STEADY STATE MODELING AND ANALYSIS OF A REVERSIBLE …

STEADY STATE MODELING AND ANALYSIS OF A REVERSIBLE SOLID OXIDE FUEL CELL BASED

ENERGY STORAGE SYSTEM

Kevin Scott

MS Thesis Defense

WSU School of Mechanical and Materials Engineering

July 2018

Motivation: Variable Renewable Energy

• Non-dispatchable Power Output

Effect of Variable Renewable Energy on Electrical Grid

Outline

• Literature Review

• Background

• System Model

• Energy Storage Application

• Conclusions

Literature Review• Many energy storage technologies are being

investigated [1,2,3,4].

• Low temperature reversible fuel cells typically have low roundtrip efficiencies of 30-40% [5,6,7].

• Hydrogen-based reSOFC typically have roundtrip efficiencies around 50-60% [8,9,10].

• Internal methanation and steam reforming can increase roundtrip efficiencies but may increase thermal gradient [11].

• Rate of internal methanation and steam reforming highly dependent on operating conditions [12,13], but can result in efficiency improvements without increasing thermal gradient [14].

Outline

• Literature Review

• Background

• System Model

• Energy Storage Application

• Conclusions

Reversible Solid Oxide Fuel Cells

12𝑂𝑂2 + 𝐻𝐻2 ↔ 𝐻𝐻2𝑂𝑂, ∆𝐻𝐻𝐻𝐻2 = −247𝐾𝐾 ⁄𝐽𝐽 𝑚𝑚𝑜𝑜𝑜𝑜

Background

reSOFC Performance

𝐸𝐸0 =∆𝐺𝐺𝑛𝑛𝑛𝑛

𝐸𝐸𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑡𝑡 = 𝐸𝐸0 +𝑅𝑅𝑢𝑢𝑇𝑇𝑛𝑛𝑛𝑛 � log 𝑃𝑃 0.5 �

∏𝑋𝑋𝑅𝑅𝑁𝑁𝑅𝑅𝑅𝑅𝑡𝑡𝑅𝑅𝑁𝑁𝑡𝑡𝑁𝑁∏𝑋𝑋𝑃𝑃𝑁𝑁𝑃𝑃𝑃𝑃𝑢𝑢𝑅𝑅𝑡𝑡𝑁𝑁

𝑉𝑉𝐹𝐹𝐹𝐹 = 𝐸𝐸𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑡𝑡 − 𝑉𝑉𝐿𝐿𝑃𝑃𝑁𝑁𝑁𝑁𝑉𝑉𝐸𝐸𝐹𝐹 = 𝐸𝐸𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑡𝑡 + 𝑉𝑉𝐿𝐿𝑃𝑃𝑁𝑁𝑁𝑁

𝑉𝑉𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑅𝑅 = 𝑗𝑗 � Ω𝐴𝐴𝐴𝐴𝑅𝑅 𝑉𝑉𝑃𝑃𝑂𝑂𝑑𝑑𝑑𝑑𝑢𝑢𝑁𝑁𝑂𝑂𝑃𝑃𝑁𝑁 =𝑅𝑅𝑢𝑢𝑇𝑇𝑛𝑛𝑛𝑛 log

