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Thermodynamics, polarizations, and intermediate temperature solid oxide fuel cell performance
CHAPTER 2
Thermodynamics, polarizations, and intermediate temperature solid oxide fuel cell performance Gurbinder Kaur1, Vishal Kumar2, Mandeep Kaur3, Gary Pickrell4 and John Mauro5 1 School of Physics and Materials Science, Thapar University, Patiala, India Sri Guru Granth Sahib World University, Fatehgarh Sahib, India Baba Banda Singh Bahadur Engineering College, Fatehgarh Sahib, India 4 Material Science and Engineering, Virginia Tech, Blacksburg, VA, United States 5 College of Earth and Mineral Sciences, The Pennsylvania State University, PA, United States 2 3
2.1 Intermediate temperature solid oxide fuel cell thermodynamics (ideal reversible) Fuel cell has an inlet for the fuel and outlet through which the gases escape the fuel cell system. The schematic of fuel cell as a reversible system is depicted in Fig. 2.1 [13]. Air and fuel enter the fuel cell separately thereby leaving the fuel nonmixed. The solid oxide fuel cell (SOFC) system shall be considered reversible. For the nonmixing of the W
Q
Fuel cell (T, p)
nf Hf
niHi
Figure 2.1 SOFC as a reversible system.
Intermediate Temperature Solid Oxide Fuel Cells DOI: https://doi.org/10.1016/B978-0-12-817445-6.00013-2
© 2020 Elsevier Inc. All rights reserved.
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Intermediate Temperature Solid Oxide Fuel Cells
fuel, the total enthalpy of the system is niHi while entering, whereas enthalpy njHj leaves the cell system. For attaining equilibrium, heat Q must be extracted reversibly from the fuel cell, and then transferred to the environment. Q is negative if heat is given by the fuel cell to the surroundings and is positive if heat is transferred to the system. Similarly, work done W is taken to be positive if work is done on the system an negative, if work is done by the system. Before proceeding further, we shall be well versed with the Gibbs free energy (ΔG). Gibbs free energy is defined as the energy available for doing external work. These days a most common term “exergy” is used, which is the total external work extracted due to volume and pressure changes. We will also use term “zero-energy point” corresponding to the elements in normal state with standard temperature (25°C) and pressure (0.1 MPa). Ð According to second law of thermodynamics, for a reversible process ( dS 5 0), the change in entropy (ΔS) is given by: ΔS 5
ΔQ T
(2.1)
where ΔH is the enthalpy change representing total thermal energy available, ΔQ is the heat energy change, and TΔS represents unavailable energy attributed to the entropy or disorder of the system. When the entropy change is negative, then the heat is given out to the surroundings whereas for the positive entropy change, heat is absorbed by the surroundings. The Gibbs free energy ΔG is equal to the reversible work done and can be written as: ΔG 5 ΔH T ΔS
(2.2)
Efficiency of a fuel cell is given as the ratio of Gibbs free energy and enthalpy change, that is, ΔG ΔH
(2.3)
ΔH 2 T ΔS T ΔS 512 ΔH ΔH
(2.4)
. η5
.
η5
Thermodynamics, polarizations, and intermediate temperature
35
The efficiency of fuel cell can be determined in terms of the fuel utilization factor (Uf) and is given by [1]: Uf 5
Fuelin 2 Fuelout Fuelin
(2.5)
At the anode, the concentration of product H2O increase with the increasing fuel utilization factor. For Intermediate temperature SOFC (IT-SOFC), the following reactions take place: Anode: Cathode:
H2 - 4H 1 1 2e2 1 O2 1 2e2 -O22 2
Net reaction (O22 ions migrate through the electrolyte and reaches anodes where is combines with hydrogen ions to form water) 2H 1 1 O22 -H2 O Therefore, as fuel utilization factor increases, the byproduct H2O also increases at the anode. The voltage gain can be defined in terms of the molar flow of electrons (ne ) is twice as that of molar flow of hydrogen ðnH2 Þ. The current (I) measures the amount of fuel spent and can be given in terms of nH2 and Faraday’s constant (F, product of charge and Avogadro’s number 5 96,485 kJ mol21) [3]: .
I 5 2 2 nH2 F
(2.6)
The reversible power (P) is given by the product of voltage (V ) and current (I): .
P 5 VI
(2.7)
The power is also defined as the product of molar flow of hydrogen flow ðnH2 Þ and Gibbs free energy, that is, .
P 5 ΔG U nH2
(2.8)
which yields [using (2.6)] .
V5
2 ΔG U nH2 ne F
(2.9)
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Intermediate Temperature Solid Oxide Fuel Cells
During fuel utilization, the fuel mixing occurs within the SOFC. The system can no longer be regarded as reversible and voltage reduction calculation is mandatory. The partial pressures needs to be taken into account for the voltage reduction. For an ideal gas, the Gibbs free energy can be treated equivalent to the work done and Eq. (2.2) can be given in terms of pressure (p) and temperature (T) (for infinitesimal changes): dS 5
dH 2 Vdp T
(2.10)
The entropy and Gibbs free energy can be written in terms of gas constant (R) and equilibrium constant (K), respectively, as ΔSðT ; pÞ 5 ΔSðT Þ R ln K
(2.11)
ΔGðT ; pÞ 5 ΔGðT Þ 1 RT ln K
(2.12)
Using (2.9) in (2.12), we obtain ΔGðT Þ 1 RT ln K nH2 V 52 ne F
(2.13)
An interesting point to be noted down is that the reaction enthalpy and entropy are assumed to vary slightly with the temperature. In contrast to this, the reversible cell voltage of cell decreases with increasing system temperature and increases with increasing system pressure.
2.2 Electro-motive force of the fuel cell Electro-motive force (EMF) of the IT-SOFC can be obtained in terms of Gibbs free energy. The Gibbs free energy of formation, ΔGf is expressed in terms of free energy of products and reactants as follows: ΔGf 5 Gf ðproductsÞ Gf ðreactantsÞ
(2.14)
For example, for the reaction, H2 1 ð1=2ÞO2 -H2 O, ΔGf is expressed as 1 ΔGf 5 ðGf ÞH2 O 2 ðGf ÞH2 1 ðGf ÞO2 (2.15) 2 But one must know that for a fuel cell the reaction mechanisms are not as simple as they appear to be. Let us address it by considering the electrons moving in external circuit. For each H2 used, two electrons pass
Thermodynamics, polarizations, and intermediate temperature
37
through the external circuit. Therefore, for 1 mole of H2 used, 2N electrons will migrate through the external circuit (NAvogadro’s number or 6.023 3 1023 atoms). This corresponds to the total charge flowing through the external circuit, which is 2Ne or 2F coulombs (F 5 Faraday’s constant 5 Ne 5 charge on one mole of electrons). For E voltage (ideal potential, open circuit voltage), the total electric work done in moving 2F charge around the cell-circuit for a fuel cell is given by Work done 5 2FE Joules
(2.16)
If we consider the system to be reversible, then there are no losses. This work done is released as Gibbs free energy and we can write Eq. (2.16) as: . .
