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Mathematical Model
to
Assess and Compare Cycle Performance
of a
Hybrid Solid Oxide Fuel Cell – Gas Turbine
to a
Gas Turbine
Larry Gray
MANE 6960H01 – Mathematical Modeling of Energy and Environmental Systems
Rensselaer Hartford
Hartford, CT
December 9, 2014
Table of Contents
Abstract ......................................................................................................................................................... 3
Introduction ................................................................................................................................................... 3
Model Description ........................................................................................................................................ 4
User Defined Inputs ...................................................................................................................................... 7
Key Outputs .................................................................................................................................................. 8
Model Validation .......................................................................................................................................... 9
Validation of the GT model ....................................................................................................................... 9
Validation of the Hybrid SOFC-GT model................................................................................................ 10
Comparison of a Hybrid SOFC-GT to Gas Turbine ................................................................................... 11
Model Improvements .................................................................................................................................. 13
Conclusions ................................................................................................................................................. 13
References ................................................................................................................................................... 14
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Abstract
Solid oxide fuel cells (SOFC) are gaining attention as an attractive technology that can convert
the chemical energy in fuel directly into electrical energy at higher conversion efficiencies with
much lower environmental impact than typical combustion engines. Solid oxide fuel cells have
the advantage of being able to operate using several different types of fuel.
SOFC operate at
higher temperature than other types of fuel cells giving SOFC an advantage in that the high
operating temperature (>800°C) allows the direct reformation of natural gas to hydrogen. This
paper presents a simple zero dimension (0-D) model that estimates the performance and
emissions for a hybrid solid oxide fuel cell – gas turbine (SOFC-GT) cycle and compares the
results to a gas turbine (GT) cycle producing equivalent power. The model is developed in MS
Excel and assesses pressures, temperatures and other cycle performance parameters for each
device module, the overall cycle efficiency and the carbon dioxide emissions for a set of user
provided input assumptions.
The overall cycle efficiency and the resulting carbon dioxide emissions were assessed for a
hybrid SOFC-GT and a GT, each having a design goal of producing 500 kW of power. For the
configurations described and compared in this paper and at the selected operating conditions, the
SOFC-GT can produce 500kW at nearly twice the overall efficiency as a stand-alone gas turbine
power unit, (56 percent compared to 28.5percent). It was also shown that the carbon dioxide
emissions produced by the SOFC-GT were 40 percent lower than the GT.
Introduction
Fuel cells are gaining interest as an attractive power generation solution as they produce
electricity at higher efficiencies with reduced emissions that in turn reduces the impact on the
environment. Water vapor and a small amount of carbon dioxide are the main products from a
fuel cell thermodynamic reaction.
Fuel cells offer high energy conversion efficiency by
converting the chemical energy in fuel directly into electricity whereas combustion or heat
engines convert the energy in the fuel to heat energy from which mechanical work is extracted
by a turbine to drive a shaft that drives a generator to produce electricity.
SOFCs have
advantages over other fuel cell types. SOFCs operate at temperatures up to 1000°C. Due to their
high operating temperature, the waste heat can be put to use in running a bottoming cycle such as
a gas turbine (GT), boosting overall efficiencies even higher than the standalone SOFC. Another
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benefit to SOFCs is their ability to be operated with a number of fuel types. SOFCs, operating at
high operating temperatures, do not require a separate fuel reformer and a variety of fuels can be
used such as natural gas, carbon monoxide, syngas (mixture of hydrogen and carbon monoxide
produced from a gasification process), as well as hydrogen. Other fuel cells require hydrogen to
be the input fuel which can be an expensive upfront process which can make the overall business
case for some fuel cell types a challenge. Other fuel cell types such as proton exchange
membranes (PEMs), phosphoric acid fuel cells (PAFCs) and molten carbonate fuel cells
(MCFCs) require expensive precious metals or use corrosive acids or hard to contain molten
materials, whereas SOFCs typically use low cost solid ceramic material as the electrolyte, such
as yttria-stabilized zirconia (YSZ) [1] [2]. The value of the simple model that will be described
in the following sections is to provide a tool that can be used to compare the cycle performance,
resulting efficiencies and the emissions produced from a hybrid SOFC-GT to that of a GT for
equivalent power production. It can also be used to run sensitivity analyses to test different
operating conditions and with some limited programming it can be used to evaluate different
configurations.
