**2. Experimental**

#### **2.1 EIS measurements**

All measurements were performed using an anode-supported microtubular SOFC(Kawakami et al., 2006) with an outer and an inner diameters of 5 mm and 3 mm, respectively. The thickness of the electrolyte was 30 *μ*m(Ueno, 2005). Anode substrate tube was made of NiO/(ZrO2)0.9

(Y2O3)0.1 (NiO/YSZ). Anode interlayer of NiO/(Ce0.9Gd0.1)O1.95 (NiO/GDC10) for low temperature operation was coated onto the anode substrate. Electrolyte was La0.8Sr0.2Ga0.8Mg0.2O2.8 (LSGM). A layer of (Ce0.6La0.4)O1.8 (LDC40) was inserted between the anode interlayer and electrolyte to prevent undesirable nickel diffusion during cell preparation at high temperature. Cathode made of (La0.6Sr0.4)(Co0.2Fe0.8)O3 (LSCF), whose length in the axial direction was 3.8 cm, was coated on the electrolyte. Geometrical electrode area was 5.9 cm2.

Figure 1 illustrates the configuration of the experimental set-up. Temperature of the quartz tube having an inner diameter of 4.6 cm was maintained at 700 ◦C with an electric furnace. Anode and cathode gas lines were supplied with mixtures of H2/N2 and O2/N2 at constant flow rates, respectively. The anode NiO was reduced to Ni by feeding H2/N2 mixture gas for two hours prior to measurements. The anode and cathode were electrically connected with the four-terminal method.

Current-voltage (I-V) curves were measured with a potentio/galvanostat (HA-320, Hokuto Denko Co., Ltd) and mass flow controllers (SEC-40, Horiba STEC) controlled by LabView 8.5 (National Instruments Inc.) on a personal computer through a data acquisition board (NI USB-6008, National Instruments Inc.). ElS measurements were carried out using a frequency response analyzer (FRA) (DS-2100/DS-266/DS-273, Ono Sokki Co Ltd.) combined with the potentio/galvanostat. EIS was carried out with two-electrode set-up without the reference electrode. An equivalent circuit presented in Fig. 2 (Barsoukov & Macdonald, 2005; McIntosh et al., 2003) is used for the complex nonlinear least square (CNLS) fitting (Barsoukov & Macdonald, 2005) of obtained impedance spectra with excluding inductive part.

Each resistance and capacitance is evaluated with a CNLS fitting program, Z-View (Scribner Inc.). In this circuit, *R*hf and *C*hf denote resistance and associated capacitance corresponding to the high frequency arc in the complex-plane plot of the impedance, respectively, *R*Ohm is the Ohmic resistance of the cell, *R*lf and *C*lf are resistance and associated capacitance for the low frequency arc, respectively.

Each one of the R-C branches dominantly represents the charge transfer process in low current density region, and mass transfer process in high current density region. In the

2 Will-be-set-by-IN-TECH

EIS with two-electrode set-up on the practical microtubular IT-SOFC was thus carried out. To evaluate the impedance variation of each part of the cell under operation, gas feeding

In addition, very few experimental studies on current distributions in a cell that lead to temperature distributions have been reported to date, although a number of computational analyses have been reported (Campanari & Iora, 2004; Costamagna & Honegger, 1998; Kanamura & Takehara, 1993; Nishino et al., 2006; Suzuki et al., 2008). Thus the current

All measurements were performed using an anode-supported microtubular SOFC(Kawakami et al., 2006) with an outer and an inner diameters of 5 mm and 3 mm, respectively. The thickness of the electrolyte was 30 *μ*m(Ueno, 2005). Anode substrate

(Y2O3)0.1 (NiO/YSZ). Anode interlayer of NiO/(Ce0.9Gd0.1)O1.95 (NiO/GDC10) for low temperature operation was coated onto the anode substrate. Electrolyte was La0.8Sr0.2Ga0.8Mg0.2O2.8 (LSGM). A layer of (Ce0.6La0.4)O1.8 (LDC40) was inserted between the anode interlayer and electrolyte to prevent undesirable nickel diffusion during cell preparation at high temperature. Cathode made of (La0.6Sr0.4)(Co0.2Fe0.8)O3 (LSCF), whose length in the axial direction was 3.8 cm, was coated on the electrolyte. Geometrical electrode

Figure 1 illustrates the configuration of the experimental set-up. Temperature of the quartz tube having an inner diameter of 4.6 cm was maintained at 700 ◦C with an electric furnace. Anode and cathode gas lines were supplied with mixtures of H2/N2 and O2/N2 at constant flow rates, respectively. The anode NiO was reduced to Ni by feeding H2/N2 mixture gas for two hours prior to measurements. The anode and cathode were electrically connected with