ΔGf 5 2 2FE E52
ΔGf 2F
(2.17)
(2.18)
The generalized form of Eq. (2.18) is regarded as the fundamental EMF equation for the fuel cell and is given by E =−
ΔG f nF
ð2:19Þ
where n gives number of electrons contributing to the cell reaction. For general cell reaction pA 1 qB-rC 1 sD, the free energy change is expressed as ΔGf 5 ΔGf 1 RT ln
½Cp ½Dq ½Ar ½Bs
(2.20)
ΔGf is the free energy change at standard temperature and pressure (STP) conditions. Using (2.19) and (2.20), we find E 5 E° 1
RT ½Ap ½Bq ln nF ½Cr ½Ds
(2.21)
Eq. (2.21) is regarded as the generalized form of Nernst equation, where [A]p[B]q is the reactant activity and [C]r[D]s is the product activity [4,5]. Therefore it is clear from Eq. (2.21) that the decrease in concentration of reactants decreases the cell potential and vice versa. The potential
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Intermediate Temperature Solid Oxide Fuel Cells
can also be obtained by calculating the entropies, heat capacities, and enthalpies. The heat capacity (Cp) can be given by Cp 5 x 1 yT 1 zT 2
(2.22)
where T is the temperature and x, y, and z are the empirical constants. The change in heat capacities of products and reactants can be obtained as or dCp 5 dx 1 dðyT Þ 1 d zT 2 (2.23) The change in enthalpy and entropy are expressed as ðT ΔCp ΔS 5 ΔS° 1 dT 298 T
(2.24)
and ΔH 5 ΔH° 1
ðT
ΔCp dT
(2.25)
298
where ΔS° and ΔH° are the changes in entropies and enthalpies at STP conditions. Eqs. (2.23)(2.25) yield following results: T 1 ΔS 5 ΔS° 1 xln (2.26) 1 yðT 2 298Þ 1 zðT 2298Þ2 298 2 1 1 ΔH 5 ΔH° 1 xðT 2 298Þ 1 ðy2298Þ2 1 zðT 2298Þ3 2 3
(2.27)
For a hydrogen fuel cell, the ΔGf is 210.3 kJ mol21 at temperature of 400°C. The open circuit voltage, ‘E’ at 400°C is given by E5
210:3 3 103 5 1:09 V 2 3 96; 485
Similarly for batteries (here alkali battery), the overall reaction is given by (ΔGf 5 2 277 kJ mol21) 2MnO2 1 Zn-ZnO 1 Mn2 O3 This yields open circuit voltage for transfer of 2e2 in outer circuit as follows: E5
277 3 103 5 1:44 V 2 3 96; 485
Thermodynamics, polarizations, and intermediate temperature
39
2.3 Cell efficiency For the cell efficiency, we need to know about the reversible and irreversible processes taking place inside the fuel cell. The process is considered to be reversible, if entire energy is converted into electrical energy without any loss. In contrast to this, the process is regarded as irreversible when some part of the energy is converted to heat, which causes losses in the system. The efficiency for an ideal reversible system is 100%, whereas for an irreversible system, the efficiency is ,100%. The lower energy of irreversible system is attributed to the losses caused by wasted energy or thrown away [6,7]. The cell efficiency is defined as the electric energy produced/mol of fuel relative to the chemical energy change of the stored fuel, that is, η5
ΔGf ΔHf
(2.28)
where ΔHf is the calorific value of fuel (enthalpy of formation). The value of ΔHf is negative, when energy is released out for the process, whereas it is negative when energy is released. The maximum possible efficiency for a fuel cell is given as η5
ΔGf 3 100 ΔHf
(2.29)
we can calculate the efficiency of a fuel cell by considering standard free energy of formation of water, that is, 1 H2 1 O2 -H2 O ðlÞ 2
ΔHf 5 2 285:84 kJ mol21
For standard conditions and at free energy of 237.3 kJ mol21, the efficiency of an ideal fuel cell is (237.30/285.84) 5 0.83 or 83%. The efficiency of a fuel cell system can also be given in terms of the maximum EMF of a cell. There are two voltages in the system, actual cell voltage (Vactual) and ideal cell voltage (Videal), that is, the actual voltage is always less than the ideal voltage attributed to the losses in the system. Therefore the efficiency becomes: η5
Useful energy 0:83 Vactual 5 ΔHf Videal
(2.30)
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Intermediate Temperature Solid Oxide Fuel Cells
Let us calculate the efficiency of a hydrogen fuel cell operating at 25° C (ΔGf 5 237.3 kJ mol21). The open circuit voltage is also regarded as ideal voltage and can be given as: Videal 5 E 5 2
ΔGf 5 1:22 V nF
However, at 200°C, the ΔGf 5 220 kJ mol21, which yields, E 5 1:14 V. The ideal value of voltage is used in Eq. (2.32) to calculate efficiency: η 5 0:83 3
V actual 1:22
. η 5 0:675 V actual The cell voltage depends on the temperature and current densities (discussed later in the chapter), thereby changing the efficiencies. For example, η changes from 8.27 at 100°C to 52% at 1000°C, when we use carbon monoxide as a fuel.
2.4 Cell losses and polarization Due to the losses in the fuel cell, the actual fuel cell performance is always lower than the ideal voltage. Fig. 2.2 depicts the ideal as well as actual performance for the fuel cell. The variation of cell voltage with current density (current/area) for high and low temperatures operating cells is demonstrated in Fig. 2.2. The initial drop in voltage is small for high temperature operating cells, whereas it is high for the cells that operate at low temperatures. In addition to this, cells operating at high temperatures depict more linear behavior. The reversible system is a “no loss system,” hence considered to be the ideal one [2]. Due to the irreversible losses in the system, the actual potential of cell gets decreased from its ideal potential. These fuel cell irreversibilities are responsible for the voltage drop of the system. There are four key factors, which are responsible for lowering down cell performance, that is, fuel crossover, activation polarization, ohmic polarization, and concentration polarization. The total voltage can be summarized as Total voltageðV Þ 5 Open circuit voltageðEÞ Losses
(2.31)
The fuel crossover losses are attributed to the fuel wastage passing through the electrolyte. Fuel crossover losses are dominant for low
Thermodynamics, polarizations, and intermediate temperature
41
Figure 2.2 Graph depicting “no loss” voltage and “actual voltage” [2].
temperature operation of the fuel cells. Ohmic losses are due to the hindrance offered to the flow of ions/electrons through electrolytes and electrodes, respectively. The change in concentration of the reactants at the electrodes surface gives rise to concentration losses. Slow reaction kinetics on the electrode surface contributes to the activation losses.
2.4.1 Activation losses The voltage drops (mainly at cathode) for the low/medium temperature fuel cells are responsible for the activation losses. In 1905, a pattern for the verification of overvoltage against current density was obtained by Tafel and is depicted in Fig. 2.3 [2,3]. The activation polarization leads to drop in voltage, which can be represented by Tafel equation as follows: 2:3RT I log ηact 5 (2.32) αnF Io where Io is the exchange current density, n is the number of electrons participating in the reaction, and F is Faraday’s constant.
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Intermediate Temperature Solid Oxide Fuel Cells
Figure 2.3 Electrochemical reactions depicted as Tafel plots.
. .
2:3RT 2:3RT logðIÞ 2 logðIo Þ αnF αnF
(2.33)
2 2:3 RT 2:3RT logðIo Þ 1 logðIÞ αnF αnF
(2.34)
ηact 5 ηact 5 .
ηact 5 A 1 B log I
(2.35)
2 2:3RT log ðIo Þ αnF
(2.36)
2:3RT αnF
(2.37)
where A5 and B5
The exchange current density Io gives the current density at which overvoltage starts rising from zero, A denotes the electron transfer coefficient proportional to the electrical energy, which influences electrochemical reaction rate and the term “B” gives the Tafel slope. The factor
Thermodynamics, polarizations, and intermediate temperature
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RT =α nF is low for the electrochemical reactions that are fast and vice versa for the fast electrochemical reactions. For fast electrochemical reaction, Io is large and vice versa. Upon rearranging Eq. (2.32), we obtain αnFηact I 5 Io exp (2.38) RT Eq. (2.38) is known as ButlerVollmer equation and is useful as an alternative for the Tafel equation. However, Tafel equation is valid only for I . Io . The activation polarization is usually dominant for the low current density because the electronic barriers need to be overcome prior to current/ion flow. The electrochemical reaction rate directly influences the activation polarization and is due to the sluggish electrode kinetics. Zero exchange current density depicts same reaction rate for the forward as well as reverse reactions. This means that the electron flow occurs at the same rate. However, when the exchange current density is low, the reaction rate becomes lower for one direction and the electrode surface becomes less active. The cell performance is affected by current density considerably, that is, upon increasing Io 10-fold, the activation polarization increases by 100 mV [for slope of 100 mV/log(Io)]. For the Tafel slope of 70 mV/log Io, the 10-fold increase in the current density yields 70 mV increase for the activation polarization. On the other hand, the activation polarization gets lowered down upon lowering the Tafel slope. Eq. (2.32) also gives a premonition that the temperature increase might raise the ηact, whereas it has been observed that Io is the controlling factor for ηact. Low value of Io yields high voltage drop, usually the value of Io is smaller at the cathode as compared with the anode in case of the hydrogen fed fuel cells. For the fuel cells with methane as a fuel, it is not possible to ignore anode overvoltage. The total voltage drop would be given by I I Voltage drop5Banode log 1 Bcathode log (2.39) Ioa Ioc Ioa is the anode exchange current density; Ioc is the cathode exchange current density; Banode/Bcathode is the anode and cathode Tafel constants, respectively. Eq. (2.39) is quite similar to Eq. (2.32). Therefore the activation polarization may be due to both the electrodes. When Io decreases then the activation polarization increases; hence, the effort should be to increase the exchange current. Catalysts can enhance the exchange current density.
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Intermediate Temperature Solid Oxide Fuel Cells
Moreover, the pressure as well as temperature increase (Fig. 2.1) would markedly enhance the current density. When the concentration of reactants as well as electrode roughness increases, it probably leads to the increase in the current density owing to the catalyst sites occupancy by the reactants.
2.4.2 Ohmic losses (resistive losses) Electrodes and electrolytes are the vital components of fuel cells. The role of electrolytes is to offer resistance to the flow of ions whereas electrodes obstructs the flow of electrons [3]. The flow of the electrons at cathodes and flow of ions through the electrolyte contribute to the ohmic losses. In addition to this, the cell interconnects and bipolar plates also yield the ohmic losses. The Ohm’s law gives potential (V) as V 5 Ir
(2.40)
where I is the current and r is the resistance. For a fuel cell, Eq. (2.40) can be written as ηohm 5 Ir 0
(2.41)
where I is the current density (in mA cm2) and r0 is the area specific resistance (ASR in kΩ cm2). ASR is the resistance corresponding to an area of 1 cm2 of the cell. The ASR comprises total resistance of the cell, that is, electronic, ionic, and contact resistance. Since the cell resistance remains constant; therefore when the current decreases, the ohmic polarization also decreases. Ohmic polarization can be decreased by reducing the electrodes separation, which lowers the resistance, thereby decreasing the ohmic polarization. In addition to this, if the electrolyte supported electrodes are used, then the electrolyte must be made very thin. It should be kept in consideration that the two electrodes should not short-circuit. The electrolyte and electrodes should possess high ionic and electronic conductivity, respectively.
