CFD Modeling and Analysis of a Planar Anode
Supported Internal Reforming Intermediate
Temperature Solid Oxide Fuel Cell Fueled with
Partially Pre-Reformed Methane
Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF SCIENCE
Major Subject: Mechanical Engineering
by
Melissa Tweedie
May, 2014
Rensselaer Polytechnic Institute
Hartford, Connecticut
Contents
ABSTRACT ...................................................................................................................... 1
1
Introduction.................................................................................................................. 1
2
Methodology ................................................................................................................ 6
3
4
5
2.1
Domain and Physical Parameters ....................................................................... 6
2.2
Operating Conditions ......................................................................................... 9
2.3
CFD Model Overview ...................................................................................... 10
2.4
Assumptions ..................................................................................................... 10
Momentum Model ..................................................................................................... 11
3.1
General Equations ............................................................................................ 11
3.2
Density and Viscosity ...................................................................................... 12
3.3
Microstructural Properties................................................................................ 13
Mass Transfer Model ................................................................................................. 15
4.1
General Equations ............................................................................................ 15
4.2
Maxwell Stefan Diffusivity .............................................................................. 16
Heat Transfer Model .................................................................................................. 17
5.1
General Equations ............................................................................................ 18
5.1.1. Flow Fields ........................................................................................... 18
5.1.2. Electrodes ............................................................................................. 20
5.1.3. Electrolyte and Interconnects ............................................................... 21
5.2
Heat Generation Source Terms ........................................................................ 21
5.2.1. Heat Generated by Reactions ............................................................... 21
5.2.2. Ohmic and Overpotential Heat Generation .......................................... 22
6
Chemical Model......................................................................................................... 23
6.1
Internal Reforming ........................................................................................... 23
6.2
Chemical Species Balance Equations .............................................................. 24
i
6.3
Reforming Kinetics .......................................................................................... 25
6.3.1. MSR Kinetics ....................................................................................... 25
6.3.2. WGS Kinetics ...................................................................................... 27
6.3.3. MCDR Kinetics.................................................................................... 27
6.3.4. DSR Kinetics........................................................................................ 28
6.3.5. Carbon Deposition ............................................................................... 29
6.4
7
Additional Chemical Model Information ......................................................... 29
Electrochemical Model .............................................................................................. 30
7.1
Approaches to Electrochemical Modeling ....................................................... 30
7.2
Electrochemical Species Balance Equations .................................................... 31
7.3
Ion and Charge Transfer................................................................................... 32
7.3.1. Electrode Backing Layers .................................................................... 33
7.3.2. Electrochemical Reaction Layers (ERL) ............................................. 34
7.3.3. Electrolyte ............................................................................................ 34
7.4
Cell Voltage ..................................................................................................... 35
7.5
Activation Losses ............................................................................................. 36
7.5.1. Electrochemical Activation Energies ................................................... 39
7.6
Ohmic Losses ................................................................................................... 39
7.7
Concentration Losses ....................................................................................... 40
8
Simulation Validation ................................................................................................ 41
9
Results........................................................................................................................ 41
9.1
Kinetics ............................................................................................................ 42
9.2
Electrochemistry .............................................................................................. 43
10 Conclusion ................................................................................................................. 43
11 Future Work ............................................................................................................... 43
12 Notation ..................................................................................................................... 43
13 References.................................................................................................................. 46
ii
14 Appendix A................................................................................................................ 51
15 Appendix B ................................................................................................................ 52
16 Appendix C ................................................................................................................ 53
17 Appendix D................................................................................................................ 54
18 Appendix E ................................................................................................................ 56
iii
List of Tables
Table 1 Types of Fuel Cells [1] ......................................................................................... 2
Table 2 Cell Dimensions ................................................................................................... 8
Table 3 Cell Materials ....................................................................................................... 8
Table 4 Cell Physical Properties and Parameters .............................................................. 8
Table 5 Model Operating Conditions ................................................................................ 9
Table 6 Simulated Fuel Feed Mole Fractions [27] ............................................................ 9
Table 7 Species Dynamic Viscosity Coefficients [29] .................................................... 13
Table 8 Ni-YSZ Anode Microstructural Characteristics in Literature ............................ 14
Table 9 Fuller Diffusion Volume [29] ............................................................................. 16
Table 10 Species Heat Capacity Coefficients [29] .......................................................... 19
Table 11 Species Thermal Conductivity Coefficients [29] ............................................. 20
Table 12 Summary of Heat Source Equations ................................................................. 22
Table 13 Summary of Chemical Species Balance Equations used in Model .................. 24
Table 14 Summary of Electrochemical Species Balance Equations used in Model ....... 32
Table 15 Summary of Charge Transfer Equations .......................................................... 35
Table 16 Summary of Effective Conductivity Equations ................................................ 40
Table 17 Simulation Validation Operating Conditions .................................................. 41
Table 20 Fuel Feed Mole Fractions from SOFC Literature ............................................ 41
Table 18 Fuel Feed Mass Fractions for Range of Pre-reformed Percentages[27] ........... 51
Table 19 Fuel Feed Mole Fractions for Range of Pre-reformed Percentages[27] ........... 51
Table 21 Sensitivity Analysis of Calculated Diffusion Coefficients ............................... 52
Table 22 Kinetic Models for SOFC MSR and WGS Reactions on Ni Catalysts ............ 54
List of Figures
Figure 1 Model Domain..................................................................................................... 7
Figure 2 Relationship between Ideal and True Cell Voltages [3] ................................... 36
Figure 3 MCDR Reaction Rate at Study Operating Conditions ...................................... 42
iv
ABSTRACT
This study considered a planar anode-supported intermediate temperature internal
reforming solid oxide fuel cell. The effects of varying pre-reforming fuel inlet conditions
were considered along with probability of carbon formation in the anode. A 2-D model
was developed containing a composite Ni-YSZ anode, YSZ electrolyte, composite LSMYSZ cathode surrounded by metal interconnects. The domain included separate defined
electrochemical reaction layers on either side of the electrolyte where chemical
reforming and electrochemical reactions simultaneously occurred. Both H2 and CO
electrochemical oxidation was considered along with the internal reforming reactions for
methane steam reforming (MSR), and water gas shift (WGS).
The CFD model consists of 5 submodels including the Navier Stokes and continuity
equations for momentum transport, Maxwell Stefan considering Knudsen diffusion for
mass transport, energy equation for heat transfer, a chemical reforming model and an
electrochemical model considering distributed charge transfer over the cell including
Butler-Volmer type kinetics.
1 Introduction
Fuel cells are promising alternative energy technologies which convert fuel and oxygen
to electricity, water and carbon dioxide. In general, a fuel cell consists of an ion
conducting electrolyte sandwiched in between two porous electrodes. Typically air or
oxygen flows over one of the electrodes (cathode) while hydrogen or a hydrogen
containing fuel flows over the other electrode (anode). In the cathode of the fuel cell, the
oxygen atoms are reduced to oxygen ions which then pass through the electrolyte. Once
they reach the anode, the oxygen ions react with the hydrogen which is oxidized to
produce both water and electrons. The water is carried out of the fuel cell in the anode
flow channel while the electrons are carried through an external circuit back to the
cathode to repeat the process.
1
There are generally six main types of fuel cells including polymer electrolyte membrane
fuel cells (PEM), phosphoric acid fuel cells (PAFC), solid oxide fuel cells (SOFC)
alkaline fuel cells (AFC), molten carbonate fuel cells (MCFC), and direct methanol fuel
cells (DMFC). Table 1 below illustrates the general characteristics of each type of fuel
cell.
Table 1 Types of Fuel Cells [1]
Fuel Cell
Temperature (oC)
Applications
Polymer Electrolyte Membrane (PEM)
60 - 100
Automotive, Transportation
Phosphoric Acid (PAFC)
175 - 220
Solid Oxide (SOFC)
600 - 1000
Alkaline (AFC)
65 - 220
Distributed generation: Grid support,
cogeneration, stand-alone
Centralized power plant, stand-alone,
cogeneration
Space program, military
Molten Carbonate (MCFC)
600 - 650
Direct Methanol (DMFC)
50 - 120
Centralized power plant, stand-alone,
cogeneration
Portable small-scale power
In this study, we will focus on solid oxide fuel cells. There are currently several
innovative SOFC products in the market from companies such as Fuel Cell Energy,
Acumentrics, and Bloom Energy. SOFCs are a class of high temperature fuel cells
operating between 600oC to 1000oC which use hydrogen or hydrocarbons as the fuel and
air as the oxidant. This type of fuel cells utilizes porous ceramic electrodes for the anode
and cathode which are separated by a solid ceramic electrolyte. The structure of the
SOFC is commonly referred to as the PEN or positive-electrode/electrolyte/negativeelectrode structure. The two primary configurations of SOFC’s are tubular and planar.
Due to limitations in the performance of tubular SOFC’s, namely that tubular stack
designs have demonstrated low specific power densities, the focus in recent years has
been optimizing the planar design configurations [2].
In the planar type of solid oxide fuel cells evaluated in this study, the general
configuration consists of an interconnect plate, an air/fuel flow channel, the positiveelectrolyte-negative electrode or PEN structure, the alternate air/fuel flow channel and
an alternate interconnect plate. The interconnect plates within a fuel cell stack are
2
typically fabricated with flow channels on either side such that only one interconnect
plate is present in the repeating cell units.
The planar type of SOFCs are typically configured in two different ways; either
electrolyte supported or electrode supported. It has been found that under the same
operating conditions, anode supported SOFCs exhibit better performance than
electrolyte supported SOFCs [3]. In the electrode supported fuel cells the electrode is the
thickest layer in the cell on which all other layers are deposited. Thus for an anode
supported planar SOFC, the anode within the PEN structure provides the structural
support for the unit cell.
Different materials have been studied for use in electrode supported solid oxide fuel cells
and the most common materials utilized today consist of a yttria-stabilized zirconia
(YSZ) electrolyte, and porous ceramic metallic composites (cermets), including
nickel/zirconia (Ni-YSZ) anode and strontium doped LaMnO3 (LSM) mixed with YSZ
composite cathode [4].
SOFC systems can be configured in several different ways according to the approach
taken in the fuel reforming process. In the case where a separate reformer adjacent to the
fuel cell is utilized to extract the hydrogen from the hydrocarbon before feeding it to the
fuel cell, this method is called indirect internal reforming (IIR). The other method is
feeding the hydrocarbon directly to the fuel cell where the reformation process takes
place on the catalyst in the anode. This method is called direct internal reforming (DIR).
DIR fuel cells are advantageous over non-DIR fuel cells in that both the fuel reforming
and electrochemical processes occur within the cell, thus a separate reformer is not
required to extract the hydrogen from the hydrocarbon fuel resulting in less fuel cell
powerplant cost and less overall footprint. They also have increased performance due to
the utilization of the waste heat from the exothermic electrochemical reaction in the
endothermic reforming process. Typical DIR fuel cells can operate at high efficiencies of
50-60%. Various studies have investigated the performance of different types of fuels
used in DIR SOFC’s [5] [6] [7] [8] [9] [10].
3
Today, the major challenges with DIR SOFCs, include material degradation, high cost of
operation, coking, reduced efficiency with higher inlet steam to carbon ratios and sulfur
intolerance.
SOFCs must operate at higher temperatures to both achieve sufficient conversion in the
internal reforming reactions, as well as be able to attain reasonable power densities.
There are several negative aspects to this higher operating temperature including high
costs of operation, degradation and cracking in the materials from thermal cycling, thus
resulting in higher maintenance costs as well as a higher cost of material fabrication due
to the need for specialty materials that can survive at the higher temperatures. The
solution in this case would be to operate SOFC’s at lower operating temperatures while
still maintaining high efficiencies. These lower temperature SOFC’s are called
intermediate temperature fuel cells (IT-SOFC). IT-SOFC’s typically operate between
550oC and 800oC (823 K and 1073K).
Another contributor to DIR SOFC material degradation is the non-uniform temperature
distribution across the cell. In a solid oxide fuel cell with direct internal reforming, the
endothermic reforming process generally occurs much faster than the exothermic
electrochemical process. This results in lower temperatures at the anode entrance, large
temperature gradients and thus thermal stress along the cell causing material cracking.
Meusinger [11] performed experiments demonstrating that a higher S/C ratio can lower
the temperature gradients in a cell however using more steam can dramatically decrease
performance. He suggested optimizing the percentage of pre-reforming to lessen the
temperature gradients in the cell. Another approach to lowering the gradients across the
cell is modifying the cell materials by impregnating the anode with copper. This copper
impregnation has been shown to both reduce temperature gradients by lowering the
operating temperature, reducing cost, and reducing carbon formation [12].
One of the other challenges with DIR SOFCs is the formation of solid carbon on the
electrode (coking) that blocks or destroys catalyst sites. The general approach to solving
4
this issue has been to increase the amount of steam in the inlet fuel. However, the more
steam that is added to the inlet fuel reduces the performance by reducing the open circuit
voltage (OCV) in the fuel cell. Fuel stream recycling has been investigated to reduce the
costs of maintaining high steam to carbon ratios, prevent the lowered OCV from the
steam, as well as preventing coking in the anode of the fuel cell [13] [14]. Alternate
catalysts have also been investigated with respect to carbon formation [15].
To maximize the performance of anode supported DIR SOFC’s while minimizing
material degradation and overall cost, the system performance including the inlet species
concentrations, inlet conditions, cell flow configurations and the thermal management
within the fuel cell along with the material optimizations mentioned previously must
also be optimized. Among the types of modeling utilized to simulate the SOFC single
cell level conditions for optimization, generally the more predominant models are of the
type where the electrochemical reactions are defined to occur at the electrode-electrolyte
interfaces which Hussain et al. [16] refers to as macro-level models. The other primary
approach, micro-level type models assume the electrochemical reactions occur
throughout the electrode and typically focus on only one electrode. However, as Hussain
et al. notes, incorporating the two types of models together enhances the predictive
capability of a cell level study. To incorporate the two types, distinctive
electrochemically reactive layers are introduced into the model between the bulk
electrode and the electrolyte in this study.
Of the researchers that have considered this novel approach using distinct
electrochemical reactive layers, Hussain et. al. [16] developed a numerical model to
predict the electrochemical performance characteristics of a DIR SOFC utilizing a
distributed charge transfer model not including CO oxidation. Ho et al. [17] developed a
numerical model to determine the electrochemical performance and temperature
distribution of an anode supported IT-DIR SOFC considering distinct electrode and
electrochemical reaction layers (ERL’s), a charge transfer approach using a modified
Nernst-Planck equation and included CO oxidation at a rate 3 times less than H2
oxidation in calculating current density. Anderssen [10] developed a 2-D CFD model of
5
a IT DIR SOFC including CO oxidation and finite electrochemical reaction layers
utilizing electrochemical kinetics presented by Suwanwarangkul [18] comparing the
effect of varying inlet fuel and air compositions, utilizations and inlet velocities on coflow cell performance. Jeon [19] developed a 2-D IT SOFC CFD model examining the
effect of parameters on performance and temperature distribution including distinct
electrochemical reaction layers using only H2 fuel.