∏𝑋𝑋𝑅𝑅𝑁𝑁𝑅𝑅𝑅𝑅𝑡𝑡𝑅𝑅𝑁𝑁𝑡𝑡𝑁𝑁,𝑏𝑏 ∏𝑋𝑋𝑃𝑃𝑁𝑁𝑃𝑃𝑃𝑃𝑢𝑢𝑅𝑅𝑡𝑡𝑁𝑁,𝑇𝑇𝑇𝑇𝑇𝑇

∏𝑋𝑋𝑅𝑅𝑁𝑁𝑅𝑅𝑅𝑅𝑡𝑡𝑅𝑅𝑁𝑁𝑡𝑡𝑁𝑁𝑇𝑇𝑇𝑇𝑇𝑇 ∏𝑋𝑋𝑃𝑃𝑁𝑁𝑃𝑃𝑃𝑃𝑢𝑢𝑅𝑅𝑡𝑡𝑁𝑁𝑏𝑏

𝐽𝐽 = 𝑅𝑅𝑁𝑁𝑁𝑁𝑃𝑃𝑃𝑃𝑟𝑟 � 2𝑛𝑛

Open Circuit Voltage

Losses

Internal Methanation and Steam Reforming

𝐶𝐶𝐻𝐻4 + 𝐻𝐻2𝑂𝑂 ⇋ 3𝐻𝐻2 + 𝐶𝐶𝑂𝑂, ∆𝐻𝐻𝐴𝐴𝑅𝑅 = 206 𝐾𝐾𝐽𝐽/𝑚𝑚𝑜𝑜𝑜𝑜𝐶𝐶𝑂𝑂 + 𝐻𝐻2𝑂𝑂 ⇋ 𝐻𝐻2 + 𝐶𝐶𝑂𝑂2, ∆𝐻𝐻𝑊𝑊𝑊𝑊𝐴𝐴 = −41 𝐾𝐾 ⁄𝐽𝐽 𝑚𝑚𝑜𝑜𝑜𝑜

Carbon Deposition Region

C H O11% 79% 10%

Selected Composition

CO2: 0.7%CO: 0.5%

CH4: 38%H2: 29%

H2O: 32%

Hydrogen Transport Membrane

• Coproduce Hydrogen Through Fuel Recovery of Fuel Cell Mode Exhaust Stream

• Equalize Partial Pressure Of Hydrogen Across Membrane

reSOFC System: Fuel Cell Mode

reSOFC System: Electrolysis Cell Mode

Outline

• Literature Review

• Background

• System Model

• Energy Storage Application

• Conclusions

Component Models

• System model created from steady state component level models

System Model

Compressor HTM

𝑛𝑛𝑜𝑜𝑜𝑜𝐹𝐹𝐻𝐻2 = 𝐴𝐴 �

𝑃𝑃𝑝𝑝𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻2

� 𝐼𝐼𝑛𝑛𝑜𝑜𝐼𝐼𝐼𝐼𝐻𝐻2 − 𝐼𝐼𝑛𝑛𝑜𝑜𝐼𝐼𝐼𝐼𝐻𝐻𝑃𝑃𝑡𝑡𝑅𝑅𝑇𝑇𝑃𝑃

𝑝𝑝𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻2− 1

𝑝𝑝𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻2 =𝛼𝛼𝐻𝐻2𝑁𝑁2

𝛼𝛼𝐻𝐻2𝑁𝑁2

+ 1𝑃𝑃𝐻𝐻𝐻𝐻𝐻𝐻𝑇𝑇2𝑁𝑁 = 𝑇𝑇1 �𝑃𝑃𝑃𝑃𝑢𝑢𝑡𝑡𝑃𝑃𝑂𝑂𝑁𝑁

Υ−1Υ

𝐻𝐻2𝑅𝑅 = 𝐻𝐻1 +𝐻𝐻2𝑁𝑁 − 𝐻𝐻1𝜂𝜂𝐹𝐹𝑃𝑃𝑂𝑂𝐶𝐶

𝑊𝑊𝐹𝐹𝑃𝑃𝑂𝑂𝐶𝐶 = 𝐻𝐻2𝑅𝑅 − 𝐻𝐻1

𝑇𝑇2𝑅𝑅 = 𝑇𝑇2𝑁𝑁 +𝐻𝐻2𝑅𝑅 − 𝐻𝐻2𝑁𝑁

𝐶𝐶𝐶𝐶𝑁𝑁

𝛼𝛼𝐻𝐻2𝑁𝑁2

5000

𝐴𝐴 0.5𝜂𝜂𝐹𝐹𝑃𝑃𝑂𝑂𝐶𝐶 80%

reSOFC Model

• Spatially Discretized Nodal Model• Symmetry• Chemical Composition• Voltage

𝑛𝑛𝑜𝑜𝑜𝑜𝐹𝐹𝐻𝐻2n+1 = 𝑛𝑛𝑜𝑜𝑜𝑜𝐹𝐹𝐻𝐻2n −𝐽𝐽𝑁𝑁2𝑛𝑛

𝑋𝑋 =𝑅𝑅𝐴𝐴𝑅𝑅𝑅𝑅𝐹𝐹𝐶𝐶𝑅𝑅𝑊𝑊𝑊𝑊𝐴𝐴

)𝑋𝑋𝑂𝑂+1 = 𝑋𝑋𝑂𝑂 − 𝐽𝐽−1𝑛𝑛(𝑋𝑋𝑂𝑂

Reformer Plate

reSOFC

Inlet Outlet

Nodal reSOFC Model (cont)