2.4.3 Concentration losses (mass transport losses) The concentration losses are also known as mass transport losses and gas transport losses. The reactant gets consumed at the electrode during an electrochemical process. The losses which occur due to concentrations gradient and reactants depletion at the electrode are known as concentration losses. The surrounding material cannot maintain the initial bulk
Thermodynamics, polarizations, and intermediate temperature
45
concentration, thereby leading to potential loss and concentration gradient. The hydrogen is fed at anode and oxygen is fed at cathode. The concentration of hydrogen and oxygen will fall on the anode and cathode, respectively, during the fuel cell operation. The reduction in concentration depends on the circulation of hydrogen/air around the anode/cathode and the current extracted from the fuel cell. Therefore the concentration changes due to the partial pressure of oxygen and hydrogen, which means the increase in concentration leads to increase in the partial pressure. If the gas pressure is reduced, it further leads to the decrease in cell voltage. Concentration losses occur mainly due to the solution of reactants into the electrolyte, slow gas diffusion into electrode pores, dissolution of products out of the electrolyte, products diffusion through the electrolyte from the electrochemical reaction site, and reactants diffusion through the electrolyte to the electrochemical reaction site. Concentration polarization causes voltage change, which can be explained with the help of Fick’s law. According to the Fick’s first law of diffusion, the current density is given by [8]: I5
nFDR ðCB 2 CS Þ t
(2.42)
where n denotes the number of electrons participating in the reaction, CB and CS are bulk and surface concentration, respectively, DR represents the diffusion coefficient of the reactants, and t is the diffusion layer thickness. I is the limited current density, which is regarded as the current density at which the rate of supply of fuel equals to the rate of use of fuel. This takes place for the zero surface concentration, that is, CS 5 0. Eq. (2.42) becomes nFDR CB t
(2.43)
nFDR CB nFDR CS 2 t t
(2.44)
I1 5 Eq. (2.42) can be re-written as I5
.
CS I 512 I1 CB
(2.45)
Two conditions are possible in the cell, one is when the current is flowing (reduction in bulk concentration) and second is zero current,
46
Intermediate Temperature Solid Oxide Fuel Cells
which is regarded as the equilibrium condition. For both the conditions, Nernst equations and potential difference can be written as ðfor I 5 0Þ
E 5 Eo 1
E 5 Eo 1
RT lnCB nF
RT lnCS nF
ηconc: 5 ΔE 5
RT CB ln nF CS
(2.46)
(2.47)
(2.48)
where ηconc. is the concentration polarization attributed to the concentration changes occurring at the electrode. From Eqs. (2.45) and (2.48), we obtain: 2 RT I ηconc: 5 ln 1 2 (2.49) nF I1 For the fuel cells operating at high temperature, concentration polarization is prominent loss. The rate of supply of hydrogen decreases when compared with its consumption rate at anode upon supplying hydrogen via reformer. This leads to the dominance of concentration polarization for the fuel cell. The water removal is also regarded as main factor for the mass transport or gas transport losses.
2.4.4 Fuel crossover losses The conduction process takes place through ion-conducting electrolyte of the fuel cell. The diffusion of fuel may occur from anode side to the cathode side. This might cause reaction of fuel with the oxygen directly at cathode in the presence of catalyst. This direct reaction would not yield any current, thereby leading to the fuel wastage, which migrates from cathode to anode via the electrolyte. Such type of fuel losses is known as the fuel crossover losses. Internal currents and fuel crossover losses are taken to be equivalent. The cell current density is not zero because of the internal current density, when the cell is in open circuit. The fuel crossover losses are summarized as 2:3RT In Hfuel 5 (2.50) ln Io αnF
Thermodynamics, polarizations, and intermediate temperature
47
where In is the internal current density. For the fuel cells operating at high temperature, there is small significance of the internal current as exchange current density I0 is quite high. In contrast to this, for the fuel cells operating at low temperatures, the internal currents and fuel crossover play imperative role.
2.4.5 Fuel losses summary for fuel cells The total effective cell voltage after the losses is given by V 5 E 2 ðTotal lossesÞ .
V 5 E ηact 1 ηohm 1 ηconc: 1 ηfuel
(2.51) (2.52)
2:3 RT I 2:3 RT In 2:3RT I V 5 E 2 Ir 2 2 1 log log ln 1 2 αnF Io αnF nF I1 Io (2.53) 0
For high temperature fuel cells, the cross over current is negligible, hence the Eq. (2.53) becomes: 2:3 RT I 2:3 RT I 0 1 (2.54) . V 5 E 2 Ir 2 log ln 1 2 αnF Io nF I1 Another approach using total polarization, to obtain the total effective voltage can also be used. Generally, the total polarization is primarily due to activation losses and concentration losses, that is, ηanode 5 ηconc: 1ηact anode (2.55) and
ηcathode 5 ηconc: 1ηact cathode
(2.56)
The equations can be generalized as ηelectrode 5 ηconc: 1ηact electrode
(2.57)
The electrode potential will also change as follows:
Velectrode 5 Eelectrode 6 ηelectrode
(2.58)
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Intermediate Temperature Solid Oxide Fuel Cells
In case of cathode, the potential decreases whereas for anode potential increases. Along with anode and cathode potentials, the cell voltage also consists of ohmic polarization, that is, . V 5 E ηconc: 1ηact cathode ηconc: 1ηact anode ir 0 (2.59) For low temperature fuel cells, the In factor is also added for the fuel cells operating at low temperatures V 5 E ηconc: 1ηact cathode ηconc: 1ηact anode ir 0 2 In (2.60) Ohmic and electrode polarizations are the main causes for fuel cell voltage loss. The current focus is to reduce the polarizations, in order to obtain maximum efficiency. By improving electrode structure, better electrocatalyst and highly conducting electrolyte high efficiency can be obtained. Therefore the ideal and actual fuel cell potentials are different due to the various losses.
2.5 Fuel cells performance Low temperature fuel cells require electrocatalysts to attain required reaction rates at cathode and anode. However, there is no such requirement of electrocatalyst for the high temperature fuel cells. The electrode reactions and their corresponding Nernst potential are listed as [1,2] Reaction 1 H2 1 O2 -H2 O 2 1 CO 1 O2 -CO2 2 1 H2 1 O2 1 CO2 -H2 O 1 CO2 2 CH4 1 2O2 -2H2 O 1 CO2
Nernst Potential 1 2 0 3 RT P H 1=2 2 A 1 ln PO 5 4ln@ E 5 Eo 1 2 PH 2 O 2F 1 2 0 3 RT P CO 1=2 A 1 ln PO 5 4ln@ E 5 Eo 1 2 PCO2 2F 1 2 0 3 RT P H2 1=2 A 1 lnPO 4ln@ E 5 Eo 1 PCO2 5 2 PH2 OðPCO Þ 2F 2 2 0 1 3 RT 4 @ PCH4 A 2 5 o 1 ln PO2 ln 2 E5E 1 PH2 O PCO 2F 2
(2.61 2.64)
where P represents the partial pressure of respective gas. There is another way to express the Nernst equation. Let us assume that the pressure on cathode and anode is same and the pressure of the system can be taken P. Then PH2 ; PO2 ; andPH2 O can be expressed as N1P, N2P, and N3P where
Thermodynamics, polarizations, and intermediate temperature
49
N1, N2, and N3 are constants dependent upon molar mass and concentration of hydrogen, oxygen, and water, respectively. Eq. (2.61) then becomes 1=2 RT N1 N2 P 1=2 o E5E 1 (2.65) 2F N3 ! 1=2 RT N N RT 1 2 1 E 5 Eo 1 ln lnðPÞ 2F 4F N3 |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 1st term
(2.66)
2nd term
The first term signifies the E.M.F. drop due to the change in concentration of reactants and products. The products concentration will rise at the cost of concentration of reactants during the fuel cell operation, which will shorten the 1st term. At the outlet, the fuel concentration and current density is low; therefore, the E.M.F. is also low near the exit. At high temperatures, the ‘RT’ term shows high Nernst voltage. The 2nd term depicts system pressure increase, which leads to the increase of the cell E.M.F. The 2nd term increases, if pressure increases from P1 to P2. For the low temperature fuel cells, the pressure increase is more significant, as high pressure reduces the losses specifically for the cathode. The performance of the fuel cell is influenced by a variety of variables like gas composition, current density, fuel utilization, pressure, temperature, etc. The system cost may get increased, if the operating parameters for a fuel cell are varied. For an ideal fuel cell system, the cost must be low along with less system weight, portability, and high power density. Upon reducing the cell stack size, both the efficiency and cell cost get reduced. Higher voltages and higher efficiencies are mandatory for stationary power applications. However, lower cell voltages or high current density is necessary for the mobile applications. The voltagepower relationship for a fuel cell is depicted in Fig. 2.4. As evident from Fig. 2.4, the higher power density and current density are yielded at low cell voltage. To obtain maximum efficiency, there should be balanced operation so as to get optimum compromise between voltage, current density, and operating cost. Temperature and pressure effect the ideal fuel cell potential, which can be derived from the Gibbs free energy and is given as
50
Intermediate Temperature Solid Oxide Fuel Cells
Figure 2.4 The graphical depiction between power density, current density, and cell voltage for a fuel cell [2].