Other recent cell level modeling includes; Iwai et al developed a numerical model of a
DIR SOFC using an equivalent circuit approach involving a volume averaging method
to examine the electrochemical performance and thermal distribution considering the
MCDR reaction with high methane and CO2 content fuel [20]. Park et al developed a 3D
numerical model of a DIR SOFC examining the effect of inlet species concentrations
and S/C ratio on chemical and electrochemical reactions and cell performance [21]. Ni et
al developed a 2D model of a DIR SOFC examining chemical kinetics approaches and
operating parameters on performance [22]. Nikooyeh developed a 3-D CFD model of a
DIR SOFC examining carbon formation and recycling of gas exhaust [14].
2 Methodology
2.1 Domain and Physical Parameters
This study considered a 2D model of a planar anode-supported IT-DIR SOFC with
composite Ni-YSZ anode and composite LSM-YSZ cathode. In the 2-D single cell
model shown in Figure 1, there are a total of nine distinct layers, each having a total
length of 100 mm and heights as defined in Table 2.
In this particular model, the active catalyst electrochemical reaction layers (ERL) were
treated as a separate layer from the electrode to replicate the location of the
electrochemically active zone at the boundaries of the electrode and electrolyte layers.
This is different from the majority of modeling which assumes the electrochemical
reactions occur as surface reactions at the interface between the electrolyte and
electrodes. This defined separate layer approach is a more accurate simulation than the
6
assumption of surface electrochemical reactions only as it has been shown that the
electrochemical reactions occur within the electrode at a distance of 10 to 50µm away
from the electrode-electrolyte interface [17] [16]. The materials and properties assumed
in the model are listed in the following tables.
Figure 1 Model Domain
7
Cell length
Cell height
Interconnect Height
Fuel channel height
Anode Backing Layer Height
Anode ERL Layer Height
Table 2 Cell Dimensions
100mm
Air channel height
3.31mm
Cathode Backing Layer Height
0.5mm
Cathode ERL Layer Height
0.6mm
Electrolyte Height
0.6mm
0.03mm
1mm
0.05mm
0.01mm
0.02mm
Table 3 Cell Materials
Anode and Cathode Interconnect
Stainless Steel
Anode Electrode and Anode ERL Layer
Ni-YSZ (Nickel - Yttria Stabilized Zirconia)
Electrolyte
YSZ (Yttria Stabilized Zirconia)
Cathode Electrode and Cathode ERL Layer
LSM-YSZ (Strontium doped Lanthanum
Manganite – Yttria Stabilized Zirconia)
The following table details the physical properties, and electrochemical/thermal
parameters assumed for the cell. More research has been performed on Ni-YSZ anode
characteristics than LSM-YSZ cathode characteristics therefore in the cases where
cathode properties or parameters were unavailable, the corresponding anode values were
utilized.
Table 4 Cell Physical Properties and Parameters
Anode
2.42 x 10 -15
0.489
Cathode
2.54 x 10 -15
0.515
Pore Diameter (µm)
Electronic/Ionic/Pore Tortuosity
0.971
7.53, 8.48, 1.80
1
7.53, 3.4, 1.80
Electronic/Ionic Volume Fraction
0.257, 0.254
0.232, 0.253
3.97x10 6 , 7.93x10 6
3.97x10 6 , 7.93x10 6
[23]
[23] [25]
[24]
[23] [25]
[24]
[26]
H2 2.9 x108,
CO 2.07 x108
H2 1.2x105
CO 1.2x105
11
450
3310
Electrolyte
2.7
470
5160
7.0x108
[7]
1.2x105
[7]
6
430
3030
Interconnect
20
550
3030
[10]
[10]
[10]
Permeability (m2)
Porosity
Electronic/Ionic Reactive Surface Area
per Unit Volume (m2/m3)
Pre-exponential factor (m2/m3)
Electrochemical Activation Energy
(J/mol)
Solid Thermal Conductivity (W/m-K)
Solid Specific Heat Capacity (J/kg-K)
Solid Density (kg/m3)
Thermal Conductivity (W/m-K)
Specific Heat Capacity (J/kg-K)
Solid Density (kg/m3)
8
[23]
[23] [24]
[10]
[10]
[10]
2.2 Operating Conditions
The model operating conditions are shown in Table 5. Many SOFC models that include
internal reforming use anode inlet conditions that neglect N2 which suggests they assume
steam pre-reforming (See Appendix A). However, Heinzel et al. notes that autothermal
reforming has more flexibility than steam reforming in the start-up time and load
changes. This flexibility may be more practical for SOFC systems although maximum
product H2 composition in autothermal reforming is lower than in steam reforming [27].
One of the benefits of SOFC’s however is that their internal reforming characteristics
can mitigate the lower reformer conversion. Thus autothermal reforming may be a more
practical alternative for the pre-reforming process for SOFC’s as long as the deleterious
effects of high temperature gradients in the fuel cell can be sufficiently mitigated. Ideally
a pre-reformer would not be required and natural gas could be fed directly to the fuel cell
however until technology improves such that the large temperature gradients caused by
internal reforming do not cause unacceptable damage to the cell, this approach is not
practical and partial pre-reforming must be used.
Table 5 Model Operating Conditions
Inlet Temperature (K)
Cathode Inlet Velocity (m/s)
Anode Inlet Velocity (m/s)
Outlet Pressure (atm)
1023
13
5
1.0
Anode Fuel Feed xi
Cathode Air Feed xi
Operating Voltage (V)
Varies
.21 O2 .79 N2
0.4 to 1.0
The anode inlet fuel was varied from 20% pre-reformed to 50% pre-reformed with the
compositions from Recknagle et al. in the table below. The composition of the fuel
which contains nitrogen is such that autothermal pre-reforming or recycling inlet
conditions can be assumed.
Table 6 Simulated Fuel Feed Mole Fractions [28]
% Pre-Reformed
50%
40%
30%
20%
H2
0.04101
0.03201
0.02269
0.01295
H2O
0.37713
0.40796
0.44169
0.47913
CO
0.08536
0.05621
0.03158
0.01272
CO2
0.16864
0.15389
0.13204
0.10111
9
CH4
0.11037
0.13244
0.15451
0.17659
N2
0.21749
0.21749
0.21749
0.21749
2.3 CFD Model Overview
The commercially available software COMSOL was used to model the domain. The
domain parameters are shown in Table 5. The 2-D computational fluid dynamics (CFD)
model consists of conservation equations for mass, momentum, species, charge and
energy. Using the Free and Porous Media Flow Module, Navier-Stokes equations are
utilized to model the flow in the anode and cathode flow channels, and the Brinkman
equations are utilized to model the flow in the porous electrodes. For the mass balances,
the Transport of Concentrated Species Module was used. It includes Maxwell-Stefan
diffusion, where species convection and kinetic rate equations were considered. The
energy balance was incorporated through the use of the Heat Transfer in Fluids Module.
For the electronic and ionic and charge balance, the appropriate balance equations were
entered manually into the mathematics module using Poisson’s equation. All other
definitions and equations utilized are manually entered as defined variables applied into
the model. More details on the equations utilized can be found in the modeling sections
of this paper.
2.4 Assumptions
1. The methane reforming and water gas shift reaction are not in equilibrium due to
the slower reforming kinetics when compared with both the shift and
electrochemical oxidation reactions therefore a kinetic model was used.
2. Steady state operation
3. Laminar flow in gas channels
4. Ideal gases
5. Convective heat transfer from fluid flowing in/out of cell
6. Electrochemical reactions are thermodynamically reversible thus the Nernst
equation applies.
10
3 Momentum Model
The Free and Porous Media Flow module was selected to model the momentum balance
and calculate the velocity fields and pressure gradients across the cell in the fuel cell
flow fields and electrodes at steady state. This program module includes the
functionality to model systems with both free and porous media flow.
3.1 General Equations
For the open flow channels in the cell including the fuel and air flow channels, the
following Continuity and Navier Stokes Equations were utilized considering
compressible flow at steady state conditions.
∇ ∙ (ρ𝐮) = 0
(1)
2
ρ𝐮 ∙ ∇𝐮 = ∇ ∙ [−p𝐈 + μ ((∇𝐮 + (∇𝐮)T ) − μ(∇ ∙ 𝐮)𝐈)] + 𝐅
3
(2)
For the flow in the porous electrodes (and corresponding electrode reaction layers) the
Stokes-Brinkman equations were utilized which neglects the initial term in the Brinkman
equations due to very low Reynolds number.
∇ ∙ (ρ𝐮) = S
(3)
μ
μ
2μ
𝐮 ( + S) = ∇ ∙ [−p𝐈 + ((∇𝐮 + (∇𝐮)T ) − (∇ ∙ 𝐮)𝐈)] + 𝐅
κ
𝜀
3𝜀
(4)
In these equations S is the mass source term (kg/m3-s) from current density in the ERLs,
F is the volume force vector, μ is viscosity, κ is permeability, u is the velocity vector, ρ is
density, I is the unit matrix. The mass source term in the electrochemically reactive layer of the
porous electrodes (ERL) can be defined as follows.
S = ∑(𝜐𝑖 )𝑀𝑖 𝑗𝑖 𝐴𝑣 /𝑛𝑒 𝐹
𝑖
11
(5)
The boundary conditions utilized in the model included: no slip conditions at the walls.
The initial conditions are ux, uy= 0 m/s, p = 101325 Pa.
3.2 Density and Viscosity
The density of the gases is dependent on temperature as defined in the Transport of
Concentrated Species module and is determined by the ideal gas model. The dynamic
viscosity μ of a mixture is dependent on both the temperature and mixture composition.
To calculate the dynamic viscosity of these low pressure mixtures, there are several
methods available varying in complexity [29]. For this study, thermodynamic data in
Table 7 was utilized in the following equations as a combination of the Wilke and
Herning & Zipperer methods.
𝜇𝑖 = 1𝑥10−7 [𝑎0 + 𝑎1 (𝑇⁄1000) + 𝑎2 (𝑇⁄1000)2 + 𝑎3 (𝑇⁄1000)3 + 𝑎4 (𝑇⁄1000)4
(6)
+ 𝑎5 (𝑇⁄1000)5 + 𝑎6 (𝑇⁄1000)6 ]
𝑁
𝑥𝑖 𝜇𝑖
𝑛
∑𝑗=1 𝑥𝑗 𝜃𝑖𝑗
(7)
𝜃𝑖𝑗 = (𝑀𝑗 /𝑀𝑖 )1/2 = 𝜃𝑗𝑖−1
(8)
𝜇𝑚𝑖𝑥𝑡𝑢𝑟𝑒 = ∑
𝑖=1
In these equations T is in Kelvin, 𝜇𝑖 is the species dynamic viscosity in Pa-s (conversion
made by multiplying 1𝑥10−7), 𝑥𝑖 is the mole fraction of species i, and 𝑀𝑖 is the
molecular weight of species i. For the binary mixture in the cathode (O2 and N2) the
following equations result:
𝜇𝑐𝑎𝑡ℎ𝑜𝑑𝑒 =
𝑥𝑂2 μ𝑂2
𝑥𝑁2 μ𝑁2
+
𝑥𝑂2 + 𝑥𝑁2 𝜃𝑂2,𝑁2 𝑥𝑁2 + 𝑥𝑂2 𝜃𝑁2,𝑂2
𝜃𝑂2,𝑁2 = (𝑀𝑁2 /𝑀𝑂2 )1/2
𝜃𝑁2,𝑂2 =
1
𝜃𝑂2,𝑁2
12
(9)
(10)
(11)
In the anode, the equation set for dynamic viscosity becomes significantly more
complicated due to the greater number of species. In total there are 6 species including:
CH4, H2, H2O, CO, CO2 and N2. For a 6 species mixture equation (8) is combined with
the following definitions.
𝜇𝑎𝑛𝑜𝑑𝑒 =
𝑥1 𝜇1 𝑥2 𝜇2 𝑥3 𝜇3 𝑥4 𝜇4 𝑥5 𝜇5 𝑥6 𝜇6
+
+
+
+
+
𝛽1
𝛽2
𝛽3
𝛽4
𝛽5
𝛽6
(12)
𝛽1 = 𝑥1 + 𝑥2 𝜃12 + 𝑥3 𝜃13 + 𝑥4 𝜃14 + 𝑥5 𝜃15 + 𝑥6 𝜃16
(13)
𝑥1
+ 𝑥2 + 𝑥3 𝜃23 + 𝑥4 𝜃24 + 𝑥5 𝜃25 + 𝑥6 𝜃26
𝜃12
𝑥1
𝑥2
𝛽3 =
+
+ 𝑥3 + 𝑥4 𝜃34 + 𝑥5 𝜃35 + 𝑥6 𝜃36
𝜃13 𝜃23
x1
x2
x3
β4 =
+
+
+ x4 + x5 𝜃45 + 𝑥6 𝜃46
𝜃14 𝜃24 𝜃34
x1
x2
x3
x4
β5 =
+
+
+
+ x5 + 𝑥6 𝜃56
𝜃15 𝜃25 𝜃35 𝜃45
x1
x2
x3
x4
x5
β6 =
+
+
+
+
+ 𝑥6
𝜃16 𝜃26 𝜃36 𝜃46 𝜃56
(14)
𝛽2 =
CH4
H2O
CO2
CO
H2
N2
O2
a0
-9.9989
-6.7541
-20.434
-4.9137
15.553
1.2719
-1.6918
(15)
(16)
(17)
(18)
Table 7 Species Dynamic Viscosity Coefficients [30]
a1
a2
a3
a4
a5
529.37
-543.82
548.11
-367.06
140.48
244.93
419.50
-522.38
348.12
-126.96
680.07
-432.49
244.22
-85.929
14.450
793.65
875.90
883.75
-572.14
208.42
299.78
-244.34
249.41
-167.51
62.966
771.45
-809.20
832.47
-553.93
206.15
889.75
-892.79
905.98
-598.36
221.64
a6
-22.920
19.591
-0.4564
-32.298
-9.9892
-32.430
-34.754
3.3 Microstructural Properties
In determining the momentum balance for the porous electrodes, the microstructure
material properties need to be defined. The porosity and the permeability are important
values
typically determined experimentally along with
other
microstructural
characteristics for the particular material. Cell performance, namely electrical
conductivity and the effective gas diffusion within an electrode depend on the pore
13
structure within the material. To determine the permeability, the Carman-Kozeny
correlation is applicable which is derived from Darcy’s Law (assuming laminar flow Re
< 2300) is used. The general form for the Carman-Kozeny equation is:
𝜅=
𝜀 3 𝑑𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 2
c(1 − 𝜀 )2
(19)
The Carman-Kozeny equation can be used in which the constant 𝑐 is set equal to 180 or
150 which are both empirical values commonly used for particles assumed to be
spherical in shape. Zhu et al [31] proposed constant 𝑐 may also be further defined in
terms of tortuosity where 𝑐 = 𝜏𝑐𝑜 and 𝑐𝑜 is a shape factor 𝑐𝑜 = 72 however the
resulting calculated permeability was one to three orders of magnitude smaller than the
reported values by Kishimoto [23] who calculated the permeability of Ni-YSZ cermet
anodes based on the experimentally determined microstructural characteristics of pore
volume (porosity), pore surface to volume ratio (S/V)pore and pore tortuosity as shown
in the equation below. The results of Kishimoto closely matched the microstructural
characteristics determined via the dual beam FIB-SEM experiment by Iwai et al [25]. A
summary of microstructural parameters in the literature, averaged where appropriate, are
presented in Table 8. The calculated permeability by Kishimoto using (S/V)pore =
4.33 x 106 and the following equation is implemented in this study.