0 = 𝑄𝑄𝐸𝐸𝐸𝐸𝑢𝑢𝑂𝑂𝑇𝑇𝑂𝑂𝐸𝐸𝑁𝑁𝑂𝑂𝑢𝑢𝑂𝑂 + 𝑄𝑄𝑁𝑁𝑁𝑁𝐴𝐴𝑂𝑂𝐹𝐹𝐹𝐹 + 𝑄𝑄𝐻𝐻𝐻𝐻 + 𝑄𝑄𝐽𝐽𝑃𝑃𝑢𝑢𝑇𝑇𝑁𝑁

𝑄𝑄𝑁𝑁𝑁𝑁𝐴𝐴𝑂𝑂𝐹𝐹𝐹𝐹 = (∆𝐻𝐻𝐻𝐻22𝐹𝐹

+ 𝑉𝑉) � 𝐽𝐽

𝑄𝑄𝐸𝐸𝐸𝐸𝑢𝑢𝑂𝑂𝑇𝑇𝑂𝑂𝐸𝐸𝑁𝑁𝑂𝑂𝑢𝑢𝑂𝑂 = ∆𝐻𝐻𝐴𝐴𝑅𝑅 � 𝑅𝑅𝐴𝐴𝑅𝑅 + ∆𝐻𝐻𝐹𝐹𝐶𝐶 � 𝑅𝑅𝐹𝐹𝐶𝐶 + ∆𝐻𝐻𝑊𝑊𝑊𝑊𝐴𝐴 � 𝑅𝑅𝑊𝑊𝑊𝑊𝐴𝐴

Thermal Balance

Cell Efficiency

𝜂𝜂𝐹𝐹𝐹𝐹 =𝑉𝑉 � 𝐽𝐽

∑(ℎ𝑂𝑂𝐼𝐼𝑛𝑛𝑜𝑜𝐼𝐼𝐼𝐼𝑂𝑂 − ℎ𝑂𝑂𝑂𝑂𝑂𝑂𝐼𝐼𝑜𝑜𝐼𝐼𝐼𝐼𝑂𝑂) + 𝑄𝑄𝐽𝐽𝑃𝑃𝑢𝑢𝑇𝑇𝑁𝑁

𝜂𝜂𝐸𝐸𝐹𝐹 =∑ �(ℎ𝑂𝑂𝐼𝐼𝑛𝑛𝑜𝑜𝐼𝐼𝐼𝐼𝑂𝑂 − ℎ𝑂𝑂𝑂𝑂𝑂𝑂𝐼𝐼𝑜𝑜𝐼𝐼𝐼𝐼𝑂𝑂

𝑉𝑉 � 𝐽𝐽 + 𝑄𝑄𝐽𝐽𝑃𝑃𝑢𝑢𝑇𝑇𝑁𝑁

Comparison to a Hydrogen-based System

𝐶𝐶𝐻𝐻4 + 2𝑂𝑂2 ↔ 2𝐻𝐻2𝑂𝑂 + 𝐶𝐶𝑂𝑂2, ∆𝐻𝐻𝐹𝐹𝐻𝐻4 = −802𝐾𝐾𝐽𝐽/𝑚𝑚𝑜𝑜𝑜𝑜4𝐻𝐻2 + 2𝑂𝑂2 ↔ 4𝐻𝐻2𝑂𝑂, 4 � ∆𝐻𝐻𝐻𝐻2 = −988𝐾𝐾𝐽𝐽/𝑚𝑚𝑜𝑜𝑜𝑜