Model Description
The model is a simple 0-D model programmed in MS Excel in a block flow format. It is
a zero dimension model in that the model doesn’t assess the internal thermo-chemical reactions
within the SOFC.
The journal article by Chinda and Brault 2012 [3] which described
mathematical modeling of SOFC-gas turbine hybrid systems was used to provide guidance in
developing the MS Excel model described here. The SOFC is treated as a “black box” because
the user is required to enter into the model the SOFC cell voltage and power density relationships
as a function of current density for a given operating temperature and pressure as a preprocessing step. A bottoming gas turbine cycle is then combined with the SOFC to produce the
hybrid SOFC-GT cycle as illustrated by the cycle flow chart shown in Figure 1.
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SOFC-GT Cycle Analysis
CH4 reforms into H2
CO intermediate species then
forming CO2
Voltage
Bus
O2
Cathode
P3
T3
Fuel Hx
Exhaust
P4, T4
ANODE
Air
(O2, N2)
Turbine
Exhaust
P7
T7
P6
T6
Turbine
Unreacted H2,
H2O, CO2
O2, N2,
O2electrolyte
P1
T1
Compressor
Fuel
(CH4)
P5, T5
Fuel from tank
P2 T2
Air
Combustor
SOFC
Air Hx
Exhaust to
Environment
Expanded
View
unreacted H2
H2O (vapor)
O2, N2, CO2
Nat gas Fuel
Anode
Hot
compressed
air
~
Generator
CATHODE
Hydrogen ions and Oxygen
ions form H2O
Combustor
Air
(O2, N2),
Less
reacted
O2
Not all Hydrogen reacts with
Oxygen in SOFC
Power Turnbine
Figure 1 – Hybrid SOFC-GT Cycle Modeled in this paper
The user then calculates the cycle performance by entering fuel and air mass flow rates, along
with a few other operational and design input parameters.
The SOFC design point is represented by a graph at the bottom of the hybrid SOFC-GT
model along with a table of the current density, the cell voltage and the power density that are
used to create the graph. See Figure 2 for the SOFC design point performance curves currently
used in the model. The MS Excel “cell” for calculating the cell voltage in the model relies on an
equation developed by fitting a second order fit through the cell voltage as a function of current
density, which needs to be manually entered into this MS Excel cell.
Figure 2 - SOFC performance curve used. Source [4]
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To
H2O, Turbine
CO2
This pre-processing work would need to be completed first in order to evaluate another operating
design point. The current SOFC design point was obtained from a Scientific Reports conference
proceeding entitled “Micro-tubular solid oxide fuel cell based on a porous yttria-stabilized
zirconia support” [4].
The SOFC cell current is calculated from a relationship to the fuel mass
flow rate as shown by the following equation:
𝐼𝑇𝑜𝑡𝑎𝑙 =
2∗𝐹∗𝑛̇ 𝐹𝑢𝑒𝑙
𝑁
∗ 𝑢𝑓
(1)
where, F= Faraday’s constant (9.6485 x 104 C/mol)
N= number of cells in SOFC stack (user input)
𝑛̇ 𝐹𝑢𝑒𝑙 = molar fuel flow rate (mol/s)
𝑢𝑓 = percentage of fuel consumed (converted) in SOFC
The number of moles of hydrogen produced from the reformation of the fuel, if any reformation
takes place, must be considered in determining the molar fuel flow rate. Natural gas is the fuel
used in this analysis and natural gas is primarily composed of methane. When methane is
reformed in the SOFC, it forms 4 moles of hydrogen, therefore once the molar fuel flow rate of
natural gas, (simplified to be represented by CH4), is calculated, then this value needs to be
multiplied by 4. The reformation chemical reactions are described later in this paper in the
section comparing the performance of the SOFC-Gt to the GT.