𝜅=
𝑉𝑝𝑜𝑟𝑒
(20)
6𝜏𝑝𝑜𝑟𝑒 (S/V)2pore
Table 8 Ni-YSZ Anode Microstructural Characteristics in Literature
𝑑𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒
𝑑𝑝𝑜𝑟𝑒
𝑉𝑝𝑜𝑟𝑒
(μ𝑚)
(μ𝑚)
or 𝜀
15
-
0.971
4.510 x 1016
to
3.132 x 10-
Source
𝜅 (𝑚2 )
*Kishimot
o [23]
2.415 x 10 -
*Iwai
[25]
Zhu
[31]
𝑉𝑒𝑙
𝑉𝑖𝑜
𝜏𝑝𝑜𝑟𝑒
𝜏𝑒𝑙
𝜏𝑖𝑜
0.489
0.257
0.254
1.80
8.13
7.11
-
0.489
0.257
0.254
1.96
6.93
9.85
0.1 –
1.2
0.35
0.23
0.42
4.5
-
-
18
14
Jeon
1.4 /
0.42 /
3.34 x 10-15
[19]
0.33
0.097
Anderssen
-11
1.76 x 10
0.34
0.30
[10]
*Indicates experimentally determined value
0.4
-
-
-
-
0.28
0.42
-
10
10
Two approaches are available for calculating the pore diameter if not available from
experiment. The first assumes that the mean pore diameter is equivalent to the hydraulic
diameter [23]. The second calculates the pore diameter as a function of the electrode
porosity and particle diameter 𝑑𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 [19].
4
(𝑆/𝑉)𝑝𝑜𝑟𝑒
(21)
2 𝜀
𝑑
3 1 − 𝜀 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒
(22)
𝑑𝑝𝑜𝑟𝑒 ≈ 𝑑ℎ =
𝑑𝑝𝑜𝑟𝑒 =
4 Mass Transfer Model
The Transport of Concentrated Species module was used to determine the flux of species
through the cell. This included use of Maxwell-Stefan diffusion considering Knudsen
diffusion along with convection and the chemical kinetic rate expressions as well as the
electrochemical reaction rate expressions.
4.1 General Equations
The following mass transport equations are utilized for a steady state system to model
species transport in the fuel cell flow fields and electrodes.
𝑁
̃𝑖𝑗 𝒅𝑗 +
𝜌𝑦𝑖 (∇ ∙ 𝒖) − ∇ ∙ (𝜌𝑦𝑖 ∑ 𝐷
𝑗
𝒅𝑗= ∇𝑥𝑗 +
1
𝑝𝑎𝑏𝑠
𝛻𝑇
𝐷𝑖𝑇 )
𝑇
⌈(𝑥𝑗 − 𝑦𝑗 )∇𝑝𝑎𝑏𝑠 ⌉
15
(23)
= 𝑅𝑖
(24)
̃ ij are the Fick diffusivity values (m2/s) calculated from the Maxwell-Stefan
Where D
diffusivity matrix values DMS
in (m2/s) as shown in reference [32], DTi are thermal
ij
diffusion coefficients (kg/m-s), and R i represents the kinetic rate expressions for species
generation or consumption in the chemical reactions (kg/(m3-s). The subscript i indicates
each unique species in consideration. For the boundary conditions in the model, no mass
flux was assumed where there was no flow of species (ie. outside the cell other than in
the flow channels and electrodes), the inflow species concentrations were predefined
mass fractions. The velocity and pressure values were derived from the momentum
balance, and the gases were assumed to be ideal to determine the mixture density.
4.2 Maxwell Stefan Diffusivity
2
To calculate the Maxwell-Stefan diffusivity values DMS
ij (m /s) for the species present in
the non-porous flow fields there are two types of equations generally used. The first type
utilizes the Chapman-Enskog theory combined with Lennard-Jones parameters [17] [21]
while the second type utilizes the Fuller expression shown below [30]. The values Vi and
Vj are specific diffusion volumes calculated in the Fuller method, 𝑇 is in K and p is in
Pa.
𝑀𝑆
𝐷𝑖𝑗
=
1.43𝑥10−2 𝑇 1.75
1/2
𝑝𝑀𝑖𝑗 [𝑉𝑖1
𝑀𝑖𝑗 =
⁄3
⁄
(25)
2
+ 𝑉𝑗 1 3 ]
2
1
1
+
𝑀𝑖 𝑀𝑗
(26)
Table 9 Fuller Diffusion Volume [30]
CH4
H2O
CO2
CO
H2
Fuller Diffusion
Volume
25.14
13.1
26.7
18.0
6.12
N2
O2
18.5
16.3
The above equation is practical for use in the flow fields however another approach
utilizing the Dusty Gas Model must be taken for diffusion in porous media. A new
diffusivity value is introduced called the effective diffusivity Deff
ij which combines the
16
standard binary diffusivity with the Knudsen diffusivity DKn
ij and introduces the effects
of pore characteristics including tortuosity and porosity [10] [16] [26].
𝑒𝑓𝑓
𝐷𝑖𝑗
𝜀
1
1
=
( 𝑀𝑆 + 𝐾𝑛 )
𝜏𝑝𝑜𝑟𝑒 𝐷𝑖𝑗
𝐷𝑖
−1
=
𝜀
𝑀𝑆 𝐾𝑛
𝐷𝑖𝑗
𝐷𝑖𝑗
(27)
𝐾𝑛
𝑀𝑆
𝜏𝑝𝑜𝑟𝑒 𝐷𝑖𝑗
+ 𝐷𝑖𝑗
2
8𝑅𝑇
𝑇
𝐾𝑛
𝐷𝑖𝑗
= 𝑟𝑝𝑜𝑟𝑒 𝑥10−4 √
= 48.5𝑥10−4 𝑑𝑝𝑜𝑟𝑒 √
̅𝑖𝑗
̅𝑖𝑗
3
𝜋𝑀
𝑀
(28)
̅𝑖𝑗 = (𝑀𝑖 + 𝑀𝑗 )/2
𝑀
(29)
In the previous equations, the pore diameter 𝑑𝑝𝑜𝑟𝑒 is in meters, the temperature of the
2
diffusing medium T is in K and DKn
ij is (m /s).
The thermal diffusion coefficient DTi is not typically included in current cell sized
modeling. There has been much work in the past to determine the thermal diffusion
coefficient and predictive equations for binary mixtures and some work on ternary
mixtures. However data for multicomponent mixtures, specifically for the gaseous
mixture in the anode, is not currently available. Based on this, the thermal diffusion
coefficient will not be considered in this study. Thus, the mass transfer equation reduces
to the following.
𝑁
(30)
̃𝑖𝑗 𝒅𝑗 ) = 𝑅𝑖
𝜌𝑦𝑖 (∇ ∙ 𝒖) − ∇ ∙ (𝜌𝑦𝑖 ∑ 𝐷
𝑗
5 Heat Transfer Model
To model the heat transfer within the cell The Heat Transfer Modules in Comsol were
utilized. Depending on the layer considered, different forms of the energy equation for
each type of domain were considered.
17
5.1 General Equations
5.1.1. Flow Fields
For the heat transfer in the gas flow fields the following form of the energy equation was
used:
𝜌𝐶𝑝 𝒖∇𝑇 − ∇(𝜆∇𝑇) = 𝑄
(31)
Where 𝐶𝑝 is the specific heat capacity (J/kg-K) at constant pressure, 𝜆 is the thermal
conductivity (W/m-K), and 𝑄 is the heat generation or source term discussed in Section
5.2. To determine the specific heat capacity of the fluid mixtures, ideal gases were
assumed such that the heat capacities were functions of temperature only per the
following equations [30].
𝐶𝑝,𝑖 =
1000
[𝑏0 + 𝑏1 (𝑇⁄1000) + 𝑏2 (𝑇⁄1000)2 + 𝑏3 (𝑇⁄1000)3 + 𝑏4 (𝑇⁄1000)4
𝑀𝑖
(32)
+ 𝑏5 (𝑇⁄1000)5 + 𝑏6 (𝑇⁄1000)6 ]
𝑁
(33)
𝐶𝑝,𝑚𝑖𝑥𝑡𝑢𝑟𝑒 = ∑ 𝑥𝑖 𝐶𝑝,𝑖
𝑖=1
In these equations 𝐶𝑝 is the specific heat (J/kg-K), T is temperature in K, and 𝑥𝑖 is the
species mole fraction. Applying this equation to the cathode of the cell yields the
following equation.
𝐶𝑝,𝑐𝑎𝑡ℎ𝑜𝑑𝑒 = 𝑥𝑂2 𝐶𝑝,𝑂2 + 𝑥𝑁2 𝐶𝑝,𝑁2
(34)
In the anode we must account for the additional species, as shown in the following
equation.
𝐶𝑝,𝑎𝑛𝑜𝑑𝑒 = 𝑥𝐶𝐻4 𝐶𝑝,𝐶𝐻4 + 𝑥𝐻2𝑂 𝐶𝑝,𝐻2𝑂 + 𝑥𝐶𝑂2 𝐶𝑝,𝐶𝑂2 + 𝑥𝐶𝑂 𝐶𝑝,𝐶𝑂 + 𝑥𝐻2 𝐶𝑝,𝐻2 + 𝑥𝑁2 𝐶𝑝,𝑁2
18
(35)
CH4
H2O
CO2
CO
H2
N2
O2
Table 10 Species Heat Capacity Coefficients [30]
b1
b2
b3
b4
-178.59
712.55
-1068.7
856.93
-41.205
146.01
-217.08
181.54
204.60
-471.33
657.88
-519.9
-8.1781
5.2062
41.974
-66.346
56.036
-150.55
199.29
-136.15
4.8987
-38.040
105.17
-113.56
-57.975
203.68
-300.37
231.72
b0
47.964
37.373
4.3669
30.429
21.157
29.027
34.850
b5
-358.75
-79.409
214.58
37.756
46.903
55.554
-91.821
b6
61.321
14.015
-35.992
-7.6538
-6.4725
-10.350
14.776
To determine the thermal conductivity of the fluid mixtures the method of Wassiljewa
was utilized [29] with the Mason and Saxena modification as suggested by Todd and
Young [30]. It is interesting to note that equations similar to those used to calculate
thermal conductivity in this study are used by Wilke in an alternate calculation for
mixture viscosity not used in this model [29]. Using the equations of Mason and Saxena,
the following applies for the thermal conductivity of the fluids.
𝜆𝑖 = 𝑐0 + 𝑐1 (𝑇⁄1000) + 𝑐2 (𝑇⁄1000)2 + 𝑐3 (𝑇⁄1000)3 + 𝑐4 (𝑇⁄1000)4
(36)
+ 𝑐5 (𝑇⁄1000)5 + 𝑐6 (𝑇⁄1000)6
𝑁
𝜆𝑚𝑖𝑥𝑡𝑢𝑟𝑒 = ∑
𝑖=1
1/2
∅𝑖𝑗 =
(37)
𝑥𝑖 𝜆𝑖
𝑛
∑𝑗=1 𝑥𝑗 ∅𝑖𝑗
[1 + (𝜇𝑖 / 𝜇𝑗 )
1/4 2
(𝑀𝑖 /𝑀𝑗 )
(38)
]
1/2
[8(1 + 𝑀𝑖 /𝑀𝑗 )]
∅𝑗𝑖 = ∅𝑖𝑗 (𝜇𝑗 𝑀𝑖 /𝜇𝑖 𝑀𝑗 )
(39)
Where 𝜆 is in W/m-K, and 𝜇𝑖 is the species dynamic viscosity in µPoise. Applying these
equations to the cathode fluid mixture yields the following equation set for the mixture
thermal conductivity.
𝜆𝑐𝑎𝑡ℎ𝑜𝑑𝑒 =
∅𝑂2,𝑁2
𝑥𝑂2 𝜆𝑂2
𝑥𝑁2 𝜆𝑁2
+
𝑥𝑂2 +𝑥𝑁2 ∅𝑂2,𝑁2 𝑥𝑂2 ∅𝑁2,𝑂2 + 𝑥𝑁2
[1 + (𝜇𝑂2 / 𝜇𝑁2 )1/2 (𝑀𝑂2 /𝑀𝑁2 )1/4 ]
=
[8(1 + 𝑀𝑂2 /𝑀𝑁2 )]1/2
19
2
(40)
(41)
∅𝑁2,𝑂2 = ∅𝑂2,𝑁2 (𝜇𝑁2 𝑀𝑂2 /𝜇𝑂2 𝑀𝑁2 )
(42)
In the anode the additional species in the gas mixture must be accounted for. In total
there are 6 possible species including: CH4, H2, H2O, CO, CO2, and N2. For a 6 species
mixture, equation (38) is combined with the following definitions to determine mixture
thermal conductivity.