Methane-based reSOFC Voltage

Outline

• Literature Review

• Background

• System Model

• Energy Storage Application

• Conclusions

Demand Profile

• Curtailment• Battery Energy Storage• reSOFC-based Energy Storage

LCOESolar $83.9

𝑀𝑀𝑊𝑊ℎWind $49.9

𝑀𝑀𝑊𝑊ℎ

Energy Storage Application

𝑃𝑃𝐶𝐶𝑁𝑁𝑂𝑂𝑅𝑅𝑁𝑁𝑃𝑃𝑂𝑂 = 𝑃𝑃𝐵𝐵𝑅𝑅𝑁𝑁𝑁𝑁𝐿𝐿𝑃𝑃𝑅𝑅𝑃𝑃𝑂𝑂 + 𝑃𝑃𝐹𝐹𝑃𝑃𝑁𝑁𝐶𝐶𝑁𝑁𝑁𝑁𝑡𝑡𝑂𝑂𝑃𝑃𝑁𝑁𝑅𝑅𝑇𝑇𝑂𝑂 + 𝑃𝑃𝑅𝑅𝑁𝑁𝑁𝑁𝑁𝑁𝑅𝑅𝑅𝑅𝐸𝐸𝑇𝑇𝑁𝑁𝑁𝑁𝑂𝑂 + 𝑃𝑃𝐴𝐴𝑡𝑡𝑃𝑃𝑁𝑁𝑅𝑅𝑆𝑆𝑁𝑁𝑂𝑂

Battery Energy Storage

𝑃𝑃𝐶𝐶𝑂𝑂𝑁𝑁𝑅𝑅𝑂𝑅𝑅𝑁𝑁𝑆𝑆𝑁𝑁𝑂𝑂 =𝐸𝐸𝐴𝐴𝑡𝑡𝑃𝑃𝑁𝑁𝑅𝑅𝑆𝑆𝑁𝑁𝑂𝑂 � 𝛼𝛼𝐶𝐶𝑂𝑂𝑁𝑁𝑅𝑅𝑂𝑅𝑅𝑁𝑁𝑆𝑆𝑁𝑁

∆𝐼𝐼

Self-Discharge

𝐸𝐸𝐴𝐴𝑡𝑡𝑃𝑃𝑁𝑁𝑅𝑅𝑆𝑆𝑁𝑁𝑂𝑂 = 𝐸𝐸𝐴𝐴𝑡𝑡𝑃𝑃𝑁𝑁𝑅𝑅𝑆𝑆𝑁𝑁𝑂𝑂−1 + (𝜂𝜂𝐴𝐴𝑡𝑡𝑃𝑃𝑁𝑁𝑅𝑅𝑆𝑆𝑁𝑁 � 𝑃𝑃𝐴𝐴𝑡𝑡𝑃𝑃𝑁𝑁𝑅𝑅𝑆𝑆𝑁𝑁𝑂𝑂 − 𝑃𝑃𝐶𝐶𝑂𝑂𝑁𝑁𝑅𝑅𝑂𝑅𝑅𝑁𝑁𝑆𝑆𝑁𝑁𝑂𝑂) � ∆𝐼𝐼

𝐸𝐸𝐴𝐴𝑡𝑡𝑃𝑃𝑁𝑁𝑅𝑅𝑆𝑆𝑁𝑁𝑂𝑂 = 𝐸𝐸𝐴𝐴𝑡𝑡𝑃𝑃𝑁𝑁𝑅𝑅𝑆𝑆𝑁𝑁𝑂𝑂−1 − (𝑃𝑃𝐴𝐴𝑡𝑡𝑃𝑃𝑁𝑁𝑅𝑅𝑆𝑆𝑁𝑁𝑂𝑂𝜂𝜂𝐴𝐴𝑡𝑡𝑃𝑃𝑁𝑁𝑅𝑅𝑆𝑆𝑁𝑁

+ 𝑃𝑃𝐶𝐶𝑂𝑂𝑁𝑁𝑅𝑅𝑂𝑅𝑅𝑁𝑁𝑆𝑆𝑁𝑁𝑂𝑂) � ∆𝐼𝐼

𝛼𝛼𝐶𝐶𝑂𝑂𝑁𝑁𝑅𝑅𝑂𝑅𝑅𝑁𝑁𝑆𝑆𝑁𝑁 4%𝑚𝑚𝑜𝑜𝑛𝑛𝐼𝐼ℎ

𝜂𝜂𝐴𝐴𝑡𝑡𝑃𝑃𝑁𝑁𝑅𝑅𝑆𝑆𝑁𝑁 90%Installation

Cost$250𝑘𝑘𝑊𝑊ℎ

O&M $4𝑀𝑀𝑊𝑊ℎ

Lifetime 10 Years

reSOFC-based Energy Storage

𝑀𝑀𝑀𝑀𝑀𝑀 𝑃𝑃𝑜𝑜𝐹𝐹𝐼𝐼𝑃𝑃𝐷𝐷𝐼𝐼𝑛𝑛𝐷𝐷𝐷𝐷𝐼𝐼𝐷𝐷 0.515

𝑊𝑊𝑐𝑐𝑚𝑚2

𝑀𝑀𝐷𝐷𝑛𝑛 𝑃𝑃𝑜𝑜𝐹𝐹𝐼𝐼𝑃𝑃𝐷𝐷𝐼𝐼𝑛𝑛𝐷𝐷𝐷𝐷𝐼𝐼𝐷𝐷 −1.3

𝑊𝑊𝑐𝑐𝑚𝑚2

Installation Cost

~$2150𝑘𝑘𝑊𝑊

O&M: FC Mode

$20𝑀𝑀𝑊𝑊ℎ

O&M: EC Mode