A companion gas turbine cycle is modeled in the same MS Excel file on a second
worksheet that allows for a comparison of the SOFC-GT cycle performance to that of a GT. The
gas turbine thermodynamic cycle can be seen in the flow chart shown in Figure 3.
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Gas Turbine Power Plant Cycle Analysis
Fuel
P3
T3
P2
T2
Turbine
Exhaust
Combustor
P5
T5
P4
T4
Air
Generator
~
P1
T1
Compressor
Turbine
Power
Turbine
Exhaust to
Environment
Figure 3 - Gas turbine cycle modeled in this paper
The advantages of programming the cycle performance into MS Excel are first, Excel
allows the user to visualize the cycle of the model by drawing out the cycle using Excel shapes,
arrows and symbols. Secondly, the performance of cycle components can be programed into
cells and block of cells in Excel further allowing the user to organize and visualize the turbo
machinery, fuel cell and combustor module performance data “hand-offs” to the next module in
the system. Thirdly, the user can test other cycle configurations with some level of programming
effort in reorganizing the calculations in each block of cells representing the gas turbine or SOFC
components. The cycle performance can be modeled with changes to a few user-defined input
assumptions. The results for overall efficiency, power production and environmental impact
from the resulting carbon dioxide emissions can be quickly assessed and compared to alternative
set of inputs allowing for quick sensitivity analyses.
The model is useful for providing first order evaluations of cycle performance for
different values of the current set of input variables, a comparison of different cycle
configurations, with additional programming required, and for evaluating the benefits of a hybrid
SOFC-GT compared to a GT.
User Defined Inputs
The following table describes the user-define inputs for both the gas turbine (GT) model and the
hybrid solid oxide fuel cell gas turbine (SOFC-GT) model. Each model operates independent
from the other, which means for a comparison at say, equivalent power production, each model’s
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input variables will have to be adjusted to achieve the same power output. It is shown in Table
1 where the user input assumptions are used in the same way in the two models.
Table 1 - User control variables for models
Variable Used in
SOFCGT





Input Variable
Tamb
Pamb
Compressor Pressure Ratio
Air mass flow rate at inlet
Fuel flow rate to SOFC
Fuel flow rate to combustor
Max Turbine Inlet Temperature
Power turbine exit pressure
LHV – primary fuel
LHV – reformed fuel
SOFC fuel utilization
Fuel Cell -single cell area
Number of fuel cells in stack
F
Units
°C
atm
none
kg/s
kg/s
kg/s
°C
atm
kJ/kg
kJ/kg
%
m2
Qty
C/mol
CP [5]
kJ/kg K


CV [5]
CP fuel for SOFC (natural gas)
kJ/kg K













GT
















kJ/kg K
ηc
ηt
 Pcombustor
ηtran
ηgen
decimal
decimal
%
decimal
decimal
Description
Temperature at inlet
Pressure at inlet
Pressure rise across compressor
Mass flow rate of air at inlet
Fuel flow rate to solid oxide fuel cell
Fuel flow rate to combustor
Max turbine design temperature
Pressure at power turbine exit plane
Lower heating value of primary fuel
Lower heating value of reformed fuel
The percent of fuel converted in SOFC
Single fuel cell active area
Number of fuel cells in the stack, in series
Faraday constant
Specific heat, constant pressure (2 entries for
air at different temperature and one for
combustion products at user specified temp)
Specific heat, constant volume ( 3 entries
similar to Cp)
Specific heat for fuel entering heat
exchanger before entering SOFC
Polytropic compressor efficiency
Polytropic turbine efficiency
Pressure loss across combustor
Turbine to compressor transmission
efficiency
Generator efficiency
Key Outputs
The GT and SOFC-GT models produce key output results that can be used to assess the cycle
performance and environmental impacts.