𝜆𝑎𝑛𝑜𝑑𝑒 =
𝑥1 𝜆1 𝑥2 𝜆2 𝑥3 𝜆3 𝑥4 𝜆4 𝑥5 𝜆5 𝑥6 𝜆6
+
+
+
+
+
𝛿1
𝛿2
𝛿3
𝛿4
𝛿5
𝛿6
(43)
𝛿1 = 𝑥1 + 𝑥2 ∅12 + 𝑥3 ∅13 + 𝑥4 ∅14 + 𝑥5 ∅15
𝜇2 𝑀1
+ 𝑥2 + 𝑥3 ∅23 + 𝑥4 ∅24 + 𝑥5 ∅25
𝜇1 𝑀2
(45)
𝜇3 𝑀1
𝜇3 𝑀2
+ 𝑥2 ∅23
+ 𝑥3 + 𝑥4 ∅34 + 𝑥5 ∅35
𝜇1 𝑀3
𝜇2 𝑀3
(46)
𝜇4 𝑀1
𝜇4 𝑀2
𝜇4 𝑀3
+ 𝑥2 ∅24
+ 𝑥3 ∅34
+ x4 + x5 ∅45
𝜇1 𝑀4
𝜇2 𝑀4
𝜇3 𝑀4
(47)
𝜇5 𝑀1
𝜇5 𝑀2
𝜇5 𝑀3
𝜇5 𝑀4
+ 𝑥2 ∅25
+ 𝑥3 ∅35
+ 𝑥4 ∅45
+ x5
𝜇1 𝑀5
𝜇2 𝑀5
𝜇3 𝑀5
𝜇4 𝑀5
(48)
𝜇6 𝑀1
𝜇6 𝑀2
𝜇6 𝑀3
𝜇6 𝑀4
𝜇6 𝑀5
+ 𝑥2 ∅26
+ 𝑥3 ∅36
+ 𝑥4 ∅46
+ x5
+ x6
𝜇1 𝑀6
𝜇2 𝑀6
𝜇3 𝑀6
𝜇4 𝑀6
𝜇4 𝑀5
(49)
𝛿2 = 𝑥1 ∅12
𝛿3 = 𝑥1 ∅13
δ4 = 𝑥1 ∅14
δ5 = 𝑥1 ∅15
δ6 = 𝑥1 ∅16
CH4
H2O
CO2
CO
H2
N2
O2
(44)
c0
0.4796
2.0103
2.8888
-0.2815
1.5040
0.3216
-0.1857
Table 11 Species Thermal Conductivity Coefficients [30]
c1
c2
c3
c4
c5
1.8732
37.413
-47.440
38.251
-17.283
-7.9139
35.922
-41.390
35.993
-18.974
-27.018
129.65
-233.29
216.83
-101.12
13.999
-23.186
36.018
-30.818
13.379
62.892
-47.190
47.763
-31.939
11.972
14.810
25.473
38.837
32.133
13.493
11.118
-7.3734
6.7130
-4.1797
1.4910
c6
3.2774
4.1531
18.698
-2.3224
-1.8954
2.2741
-0.2278
5.1.2. Electrodes
In the electrodes (both the backing layers and ERLs) a modified version of the heat
equation is utilized which introduces the values of effective thermal conductivity and
20
effective specific heat capacity. These values are introduced to account for the electrode
porosity [16] [22] [10]. The heat generation source term 𝑄 is discussed in Section 5.2
𝑒𝑓𝑓
𝜌𝐶𝑝 𝒖∇𝑇 − ∇(𝜆𝑒𝑓𝑓 ∇𝑇) = 𝑄
(50)
𝜆𝑒𝑓𝑓 = 𝜀𝜆𝑓𝑙𝑢𝑖𝑑 + (1 − 𝜀)𝜆𝑠𝑜𝑙𝑖𝑑
(51)
𝑒𝑓𝑓
(52)
𝐶𝑝
= 𝜀𝐶𝑝,𝑓𝑙𝑢𝑖𝑑 + (1 − 𝜀)𝐶𝑝,𝑠𝑜𝑙𝑖𝑑
The subscript “fluid” is the calculated thermal conductivity and specific heat capacity of the
fluid mixture in the anode or cathode electrode using the methods described in the previous
section. The subscript “solid” indicates the thermal conductivity or specific heat capacity of the
solid phase of the anode or cathode provided in specified model parameters.
5.1.3. Electrolyte and Interconnects
The following form of the heat equation considering conduction heat transfer is used for
the electrolyte and interconnects where the thermal conductivity is a pre-defined value
taken from the literature.
−∇(𝜆∇𝑇) = 𝑄
(53)
5.2 Heat Generation Source Terms
5.2.1. Heat Generated by Reactions
The heat generated from the chemical reactions is entered into the model for the anode
flowfield, backing layer and ERL with the general form shown below. In these equation
𝑄 is in W/m3, ∆𝐻298 is in J/mol and 𝑟̇𝑟𝑥𝑛 in mol/m3-s.
𝑟𝑥𝑛
𝑟𝑥𝑛
∑ − (∆𝐻298 ∗ 𝑟̇𝑟𝑥𝑛 )
𝑗
21
(54)
The heat generated from the electrochemical reactions applies only in the
electrochemically reactive layers (ERLs) of the cell. The following is the general form
for these equations to be applied separately in the anode and cathode ERLs of the cell
where s = a or c (anode or cathode) [10].
∑ −𝑇∆𝑆𝑖
𝑖
𝑗𝑠 𝐴𝑣
(55)
𝑛𝑒,𝑖 𝐹
The reaction entropies
5.2.2. Ohmic and Overpotential Heat Generation
Joule heating or ohmic heat generation in SOFC modeling is heat generation from
resistance to ion or electron flow in the cell. Therefore these heat source terms apply in
each electrode (backing layer and ERL) as well as the electrolyte. The general form of
the Joule heating and overpotential heating equations respectively are shown below and
applied to the appropriate layers where s = a or c (anode or cathode) [16] [10]. In the
electrodes the calculated conductivity from Section 7.6 is utilized in the denominator of
the joule heating equation.
2
2
𝑗𝑖𝑜,𝑠
𝑗𝑒𝑙,𝑠
+
𝜎𝑒𝑙,𝑠 𝜎𝑖𝑜,𝑠
(56)
𝜂𝑎𝑐𝑡,𝑠 𝑗𝑠 𝐴𝑣
(57)
The summary of heat source terms and their respective domains is shown in the table
below.
Table 12 Summary of Heat Source Equations
Anode Flow Field
Anode Backing
Layer
𝑊𝐺𝑆
𝑄 = (−∆𝐻298
∗ 𝑟̇ 𝑊𝐺𝑆 )
𝑀𝑆𝑅
𝑊𝐺𝑆
𝑄 = (−∆𝐻298
∗ 𝑟̇ 𝑀𝑆𝑅 ) + (−∆𝐻298
∗ 𝑟̇ 𝑊𝐺𝑆 ) +
22
(58)
2
𝑗𝑒𝑙,𝑎
𝑒𝑓𝑓
𝜎𝑒𝑙,𝑏𝑎
(59)
𝑀𝑆𝑅
𝑊𝐺𝑆
𝑄 = (−∆𝐻298
∗ 𝑟̇ 𝑀𝑆𝑅 ) + (−∆𝐻298
∗ 𝑟̇ 𝑊𝐺𝑆 ) +
Anode ERL
+ 𝑇∆𝑆𝐻2
𝑗𝑎 𝐴𝑣
2𝐹
+ 𝑇∆𝑆𝐶𝑂
𝑗𝑎 𝐴𝑣
2𝐹
2
𝑗𝑒𝑙,𝑎
𝑒𝑓𝑓
𝜎𝑒𝑙,𝑎
+
2
𝑗𝑖𝑜,𝑎
𝑒𝑓𝑓
𝜎𝑖𝑜,𝑎
+ 𝜂𝑎𝑐𝑡,𝑎,𝐻2 𝑗𝑎 𝐴𝑣
(60)
+ 𝜂𝑎𝑐𝑡,𝑎,𝐶𝑂 𝑗𝑎 𝐴𝑣
𝑄=
Electrolyte
𝑄=
Cathode ERL
2
𝑗𝑖𝑜
𝜎𝑖𝑜
2
𝑗𝑒𝑙,𝑐
2
𝑗𝑖𝑜,𝑐
𝑗𝑐 𝐴𝑣
𝜎𝑒𝑙,𝑐
𝜎𝑖𝑜,𝑐
4𝐹
+
𝑒𝑓𝑓
+ 𝑇∆𝑆𝑂2
𝑒𝑓𝑓
Cathode Backing
𝑄=
Layer
Cathode Flow
Field
Interconnects
(61)
+ 𝜂𝑎𝑐𝑡,𝑐,𝑂2 𝑗𝑐 𝐴𝑣
2
𝑗𝑒𝑙,𝑐
𝑒𝑓𝑓
𝜎𝑒𝑙,𝑐
(62)
(63)
Q=0
(64)
Q=0
(65)
6 Chemical Model
6.1 Internal Reforming
When direct internal reforming of natural gas occurs in a fuel cell the methane is
converted via the catalyzed methane steam reforming reaction (MSR) (66), while
simultaneously the slightly exothermic water gas shift reaction (WGS) (67) occurs. Most
of the modeling currently available in the literature today only considers these two
reactions. Several researchers have also investigated the inclusion the methane carbon
dioxide reaction MCDR (68) [22]. This study considers the three as reactions shown
below.
𝐶𝐻4 + 𝐻2 𝑂 ↔ 3𝐻2 + 𝐶𝑂
∆𝐻298 = 206.1 𝑘𝐽/𝑚𝑜𝑙
(66)
𝐶𝑂 + 𝐻2 𝑂 ↔ 𝐻2 + 𝐶𝑂2
∆𝐻298 = −41.2 𝑘𝐽/𝑚𝑜𝑙
(67)
∆𝐻298 = 247 𝑘𝐽/𝑚𝑜𝑙
(68)
𝐶𝐻4 + 𝐶𝑂2 ↔ 2𝐻2 + 2𝐶𝑂
Within an internal reforming SOFC, the catalyzed methane steam reforming reaction
occurs at the surface of the catalyst throughout the anode electrode (backing layer and
23
ERL). The water gas shift reaction is assumed to take place both in the anode flow
channel and in the anode electrode. The methane carbon dioxide reaction would take
place in only the electrode.
Due to the fuel flexibility of DIR SOFCs, it is worth noting that for a generic
hydrocarbon gas CxHy the MSR can be written:
𝑦
𝐶𝑥 𝐻𝑦 + 𝑥𝐻2 𝑂 ↔ ( + 𝑥) 𝐻2 + 𝑥𝐶𝑂
2
(69)
6.2 Chemical Species Balance Equations
Based on the balanced reforming equations above, the rates of production and
consumption of each species, R rxn,i (kg⁄m3 s) is shown in Table 13 for the different
reactions where 𝑟̇𝑟𝑥𝑛 is given in(mol⁄m3 s).
𝑅𝑟𝑥𝑛,𝑅𝑒𝑎𝑐𝑡𝑎𝑛𝑡𝑠 = (−𝜐𝑖 )𝑀𝑖 𝑟̇𝑟𝑥𝑛 /1000
(70)
𝑅𝑟𝑥𝑛,𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑠 = (𝜐𝑖 ) 𝑀𝑖 𝑟̇𝑟𝑥𝑛 /1000
(71)
Table 13 Summary of Chemical Species Balance Equations used in Model
Description
Reactions
MSR
𝐶𝐻4 + 𝐻2 𝑂 ↔ 3𝐻2 + 𝐶𝑂
WGS
𝐶𝑂 + 𝐻2 𝑂 ↔ 𝐻2 + 𝐶𝑂2
MCDR
𝐶𝐻4 + 𝐶𝑂2 ↔ 2𝐻2 + 2𝐶𝑂
Species Balance Equations (kg/m3 ∙ s)
−𝑀𝐶𝐻4 𝑟̇𝑀𝑆𝑅 −𝑀𝐶𝐻4 𝑟̇𝑀𝐶𝐷𝑅
+
1000
1000
−𝑀𝐻2𝑂 𝑟̇𝑀𝑆𝑅 −𝑀𝐻2𝑂 𝑟̇𝑊𝐺𝑆
𝑅𝐻2𝑂 =
+
1000
1000
3𝑀𝐻2 𝑟̇𝑀𝑆𝑅 𝑀𝐻2 𝑟̇𝑊𝐺𝑆 2𝑀𝐻2 𝑟̇𝑀𝐶𝐷𝑅
𝑅𝐻2 =
+
+
1000
1000
1000
𝑀𝐶𝑂 𝑟̇𝑀𝑆𝑅 −𝑀𝐶𝑂 𝑟̇𝑊𝐺𝑆 2𝑀𝐶𝑂 𝑟̇𝑀𝐶𝐷𝑅
𝑅𝐶𝑂 =
+
+
1000
1000
1000
𝑀𝐶𝑂2 𝑟̇𝑊𝐺𝑆 −𝑀𝐶𝑂2 𝑟̇𝑀𝐶𝐷𝑅
𝑅𝐶𝑂2 =
+
1000
1000
𝑅𝐶𝐻4 =
24
6.3 Reforming Kinetics
In order to most accurately predict the concentrations of species along the cell length it is
necessary to obtain kinetic rate equations or appropriate reaction mechanism models for
all the reforming reactions above.
6.3.1. MSR Kinetics
According to Mogensen et al. [33] and Nagel et al. [7], the wide variation in kinetic
equations formulated for the steam reforming reactions are due to the different operating
conditions of the experiments that have been previously performed as well as conditions
not being allowed to reach steady state before data is taken. Nagel [7] notes that most of
the discrepancies in the kinetic equations are in the reaction orders for water. It is noted
that S/C ratios << 2 yield positive reaction orders, S/C ratios ~ 2 yield a zero reaction
order and larger S/C ratios > 2 yield a negative reaction order in the MSR steam
reforming kinetic equations. There are also studies which utilize the elementary steps of
the reactions to predict the reaction kinetics [34]. However for simplicity, this model will
focus on use of a rate equation for each of the reforming reactions noted above.
There are three types of kinetic expressions currently used for the MSR which include;
General Langmuir-Hinshelwood kinetics, First order reaction in methane, and Power law
expressions derived from data fitting [33]. In order to determine the appropriate kinetic
models for this study, the available models in the literature were reviewed.
In general Langmuir-Hinshelwood kinetic models (Type 1) for the reforming reaction
focus on the rate determining steps in species surface reactions and the generation of
kinetic equations is a direct result of which mechanistic reaction steps are assumed or
determined to be rate determining steps. In 1989 Xu and Froment proposed a steam
reforming reaction rate that was based on the rate determining step of the reaction of
adsorbed carbon and oxygen species utilizing the partial pressures of methane, water and
hydrogen [35] and their work has been utilized in many studies [21]. Similar to Xu and
Froment, Lehnert [36] and Haberman [37] also include a first order dependence on
25
water. According to Mogensen et. al. [33], the presence and thus the effect of the partial
pressure value for water in the numerator of these equations is not commonly observed
in experiments. Newer studies have been performed that identify the rate limiting step as
the dissociative adsorption of methane with a reaction order of 1 which is a generally
agreed upon approach. However there is disagreement on whether other rate limiting
steps in the mechanism should contribute to the kinetic models. Mogensen et al. also
notes that these kinetic equations are subject to the operating conditions as well since
different reaction steps become rate controlling depending on operating temperatures.