Current-voltage (I-V) curves were measured with a potentio/galvanostat (HA-320, Hokuto Denko Co., Ltd) and mass flow controllers (SEC-40, Horiba STEC) controlled by LabView 8.5 (National Instruments Inc.) on a personal computer through a data acquisition board (NI USB-6008, National Instruments Inc.). ElS measurements were carried out using a frequency response analyzer (FRA) (DS-2100/DS-266/DS-273, Ono Sokki Co Ltd.) combined with the potentio/galvanostat. EIS was carried out with two-electrode set-up without the reference electrode. An equivalent circuit presented in Fig. 2 (Barsoukov & Macdonald, 2005; McIntosh et al., 2003) is used for the complex nonlinear least square (CNLS) fitting (Barsoukov & Macdonald, 2005) of obtained impedance spectra with excluding inductive

Each resistance and capacitance is evaluated with a CNLS fitting program, Z-View (Scribner Inc.). In this circuit, *R*hf and *C*hf denote resistance and associated capacitance corresponding to the high frequency arc in the complex-plane plot of the impedance, respectively, *R*Ohm is the Ohmic resistance of the cell, *R*lf and *C*lf are resistance and associated capacitance for the

Each one of the R-C branches dominantly represents the charge transfer process in low current density region, and mass transfer process in high current density region. In the

distributions are estimated by using the overpotentials evaluated with EIS.

conditions for the anode and cathode were varied.

**2. Experimental**

area was 5.9 cm2.

part.

the four-terminal method.

low frequency arc, respectively.

**2.1 EIS measurements**

tube was made of NiO/(ZrO2)0.9

Fig. 1. Experimental set-up of the microtubular SOFC.

present chapter, the equivalent circuit is not separated into the charge and mass transfer processes since the complex plane plots exhibited only two arcs, whose behavior should be analyzed with simple one R-C branch prior to appropriate separations of overlapping arcs and equivalent circuit.

The Nernst loss by the partial pressure gradient of hydrogen and oxygen ascribed to their consumption leads to current distribution in the axial direction. Ohmic resistance in the anode and cathode electrodes is also attributed to this current distribution. However, the author use the above equivalent circuit for uniform current distribution to obtain average behavior over the axial direction of the cell.

As a result, the variations in these circuit parameters are obtained in accordance with current densities, and anode and cathode gas-feed conditions. In addition, the impedance of the mass transfer process can be analyzed as that of the finite length diffusion (Nakajima et al., 2010).

Fig. 2. Equivalent circuit of an SOFC.

frequency arc is small. In the previous study, the high and low frequency arcs have been found to be attributed to the cathode and anode reactions, respectively, in the low and medium current regions, from the variation of the arcs with anode and cathode gas feeding conditions (Nakajima et al., 2010). Hence anode and cathode impedances can be separated for a cell with two electrode set-up (without the reference electrode) by frequency domain when the time constant (relaxation time) sufficiently differs between the anode and cathode as the present

<sup>289</sup> Electrochemical Impedance Spectroscopy Study

*Z*' (Ω cm2 )

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8


Fig. 5. Cell voltage and overpotential map of the SOFC. Cathode: Dried air of 1000 cm3min−1. Anode: H2/N2 = 40/40 cm3min−1. *U*<sup>f</sup> = 57% and *U*ox = 5.7% at 0.5 A cm−2.

Current density (A cm-2)

Fig. 4. Complex-plane plots of the SOFC at 0.51 A cm−<sup>2</sup> under different H2 partial pressures.

The resistances obtained by the CNLS fitting are numerically integrated according to Eq. 2. Then overpotential at each part is averagely obtained as illustrated in Fig. 5 with I-V curves. In this case, *η*hf is the cathode overpotential, *η*c, *η*Ohm is the Ohmic overpotential, and *η*lf is the anode overpotential, *η*a. The I-V curves and voltages evaluated by subtracting the sum of the overpotentials from the open circuit voltage are in good agreement. Each overpotential is

0 0.2 0.4 0.6 0.8 1.0 1.2

100 Hz

H2 /N2

H2 /N2 : 40/80

H2 /N2

H2 /N2

: 40/40 cm-3 min-1

: 40/120

of the Mass Transfer in an Anode-Supported Microtubular Solid Oxide Fuel Cell

 : 40/160 1 Hz

Cathode: Dried air of 1000 cm3min−1. *U*<sup>f</sup> = 58% and *U*ox = 5.8%.

