Figure 2.4 depicts the temperature distribution of the cell calculated both numerically method and using FlexPDE software at a cell voltage of 0.8V. In this numerical method, the grid dimension depends on the mole flow rates of the anode gas and the cathode gas, and must increase as the mole flow rate falls to avoid the negative values in the finite difference equations [47]. In this comparison case, the numerical program sets the grid dimensions to be 400 400× with a convergence criterion of 1 10× −5, and FlexPDE uses 1972 elements with a convergence criteria of