Thus, in order to utilize the Langmuir-Hinshelwood kinetic models, it is suggested to use
a model that was developed with similar operating conditions as the experiment or
model under development.
First order kinetic models (Type 2) are Langmuir-Hinshelwood kinetic models
considering only the methane dissociative adsorption as the rate determining step [12]
[38]. One of the most commonly utilized MSR kinetic equations used in Ni-YSZ SOFC
studies is that proposed by Achenbach [38] shown below. Although considering only
methane simplifies the equation and eliminates the concern for inconsistencies due to
additional rate limiting steps needing consideration, Mogensen suggests these rate
equations are only valid at high temperatures and low pressures.
𝑟̇𝑀𝑆𝑅 (𝑚𝑜𝑙 𝑚−3 𝑠 −1 ) = Av 𝑘1 𝑝𝐶𝐻4 𝑒𝑥𝑝 (−
𝐽
𝐸𝐴
)
𝑅𝑇
(72)
𝑘1 = 4274 𝑚𝑜𝑙 𝑠 −1 𝑚−2 𝑏𝑎𝑟 −1
𝐸𝐴 = 82000 𝑚𝑜𝑙
In the last set and most mathematically simple kinetic models commonly proposed for
the reforming reaction, a power law equation (Type 3) is fit to individual experimental
conditions by measuring catalytic reaction rates [39]. The general form for the power
law equation is [33]:
𝛽
𝛾
𝛼
𝛿
𝜀
−𝑟̇𝐶𝐻4 = 𝑘 𝑝𝐶𝐻4
𝑝𝐻2𝑂 𝑝𝐻2 𝑝𝐶𝑂2
𝑝𝐶𝑂
𝑒𝑥𝑝 (−
26
𝐸𝑎
)
𝑅𝑇
(73)
In Appendix B, some commonly used rate equations for Ni-YSZ MSR and WGS are
listed.
The equation developed by Haberman and Young was utilized in this study for the MSR
steam reforming kinetics in alignment with the use of the matching equation for the
WGS reaction. Partial pressure in this equation is entered as Pa.
𝑟̇𝑀𝑆𝑅 (𝑚𝑜𝑙 𝑚
−3 −1
𝑠
) = 2395 exp (−
231266
𝑅𝑇
) (𝑝𝐶𝐻4 𝑝𝐻2 𝑂 −
𝑝𝐶𝑂 𝑝𝐻
2
𝐾𝑒𝑞,1
3
)
(74)
𝐾𝑒𝑞,1 = 1.0267 ∗ 1010 𝑒𝑥𝑝(−0.2513𝑍4 + 0.3665𝑍3 + 0.5810𝑍2 − 27.134𝑍 + 3.2770)
6.3.2. WGS Kinetics
For the water gas shift reaction (WGS) less work has been performed to determine the
optimal kinetic equation. Many authors assume the shift reaction is at equilibrium. In
Appendix B, some commonly used kinetic rate equations for Ni-YSZ WGS are listed to
account for a non-equilibrium state. This study utilizes the equation presented by
Haberman and Young for WGS kinetics [37]. Partial pressures in this equation are
entered as Pa.
𝑟̇𝑊𝐺𝑆 (𝑚𝑜𝑙 𝑚−3 𝑠 −1 ) = 0.0171 exp (−
𝑝𝐻 𝑝𝐶𝑂2
103191
) (𝑝𝐻2 𝑂 𝑝𝐶𝑂 − 2
)
𝑅𝑇
𝐾𝑒𝑞,2
(75)
𝐾𝑒𝑞,2 = 𝑒𝑥𝑝(−0.2935𝑍 3 + 0.6351𝑍 2 + 4.1788𝑍 + 0.3169)
6.3.3. MCDR Kinetics
MCDR kinetic rate equations with regards to Ni-YSZ materials are difficult to find in
the literature and there are few fuel cell modeling studies considering the MCDR
reaction. In their model, Ni utilizes a Languir-Hinshelwood type equation that was taken
from experimental data of CO2 reforming of methane on Ru/Al2O3 catalyzed metallic
foam absorber [22]. Partial pressures in this equation are entered as atm.
27
𝑟̇𝑀𝐶𝐷𝑅 (𝑚𝑜𝑙 𝑚−3 𝑠 −1 ) = 1.17𝑥107 exp (−
𝐾𝐶𝑂2 𝐾𝐶𝐻4 𝑝𝐶𝑂2 𝑝𝐶𝐻4
83498
)
𝑅𝑇 (1 + 𝐾𝐶𝑂2 ∗ 𝑝𝐶𝑂2 + 𝐾𝐶𝐻4 ∗ 𝑝𝐶𝐻4 )2
(76)
49220
𝐾𝐶𝑂2 (𝑎𝑡𝑚−1 ) = 3.11𝑥10−3 𝑒𝑥𝑝 (
)
𝑅𝑇
16054
𝐾𝐶𝐻4 (𝑎𝑡𝑚−1 ) = 0.653 𝑒𝑥𝑝 (
)
𝑅𝑇
Verykios developed a kinetic equation based on dry reforming of methane over
Ni/La2O3 catalyst [40]. Partial pressures in this equation are in kPa.
𝑟̇𝑀𝐶𝐷𝑅 (𝑚𝑜𝑙/(𝑚3 ∗ 𝑠)) =
𝐾1 𝑘2 𝐾3 𝑘4 𝑝𝐶𝑂2 𝑝𝐶𝐻4
𝐾1 𝑘2 𝐾3 𝑝𝐶𝑂2 𝑝𝐶𝐻4 + 𝐾1 𝑘2 𝑝𝐶𝐻4 + 𝐾3 𝑘4 𝑝𝐶𝑂2
𝐾1 𝑘2 (𝑚𝑜𝑙 𝑔−1 𝑠 −1 𝑘𝑃𝑎−1 ) = 2.61𝑥10−3 𝑒𝑥𝑝 (−
(77)
4300
)
𝑇
8700
)
𝑇
−7500
𝑘4 (𝑚𝑜𝑙 𝑔−1 𝑠 −1 ) = 5.35𝑥10−1 𝑒𝑥𝑝 (
)
𝑇
𝐾3 (𝑘𝑃𝑎−1 ) = 5.17𝑥10−5 𝑒𝑥𝑝 (
6.3.4. DSR Kinetics
Another reaction that could potentially occur in internal reforming SOFCs is the direct
steam reforming or methanation reaction (DSR). It is well known that the methanation
reaction will only occur at temperatures below 675oC and thus this reaction is not
included in most modeling efforts. This assumption may not be correct however with the
large temperature gradients in the cell. When the DSR or methanation reaction is
included, the most popular kinetics rate equations are of the Languir-Hinshelwood type
by Xu and Froment [35] [21] and Hou and Hughes [41]. For simplification this study
does not include the DSR reaction which is shown below for reference.
𝐶𝐻4 + 2𝐻2 𝑂 ↔ 4𝐻2 + 𝐶𝑂2
28
(78)
6.3.5. Carbon Deposition
One of the major challenges in operating internal reforming solid oxide fuel cells is
coking, or carbon formation at the anode inlet. This is detrimental to SOFC performance
as the deposition of carbon particles (coking) on the anode surface can deactivate and
block the catalyst reducing cell performance, impede gas flow and put additional
mechanical stresses on the electrode. The governing reactions for carbon formation in
the fuel cell are via the methane cracking reaction (79) and the Boudouard reaction (80)
with reaction (81) being another probable pathway for carbon formation in the cell.
𝐶𝐻4 ↔ 2𝐻2 + 𝐶
(79)
2𝐶𝑂 ↔ 𝐶𝑂2 + 𝐶
(80)
𝐶𝑂 + 𝐻2 ↔ 𝐻2 𝑂 + 𝐶
(81)
To determine if the cell operating conditions were conducive to carbon formation the
following relationships can be assessed [42] where pressures are entered in Pa.
𝛼𝑐𝑎𝑟𝑏𝑜𝑛,𝐶𝐻4 = 4.161𝑥1010 𝑒𝑥𝑝 (−
10614 𝑝𝐶𝐻4
) 2
𝑇
𝑝𝐻2
2
20634 𝑝𝐶𝑂
𝛼𝑐𝑎𝑟𝑏𝑜𝑛,𝐶𝑂 = 5.744𝑥10−9 𝑒𝑥𝑝 (
)
𝑇
𝑝𝐶𝑂2
𝛼𝑐𝑎𝑟𝑏𝑜𝑛,𝐶𝑂−𝐻2 = 3.173𝑥10−7 𝑒𝑥𝑝 (
16318 𝑝𝐶𝑂 𝑝𝐻2
)
𝑇
𝑝𝐻2 𝑂
(82)
(83)
(84)
If the value of 𝛼𝑐𝑎𝑟𝑏𝑜𝑛 is greater than one the system is not at equilibrium and carbon can form
in the anode. If it is equal to one the system is at thermodynamic equilibrium and below one
carbon formation cannot occur.
6.4 Additional Chemical Model Information
Within the model it was necessary to convert from specified inlet mole fractions Xi to
mass fractions Yi then species partial pressures Pi in mixtures. Ideal gasses were
assumed and the following equations were utilized for the conversions.
29
𝑥𝑖 𝑀𝑖
)
𝑁
∑𝑖=1 𝑥𝑖 𝑀𝑖
(85)
𝑌𝑖
1
( 𝑁
)
𝑀𝑖 ∑𝑖=1 𝑦𝑖 /𝑀𝑖
(86)
𝑦𝑖 = (
𝑥𝑖 =
𝑝𝑖 = 𝑥𝑖 𝑝
(87)
7 Electrochemical Model
Within the fuel cell an electrochemical reaction occurs in which voltage and current are
produced when the anode is supplied with a hydrocarbon and the cathode is supplied
with oxygen usually in the form of air. The oxygen reacts with the catalyst to produce
oxygen ions which migrate through the electrolyte to the anode. On the anode, hydrogen
or carbon monoxide in the fuel stream reacts with the oxide ions (O2-), producing either
water or carbon dioxide while depositing electrons onto the anode. These electrons pass
through the electrode externally to the fuel cell through the load then return to the
cathode.
7.1 Approaches to Electrochemical Modeling
In the literature there are two common approaches utilized in modeling the
electrochemistry of a solid oxide fuel cell. These include 1) the distributed charge
transfer approach and 2) the stepwise subtractive polarization approach. In approach 1,
which is utilized in this study, the charge continuity equation is combined with Ohm’s
law for a balance on the electrochemically active layers of the cell and the potential
gradient across the cell is locally calculated with the ground (V=0) and cell voltage
(V=Ecell) are boundary conditions at the anode and cathode current collection points. The
Butler-Volmer equation is then used to calculate the cell current density as a function of
the overpotential. The overpotential is calculated from the local potentials in the cell
determined from the charge continuity balances [16] [10] [31].
30
In approach 2 the cell voltage is defined as a function of the reversible voltage and the
combined overpotentials as shown below. Typically, the Butler-Volmer equation is
rearranged in terms of the activation overpotential such that it is a function of current
density, and equations identified to calculate the concentration and ohmic overpotential
contributions. Following this, the equation below can be rearranged as a function of
current density. When half cell potentials are considered such as in H2 and CO oxidation
the cell voltage in approach 2 is related by the equation (89) below [20] [21] [22] [10].
𝑟𝑒𝑣
𝐸𝑐𝑒𝑙𝑙 = 𝐸𝑐𝑒𝑙𝑙
− 𝜂𝑜ℎ𝑚 − 𝜂𝑐𝑜𝑛𝑐 − 𝜂𝑎𝑐𝑡
(88)
𝑟𝑒𝑣
𝐸𝑐𝑒𝑙𝑙 = 𝐸𝑎,𝐻2
− 𝜂𝑎𝑐𝑡,𝐻2 − 𝜂𝑎𝑐𝑡,𝑂2 − 𝜂𝑜ℎ𝑚 − 𝜂𝑐𝑜𝑛𝑐
(89)
𝑟𝑒𝑣
= 𝐸𝑎,𝐶𝑂
− 𝜂𝑎𝑐𝑡,𝐶𝑂 − 𝜂𝑎𝑐𝑡,𝑂2 − 𝜂𝑜ℎ𝑚 − 𝜂𝑐𝑜𝑛𝑐
7.2 Electrochemical Species Balance Equations
Each of electrochemical reaction equations considered in the cell model are shown
below. It has been found that the overall electrochemical reaction rate can vary up to
50% with the oxidation of CO when compared with only the oxidation of hydrogen [21].
For completeness, carbon monoxide oxidation was included in this study.
The following reduction reaction occurs at the cathode ERL of the fuel cell.
1
𝑂 + 2𝑒 − → 𝑂2−
2 2
(90)
And the oxidation reactions occurring at the anode ERL of the fuel cell are:
𝐻2 + 𝑂2− → 𝐻2 𝑂 + 2𝑒 −
(91)
𝐶𝑂 + 𝑂2− → 𝐶𝑂2 + 2𝑒 −
(92)
The overall electrochemical reaction in this study is thus:
𝐻2 + 𝐶𝑂 + 𝑂2 → 𝐻2 𝑂 + 𝐶𝑂2
(93)
This paper does not consider the following oxidation reaction of methane in the anode.
𝐶𝐻4 + 4𝑂2− → 2𝐻2 𝑂 + 𝐶𝑂2 + 8𝑒 −
31
(94)
The rates of species consumption due to electrochemical reaction can be represented by
the following equations with the final equation set utilized in this model shown in Table
14.