The models produce results for overall thermal
efficiency, electrical power production and the resulting carbon dioxide emissions. The models
also produce two key parameters that need to be watched in terms of keeping them within the
limits. The first one is the turbine inlet temperature. The maximum, not-to-exceed, temperature
is entered as an input variable. This temperature is the design thermal limit for the blades and
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vanes material in the turbine. The coding in the models to do not constrain the internal modeled
parameters to stay within these limits, which could form the basis for follow-on programming
updates to the model.
However, the models do return results to an orange outlined box
indicating thermal exceedance in a red shaded cell and text and the user then must make the
appropriate adjustments. The other key watch parameter is a check on the fuel to air ratio such
that the mixture is within the flammability limits. Again the user would have to make the
appropriate adjustments to resolve any exceedance of the limits.
Model Validation
Validation of the GT model
The gas turbine model was validated by way of comparison to an example in B. K. Hodge’s
“Alternative Energy Systems and Applications.”
Example 5.5, page 99, of Chapter 5
(Combustion Turbines) [6] provides input assumptions shown in Table 2, which were entered
into the MS Excel gas turbine model.
Table 2 – Input assumptions from Example 5.5 in Hodge’s “Alternative Energy Systems and Applications”
Tamb
Pamb
Isentropic compressor efficiency
Compressor Pressure Ratio
Turbine Inlet Temperature
Combustor pressure loss
Fuel
Power turbine exit pressure
Isentropic turbine efficiency
Generator efficiency
Fuel-to-air ratio
30º C
97 kPa (0.957 atm)
0.84
5.5
1000º C
3%
Natural gas (LHV=47,100 kJ/kg)
100 kPa (0.987 atm)
0.88
0.98
0.17
The resulting cycle efficiency and heat rate from the analysis in Example 5.5 (B. K. Hodge) are
shown in Table 3 along with the results from the MS Excel gas turbine model.
Table 3 – Comparison of cycle efficiency and heat rate from two gas turbine models
Output parameter
Overall cycle efficiency
Heat rate (kJ/kWh)
B. K. Hodge
Example 5.5
MS Excel GT model
25.6%
24.0%
14,349
15,324
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Overall the MS Excel model results compare favorably with the B. K. Hodge example. The
differences between the two set of results is about 7 percent.
Validation of the Hybrid SOFC-GT model
From a review of literature of hybrid SOFC-GT systems, there are a number of different
configurations that have been modeled. A paper by R. Kandepu et al. [7], described a model of a
hybrid SOFC-GT system similar in configuration to the SOFC-GT presented in this paper. The
assumptions provided in the Kandepu paper to evaluate the performance of the SOFC-GT are
listed below, in Table 4, and are applied to the SOFC-GT model presented in this paper. Where
a required assumption was not made known in the Kandepu paper, an assumed value for that
assumption was made and noted by shading the cell of the table in yellow.
Table 4 Assumptions from Kandepu et al. SOFC-GT model performance evaluation. Unknown assumptions highlighted in
yellow. Assumptions used to validate SOFC-GT model presented in this paper
Tamb
Pamb
Airflow mass flow rate
Compressor ratio
Power turbine exit pressure
Fuel LHV (natural gas)
Fuel mass flow rate
SOFC fuel utilization
Cell area
Number of cells in stack
SOFC Current
SOFC cell voltage
SOFC temperature
Isentropic compressor efficiency
Isentropic turbine efficiency
Pressure loss across combustor
Generator efficiency
25º C
1 atm
0.445 kg/s
2.5
1 atm
47,100 kJ/kg
0.007 kg/s
85%
0.0464 m2
1160
250 A
0.657 volts
1350 K
0.84
0.86
2%
0.98
Table 5 presents the power production results for the Kandepu SOFC-GT model and the SOFCGT model presented in this paper for the set of assumptions listed in Table 4.
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Table 5 – Performance results from the comparison of the SOFC-GTmodel in this paper to the Kandepu model
Performance Results
SOFC Power
SOFC efficiency
Generator Power
System efficiency
Kandepu et al. model
191 kW
Not quoted
87 kW
Not quoted
SOFC-GT model (this paper)
189 kW
57%
89 kW
84%
The power produced by the SOFC stack and the power produced by the generator driven by the
GT of the hybrid system are very close between the two models.