@E @P
5 T
ΔV nF
(2.67)
For example, for the reaction H2 1 12 O2 -H2 O, the change in entropy is negative, which decreases the reversible potential with temperature increase. For this reaction, the change in volume is negative, which leads to reversible voltage increase for the cell with pressure increase. The solubility of the gas, reactant partial pressures, and mass transfer reacts also gets increased upon increasing reactant partial pressure. At elevated pressures in the fuel cell system, the evaporation rate of electrolyte decreases and the efficiency of the fuel cell also increases. One of the major factors that decide fuel cell efficiencies is utilization of reactants and gas composition. The utilization of fuel is given by Ufuel 5
Fuelin 2 Fuelout Fuelin
(2.68)
From Eq. (2.68), it is evident that for higher concentration of oxygen gas and fuel, the fuel cell will yield higher cell voltages and hence high efficiencies. The fuel consumption is determined by the difference
Thermodynamics, polarizations, and intermediate temperature
51
between fuel flow rate at the outlet and fuel provided at the inlet. Factors like leakage of fuels and chemical reaction may lead to more fuel consumption, hence increasing the apparent fuel utilization. The gas composition varies from inlet to outlet, thus reducing cell voltages. The reduction in voltage is due to the lower electrode potential (from Nernst equation) attributed to good ionic conductivity of the electrodes. The values for the cell voltage cannot be greater than the minimum Nernst potential. The fuel cell performance is also influenced by the impurities and current density. The activation polarization is responsible for decreasing the cell voltage during the early stages of cell operation. Upon increasing the current densities, the concentration losses dominate and cell performance decreases sharply. Ohmic losses are also present attributed to the internal resistances of the fuel cell.
2.6 Solid oxide fuel cell performance During the SOFC development, the main points of consideration are the cost issues, fabrication of material components, which are able to withstand the mechanical pressure, efficiency enhancement of the cell and ability to provide the power at low as well as intermediate temperatures [911]. The cell costs can be reduced drastically by lowering down the cell operating temperatures. In addition to this, for the intermediate operating range of 600°C800°C, the precious metals are good substitute for the ceramic materials. The SOFC efficiency depends on the methods of fabrication for instance, the pyrolysis of metallic soap slurry process, in which thin films of YSZ surround the NiO particles, has highly enhanced the adhesion of YSZ electrolyte and anode. In the same way, the cell cost can be reduced by almost 70%, when mixed lanthanides are used as the electrodes. However, the mixed lanthanides electrodes also reduce cell efficiency by 8%. Due to high ionic charge carriers and low activation energy, perovskite materials are potential candidates as electrolytes. In order to maintain the cell performance at low temperatures, the materials consisting of mixed conduction (ionic 1 electronic) are preferred as electrodes over pure electronic conductors. Thus we can conclude that the SOFC performance depends upon various factors, which include temperature, pressure, current densities, impurities, etc. Although the elevated temperatures of SOFC provide low open circuit voltage but they also reduce the polarization. Following section focuses on SOFC performance with pressure, temperature, and other factors.
52
Intermediate Temperature Solid Oxide Fuel Cells
2.6.1 Effect of utilization and gas composition During fuel reformation process, SOFC does not require any reforming catalyst since anode acts as catalyst. Also the internal reformation of fuel gases is allowed at the high temperatures. CO2 reformation from the fuel stream is no longer required as the oxygen is utilized by cathode only for the SOFC. H2, CO, and H2O; constitute fuel gas; H2 pressure as well as concentration highly influences the performance of SOFC [12,13]. This indicates that when hydrogen content is increased, the potential increases and the complete oxidation range gets extended to the higher value. Moreover, at high temperatures, the water gas shift reaction [CO 1 H2O-CO2 1 H2] is not favored. The theoretical potential of H2/O2 reaction shows increased value, whereas CO/O2 reaction potential increases for the elevated temperatures of SOFC. O/C and H/C atom ratio defines the fuel composition, that is, for no hydrogen H/C 5 O, pure CO2 yields O/C 5 2, whereas pure CO gives O/C 5 1. The cell voltages are described by the fuel gas composition, that is, for a fuel gas composition of 1.5% H2/3% CO/75.5% CO2/20% H2O, the current density is 100 mA cm22, whereas for 97% CO/3% H2O yields current density of 170 mA cm22. The composition comprising higher hydrogen (97% H2/3% H2O) content yields current density of 220 mA cm22. Additionally, for the composition with less H2 and CO content, the cell performance drops down attributed to the higher concentration polarization. Pure oxygen is used as fuel at the cathode, which enhances the cell performance. Upon increasing current density, the open circuit voltage difference between air and pure oxygen increases, due to the increased concentration polarization. At T 5 100°C, the voltage gain is given by ΔVcathode 5 92 log
ðPO2 Þ2 ðPO2 Þ1
(2.69)
Fig. 2.5 shows the fuel utilization effect on the cell voltage for pure O2 as oxidant and 2.4% H2/22% CO/11% H2O as reference fuel. When the temperature is constant, the cell voltage gets increased upon using pure O2 as oxidant. Upon increasing the fuel utilization, the cell voltage decreases. It is evident that at higher temperatures of 1000°C, the cell voltage decrease is steep as compared with the decrease observed in temperature 800°C900°C.
Thermodynamics, polarizations, and intermediate temperature
53
Figure 2.5 Variation in cell voltage with respect to the fuel utilization and temperature [2].
Figure 2.6 Cell voltage variation with respect to current density for the two-cell stack [2].
2.6.2 Effect of pressure and temperature When the temperature is increased, the decreased ohmic polarization results in increased current density. Fig. 2.6 shows the variation of cell voltage with the current density for the two-cell stack. As evident from Fig. 2.6, the increased current density yields decreased cell voltage; however, the decrease is sharp at low temperatures.
54
Intermediate Temperature Solid Oxide Fuel Cells
Figure 2.7 AES cell performance at different pressures (1000°C) (89% H2/11% H2O,O22) [1].
Like molten carbonate fuel cell and phosphoric acid fuel cell, SOFC performance also gets boosted upon increase in pressure. At 1000°C, approximated cell performance is given by ΔV ðmVÞ 5 59log
ðP2 Þ ðP1 Þ
(2.70)
The cell voltage variation with current density and pressures is depicted in Fig. 2.7 [listed for air electrode supported (AES) cells, Siemens Westing house corporation].
2.6.3 Effect of impurities, current density and cell life At T 5 1000°C, the relationship between current density and the cell voltage is depicted by the following equation: VCD 5 2 0:73ΔJ
(2.71)
The performance of single SOFC cell at reducing temperatures is given in Fig. 2.8. When current density is increased, concentration losses as well as ohmic activation also increases. It has been observed that the AES tubular SOFC yields 33% increase in power density, SOFC when compared with the tubular design calcia stabilized supported with porous
55
Thermodynamics, polarizations, and intermediate temperature
1.4
1.4 800ºC 750ºC
Cell voltage (V)
1.0
1.2
1.0
700ºC 650ºC
0.8
0.8
0.6
0.6
0.4
0.4
Power density (W cm–2)
1.2
600ºC 0.2
0.2
0 0
0.5
1.0
1.5
2.0
2.5
0 3.0
Current density (A cm–2)
Figure 2.8 SOFC performance (single cell) at reduced temperatures [14].
support [14]. HCl, ammonia, and hydrogen sulfide are the most commonly found impurities in the coal gas. Up to 1 ppm HCl and 5000 ppm of ammonia, no noticeable cell degradation could be observed whereas even 1 ppm of H2S leads to huge performance degradation. The original performance of the cell can be resumed upon H2S removal from the fuel stream.
References [1] J.H. Hirschenhofer, D.B. Stauffer, R.R. Engleman, M.G. Klett, Fuel Cell Handbook, fourth edition, 1998. [2] G. Kaur, Solid Oxide Fuel Cells Components—Interfacial Compatibility of SOFC Glass Seal, Springer, NewYork, NY, 2015. [3] J. Larminie, A. Dicks, Fuel Cells System Explained, second ed., John Wiley and Sons, England, 2003. [4] S.C. Singhal, K. Kendall, High Temperature Solid-Oxide Fuel Cells: Fundamentals, Elsevier, New York, NY, 2000. [5] S.M. Haile, Fuel cell materials and components, Acta Mater. 51 (2003) 59816000. [6] H.R. Kunz, L.A. Murphy, in: J.R. Selman, H.C. Maru (Eds.), Proceedings of the Symposium on Electrochemical Modeling of Battery, Fuel Cell, and Photoenergy Conversion Systems, 379, The Electrochemical Society Inc, Pennington, NJ, 1986. [7] S.N. Simons, R.B. King, P.R. Prokopius, Figure 1, p. 46 in: E.H. Camara (Ed.), Symposium Proceedings Fuel Cells Technology Status and Applications, 45, Institute of Gas Technology, Chicago, IL, 1982.
56
Intermediate Temperature Solid Oxide Fuel Cells
[8] V. Raghvan, Materials Science and Engineering, 5th edition, 2008. [9] E.J. Cairns, H.A. Liebhafsky, Energy Convers. 63 (1969) 9. [10] A.P. Fickett, in: J.D.E. McIntyre, S. Srinivasan, F.G. Will (Eds.), Proceedings of the Symposium on Electrode Materials and Processes for Energy Conversion and Storage, The Electrochemical Society Inc, Pennington, NJ, 1977, p. 546. [11] A.J. Appleby, J. Electroanal, Chem. 118 (1981) 31. [12] J. Huff, Status of fuel cell technologies, in Fuel Cell Seminar Abstracts, 1986 National Fuel Cell Seminar, Tucson, AZ, October 1986. [13] K. Strasser, in 26th Intersociety Energy Conversion Engineering Conference Proceedings, Volume 3, Conversion Technologies/Electrochemical Conversion, Boston, MA, August 49, 1991, published by Society of Automotive Engineers Inc., Warrendale, PA, 1991. [14] N.Q. Minh, Solid oxide fuel cell technology—features and applications, Solid State Ionics 174 (1-4) (2004) 271277.