$10𝑀𝑀𝑊𝑊ℎ

Hydrogen Cost

$2𝑘𝑘𝑘𝑘

Lifetime 9 Years

𝑆𝑆𝐿𝐿𝑂𝑂𝑂𝑂𝑂𝑂𝑡𝑡 = 𝑆𝑆𝐸𝐸𝑁𝑁𝑃𝑃1 + 𝑆𝑆𝐻𝐻𝑅𝑅𝑟𝑟1 − 𝑆𝑆𝐸𝐸𝑁𝑁𝑃𝑃2

Comparison Methodology

𝐿𝐿𝐶𝐶𝑂𝑂𝐸𝐸 =𝐼𝐼𝐶𝐶 + ∑𝑡𝑡

𝑂𝑂𝑀𝑀1 + 𝑃𝑃 𝑡𝑡

∑𝑡𝑡𝐸𝐸𝑡𝑡 � 1 − 𝑅𝑅 𝑡𝑡

1 + 𝑃𝑃 𝑡𝑡

𝛽𝛽𝐴𝐴𝑃𝑃𝑇𝑇𝑅𝑅𝑁𝑁 =𝐸𝐸𝐴𝐴𝑃𝑃𝑇𝑇𝑅𝑅𝑁𝑁

𝐸𝐸𝐴𝐴𝑃𝑃𝑇𝑇𝑅𝑅𝑁𝑁 + 𝐸𝐸𝑊𝑊𝑂𝑂𝑁𝑁𝑃𝑃

𝑃𝑃𝐼𝐼𝑛𝑛𝑅𝑅𝑁𝑁𝑁𝑁𝑁𝑁𝑅𝑅𝑅𝑅𝐸𝐸𝑇𝑇𝑁𝑁 =∑𝑃𝑃𝑅𝑅𝑁𝑁𝑁𝑁𝑁𝑁𝑅𝑅𝑅𝑅𝐸𝐸𝑇𝑇𝑁𝑁𝑂𝑂 + 𝑃𝑃𝐴𝐴𝑡𝑡𝑃𝑃𝑁𝑁𝑅𝑅𝑆𝑆𝑁𝑁𝑂𝑂

∑𝑃𝑃𝐶𝐶𝑁𝑁𝑂𝑂𝑅𝑅𝑁𝑁𝑃𝑃𝑂𝑂

Optimization Methodology

𝛿𝛿𝐿𝐿𝐹𝐹𝑂𝑂𝐸𝐸𝛿𝛿𝑃𝑃𝑁𝑁𝑁𝑁

=𝐿𝐿𝐶𝐶𝑂𝑂𝐸𝐸𝐸𝐸𝐴𝐴 − 𝐿𝐿𝐶𝐶𝑂𝑂𝐸𝐸0𝑃𝑃𝐼𝐼𝑛𝑛𝐸𝐸𝐴𝐴 − 𝑃𝑃𝐼𝐼𝑛𝑛0

Energy Storage Sizing Solar Capacity Factor

𝛿𝛿𝐿𝐿𝐹𝐹𝑂𝑂𝐸𝐸𝛿𝛿𝑃𝑃𝑁𝑁𝑁𝑁

=𝐿𝐿𝐶𝐶𝑂𝑂𝐸𝐸𝑂𝑂 − 𝐿𝐿𝐶𝐶𝑂𝑂𝐸𝐸𝑂𝑂−1𝑃𝑃𝐼𝐼𝑛𝑛𝑂𝑂 − 𝑃𝑃𝐼𝐼𝑛𝑛𝑂𝑂−1

Grid-based Search

Solar Capacity Factor

Installed Capacity

Curtailment

Battery Energy Storage

reSOFC-based Energy Storage

Optimal reSOFC Energy Storage System Efficiency

Optimal LCOE Comparison

Outline

• Literature Review

• Background

• System Model

• Energy Storage Application

• Conclusions

Conclusions

• Internal methanation and steam reforming improve system performance.– Higher System Efficiency

– Lower Thermoneutral Voltage

• Cost competitive with other energy storage technologies.