Although not all the
assumptions were made known in the Kandepu paper and with a reasonable choice of values for
those assumptions where the information was not provided, the models did produce similar
power production results to each other.
Comparison of a Hybrid SOFC-GT to Gas Turbine
In this section of the report, two alternative configurations for an auxiliary power unit have been
compared. The performance of a SOFC with a bottoming gas turbine cycle (SOFC-GT) was
compared to a gas turbine auxiliary power unit. The input assumptions for the two power units
have been chosen such that a target power generation of 500 kW is achieved. The input
assumptions for the two units are as shown in Table 6.
Table 6 - Input assumptions for modeling two alternative auxiliary power units to achieve 500 kW of power generation
Assumption
Input Variable
Tamb
Pamb
Compressor Pressure Ratio
Air mass flow rate at inlet
Fuel flow rate to SOFC
Fuel flow rate to combustor
Max Turbine Inlet Temperature
Power turbine exit pressure
LHV – primary fuel
LHV – reformed fuel
SOFC fuel utilization
Fuel Cell -single cell area
Number of fuel cells in stack
F
ηc (polytropic efficiency = 0.875 entered)
Units
°C
atm
none
kg/s
kg/s
kg/s
°C
atm
kJ/kg
kJ/kg
%
m2
Qty
C/mol
GT
25
1
6.0
1.23
n/a
0.0338
1600
1
47,100
decimal
0.84
SOFC-GT
25
1
6.0
0.28
0.0192
0.0019
1600
1
47,100
120,210
80%
0.025
3600
96,485
0.84
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Assumption
Input Variable
Units
ηt (polytropic efficiency = 0.864 entered)
 Pcombustor
ηtran
ηgen
decimal
%
decimal
decimal
GT
0.88
3%
0.99
0.98
SOFC-GT
0.88
3%
0.99
0.98
The fuel used for both units is natural gas. As a result of the high operating temperature of the
SOFC, the natural gas is reformed to hydrogen which then reacts with the oxygen ions at the
anode of the SOFC to form water vapor. Not all of the fuel is consumed. For this modeling
exercise, it is assumed that 80 percent of the hydrogen reacts with the oxygen ions and the
remainder of the hydrogen is burned in the combustor of the bottoming gas turbine cycle. The
natural gas reformation in the SOFC is depicted by the following 2 equilibrium reactions [8] [9]:
𝐶𝐻4 + 𝐻2 𝑂 → 𝐶𝑂 + 3𝐻2
(Reforming)
(2)
𝐶𝑂 + 𝐻2 𝑂 → 𝐶𝑂2 + 𝐻2
(Water-Gas Shift)
(3)
The electrochemical reaction of the hydrogen with the oxygen ions at the anode side of the fuel
cell is described by the following reaction:
𝐻2 + 𝑂2− → 𝐻2 𝑂 + 2𝑒 −
(electrochemical)
(4)
The performance results from the respective models are shown in Table 7, along with the environmental
performance as measured by the amount of carbon dioxide emissions.
Table 7 – Cycle and Environmental Performance Comparison of the Auxiliary Power Units
Results
Cycle Metrics
Power generated
Overall cycle efficiency
Heat Rate
Actual Turbine Inlet Temp.
Environmental Performance
CO2
CO2
Units
kW
%
kJ/kWh
°C
GT
500
28.5%
12,872
1433
SOFC-GT
500
56%
6,618
1517
kg/hr
kg/kWh
334.6
0.67
190.1
0.38
From Table 7, it can be seen that the SOFC-GT model as previously described and with the input
assumptions as outlined in Table 6, produce power with an overall cycle efficiency about 2 times
higher than the standalone gas turbine cycle. The solid oxide fuel cell converts the stored
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chemical energy in the fuel directly into electricity which is a more efficient energy conversion
process than can be achieved through the Brayton cycle of the gas turbine. By combining the
two cycles, the waste heat produced by the SOFC can be used in the gas turbine bottoming cycle
to further produce energy and improving the efficiency of the overall combined cycle. The
environmental impact as measured by the CO2 emissions is reduced by over 40 percent, (0.38
kg/kWh for the SOFC-GT compared to 0.67 kg/kWh of CO2 emissions for the GT).