Thermodynamics, polarizations, and intermediate temperature solid oxide fuel cell performance Gurbinder Kaur1, Vishal Kumar2, Mandeep Kaur3, Gary Pickrell4 and John Mauro5 1 School of Physics and Materials Science, Thapar University, Patiala, India Sri Guru Granth Sahib World University, Fatehgarh Sahib, India Baba Banda Singh Bahadur Engineering College, Fatehgarh Sahib, India 4 Material Science and Engineering, Virginia Tech, Blacksburg, VA, United States 5 College of Earth and Mineral Sciences, The Pennsylvania State University, PA, United States 2 3
2.1 Intermediate temperature solid oxide fuel cell thermodynamics (ideal reversible) Fuel cell has an inlet for the fuel and outlet through which the gases escape the fuel cell system. The schematic of fuel cell as a reversible system is depicted in Fig. 2.1 [13]. Air and fuel enter the fuel cell separately thereby leaving the fuel nonmixed. The solid oxide fuel cell (SOFC) system shall be considered reversible. For the nonmixing of the W
Q
Fuel cell (T, p)
nf Hf
niHi
Figure 2.1 SOFC as a reversible system.
Intermediate Temperature Solid Oxide Fuel Cells DOI: https://doi.org/10.1016/B978-0-12-817445-6.00013-2
© 2020 Elsevier Inc. All rights reserved.
33
34
Intermediate Temperature Solid Oxide Fuel Cells
fuel, the total enthalpy of the system is niHi while entering, whereas enthalpy njHj leaves the cell system. For attaining equilibrium, heat Q must be extracted reversibly from the fuel cell, and then transferred to the environment. Q is negative if heat is given by the fuel cell to the surroundings and is positive if heat is transferred to the system. Similarly, work done W is taken to be positive if work is done on the system an negative, if work is done by the system. Before proceeding further, we shall be well versed with the Gibbs free energy (ΔG). Gibbs free energy is defined as the energy available for doing external work. These days a most common term “exergy” is used, which is the total external work extracted due to volume and pressure changes. We will also use term “zero-energy point” corresponding to the elements in normal state with standard temperature (25°C) and pressure (0.1 MPa). Ð According to second law of thermodynamics, for a reversible process ( dS 5 0), the change in entropy (ΔS) is given by: ΔS 5
ΔQ T
(2.1)
where ΔH is the enthalpy change representing total thermal energy available, ΔQ is the heat energy change, and TΔS represents unavailable energy attributed to the entropy or disorder of the system. When the entropy change is negative, then the heat is given out to the surroundings whereas for the positive entropy change, heat is absorbed by the surroundings. The Gibbs free energy ΔG is equal to the reversible work done and can be written as: ΔG 5 ΔH T ΔS
(2.2)
Efficiency of a fuel cell is given as the ratio of Gibbs free energy and enthalpy change, that is, ΔG ΔH
(2.3)
ΔH 2 T ΔS T ΔS 512 ΔH ΔH
(2.4)
. η5
.
η5
Thermodynamics, polarizations, and intermediate temperature
35
The efficiency of fuel cell can be determined in terms of the fuel utilization factor (Uf) and is given by [1]: Uf 5
Fuelin 2 Fuelout Fuelin
(2.5)
At the anode, the concentration of product H2O increase with the increasing fuel utilization factor. For Intermediate temperature SOFC (IT-SOFC), the following reactions take place: Anode: Cathode:
H2 - 4H 1 1 2e2 1 O2 1 2e2 -O22 2
Net reaction (O22 ions migrate through the electrolyte and reaches anodes where is combines with hydrogen ions to form water) 2H 1 1 O22 -H2 O Therefore, as fuel utilization factor increases, the byproduct H2O also increases at the anode. The voltage gain can be defined in terms of the molar flow of electrons (ne ) is twice as that of molar flow of hydrogen ðnH2 Þ. The current (I) measures the amount of fuel spent and can be given in terms of nH2 and Faraday’s constant (F, product of charge and Avogadro’s number 5 96,485 kJ mol21) [3]: .
I 5 2 2 nH2 F
(2.6)
The reversible power (P) is given by the product of voltage (V ) and current (I): .
P 5 VI
(2.7)
The power is also defined as the product of molar flow of hydrogen flow ðnH2 Þ and Gibbs free energy, that is, .
P 5 ΔG U nH2
(2.8)
which yields [using (2.6)] .
V5
2 ΔG U nH2 ne F
(2.9)
36
Intermediate Temperature Solid Oxide Fuel Cells
During fuel utilization, the fuel mixing occurs within the SOFC. The system can no longer be regarded as reversible and voltage reduction calculation is mandatory. The partial pressures needs to be taken into account for the voltage reduction. For an ideal gas, the Gibbs free energy can be treated equivalent to the work done and Eq. (2.2) can be given in terms of pressure (p) and temperature (T) (for infinitesimal changes): dS 5
dH 2 Vdp T
(2.10)
The entropy and Gibbs free energy can be written in terms of gas constant (R) and equilibrium constant (K), respectively, as ΔSðT ; pÞ 5 ΔSðT Þ R ln K
(2.11)
ΔGðT ; pÞ 5 ΔGðT Þ 1 RT ln K
(2.12)
Using (2.9) in (2.12), we obtain ΔGðT Þ 1 RT ln K nH2 V 52 ne F
(2.13)
An interesting point to be noted down is that the reaction enthalpy and entropy are assumed to vary slightly with the temperature. In contrast to this, the reversible cell voltage of cell decreases with increasing system temperature and increases with increasing system pressure.
2.2 Electro-motive force of the fuel cell Electro-motive force (EMF) of the IT-SOFC can be obtained in terms of Gibbs free energy. The Gibbs free energy of formation, ΔGf is expressed in terms of free energy of products and reactants as follows: ΔGf 5 Gf ðproductsÞ Gf ðreactantsÞ
(2.14)
For example, for the reaction, H2 1 ð1=2ÞO2 -H2 O, ΔGf is expressed as 1 ΔGf 5 ðGf ÞH2 O 2 ðGf ÞH2 1 ðGf ÞO2 (2.15) 2 But one must know that for a fuel cell the reaction mechanisms are not as simple as they appear to be. Let us address it by considering the electrons moving in external circuit. For each H2 used, two electrons pass
Thermodynamics, polarizations, and intermediate temperature
37
through the external circuit. Therefore, for 1 mole of H2 used, 2N electrons will migrate through the external circuit (NAvogadro’s number or 6.023 3 1023 atoms). This corresponds to the total charge flowing through the external circuit, which is 2Ne or 2F coulombs (F 5 Faraday’s constant 5 Ne 5 charge on one mole of electrons). For E voltage (ideal potential, open circuit voltage), the total electric work done in moving 2F charge around the cell-circuit for a fuel cell is given by Work done 5 2FE Joules
(2.16)
If we consider the system to be reversible, then there are no losses. This work done is released as Gibbs free energy and we can write Eq. (2.16) as: . .