𝑅𝑒𝑟𝑥𝑛,𝑅𝑒𝑎𝑐𝑡𝑎𝑛𝑡𝑠 = (−𝜐𝑖 )𝑀𝑖 𝑗𝑖 𝐴𝑣 /𝑛𝑒 𝐹
(95)
𝑅𝑒𝑟𝑥𝑛,𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑠 = (𝜐𝑖 ) 𝑀𝑖 𝑗𝑖 𝐴𝑣 /𝑛𝑒 𝐹
(96)
Table 14 Summary of Electrochemical Species Balance Equations used in Model
Species Balance Equations (kg/m3 ∙ s)
Description
Reactions
O2 Red
𝑂2 + 4𝑒 − → 2𝑂2−
H2 Ox
𝐻2 + 𝑂2− → 𝐻2 𝑂 + 2𝑒 −
CO Ox
𝐶𝑂 + 𝑂2− → 𝐶𝑂2 + 2𝑒 −
Overall
𝐻2 + 𝐶𝑂 + 𝑂2 → 𝐻2 𝑂 + 𝐶𝑂2
𝑅𝑂2 =
−𝑀𝑂2 𝑗𝑂2 𝐴𝑣
4𝐹(1000)
𝑅𝐻2𝑂 =
𝑀𝐻2𝑂 𝑗𝐻2 𝐴𝑣
2𝐹(1000)
𝑅𝐻2 = −
𝑀𝐻2 𝑗𝐻2 𝐴𝑣
2𝐹(1000)
𝑅𝐶𝑂 =
−𝑀𝐶𝑂 𝑗𝐶𝑂 𝐴𝑣
2𝐹(1000)
𝑅𝐶𝑂2 =
𝑀𝐶𝑂2 𝑗𝐶𝑂 𝐴𝑣
2𝐹(1000)
7.3 Ion and Charge Transfer
For a conducting material, the general equation for the continuity of current and
applicable form of Ohm’s Law combine into the following charge balance equation for
the electron and ion conducting phases in a fuel cell [16].
𝜕𝜌
+ ∇ ∙ 𝒋𝒔 = 𝑅𝑠
𝜕𝑡
Where,
𝜕𝜌
𝜕𝑡
(97)
represents the time dependent charge density, 𝒋𝒔 the current density of the cell
layer where the subscript s = a, c, or e for anode or cathode and electrolyte and 𝑅𝑠 the
faradaic charge transfer rate (electrical current density source/sink term) for layer s
where 𝑅𝑠 = 𝑗𝑠 𝐴𝑣 and 𝑗𝑠 is the current density (A/m2). Assuming a steady state case (no
time derivative) then substituting Ohms law into the continuity equation, it can be
32
rewritten in terms of effective conductivity (𝜎𝑠𝑒𝑓𝑓 ) and potential (𝜙𝑠 ) for application
towards ionic (io) or electronic (el) potentials in the electrochemically active layers of
the fuel cell [16].
𝑒𝑓𝑓
∇ ∙ (−𝜎𝑠 ∇𝜙𝑠 ) = 𝑗𝑠 𝐴𝑣
(98)
Utilizing the electrochemically active surface area to volume ratio 𝐴𝑣 (m2/m3) is one of
two main methods to account for the conducting particle characteristics in the equation
above while ensuring the right hand side of the equation reflects a volumetric current
density source or sink term. The other approach is to multiply the exchange current or
current density which can also be calculated in (A/m) by the triple phase boundary
length 𝑙 𝑇𝑃𝐵 (m/m3). Both of these parameters 𝐴𝑣 and 𝑙 𝑇𝑃𝐵 , are a function of conducting
particle micro characteristics such as particle radii, volume fraction of particles in
reactive layer, particle coordination numbers, and reactive layer porosity. Both of these
equations are presented in Shi et al. [26]. The conducting particle characteristics are
included in this study via use of the electrochemically active surface are per unit volume
and presented by Kishimoto [23].
7.3.1. Electrode Backing Layers
In the electrode backing layers, there is neither ion transfer, nor electrochemical reaction,
only transfer of electrons via conduction which can be modeled with the following
equations.
𝑒𝑓𝑓
(99)
𝑒𝑓𝑓
(100)
∇ ∙ (−𝜎𝑒𝑙,𝑏𝑎 ∇𝜙𝑒𝑙,𝑎 ) = 0
∇ ∙ (−𝜎𝑒𝑙,𝑏𝑐 ∇𝜙𝑒𝑙,𝑐 ) = 0
𝑒𝑓𝑓
𝑒𝑓𝑓
In these equations, 𝜎𝑒𝑙,𝑏𝑎
and 𝜎𝑒𝑙,𝑏𝑐
are the anode and cathode backing layer effective
electrical conductivities, 𝜙𝑒𝑙,𝑎 and 𝜙𝑒𝑙,𝑐 are the anode and cathode electrical potentials.
The electronic potentials are the dependent variables assigned to the different domains,
so it is not necessary to identify the electrode with a subscript as shown in Table 15.
33
7.3.2. Electrochemical Reaction Layers (ERL)
One of the assumptions in this model is that the electrochemical reactions occur within a
defined electrochemical reaction layer (ERL) adjacent on either side of the electrolyte.
To satisfy this, not only is electron transfer occurring in the ERL but there is a transfer of
ionic species participating in the electrochemical reaction. The mechanism of electron
and ion transfer through the electrodes is modeled by the following equations [16] [18]
[28].
𝑒𝑓𝑓
(101)
𝑒𝑓𝑓
(102)
∇ ∙ (−𝜎𝑒𝑙,𝑎 ∇𝜙𝑒𝑙,𝑎 ) = 𝑗𝑎 𝐴𝑣
∇ ∙ (−𝜎𝑖𝑜,𝑎 ∇𝜙𝑖𝑜,𝑎 ) = 𝑗𝑎 𝐴𝑣
𝑒𝑓𝑓
(103)
𝑒𝑓𝑓
(104)
∇ ∙ (−𝜎𝑒𝑙,𝑐 ∇𝜙𝑒𝑙,𝑐 ) = −𝑗𝑐 𝐴𝑣
∇ ∙ (−𝜎𝑖𝑜,𝑐 ∇𝜙𝑖𝑜,𝑐 ) = −𝑗𝑐 𝐴𝑣
𝑒𝑓𝑓
𝑒𝑓𝑓
In these equations, 𝜎𝑖𝑜,𝑎
and 𝜎𝑖𝑜,𝑐
are the anode and cathode reaction layer effective ionic
conductivities, 𝜙𝑖𝑜,𝑎 and 𝜙𝑖𝑜,𝑐 are the anode and cathode reaction layer ionic potentials.
The ionic and electronic potentials are the dependent variables assigned to the different
domains, so it is not necessary to identify the electrode with a subscript as shown in
Table 15.
7.3.3. Electrolyte
The electrolyte layer of a SOFC is a dense solid that conducts the oxide ions from the
cathode to the anode. The applicable ion transfer equation for this layer is:
∇ ∙ (−𝜎𝑖𝑜,𝑒 ∇𝜙𝑖𝑜,𝑒 ) = 0
(105)
In this equation, 𝜎𝑖𝑜,𝑒 and 𝜙𝑖𝑜,𝑒 are the electrolyte ionic conductivity and electrolyte ionic
potential. The final form of this equation is shown in Table 15.
34
Table 15 Summary of Charge Transfer Equations
𝑒𝑓𝑓
(106)
𝑒𝑓𝑓
(107)
∇ ∙ (−𝜎𝑒𝑙,𝑏𝑎 ∇𝜙𝑒𝑙 ) = 0
Electrode Backing Layers
∇ ∙ (−𝜎𝑒𝑙,𝑏𝑐 ∇𝜙𝑒𝑙 ) = 0
Anode ERL
𝑒𝑓𝑓
∇ ∙ (−𝜎𝑒𝑙,𝑎 ∇𝜙𝑒𝑙 ) = 𝑗𝑎 𝐴𝑣
𝑒𝑓𝑓
∇ ∙ (−𝜎𝑖𝑜 ∇𝜙𝑖𝑜 ) = 𝑗𝑎 𝐴𝑣
Cathode ERL
(108)
(109)
𝑒𝑓𝑓
(110)
∇ ∙ (−𝜎𝑖𝑜 ∇𝜙𝑖𝑜 ) = −𝑗𝑐 𝐴𝑣
𝑒𝑓𝑓
(111)
∇ ∙ (−𝜎𝑖𝑜 ∇𝜙𝑖𝑜 ) = 0
(112)
∇ ∙ (−𝜎𝑒𝑙,𝑐 ∇𝜙𝑒𝑙 ) = −𝑗𝑐 𝐴𝑣
Electrolyte
7.4 Cell Voltage
The open circuit or reversible Nernst voltage for any fuel cell is the theoretical
maximum voltage the cell could achieve given a specific set of operating conditions. The
true voltage of the electrochemical cell will not however be equivalent to the open
circuit voltage during operation. The true cell voltage 𝐸𝑐𝑒𝑙𝑙 , is the open circuit voltage
𝐸 𝑟𝑒𝑣 , minus the internal cell resistances and losses. As the current is drawn from a cell,
the voltage will drop due to the presence of ohmic, concentration and activation losses.
Each of these losses contributes to the total heat produced in the fuel cell and there are
many approaches in the literature on how to calculate these losses or polarizations. A
pictoral representation of the relationship between ideal open circuit voltage and true cell
voltage is shown in Figure 2.
35
Figure 2 Relationship between Ideal and True Cell Voltages [3]
The cell voltage is satisfied in this model by setting boundary conditions such that 𝜙𝑒𝑙 =
0 in the cathode backing layer (where it interfaces with the anode flow channel) and
setting 𝜙𝑒𝑙 = 𝐸𝑐𝑒𝑙𝑙 in the anode backing layer of the cell where 𝐸𝑐𝑒𝑙𝑙 is varied from -1 to
-0.4 [18]. Thus, in terms of local potentials, the cell voltage can be written as:
𝐸𝑐𝑒𝑙𝑙 = 𝜙𝑒𝑙,𝑐 − 𝜙𝑒𝑙,𝑎
(113)
7.5 Activation Losses
The potential of an electrode directly affects the kinetics of the surface reactions. In
electrochemical reactions the activation energy not only includes thermal energy barriers
as such in chemical reactions, it must also overcome an electric potential barrier. At low
current density and operating temperatures the activation losses may significantly affect
the total voltage of the cell as the reactants must overcome this activation energy barrier
for the reactions to proceed at the electrodes.
There are two general approaches to determining the relationship between the current
density drawn to the activation overpotential (also referred to as activation polarization).
The first method is more complex and involves consideration of the detailed multistep
36
elementary reactions on the catalyst for each overall oxidation and reduction equation.
As an example, the overall oxidation of methane on nickel catalyst can be broken down
into 42 separate irreversible reactions to be considered [34]. For simplicity this study
uses the second method.
The second method assumes a single charge transfer reaction or rate limiting reaction
step and is described by using the well known Butler-Volmer form of the currentoverpotential equation where the general form written in terms of current density (𝑗 =
𝑖/𝐴) is shown below along with the commonly used form for exchange current density
[7] [43].
𝛼𝑎 𝐹 𝜂𝑎𝑐𝑡
−𝛼𝑐 𝐹 𝜂𝑎𝑐𝑡
𝑗 = 𝑗𝑜 [exp (
) − 𝑒𝑥𝑝 (
)]
𝑅𝑇
𝑅𝑇
𝑚
𝑛
𝑝𝑜𝑥
𝑝𝑟𝑒𝑑
𝐸𝑎
𝑗𝑜 = 𝛾 (
) (
) 𝑒𝑥𝑝 ( )
𝑝𝑟𝑒𝑓
𝑝𝑟𝑒𝑓
𝑅𝑇
(114)
(115)
Where 𝑗 is current density (A/m2), 𝑗𝑜 is the exchange current density (A/m2), 𝜂𝑎𝑐𝑡 is the
cell activation overpotential (V), F is Farady’s constant, 𝛾 is the pre-exponential exchange
current value, 𝑚 𝑜𝑟 𝑛 the reaction order of the oxidized or reduced species in the
respective reaction, 𝐸𝑎 is the activation energy for the electrochemical reaction (J/mol). In the
general Butler-Volmer equation above, α is the transfer coefficient or symmetry factor
for each electrode and when applied to half cell reactions becomes the forward and
backward reaction symmetry factors. The transfer coefficient is used to determine the
contribution of the anode and cathode currents to the total current.
To solve for the cell current density the local activation potentials also need to be
defined. The general form for activation potential in an electrode can be defined as
follows where 𝐸𝑒𝑞 is the local (s = anode or cathode) electric potential at equilibrium.
𝜂𝑎𝑐𝑡 = (𝜙𝑒𝑙 − 𝜙𝑖𝑜 ) − 𝐸𝑒𝑞
37
(116)
The reversible Nernst potential will be utilized in lieu of the equilibrium or reference
state potential [17] [18] [43].
Now that the general forms of the Butler-Volmer, exchange current density and
activation overpotential equations have been identified they will be applied to each of
the three half cell reactions in the ERL’s where ji is the half cell local faradaic current
density (A/m2), j0,s,i are the half cell exchange current densities for the each species.
For hydrogen oxidation the following equations are used in this study.
2𝐹 𝜂𝑎𝑐𝑡,𝑎,𝐻2
−𝐹 𝜂𝑎𝑐𝑡,𝑎,𝐻2
𝑗𝐻2 = 𝑗𝑜,𝑎,𝐻2 [exp (
) − 𝑒𝑥𝑝 (
)]
𝑅𝑇
𝑅𝑇
𝑗𝑜,𝑎,𝐻2
𝑝𝐻2 𝑝𝐻2𝑂
= 𝛾𝑎, (
)(
)
𝑝𝑟𝑒𝑓
𝑝𝑟𝑒𝑓
−0.5
𝑒𝑥𝑝 (−
𝐸𝑎,𝑎
)
𝑅𝑇
𝑟𝑒𝑣
𝜂𝑎𝑐𝑡,𝑎,𝐻2 = 𝜙𝑒𝑙,𝑎 − 𝜙𝑖𝑜,𝑎 − 𝐸𝑎,𝐻2
𝑟𝑒𝑣
𝑜
𝐸𝑎,𝐻2
= −𝐸𝐻2
−
(117)
(118)
(119)
𝑝𝐻 𝑂
𝑅𝑇
ln ( 2 )
2𝐹
𝑝𝐻2
(120)
𝑜
𝐸𝐻2
= 1.253 − 2.4516 ∙ 10−4 𝑇
For carbon monoxide oxidation the following equations are used in this study.
2𝐹 𝜂𝑎𝑐𝑡,𝑎,𝐶𝑂
−𝐹 𝜂𝑎𝑐𝑡,𝑎,𝐶𝑂
𝑗𝐶𝑂 = 𝑗𝑜,𝑎,𝐶𝑂 [exp (
) − 𝑒𝑥𝑝 (
)]
𝑅𝑇
𝑅𝑇
−0.5
𝑗𝑜,𝑎,𝐶𝑂
𝑝𝐶𝑂
𝑝𝐶𝑂2
= 𝛾𝑎,𝐶𝑂 (
)(
)
𝑝𝑟𝑒𝑓 𝑝𝑟𝑒𝑓
𝑒𝑥𝑝 (−
𝐸𝑎,𝑎
)
𝑅𝑇
𝑟𝑒𝑣
𝜂𝑎𝑐𝑡,𝑎,𝐶𝑂 = 𝜙𝑒𝑙,𝑎 − 𝜙𝑖𝑜,𝑎 − 𝐸𝑎,𝐶𝑂
𝑟𝑒𝑣
𝑜
𝐸𝑎,𝐶𝑂
= −𝐸𝐶𝑂
−
𝑅𝑇
𝑝𝐶𝑂2
ln (
)
2𝐹
𝑝𝐶𝑂
𝑜
𝐸𝐶𝑂
= 1.46713 − 4.527 ∙ 10−4 𝑇
For oxygen reduction the following equations are used in this study.