• Optimal sizing does not result in a sudden change in strategy.

References

Hydrogen Production by reSOFC System

Methane-based reSOFC Hydrogen-based reSOFC

Chemical Composition Electrolysis Cell Mode

Methane-based reSOFC Hydrogen-based reSOFC

Chemical Composition Fuel Cell Mode

Optimal Solar Capacity Fraction Comparison

Effect of the HTM

Nodal reSOFC Model (cont)

Chemical Composition Found Using Newton’s Method

𝑋𝑋 =𝑅𝑅𝐴𝐴𝑅𝑅𝑅𝑅𝐹𝐹𝐶𝐶𝑅𝑅𝑊𝑊𝑊𝑊𝐴𝐴

𝑛𝑛(𝑀𝑀) =𝐾𝐾𝐴𝐴𝑡𝑡𝑁𝑁𝑅𝑅𝑂𝑂𝑝𝑝𝐹𝐹𝐻𝐻4𝑝𝑝𝐻𝐻2𝑂𝑂 − 𝑝𝑝𝐹𝐹𝑂𝑂𝑝𝑝𝐻𝐻2

3

𝐾𝐾𝐹𝐹𝐶𝐶𝑝𝑝𝐹𝐹𝐻𝐻4 − 𝑝𝑝𝐹𝐹𝑝𝑝𝐻𝐻22

𝐾𝐾𝑊𝑊𝑊𝑊𝐴𝐴𝑝𝑝𝐹𝐹𝑂𝑂𝑝𝑝𝐻𝐻2𝑂𝑂 − 𝑝𝑝𝐻𝐻2𝑝𝑝𝐹𝐹𝑂𝑂2

𝐽𝐽 =𝜕𝜕𝑛𝑛𝜕𝜕𝑋𝑋1

𝜕𝜕𝑛𝑛𝜕𝜕𝑋𝑋2

𝜕𝜕𝑛𝑛𝜕𝜕𝑋𝑋3

)𝑋𝑋𝑂𝑂+1 = 𝑋𝑋𝑂𝑂 − 𝐽𝐽−1𝑛𝑛(𝑋𝑋𝑂𝑂

𝐽𝐽 = 𝑅𝑅𝑁𝑁𝑁𝑁𝑃𝑃𝑃𝑃𝑟𝑟 � 2𝑛𝑛𝑈𝑈𝐴𝐴𝑡𝑡𝑁𝑁𝑅𝑅𝑂𝑂 =

𝑅𝑅𝑁𝑁𝑁𝑁𝑃𝑃𝑃𝑃𝑟𝑟𝐼𝐼𝑛𝑛𝑜𝑜𝐼𝐼𝐼𝐼𝐻𝐻2𝑂𝑂

𝑈𝑈𝐹𝐹𝑢𝑢𝑁𝑁𝑇𝑇 =𝑅𝑅𝑁𝑁𝑁𝑁𝑃𝑃𝑃𝑃𝑟𝑟

4 � 𝐼𝐼𝑛𝑛𝑜𝑜𝐼𝐼𝐼𝐼𝐹𝐹𝐻𝐻4 + 𝐼𝐼𝑛𝑛𝑜𝑜𝐼𝐼𝐼𝐼𝐹𝐹𝑂𝑂 + 𝐼𝐼𝑛𝑛𝑜𝑜𝐼𝐼𝐼𝐼𝐻𝐻2

Current Distribution Found Iteratively

𝑛𝑛𝑜𝑜𝑜𝑜𝐹𝐹𝐻𝐻2𝑂𝑂n+1 = 𝑛𝑛𝑜𝑜𝑜𝑜𝐹𝐹𝐻𝐻2𝑂𝑂n +𝐽𝐽𝑁𝑁2𝑛𝑛 𝑛𝑛𝑜𝑜𝑜𝑜𝐹𝐹𝐻𝐻2n+1 = 𝑛𝑛𝑜𝑜𝑜𝑜𝐹𝐹𝐻𝐻2n −

𝐽𝐽𝑁𝑁2𝑛𝑛