ΔGf 5 2 2FE E52
ΔGf 2F
(2.17)
(2.18)
The generalized form of Eq. (2.18) is regarded as the fundamental EMF equation for the fuel cell and is given by E =−
ΔG f nF
ð2:19Þ
where n gives number of electrons contributing to the cell reaction. For general cell reaction pA 1 qB-rC 1 sD, the free energy change is expressed as ΔGf 5 ΔGf 1 RT ln
½Cp ½Dq ½Ar ½Bs
(2.20)
ΔGf is the free energy change at standard temperature and pressure (STP) conditions. Using (2.19) and (2.20), we find E 5 E° 1
RT ½Ap ½Bq ln nF ½Cr ½Ds
(2.21)
Eq. (2.21) is regarded as the generalized form of Nernst equation, where [A]p[B]q is the reactant activity and [C]r[D]s is the product activity [4,5]. Therefore it is clear from Eq. (2.21) that the decrease in concentration of reactants decreases the cell potential and vice versa. The potential
38
Intermediate Temperature Solid Oxide Fuel Cells
can also be obtained by calculating the entropies, heat capacities, and enthalpies. The heat capacity (Cp) can be given by Cp 5 x 1 yT 1 zT 2
(2.22)
where T is the temperature and x, y, and z are the empirical constants. The change in heat capacities of products and reactants can be obtained as or dCp 5 dx 1 dðyT Þ 1 d zT 2 (2.23) The change in enthalpy and entropy are expressed as ðT ΔCp ΔS 5 ΔS° 1 dT 298 T
(2.24)
and ΔH 5 ΔH° 1
ðT
ΔCp dT
(2.25)
298
where ΔS° and ΔH° are the changes in entropies and enthalpies at STP conditions. Eqs. (2.23)(2.25) yield following results: T 1 ΔS 5 ΔS° 1 xln (2.26) 1 yðT 2 298Þ 1 zðT 2298Þ2 298 2 1 1 ΔH 5 ΔH° 1 xðT 2 298Þ 1 ðy2298Þ2 1 zðT 2298Þ3 2 3
(2.27)
For a hydrogen fuel cell, the ΔGf is 210.3 kJ mol21 at temperature of 400°C. The open circuit voltage, ‘E’ at 400°C is given by E5
210:3 3 103 5 1:09 V 2 3 96; 485
Similarly for batteries (here alkali battery), the overall reaction is given by (ΔGf 5 2 277 kJ mol21) 2MnO2 1 Zn-ZnO 1 Mn2 O3 This yields open circuit voltage for transfer of 2e2 in outer circuit as follows: E5
277 3 103 5 1:44 V 2 3 96; 485
Thermodynamics, polarizations, and intermediate temperature
39
2.3 Cell efficiency For the cell efficiency, we need to know about the reversible and irreversible processes taking place inside the fuel cell. The process is considered to be reversible, if entire energy is converted into electrical energy without any loss. In contrast to this, the process is regarded as irreversible when some part of the energy is converted to heat, which causes losses in the system. The efficiency for an ideal reversible system is 100%, whereas for an irreversible system, the efficiency is ,100%. The lower energy of irreversible system is attributed to the losses caused by wasted energy or thrown away [6,7]. The cell efficiency is defined as the electric energy produced/mol of fuel relative to the chemical energy change of the stored fuel, that is, η5
ΔGf ΔHf
(2.28)
where ΔHf is the calorific value of fuel (enthalpy of formation). The value of ΔHf is negative, when energy is released out for the process, whereas it is negative when energy is released. The maximum possible efficiency for a fuel cell is given as η5
ΔGf 3 100 ΔHf
(2.29)
we can calculate the efficiency of a fuel cell by considering standard free energy of formation of water, that is, 1 H2 1 O2 -H2 O ðlÞ 2
ΔHf 5 2 285:84 kJ mol21
For standard conditions and at free energy of 237.3 kJ mol21, the efficiency of an ideal fuel cell is (237.30/285.84) 5 0.83 or 83%. The efficiency of a fuel cell system can also be given in terms of the maximum EMF of a cell. There are two voltages in the system, actual cell voltage (Vactual) and ideal cell voltage (Videal), that is, the actual voltage is always less than the ideal voltage attributed to the losses in the system. Therefore the efficiency becomes: η5
Useful energy 0:83 Vactual 5 ΔHf Videal
(2.30)
40
Intermediate Temperature Solid Oxide Fuel Cells
Let us calculate the efficiency of a hydrogen fuel cell operating at 25° C (ΔGf 5 237.3 kJ mol21). The open circuit voltage is also regarded as ideal voltage and can be given as: Videal 5 E 5 2
ΔGf 5 1:22 V nF
However, at 200°C, the ΔGf 5 220 kJ mol21, which yields, E 5 1:14 V. The ideal value of voltage is used in Eq. (2.32) to calculate efficiency: η 5 0:83 3
V actual 1:22
. η 5 0:675 V actual The cell voltage depends on the temperature and current densities (discussed later in the chapter), thereby changing the efficiencies. For example, η changes from 8.27 at 100°C to 52% at 1000°C, when we use carbon monoxide as a fuel.
2.4 Cell losses and polarization Due to the losses in the fuel cell, the actual fuel cell performance is always lower than the ideal voltage. Fig. 2.2 depicts the ideal as well as actual performance for the fuel cell. The variation of cell voltage with current density (current/area) for high and low temperatures operating cells is demonstrated in Fig. 2.2. The initial drop in voltage is small for high temperature operating cells, whereas it is high for the cells that operate at low temperatures. In addition to this, cells operating at high temperatures depict more linear behavior. The reversible system is a “no loss system,” hence considered to be the ideal one [2]. Due to the irreversible losses in the system, the actual potential of cell gets decreased from its ideal potential. These fuel cell irreversibilities are responsible for the voltage drop of the system. There are four key factors, which are responsible for lowering down cell performance, that is, fuel crossover, activation polarization, ohmic polarization, and concentration polarization. The total voltage can be summarized as Total voltageðV Þ 5 Open circuit voltageðEÞ Losses
(2.31)
The fuel crossover losses are attributed to the fuel wastage passing through the electrolyte. Fuel crossover losses are dominant for low
Thermodynamics, polarizations, and intermediate temperature
41
Figure 2.2 Graph depicting “no loss” voltage and “actual voltage” [2].
temperature operation of the fuel cells. Ohmic losses are due to the hindrance offered to the flow of ions/electrons through electrolytes and electrodes, respectively. The change in concentration of the reactants at the electrodes surface gives rise to concentration losses. Slow reaction kinetics on the electrode surface contributes to the activation losses.
2.4.1 Activation losses The voltage drops (mainly at cathode) for the low/medium temperature fuel cells are responsible for the activation losses. In 1905, a pattern for the verification of overvoltage against current density was obtained by Tafel and is depicted in Fig. 2.3 [2,3]. The activation polarization leads to drop in voltage, which can be represented by Tafel equation as follows: 2:3RT I log ηact 5 (2.32) αnF Io where Io is the exchange current density, n is the number of electrons participating in the reaction, and F is Faraday’s constant.
42
Intermediate Temperature Solid Oxide Fuel Cells
Figure 2.3 Electrochemical reactions depicted as Tafel plots.
. .
2:3RT 2:3RT logðIÞ 2 logðIo Þ αnF αnF
(2.33)
2 2:3 RT 2:3RT logðIo Þ 1 logðIÞ αnF αnF
(2.34)
ηact 5 ηact 5 .
ηact 5 A 1 B log I
(2.35)
2 2:3RT log ðIo Þ αnF
(2.36)
2:3RT αnF
(2.37)
where A5 and B5
The exchange current density Io gives the current density at which overvoltage starts rising from zero, A denotes the electron transfer coefficient proportional to the electrical energy, which influences electrochemical reaction rate and the term “B” gives the Tafel slope. The factor
Thermodynamics, polarizations, and intermediate temperature
43
RT =α nF is low for the electrochemical reactions that are fast and vice versa for the fast electrochemical reactions. For fast electrochemical reaction, Io is large and vice versa. Upon rearranging Eq. (2.32), we obtain αnFηact I 5 Io exp (2.38) RT Eq. (2.38) is known as ButlerVollmer equation and is useful as an alternative for the Tafel equation. However, Tafel equation is valid only for I . Io . The activation polarization is usually dominant for the low current density because the electronic barriers need to be overcome prior to current/ion flow. The electrochemical reaction rate directly influences the activation polarization and is due to the sluggish electrode kinetics. Zero exchange current density depicts same reaction rate for the forward as well as reverse reactions. This means that the electron flow occurs at the same rate. However, when the exchange current density is low, the reaction rate becomes lower for one direction and the electrode surface becomes less active. The cell performance is affected by current density considerably, that is, upon increasing Io 10-fold, the activation polarization increases by 100 mV [for slope of 100 mV/log(Io)]. For the Tafel slope of 70 mV/log Io, the 10-fold increase in the current density yields 70 mV increase for the activation polarization. On the other hand, the activation polarization gets lowered down upon lowering the Tafel slope. Eq. (2.32) also gives a premonition that the temperature increase might raise the ηact, whereas it has been observed that Io is the controlling factor for ηact. Low value of Io yields high voltage drop, usually the value of Io is smaller at the cathode as compared with the anode in case of the hydrogen fed fuel cells. For the fuel cells with methane as a fuel, it is not possible to ignore anode overvoltage. The total voltage drop would be given by I I Voltage drop5Banode log 1 Bcathode log (2.39) Ioa Ioc Ioa is the anode exchange current density; Ioc is the cathode exchange current density; Banode/Bcathode is the anode and cathode Tafel constants, respectively. Eq. (2.39) is quite similar to Eq. (2.32). Therefore the activation polarization may be due to both the electrodes. When Io decreases then the activation polarization increases; hence, the effort should be to increase the exchange current. Catalysts can enhance the exchange current density.
44
Intermediate Temperature Solid Oxide Fuel Cells
Moreover, the pressure as well as temperature increase (Fig. 2.1) would markedly enhance the current density. When the concentration of reactants as well as electrode roughness increases, it probably leads to the increase in the current density owing to the catalyst sites occupancy by the reactants.
2.4.2 Ohmic losses (resistive losses) Electrodes and electrolytes are the vital components of fuel cells. The role of electrolytes is to offer resistance to the flow of ions whereas electrodes obstructs the flow of electrons [3]. The flow of the electrons at cathodes and flow of ions through the electrolyte contribute to the ohmic losses. In addition to this, the cell interconnects and bipolar plates also yield the ohmic losses. The Ohm’s law gives potential (V) as V 5 Ir
(2.40)
where I is the current and r is the resistance. For a fuel cell, Eq. (2.40) can be written as ηohm 5 Ir 0
(2.41)
where I is the current density (in mA cm2) and r0 is the area specific resistance (ASR in kΩ cm2). ASR is the resistance corresponding to an area of 1 cm2 of the cell. The ASR comprises total resistance of the cell, that is, electronic, ionic, and contact resistance. Since the cell resistance remains constant; therefore when the current decreases, the ohmic polarization also decreases. Ohmic polarization can be decreased by reducing the electrodes separation, which lowers the resistance, thereby decreasing the ohmic polarization. In addition to this, if the electrolyte supported electrodes are used, then the electrolyte must be made very thin. It should be kept in consideration that the two electrodes should not short-circuit. The electrolyte and electrodes should possess high ionic and electronic conductivity, respectively.