38
(121)
(122)
(123)
(124)
𝑗𝑂2 = 𝑗𝑜,𝑐,𝑂2 [exp (
𝑗𝑜,𝑐,𝑂2
−2𝐹 𝜂𝑎𝑐𝑡,𝑐,𝑂2
2𝐹 𝜂𝑎𝑐𝑡,𝑐,𝑂2
) − 𝑒𝑥𝑝 (
)]
𝑅𝑇
𝑅𝑇
𝑝𝑂2
= 𝛾𝑐 (
)
𝑝𝑟𝑒𝑓
0.25
𝑒𝑥𝑝 (−
𝐸𝑎,𝑐
)
𝑅𝑇
𝑟𝑒𝑣
𝜂𝑎𝑐𝑡,𝑐,𝑂2 = 𝜙𝑒𝑙,𝑐 − 𝜙𝑖𝑜,𝑐 − 𝐸𝑐,𝑂2
𝑟𝑒𝑣
𝐸𝑐,𝑂2
=
𝑅𝑇
𝑝𝑂2
ln (
)
4𝐹
𝑝𝑟𝑒𝑓
(125)
(126)
(127)
(128)
The current density of the cell can be evaluated as either the ion current through the
electrolyte or by the charge transfer rates in either the anode or cathode yielding the
following relations.
𝑗𝑐 = 𝑗𝑂2
𝑗𝑎 = 𝑗𝐻2 + 𝑗𝐶𝑂
(129)
7.5.1. Electrochemical Activation Energies
Varying values have been used to represent the pre-exponential exchange current value 𝛾
and the electrochemical activation energies 𝐸𝑎 utilized in the exchange current density
equation. Nagel et al. performed a sensitivity analysis and found that the activation
energy had a considerable effect on both cell temperature distributions and power output
[7]. The pre-exponential values and 𝐸𝑎 utilized in this study was chosen to most closely
match experimental data in the literature.
7.6 Ohmic Losses
The ohmic losses in a fuel cell are due to the ionic resistance in the electrolyte combined
with the resistance for the electrons passing through the electrodes and current
collectors. Ohmic losses through an electrolyte can be reduced by decreasing the
thickness of the electrolyte or increasing its ionic conductivity. The ohmic loss which is
the potential difference across the electrolyte is included in this study via the charge
continuity equations as the effective conductivity term presented previously for each
layer. To calculate the effective conductivities the following Arrhenius form equations
are utilized [10].
39
𝜎𝑒𝑙,𝑎 =
𝜎𝑒𝑙,𝑐 =
95𝑥106
𝑇
42𝑥106
𝑇
exp (−
exp (−
1150
) for Ni-YSZ
(130)
) for LSM-YSZ
(131)
10300
(132)
𝑇
1200
𝑇
𝜎𝑖𝑜 = 3.34 𝑥104 exp (−
𝑇
) for YSZ
The temperature based calculated electronic and ionic conductivities above are utilized
to calculate the effective conductivity values for use in the charge transfer model. The
symbol 𝑉𝑒𝑙 𝑜𝑟 𝑖𝑜 is the volume fraction of electron or ion conducting particles [16].
Table 16 Summary of Effective Conductivity Equations
𝑒𝑓𝑓
𝜎𝑒𝑙,𝑏𝑎 = 𝜎𝑒𝑙,𝑎 (
Electrode Backing Layers
1 − 𝜀𝑎
)
𝜏𝑒𝑙,𝑎
1 − 𝜀𝑐
𝑒𝑓𝑓
𝜎𝑒𝑙,𝑏𝑐 = 𝜎𝑒𝑙,𝑐 (
)
𝜏𝑒𝑙,𝑐
Anode ERL
1 − 𝜀𝑎
𝑒𝑓𝑓
(134)
𝜎𝑒𝑙,𝑎 = 𝜎𝑒𝑙,𝑎 (
) 𝑉𝑒𝑙,𝑎
𝜏𝑒𝑙,𝑎
(135)
1 − 𝜀𝑎
) 𝑉𝑖𝑜,𝑎
𝜏𝑖𝑜,𝑎
(136)
1 − 𝜀𝑐
) 𝑉𝑒𝑙,𝑐
𝜏𝑒𝑙,𝑐
(137)
𝑒𝑓𝑓
𝜎𝑖𝑜,𝑎 = 𝜎𝑖𝑜 (
Cathode ERL
(133)
𝑒𝑓𝑓
𝜎𝑒𝑙,𝑐 = 𝜎𝑒𝑙,𝑐 (
𝑒𝑓𝑓
1 − 𝜀𝑐
𝜎𝑖𝑜,𝑐 = 𝜎𝑖𝑜 (
) 𝑉𝑖𝑜,𝑐
𝜏𝑖𝑜,𝑐
(138)
𝜎𝑖𝑜
(139)
Electrolyte
7.7 Concentration Losses
In addition to the activation losses, the concentration losses at each electrode must be
considered. Concentration losses are due to the physical variation in species from the
flow channels to the ERL where the electrochemical reaction occurs. This location is
assumed to be at the interface between the electrolyte and the electrode. They can be due
to diffusion of species through the cell surfaces from the bulk flow path or the transport
of species through the electrodes. The losses in the cathode are typically small when
compared to losses at the anode. In this model, the concentrations at the boundary
40
between the electrodes and electrolyte (ERL) are handled by creating separate
electrochemical reaction layers. Within these layers the partial pressures are calculated
for the present species using the Maxwell Stefan approach outlined in section 4 Mass
Transfer. Based on this approach, including the concentration overpotential term in
calculation of the cell potential is not required [21] [22].
8 Simulation Validation
In order to ensure the accuracy of the model, a validation test was performed using the
conditions outlined in Table 17.
Table 17 Simulation Validation Operating Conditions
Inlet Temperature T
Operating Pressure P
1023 K
1.0 bar
Air feed molar fractions
Fuel Feed molar
fractions
Cathode Inlet Velocity
Anode Inlet Velocity
Operating Voltage
13 m/s
5 m/s
0.7 V
Steam/Carbon Ratio
Initial fuel feed
.21 O2 .79 N2
.171 CH4, .029 CO,
.493 H2O, .263 H2,
.044 CO2
2
30% pre-reformed
Fuel recycle ratio
Fuel Utilization Ufuel
75%
Below are various species compositions found in the SOFC literature.
H2
H2O
CO
CO2
CH4
N2
Table 18 Fuel Feed Mole Fractions from SOFC Literature
30%
0.2686
0.4875
0.0240
0.0491
0.1707
0
[17]
30%
0.263
0.493
0.029
0.044
0.171
0
[36]
9 Results
The results of the study are separated into two primary analysis sections. The first
considers the model without electrochemistry and studies the chemical reaction kinetics,
species distribution and temperature distributions. The second considers the previous
41
model combined with the electrochemical sub-model and studies the species distribution,
temperature distribution, distributed current density, polarization curves, and reviews the
probability of carbon formation in the cell.
9.1 Kinetics
A preliminary study was performed to determine the rate of the MCDR reaction at the
simulation operating conditions. It was found that at the simulation conditions outlined
in Table 5 and Table 6 with the calculated reaction rates shown in Figure 3 the reaction
rate is small enough to be neglected due to the low amounts of CH4 and CO2 in the cell.
0.0010
MCDR Reaction Rate mol/m3s
0.0009
0.0008
0.0007
0.0006
Ni/La2O3
0.0005
Ru/Al2O3
0.0004
0.0003
0.0002
0.0001
0.0000
50%
40%
30%
20%
% Pre-Reformed Fuel
Figure 3 MCDR Reaction Rate at Study Operating Conditions
Rate of MSR Reaction
Rate of WGS Reaction
Species Composition (No Electrochemistry)
Temperature
42
9.2 Electrochemistry
Species Composition (incl Electrochemistry)
Temperature
Current Density in Flow Direction
Potential vs Current Density
Probability of Carbon Formation
10 Conclusion
11 Future Work
Inclusion of elementary reaction mechanisms or chemical kinetics equation that
considers reaction surface area inside porous anode, use of alternative fuels, modifying
anode with materials to reduce carbon formation, scale up to stack and system basis.
12 Notation
𝑎
Anode (subscript)
α
Transfer symmetry coefficient
𝐴𝑣
Electrochemically reactive surface area per unit volume (m2/m3)
ba
Anode backing layer (subscript)
bc
Cathode backing layer (subscript)
𝐶𝑝
Specific Heat Capacity (J/kg-K)
c
Cathode (subscript)
𝑑𝑝𝑜𝑟𝑒
𝑀𝑆
𝐷𝑖𝑗
e
Pore Diameter
Maxwell-Stefan Diffusivity
Electrolyte (subscript)
43
el
Electrical (subscript) used with potentials
𝐸𝑎
Activation Energy
𝑟𝑒𝑣
𝐸𝑐𝑒𝑙𝑙
Reversible Nernst Open Circuit Cell Potential (V)
𝑟𝑒𝑣
𝐸𝑠,𝑖
Reversible Nernst Half Cell Potential for species i (s = a, c) (V)
𝐸𝑜
Standard Potential (V)
𝐸𝑐𝑒𝑙𝑙
Actual Cell Potential (V)
ERL
Electrochemical Reaction Layer
𝜀
Porosity
F
Faraday’s Constant (9.64853 x 104 C/mol)
io
Ionic (subscript) used with potentials
𝑗
Current Density (A/m2)
𝑗𝑜
Exchange current density
𝑘𝑜
Pre-exponential factor
𝐾𝑝
Equilibrium constant
κ
Permeability of porous medium (m2)
𝜆
Thermal Conductivity (W/m-K)
𝑀i
Molecular weight of species i
𝑛𝑒
Number of electrons transferred in rate limiting step
𝜂𝑎𝑐𝑡
Activation Overpotential
𝜎
Conductivity (S/m)
𝑝𝑖
Partial pressure of species i
𝜌
Density (kg/m3)
R
Gas Constant (8.3145 J/mol-K)
S/C
Steam to carbon ratio: molar ratio of steam to atomic carbon in fuel
𝜏
Tortuosity
T
Temperature
𝜙𝑠
Actual half-cell or electrolyte potential (s = a, c, e) (V)
𝐮
Velocity Vector (m/s)
44
μ
Dynamic viscosity (Pa-s)
𝜐𝑖
Stoichiometric coefficient of species i
𝑦𝑖
Mass fraction of species i
Z
Chemical reaction index value [Z=1000/(T(K)-1)]
45
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50
14 Appendix A
Below are species compositions for reformed methane fuel from 100% pre-reforming
prior to entering the SOFC fuel cell to 20% pre-reforming based on the data provided by
Recknagle et al. from the Pacific Northwest Laboratory [28]. The mass fractions are
shown in the first table and the corresponding mole fractions in the second table.