Model Improvements
As part of a future work program, the model could be improved to give it more capability by
including a module to calculate the SOFC cell voltage and power density as a function of current
density, thereby eliminating the current pre-processing work of coding in a unique set of cell
voltage and power density curves for a given operating temperature and pressure. This would
give the model more flexibility allowing for a larger envelope of operating conditions for which
to calculate the SOFC-GT performance and efficiency. It would also allow for a wider range of
sensitivity analyses.
Conclusions
A simple MS Excel spreadsheet model has been presented and described in this paper that allows
the user to perform limited cycle performance comparisons of a hybrid solid oxide fuel cell with
a bottoming gas turbine cycle to a standalone gas turbine cycle for generating power as an
auxiliary power unit. The spreadsheet model treats the SOFC as a “black box” and some preprocessing work must be completed before the spreadsheet model can be used for cycle
performance analyses. The first step is to develop the cell voltage and power density
relationships as a function of current density for the SOFC for a given operating temperature and
pressure, which then must be programmed into the spreadsheet model for the cell voltage
calculation. Once a few input assumptions are entered, the model is then useful in calculating
the power that can be generated, the overall efficiency and the environmental performance. Its
usefulness comes into play as an optimizing tool through sensitivity analyses of key input
variables. It shows how a hybrid configuration of a SOFC with a bottoming gas turbine cycle
that takes advantage of the waste heat from the SOFC can significantly improve the efficiency,
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(on the order of doubling), over a standalone gas turbine cycle used for energy production and
significantly reducing the emissions of carbon dioxide into the atmosphere.
References
[1] Bloom Energy, "Solid Oxide Fuel Cells," [Online]. Available: http://www.bloomenergy.com/fuelcell/solid-oxide/. [Accessed 18 October 2014].
[2] B. K. Hodge, "Chapter 10 Fuel Cells," in Alternative Energy Systems and Applications, Hoboken, NJ,
John Wiley & Sons, Inc, 2010, pp. 249-266.
[3] P. Chinda and P. Brault, "The hybrid solid oxide fuel cell (SOFC) and gas turbine (GT) systems
steady state modeling," International Journal of Hydrogen Energy, vol. 37, pp. 9237 - 9248, 2012.
[4] D. Panthi and A. Tsutsumi, "Micro-tubular solid oxide fuel cell based on a porous yttria-stabilized
zirconia support," Scientific Reports, 29 August 2014. [Online]. Available:
http://www.nature.com/srep/2014/140829/srep05754/fig_tab/srep05754_F5.html. [Accessed 2
November 2014].
[5] "Engineering Toolbox Specific Heat Capacity Gases," [Online]. Available:
http://www.engineeringtoolbox.com/specific-heat-capacity-gases-d_159.htmlEn. [Accessed 7
November 2014].
[6] B. K. Hodge, "Chapter 5 Combustion Turbines," in Alternative Energy Systems and Applications,
Hoboken, NJ, John Wiley & Sons, Inc, 2010, pp. 99 - 100.
[7] R. Kandepu, B. A. Foss and L. Imsland, "Integrated modeling and control of a load-connected SOFCGT autonomous power system," in American Control Conference, Minneapolis, MN, 2006.
[8] J. Pirkandi, M. Ghassemi, M. H. Hamedi and R. Mohammadi, "Electrochemical and thermodynamic
modeling of a CHP system using tubular solid oxide fuel cell (SOFC-CHP)," Journal of Cleaner
Production, Vols. 29-30, pp. 151 - 162, 2012.
[9] W. Jiang, R. Fang, R. A. Dougal and J. A. Khan, "Thermoelectric Model of a Tubular SOFC for
Dynamic Simulation," Journal of Energy Resources Technology, vol. 130, pp. 1-10, June 2008.
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