2.4.3 Concentration losses (mass transport losses) The concentration losses are also known as mass transport losses and gas transport losses. The reactant gets consumed at the electrode during an electrochemical process. The losses which occur due to concentrations gradient and reactants depletion at the electrode are known as concentration losses. The surrounding material cannot maintain the initial bulk
Thermodynamics, polarizations, and intermediate temperature
45
concentration, thereby leading to potential loss and concentration gradient. The hydrogen is fed at anode and oxygen is fed at cathode. The concentration of hydrogen and oxygen will fall on the anode and cathode, respectively, during the fuel cell operation. The reduction in concentration depends on the circulation of hydrogen/air around the anode/cathode and the current extracted from the fuel cell. Therefore the concentration changes due to the partial pressure of oxygen and hydrogen, which means the increase in concentration leads to increase in the partial pressure. If the gas pressure is reduced, it further leads to the decrease in cell voltage. Concentration losses occur mainly due to the solution of reactants into the electrolyte, slow gas diffusion into electrode pores, dissolution of products out of the electrolyte, products diffusion through the electrolyte from the electrochemical reaction site, and reactants diffusion through the electrolyte to the electrochemical reaction site. Concentration polarization causes voltage change, which can be explained with the help of Fick’s law. According to the Fick’s first law of diffusion, the current density is given by [8]: I5
nFDR ðCB 2 CS Þ t
(2.42)
where n denotes the number of electrons participating in the reaction, CB and CS are bulk and surface concentration, respectively, DR represents the diffusion coefficient of the reactants, and t is the diffusion layer thickness. I is the limited current density, which is regarded as the current density at which the rate of supply of fuel equals to the rate of use of fuel. This takes place for the zero surface concentration, that is, CS 5 0. Eq. (2.42) becomes nFDR CB t
(2.43)
nFDR CB nFDR CS 2 t t
(2.44)
I1 5 Eq. (2.42) can be re-written as I5
.
CS I 512 I1 CB
(2.45)
Two conditions are possible in the cell, one is when the current is flowing (reduction in bulk concentration) and second is zero current,
46
Intermediate Temperature Solid Oxide Fuel Cells
which is regarded as the equilibrium condition. For both the conditions, Nernst equations and potential difference can be written as ðfor I 5 0Þ
E 5 Eo 1
E 5 Eo 1
RT lnCB nF
RT lnCS nF
ηconc: 5 ΔE 5
RT CB ln nF CS
(2.46)
(2.47)
(2.48)
where ηconc. is the concentration polarization attributed to the concentration changes occurring at the electrode. From Eqs. (2.45) and (2.48), we obtain: 2 RT I ηconc: 5 ln 1 2 (2.49) nF I1 For the fuel cells operating at high temperature, concentration polarization is prominent loss. The rate of supply of hydrogen decreases when compared with its consumption rate at anode upon supplying hydrogen via reformer. This leads to the dominance of concentration polarization for the fuel cell. The water removal is also regarded as main factor for the mass transport or gas transport losses.
2.4.4 Fuel crossover losses The conduction process takes place through ion-conducting electrolyte of the fuel cell. The diffusion of fuel may occur from anode side to the cathode side. This might cause reaction of fuel with the oxygen directly at cathode in the presence of catalyst. This direct reaction would not yield any current, thereby leading to the fuel wastage, which migrates from cathode to anode via the electrolyte. Such type of fuel losses is known as the fuel crossover losses. Internal currents and fuel crossover losses are taken to be equivalent. The cell current density is not zero because of the internal current density, when the cell is in open circuit. The fuel crossover losses are summarized as 2:3RT In Hfuel 5 (2.50) ln Io αnF
Thermodynamics, polarizations, and intermediate temperature
47
where In is the internal current density. For the fuel cells operating at high temperature, there is small significance of the internal current as exchange current density I0 is quite high. In contrast to this, for the fuel cells operating at low temperatures, the internal currents and fuel crossover play imperative role.
2.4.5 Fuel losses summary for fuel cells The total effective cell voltage after the losses is given by V 5 E 2 ðTotal lossesÞ .
V 5 E ηact 1 ηohm 1 ηconc: 1 ηfuel
(2.51) (2.52)
2:3 RT I 2:3 RT In 2:3RT I V 5 E 2 Ir 2 2 1 log log ln 1 2 αnF Io αnF nF I1 Io (2.53) 0
For high temperature fuel cells, the cross over current is negligible, hence the Eq. (2.53) becomes: 2:3 RT I 2:3 RT I 0 1 (2.54) . V 5 E 2 Ir 2 log ln 1 2 αnF Io nF I1 Another approach using total polarization, to obtain the total effective voltage can also be used. Generally, the total polarization is primarily due to activation losses and concentration losses, that is, ηanode 5 ηconc: 1ηact anode (2.55) and
ηcathode 5 ηconc: 1ηact cathode
(2.56)
The equations can be generalized as ηelectrode 5 ηconc: 1ηact electrode
(2.57)
The electrode potential will also change as follows:
Velectrode 5 Eelectrode 6 ηelectrode
(2.58)
48
Intermediate Temperature Solid Oxide Fuel Cells
In case of cathode, the potential decreases whereas for anode potential increases. Along with anode and cathode potentials, the cell voltage also consists of ohmic polarization, that is, . V 5 E ηconc: 1ηact cathode ηconc: 1ηact anode ir 0 (2.59) For low temperature fuel cells, the In factor is also added for the fuel cells operating at low temperatures V 5 E ηconc: 1ηact cathode ηconc: 1ηact anode ir 0 2 In (2.60) Ohmic and electrode polarizations are the main causes for fuel cell voltage loss. The current focus is to reduce the polarizations, in order to obtain maximum efficiency. By improving electrode structure, better electrocatalyst and highly conducting electrolyte high efficiency can be obtained. Therefore the ideal and actual fuel cell potentials are different due to the various losses.
2.5 Fuel cells performance Low temperature fuel cells require electrocatalysts to attain required reaction rates at cathode and anode. However, there is no such requirement of electrocatalyst for the high temperature fuel cells. The electrode reactions and their corresponding Nernst potential are listed as [1,2] Reaction 1 H2 1 O2 -H2 O 2 1 CO 1 O2 -CO2 2 1 H2 1 O2 1 CO2 -H2 O 1 CO2 2 CH4 1 2O2 -2H2 O 1 CO2
Nernst Potential 1 2 0 3 RT P H 1=2 2 A 1 ln PO 5 4ln@ E 5 Eo 1 2 PH 2 O 2F 1 2 0 3 RT P CO 1=2 A 1 ln PO 5 4ln@ E 5 Eo 1 2 PCO2 2F 1 2 0 3 RT P H2 1=2 A 1 lnPO 4ln@ E 5 Eo 1 PCO2 5 2 PH2 OðPCO Þ 2F 2 2 0 1 3 RT 4 @ PCH4 A 2 5 o 1 ln PO2 ln 2 E5E 1 PH2 O PCO 2F 2
(2.61 2.64)
where P represents the partial pressure of respective gas. There is another way to express the Nernst equation. Let us assume that the pressure on cathode and anode is same and the pressure of the system can be taken P. Then PH2 ; PO2 ; andPH2 O can be expressed as N1P, N2P, and N3P where
Thermodynamics, polarizations, and intermediate temperature
49
N1, N2, and N3 are constants dependent upon molar mass and concentration of hydrogen, oxygen, and water, respectively. Eq. (2.61) then becomes 1=2 RT N1 N2 P 1=2 o E5E 1 (2.65) 2F N3 ! 1=2 RT N N RT 1 2 1 E 5 Eo 1 ln lnðPÞ 2F 4F N3 |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 1st term
(2.66)
2nd term
The first term signifies the E.M.F. drop due to the change in concentration of reactants and products. The products concentration will rise at the cost of concentration of reactants during the fuel cell operation, which will shorten the 1st term. At the outlet, the fuel concentration and current density is low; therefore, the E.M.F. is also low near the exit. At high temperatures, the ‘RT’ term shows high Nernst voltage. The 2nd term depicts system pressure increase, which leads to the increase of the cell E.M.F. The 2nd term increases, if pressure increases from P1 to P2. For the low temperature fuel cells, the pressure increase is more significant, as high pressure reduces the losses specifically for the cathode. The performance of the fuel cell is influenced by a variety of variables like gas composition, current density, fuel utilization, pressure, temperature, etc. The system cost may get increased, if the operating parameters for a fuel cell are varied. For an ideal fuel cell system, the cost must be low along with less system weight, portability, and high power density. Upon reducing the cell stack size, both the efficiency and cell cost get reduced. Higher voltages and higher efficiencies are mandatory for stationary power applications. However, lower cell voltages or high current density is necessary for the mobile applications. The voltagepower relationship for a fuel cell is depicted in Fig. 2.4. As evident from Fig. 2.4, the higher power density and current density are yielded at low cell voltage. To obtain maximum efficiency, there should be balanced operation so as to get optimum compromise between voltage, current density, and operating cost. Temperature and pressure effect the ideal fuel cell potential, which can be derived from the Gibbs free energy and is given as
50
Intermediate Temperature Solid Oxide Fuel Cells
Figure 2.4 The graphical depiction between power density, current density, and cell voltage for a fuel cell [2].