Table 19 Fuel Feed Mass Fractions for Range of Pre-reformed Percentages [28]
100%
90%
80%
70%
60%
50%
40%
30%
20%
H2
0.538
0.503
0.465
0.423
0.376
0.324
0.264
0.196
0.118
H2O
0.181
0.205
0.232
0.262
0.295
0.333
0.377
0.428
0.488
CO
0.127
0.112
0.096
0.08
0.064
0.049
0.033
0.02
0.008
CO2
0.052
0.056
0.059
0.061
0.062
0.061
0.058
0.052
0.042
CH4
0
0.019
0.039
0.06
0.084
0.11
0.137
0.168
0.202
0.101
0.105
0.109
0.114
0.118
0.124
0.129
0.136
0.142
N2
Table 20 Fuel Feed Mole Fractions for Range of Pre-reformed Percentages [28]
100%
90%
80%
70%
60%
50%
40%
30%
20%
H2
0.08298
0.07487
0.06664
0.05828
0.04975
0.04101
0.03201
0.02269
0.01295
H2O
0.24993
0.27283
0.29676
0.32193
0.34861
0.37713
0.40796
0.44169
0.47913
CO
0.27298
0.23151
0.19164
0.15369
0.11809
0.08536
0.05621
0.03158
0.01272
CO2
0.17663
0.18123
0.18332
0.18238
0.17777
0.16864
0.15389
0.13204
0.10111
CH4
0
0.02207
0.04415
0.06622
0.08829
0.11037
0.13244
0.15451
0.17659
0.21749
0.21749
0.21749
0.21749
0.21749
0.21749
0.21749
0.21749
0.21749
N2
51
15 Appendix B
Table 21 Sensitivity Analysis of Calculated Diffusion Coefficients
Temperature
(K)
1073.15
873.15
1273.15
1073.15
1073.15
1073.15
1073.15
1073.15
Porosity
0.5
0.5
0.5
0.3
0.3
0.5
0.5
0.5
Tortuosity
2
2
2
2
5
5
2
2
Pore
Diameter
(µm)
1
1
1
1
1
1
1
2
Pressure
(Pa)
101325
101325
101325
101325
101325
101325
202650
101325
CH4-CO
2.041E-4
1.423E-4
2.752E-4
2.041E-4
2.041E-4
2.041E-4
1.020E-4
2.041E-4
CH4-H2O
2.466E-4
1.719E-4
3.326E-4
2.466E-4
2.466E-4
2.466E-4
1.233E-4
2.466E-4
CH4-H2
6.626E-4
4.619E-4
8.936E-4
6.626E-4
6.626E-4
6.626E-4
3.313E-4
6.626E-4
CH4-CO2
1.672E-4
1.165E-4
2.255E-4
1.672E-4
1.672E-4
1.672E-4
8.360E-5
1.672E-4
CO-H2O
2.447E-4
1.706E-4
3.300E-4
2.447E-4
2.447E-4
2.447E-4
1.224E-4
2.447E-4
CO-H2
7.395E-4
5.154E-4
9.972E-4
7.395E-4
7.395E-4
7.395E-4
3.697E-4
7.395E-4
CO-CO2
1.542E-4
1.075E-4
2.080E-4
1.542E-4
1.542E-4
1.542E-4
7.712E-5
1.542E-4
H2O-H2
8.509E-4
5.931E-4
1.147E-3
8.509E-4
8.509E-4
8.509E-4
4.254E-4
8.509E-4
H2O- CO2
1.965E-4
1.370E-4
2.650E-4
1.965E-4
1.965E-4
1.965E-4
9.825E-5
1.965E-4
H2-CO2
6.230E-4
4.343E-4
8.402E-4
6.230E-4
6.230E-4
6.230E-4
3.115E-4
6.230E-4
O2 -N2
1.936E-4
1.350E-4
2.611E-4
1.936E-4
1.936E-4
1.936E-4
9.680E-5
1.936E-4
CH4-CO
8.462E-9
7.632E-9
9.217E-9
5.077E-9
2.031E-9
3.385E-9
8.460E-9
1.692E-8
CH4-H2O
9.624E-9
8.680E-9
1.048E-8
5.774E-9
2.310E-9
3.850E-9
9.622E-9
1.924E-8
CH4-H2
1.322E-8
1.192E-8
1.440E-8
7.931E-9
3.172E-9
5.287E-9
1.322E-8
2.643E-8
CH4-CO2
7.247E-9
6.537E-9
7.894E-9
4.348E-9
1.739E-9
2.899E-9
7.246E-9
1.449E-8
CO-H2O
8.279E-9
7.467E-9
9.018E-9
4.967E-9
1.987E-9
3.312E-9
8.278E-9
1.656E-8
CO-H2
1.025E-8
9.246E-9
1.117E-8
6.150E-9
2.460E-9
4.100E-9
1.025E-8
2.050E-8
CO-CO2
6.618E-9
5.969E-9
7.209E-9
3.971E-9
1.588E-9
2.647E-9
6.617E-9
1.323E-8
H2O-H2
1.255E-8
1.132E-8
1.367E-8
7.530E-9
3.012E-9
5.020E-9
1.255E-8
2.510E-8
H2O- CO2
7.131E-9
6.432E-9
7.768E-9
4.279E-9
1.712E-9
2.853E09
7.130E-9
1.426E-8
H2-CO2
8.279E-9
7.468E-9
9.018E-9
4.968E-9
1.987E-9
3.312E09
8.279E-9
1.656E-8
𝑀𝑆
𝐷𝑖𝑗
𝑒𝑓𝑓
𝐷𝑖𝑗
52
7.250E-9
6.539E-9
7.897E-9
4.350E-9
1.740E-9
2.900E09
7.249E-09
1.450E-8
CH4-CO
3.385E-8
3.054E-8
3.687E-8
3.385E-8
3.385E-8
3.385E-8
3.385E-8
6.771E-8
CH4-H2O
3.850E-8
3.473E-8
4.194E-8
3.850E-8
3.850E-8
3.850E-8
3.850E-8
7.700E-8
CH4-H2
5.287E-8
4.769E-8
5.759E-8
5.287E-8
5.287E-8
5.287E-8
5.287E-8
1.057E-7
CH4-CO2
2.899E-8
2.615E-8
3.158E-8
2.899E-8
2.899E-8
2.899E-8
2.899E-8
5.799E-8
CO-H2O
3.312E-8
2.987E-8
3.607E-8
3.312E-8
3.312E-8
3.312E-8
3.312E-8
6.624E-8
CO-H2
4.100E-8
3.699E-8
4.466E-8
4.100E-8
4.100E-8
4.100E-8
4.100E-8
8.201E-8
CO-CO2
2.648E-8
2.388E-8
2.884E-8
2.648E-8
2.648E-8
2.648E-8
2.648E-8
5.295E-8
H2O-H2
5.020E-8
4.528E-8
5.468E-8
5.020E-8
5.020E-8
5.020E-8
5.020E-8
1.004E-7
H2O- CO2
2.853E-8
2.573E-8
3.108E-8
2.853E-8
2.853E-8
2.853E-8
2.853E-8
5.706E-8
H2-CO2
3.312E-8
2.987E-8
3.607E-8
3.312E-8
3.312E-8
3.312E-8
3.312E-8
6.624E-8
O2 -N2
2.900E-8
2.616E-8
3.159E-8
2.900E-8
2.900E-8
2.900E-8
2.900E-8
5.801E-8
O2 -N2
𝑘𝑛
𝐷𝑖𝑗
16 Appendix C
53
17 Appendix D
Table 22 Kinetic Models for SOFC MSR and WGS Reactions on Ni Catalysts
Equations
+
−
3
𝑟̇𝑀𝑆𝑅 = 𝑘𝑀𝑆𝑅
𝑝𝐶𝐻4 𝑝𝐻2𝑂 − 𝑘𝑀𝑆𝑅
𝑝𝐶𝑂 𝑝𝐻2
T
(oC)
750 900
P
(bar)
1.5
750 900
1.5
800 900
1
300 575
3 - 15
S/C
1
+
−
𝑟̇𝑊𝐺𝑆 = 𝑘𝑊𝐺𝑆
𝑝𝐶𝑂 𝑝𝐻2𝑂 − 𝑘𝑊𝐺𝑆
𝑝𝐶𝑂2 𝑝𝐻2
𝑟̇𝑀𝑆𝑅 = 𝑘1 (𝑝𝐶𝐻4 𝑝𝐻2𝑂 −
𝑝𝐶𝑂 𝑝𝐻2 3
𝐾𝑒𝑞,1
)
𝑘1 = 2395 exp(−231266/𝑅𝑇)
3
𝐾𝑒𝑞,1 = 1.0267 ∗ 1010 𝑒𝑥𝑝(−0.2513𝑍 4 + 0.3665𝑍 3 + 0.5810𝑍 2 − 27.134𝑍 + 3.2770)
𝑟̇𝑊𝐺𝑆 = 𝑘2 (𝑝𝐻2𝑂 𝑝𝐶𝑂 −
𝑝𝐻2 𝑝𝐶𝑂2
) 𝑚𝑜𝑙 𝑚−3 𝑠 −1
𝐾𝑒𝑞,2
Material
Type
50%wt ZrO2, 50%wt Ni
2 mm thick cermet (CH4
reforming zone 0.15 to
0.3mm)
50%wt ZrO2, 50%wt Ni
2 mm thick cermet (CH4
reforming zone 0.15 to
0.3mm)
Note: Experimental Data
used from Lehnert et al.
Type 1
30%wt ZrO2, 70%wt Ni
10 µm thick anode on 2mm
thick YSZ disk
Tubular reactor of 15.2%
Ni and MgAl2O4 catalyst
Type 1
[36]
Type 1
[37]
𝑘2 = 0.0171 exp(−103191/𝑅𝑇) 𝑚𝑜𝑙 𝑚−3 𝑃𝑎 −2 𝑠 −1
𝐾𝑒𝑞,2 = 𝑒𝑥𝑝(−0.2935𝑍 3 + 0.6351𝑍 2 + 4.1788𝑍 + 0.3169)
𝑟̇𝑀𝑆𝑅 = 𝑘𝑎𝑑 𝑝𝐶𝐻4 (1 −
𝑟̇𝑀𝑆𝑅 = 𝑘1 (
𝑘𝑎𝑑 𝑝𝐻2 𝑝𝐶𝐻4
)
𝑘𝑟 𝐾𝐻2𝑂 𝑝𝐻2𝑂
𝑝𝐶𝐻4 𝑝𝐻2𝑂 𝑝𝐶𝑂 𝑝𝐻2 0.5
−
) /𝐷𝐸𝑁 2
𝑝𝐻2 2.5
𝐾𝑒𝑞,1
17
𝐾𝑒𝑞,1 = 1.198 ∗ 10
𝑟̇𝑊𝐺𝑆 = 𝑘2 (
exp(−26830/𝑇)
𝑝𝐶𝑂 𝑝𝐻2𝑂 𝑝𝐶𝑂2
−
) /𝐷𝐸𝑁 2
𝑝𝐻2
𝐾𝑒𝑞,2
𝐾𝑒𝑞,2 = 1.767 ∗ 10−2 exp(4400/𝑇)
𝑟̇𝐷𝑆𝑅 = 𝑘3 (
𝑝𝐶𝐻4 𝑝𝐻2 2𝑂 𝑝𝐶𝑂2 𝑝𝐻2 0.5
−
) /𝐷𝐸𝑁 2
𝑝𝐻2 3.5
𝐾𝑒𝑞,3
54
0-2
3, 5
[44]
Type 1
[35]
𝐾𝑒𝑞,3 = 2.117 ∗ 1015 exp(−22430/𝑇)
𝐷𝐸𝑁 = 1 + 𝐾𝐶𝑂 𝑝𝐶𝑂 + 𝐾𝐻2 𝑝𝐻2 + 𝐾𝐶𝐻4 𝑝𝐶𝐻4 + 𝐾𝐻2𝑂 𝑝𝐻2𝑂 /𝑝𝐻2
𝑟̇𝑀𝑆𝑅 = 𝑘1 (
𝑝𝐶𝐻4 𝑝𝐻2𝑂 0.5
𝑝𝐶𝑂 𝑝𝐻2 3
) (1 −
) /𝐷𝐸𝑁 2
1.5
𝑝𝐻2
𝑝𝐶𝐻4 𝑝𝐻2𝑂 𝐾𝑒𝑞,1
17
𝐾𝑒𝑞,1 = 1.198 ∗ 10
𝑟̇𝑊𝐺𝑆 = 𝑘2 (
598 823
1.2 6
4-7
Tubular reactor of 8385%wt Al2O, 15-17%wt Ni
Type 1
700 940
1.1 2.8
2.6 8
80%wt ZrO2, 20%wt Ni
1.4mm thick cermet
Type 2
650 950
-
2
35%wt ZrO2, 65%wt Ni
40 µm thick anode
Type 2
854 907
1
Ni- ZrO2 50 µm thick anode
Type 3
[41]
exp(−26830/𝑇)
𝑝𝐶𝑂 𝑝𝐻2𝑂 0.5
𝑝𝐶𝑂2 𝑝𝐻2
) (1 −
) /𝐷𝐸𝑁 2
𝑝𝐻2 0.5
𝑝𝐶𝑂 𝑝𝐻2𝑂 𝐾𝑒𝑞,2
𝐾𝑒𝑞,2 = 1.767 ∗ 10−2 exp(4400/𝑇)
𝑟̇𝐷𝑆𝑅 = 𝑘3 (
𝑝𝐶𝐻4 𝑝𝐻2𝑂
𝑝𝐶𝑂2 𝑝𝐻2 4
)
(1
−
) /𝐷𝐸𝑁 2
𝑝𝐻2 1.75
𝑝𝐶𝐻4 𝑝𝐻2𝑂 2 𝐾𝑒𝑞,3
𝐾𝑒𝑞,3 = 2.117 ∗ 1015 exp(−22430/𝑇)
𝐷𝐸𝑁 = 1 + 𝐾𝐶𝑂 𝑝𝐶𝑂 + 𝐾𝐻2 𝑝𝐻2 0.5 + 𝐾𝐻2𝑂 𝑝𝐻2 𝑂 /𝑝𝐻2
𝑝𝐶𝑂 𝑝𝐻3 2
𝐸𝐴
) 𝑒𝑥𝑝 (− )
𝑝𝐶𝐻4 𝑝𝐻2𝑂 𝐾𝑒𝑞
𝑅𝑇
𝑟̇𝑀𝑆𝑅 = 𝑘1 𝑝𝐶𝐻4 (1 −
𝐸𝐴 = 82000
𝐽
[38]
𝑘1 = 4274𝑚𝑜𝑙 𝑠 −1 𝑚 −2 𝑏𝑎𝑟 −1
𝑚𝑜𝑙
𝑟̇𝑀𝑆𝑅 = 𝑘1 𝑝𝐶𝐻4 (1 −
3
𝑝𝐶𝑂 𝑝𝐻
2
𝑝𝐶𝐻4 𝑝𝐻2 𝑂 𝐾𝑒𝑞
) 𝑒𝑥𝑝 (−
𝐸𝐴
𝑅𝑇
) 𝐸𝐴 = 63300
𝐽
𝑚𝑜𝑙
𝑘1 =
[12]
.00498 𝑚𝑜𝑙 −1 𝑠 −1 𝑚−2 𝑃𝑎 −1
𝛽
𝛼
−𝑟𝑀𝑆𝑅 = 𝑘 𝑝𝐶𝐻
𝑝
𝑒𝑥𝑝 (−
4 𝐻2 𝑂
𝐸𝑎 = 95 ± 2
𝑘𝐽
𝑚𝑜𝑙
𝐸𝑎
𝑅𝑇
)
𝛼 = 0.85 ± 0.05
𝛽 = −0.35 ± 0.04
𝑘 = 8542 𝑚𝑜𝑙 𝑠 −1 𝑚−2 𝑏𝑎𝑟 −1
55
1.53 2.5
[39]
18 Appendix E
Common Mathematical Shorthand used in this paper:
∇ Del/Nabla Operator
𝜕𝑇
𝜕𝑇
𝜕𝑇
1. ∇𝑇 : Gradient of a scalar ∇𝑇 = 𝜕𝑥 𝐱̂ + 𝜕𝑦 𝐲̂ + 𝜕𝑧 𝐳̂ where 𝐱̂, 𝐲̂, 𝐳̂ are directional unit vectors
2. ∇ ∙ 𝒖 : Divergence of a vector ∇ ∙ 𝒖 =
3. ∇ ∙ (ρ𝐮) : Scalar and vector product
𝜕𝑢𝑥
𝜕𝑥
+
𝜕𝑢𝑦
𝜕𝑦
+
𝜕𝑢𝑧
𝜕𝑧
∇ ∙ (ρ𝐮) = ρ(∇ ∙ 𝐮) + 𝐮 ∙ (∇ρ)
Comsol Notes:
1. Utilizing the equations mentioned in this study results in a highly nonlinear, highly
coupled system. A segregated solver was introduced to handle the solution.
2. Challenges were discovered when the MSR, WGS and MCDR reactions were added
to the model. Errors would occur in Comsol when the model would previously
converge without the reactions included. To solve this several steps were taken
including;
a.
Finer mesh
b. Addition of a continuation parameter
i. This slowly introduces the reactions into the model. Add a parameter to
the global parameters list and set it equal to 1 (For example ‘kc’).
Multiply this parameter by the reaction rate (‘kc’ x ‘rate’). Under
‘Study 1’, select the stationary step you are computing. Check
‘continuation’ and select your added continuation parameter then enter
a range of values (in this case 0, 1e-3, 1e-2, 1e-1, 1). Insert a parametric
step under the respective stationary solver, then in the drop down menu,
select the respective stationary step.
3. Memory usage errors, not enough memory on computer.
a.
One option is to select the solution method of the segregated step
or fully coupled approach that is using excessive memory (direct or
iterative). Check “out of core”. This approach will use less memory
resulting in longer computational times.
56