@E @P
5 T
ΔV nF
(2.67)
For example, for the reaction H2 1 12 O2 -H2 O, the change in entropy is negative, which decreases the reversible potential with temperature increase. For this reaction, the change in volume is negative, which leads to reversible voltage increase for the cell with pressure increase. The solubility of the gas, reactant partial pressures, and mass transfer reacts also gets increased upon increasing reactant partial pressure. At elevated pressures in the fuel cell system, the evaporation rate of electrolyte decreases and the efficiency of the fuel cell also increases. One of the major factors that decide fuel cell efficiencies is utilization of reactants and gas composition. The utilization of fuel is given by Ufuel 5
Fuelin 2 Fuelout Fuelin
(2.68)
From Eq. (2.68), it is evident that for higher concentration of oxygen gas and fuel, the fuel cell will yield higher cell voltages and hence high efficiencies. The fuel consumption is determined by the difference
Thermodynamics, polarizations, and intermediate temperature
51
between fuel flow rate at the outlet and fuel provided at the inlet. Factors like leakage of fuels and chemical reaction may lead to more fuel consumption, hence increasing the apparent fuel utilization. The gas composition varies from inlet to outlet, thus reducing cell voltages. The reduction in voltage is due to the lower electrode potential (from Nernst equation) attributed to good ionic conductivity of the electrodes. The values for the cell voltage cannot be greater than the minimum Nernst potential. The fuel cell performance is also influenced by the impurities and current density. The activation polarization is responsible for decreasing the cell voltage during the early stages of cell operation. Upon increasing the current densities, the concentration losses dominate and cell performance decreases sharply. Ohmic losses are also present attributed to the internal resistances of the fuel cell.
2.6 Solid oxide fuel cell performance During the SOFC development, the main points of consideration are the cost issues, fabrication of material components, which are able to withstand the mechanical pressure, efficiency enhancement of the cell and ability to provide the power at low as well as intermediate temperatures [911]. The cell costs can be reduced drastically by lowering down the cell operating temperatures. In addition to this, for the intermediate operating range of 600°C800°C, the precious metals are good substitute for the ceramic materials. The SOFC efficiency depends on the methods of fabrication for instance, the pyrolysis of metallic soap slurry process, in which thin films of YSZ surround the NiO particles, has highly enhanced the adhesion of YSZ electrolyte and anode. In the same way, the cell cost can be reduced by almost 70%, when mixed lanthanides are used as the electrodes. However, the mixed lanthanides electrodes also reduce cell efficiency by 8%. Due to high ionic charge carriers and low activation energy, perovskite materials are potential candidates as electrolytes. In order to maintain the cell performance at low temperatures, the materials consisting of mixed conduction (ionic 1 electronic) are preferred as electrodes over pure electronic conductors. Thus we can conclude that the SOFC performance depends upon various factors, which include temperature, pressure, current densities, impurities, etc. Although the elevated temperatures of SOFC provide low open circuit voltage but they also reduce the polarization. Following section focuses on SOFC performance with pressure, temperature, and other factors.
52
Intermediate Temperature Solid Oxide Fuel Cells
2.6.1 Effect of utilization and gas composition During fuel reformation process, SOFC does not require any reforming catalyst since anode acts as catalyst. Also the internal reformation of fuel gases is allowed at the high temperatures. CO2 reformation from the fuel stream is no longer required as the oxygen is utilized by cathode only for the SOFC. H2, CO, and H2O; constitute fuel gas; H2 pressure as well as concentration highly influences the performance of SOFC [12,13]. This indicates that when hydrogen content is increased, the potential increases and the complete oxidation range gets extended to the higher value. Moreover, at high temperatures, the water gas shift reaction [CO 1 H2O-CO2 1 H2] is not favored. The theoretical potential of H2/O2 reaction shows increased value, whereas CO/O2 reaction potential increases for the elevated temperatures of SOFC. O/C and H/C atom ratio defines the fuel composition, that is, for no hydrogen H/C 5 O, pure CO2 yields O/C 5 2, whereas pure CO gives O/C 5 1. The cell voltages are described by the fuel gas composition, that is, for a fuel gas composition of 1.5% H2/3% CO/75.5% CO2/20% H2O, the current density is 100 mA cm22, whereas for 97% CO/3% H2O yields current density of 170 mA cm22. The composition comprising higher hydrogen (97% H2/3% H2O) content yields current density of 220 mA cm22. Additionally, for the composition with less H2 and CO content, the cell performance drops down attributed to the higher concentration polarization. Pure oxygen is used as fuel at the cathode, which enhances the cell performance. Upon increasing current density, the open circuit voltage difference between air and pure oxygen increases, due to the increased concentration polarization. At T 5 100°C, the voltage gain is given by ΔVcathode 5 92 log
ðPO2 Þ2 ðPO2 Þ1
(2.69)
Fig. 2.5 shows the fuel utilization effect on the cell voltage for pure O2 as oxidant and 2.4% H2/22% CO/11% H2O as reference fuel. When the temperature is constant, the cell voltage gets increased upon using pure O2 as oxidant. Upon increasing the fuel utilization, the cell voltage decreases. It is evident that at higher temperatures of 1000°C, the cell voltage decrease is steep as compared with the decrease observed in temperature 800°C900°C.
Thermodynamics, polarizations, and intermediate temperature
53
Figure 2.5 Variation in cell voltage with respect to the fuel utilization and temperature [2].
Figure 2.6 Cell voltage variation with respect to current density for the two-cell stack [2].
2.6.2 Effect of pressure and temperature When the temperature is increased, the decreased ohmic polarization results in increased current density. Fig. 2.6 shows the variation of cell voltage with the current density for the two-cell stack. As evident from Fig. 2.6, the increased current density yields decreased cell voltage; however, the decrease is sharp at low temperatures.
54
Intermediate Temperature Solid Oxide Fuel Cells
Figure 2.7 AES cell performance at different pressures (1000°C) (89% H2/11% H2O,O22) [1].
Like molten carbonate fuel cell and phosphoric acid fuel cell, SOFC performance also gets boosted upon increase in pressure. At 1000°C, approximated cell performance is given by ΔV ðmVÞ 5 59log
ðP2 Þ ðP1 Þ
(2.70)
The cell voltage variation with current density and pressures is depicted in Fig. 2.7 [listed for air electrode supported (AES) cells, Siemens Westing house corporation].
2.6.3 Effect of impurities, current density and cell life At T 5 1000°C, the relationship between current density and the cell voltage is depicted by the following equation: VCD 5 2 0:73ΔJ
(2.71)
The performance of single SOFC cell at reducing temperatures is given in Fig. 2.8. When current density is increased, concentration losses as well as ohmic activation also increases. It has been observed that the AES tubular SOFC yields 33% increase in power density, SOFC when compared with the tubular design calcia stabilized supported with porous
55
Thermodynamics, polarizations, and intermediate temperature
1.4
1.4 800ºC 750ºC
Cell voltage (V)
1.0
1.2
1.0
700ºC 650ºC
0.8
0.8
0.6
0.6
0.4
0.4
Power density (W cm–2)
1.2
600ºC 0.2
0.2
0 0
0.5
1.0
1.5
2.0
2.5
0 3.0
Current density (A cm–2)
Figure 2.8 SOFC performance (single cell) at reduced temperatures [14].
support [14]. HCl, ammonia, and hydrogen sulfide are the most commonly found impurities in the coal gas. Up to 1 ppm HCl and 5000 ppm of ammonia, no noticeable cell degradation could be observed whereas even 1 ppm of H2S leads to huge performance degradation. The original performance of the cell can be resumed upon H2S removal from the fuel stream.
References [1] J.H. Hirschenhofer, D.B. Stauffer, R.R. Engleman, M.G. Klett, Fuel Cell Handbook, fourth edition, 1998. [2] G. Kaur, Solid Oxide Fuel Cells Components—Interfacial Compatibility of SOFC Glass Seal, Springer, NewYork, NY, 2015. [3] J. Larminie, A. Dicks, Fuel Cells System Explained, second ed., John Wiley and Sons, England, 2003. [4] S.C. Singhal, K. Kendall, High Temperature Solid-Oxide Fuel Cells: Fundamentals, Elsevier, New York, NY, 2000. [5] S.M. Haile, Fuel cell materials and components, Acta Mater. 51 (2003) 59816000. [6] H.R. Kunz, L.A. Murphy, in: J.R. Selman, H.C. Maru (Eds.), Proceedings of the Symposium on Electrochemical Modeling of Battery, Fuel Cell, and Photoenergy Conversion Systems, 379, The Electrochemical Society Inc, Pennington, NJ, 1986. [7] S.N. Simons, R.B. King, P.R. Prokopius, Figure 1, p. 46 in: E.H. Camara (Ed.), Symposium Proceedings Fuel Cells Technology Status and Applications, 45, Institute of Gas Technology, Chicago, IL, 1982.
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Intermediate Temperature Solid Oxide Fuel Cells
[8] V. Raghvan, Materials Science and Engineering, 5th edition, 2008. [9] E.J. Cairns, H.A. Liebhafsky, Energy Convers. 63 (1969) 9. [10] A.P. Fickett, in: J.D.E. McIntyre, S. Srinivasan, F.G. Will (Eds.), Proceedings of the Symposium on Electrode Materials and Processes for Energy Conversion and Storage, The Electrochemical Society Inc, Pennington, NJ, 1977, p. 546. [11] A.J. Appleby, J. Electroanal, Chem. 118 (1981) 31. [12] J. Huff, Status of fuel cell technologies, in Fuel Cell Seminar Abstracts, 1986 National Fuel Cell Seminar, Tucson, AZ, October 1986. [13] K. Strasser, in 26th Intersociety Energy Conversion Engineering Conference Proceedings, Volume 3, Conversion Technologies/Electrochemical Conversion, Boston, MA, August 49, 1991, published by Society of Automotive Engineers Inc., Warrendale, PA, 1991. [14] N.Q. Minh, Solid oxide fuel cell technology—features and applications, Solid State Ionics 174 (1-4) (2004) 271277.