Cell voltage

OCV - *η*Ohm OCV - *η*Ohm - *η*<sup>c</sup> OCV - *η*Ohm - *η*<sup>c</sup>

OCV

*Z* " (Ω cm2

presented in Figs. 6 and 7.

0

0.2

0.4

0.6

Voltage (V)

0.8

1.0

1.2

)


0.1 Hz

*η*Ohm

*η*c

*η*a

cell.

#### **2.2 Temperature measurements**

During the measurements, anode and cathode were fed upward with mixtures of H2/N2 and dried air at constant flow rates with current density, respectively. Temperatures at the upper, middle, and lower parts in the axial direction of the anode and cathode surfaces were measured by thermocouples. The cathode side thermocouple tip was fixed with silver paste and wire to retain contact and to minimize radiation heat transfer at the thermocouple with their low emissivity. The changes of the cell voltages by the installation of the thermocouples were less than 3%.

#### **3. Results and discussion**

#### **3.1 Evaluation of overpotentials from EIS spectra**

Figure 3 shows the I-V curves for the different anode fed gas flow rates and gas compositions. The performance of this type of the cell significantly depends on the fuel utilization and the partial pressure, indicating the effect of the fuel mass transfer.

Fig. 3. I-V curves of the SOFC under different (a) anode gas flow rates and (b) partial pressures. Cathode: Dried air of 1000 cm3min−1. *U*ox = 5.7% at 0.5 A cm−2.

The resistances in the equivalent circuit can be written as the derivatives of the anode, Ohmic, and cathode overpotentials because EIS measures voltage drops in an infinitesimal interval of the current (Konomi & Saho, 2006; Nakajima et al., 2005). The resistances are non-Ohmic resistance and depend on current density (Nakajima et al., 2006). Overpotentials are hence calculated by integrating each resistance with respect to current as follows.

$$R\_{\chi}(I) = \frac{\partial \eta\_{\chi}}{\partial I} \tag{1}$$

where x = hf, Ohm, lf. Each overpotential is then evaluated by integration of those resistances as follows.

$$
\eta\_{\mathbf{x}} = \int R\_{\mathbf{x}}(I) dI \tag{2}
$$

Figure 4 shows the complex plane plots by EIS, where low frequency arc becomes large according to a decrease in hydrogen partial pressure by variation of the anode gas flow rate from H2/N2 = 40/40 cm<sup>3</sup> min−<sup>1</sup> (1atm, 25 ◦C) to 40/160 cm<sup>3</sup> min−1. Change of high 4 Will-be-set-by-IN-TECH

During the measurements, anode and cathode were fed upward with mixtures of H2/N2 and dried air at constant flow rates with current density, respectively. Temperatures at the upper, middle, and lower parts in the axial direction of the anode and cathode surfaces were measured by thermocouples. The cathode side thermocouple tip was fixed with silver paste and wire to retain contact and to minimize radiation heat transfer at the thermocouple with their low emissivity. The changes of the cell voltages by the installation of the thermocouples

Figure 3 shows the I-V curves for the different anode fed gas flow rates and gas compositions. The performance of this type of the cell significantly depends on the fuel utilization and the

Cell voltage (V)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Current density (A cm-2

*<sup>∂</sup><sup>I</sup>* (1)

*R*x(*I*)*dI* (2)

0 0.2 0.4 0.6 0.8 1.0

H2 /N2

H2 /N2 : 40/80

H2 /N2 : 40/120

H2 /N2 : 40/160

)

: 40/40 cm-3 min-1

**2.2 Temperature measurements**

were less than 3%.

Cell voltage (V)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

as follows.

**3. Results and discussion**

H2

H2

H2

/N2 : 80/80 (*U*<sup>f</sup>

/N2 :120/120 (*U*<sup>f</sup>

/N2 : 40/40 cm-3 min-1 (*U*<sup>f</sup>

**3.1 Evaluation of overpotentials from EIS spectra**

partial pressure, indicating the effect of the fuel mass transfer.

= 28 % at 0.5 A cm-2)

= 19 % at 0.5 A cm-2)

= 57 % at 0.5 A cm-2)

Current density (A cm-2

0 0.5 1.0 1.5

)

pressures. Cathode: Dried air of 1000 cm3min−1. *U*ox = 5.7% at 0.5 A cm−2.

calculated by integrating each resistance with respect to current as follows.

Fig. 3. I-V curves of the SOFC under different (a) anode gas flow rates and (b) partial

The resistances in the equivalent circuit can be written as the derivatives of the anode, Ohmic, and cathode overpotentials because EIS measures voltage drops in an infinitesimal interval of the current (Konomi & Saho, 2006; Nakajima et al., 2005). The resistances are non-Ohmic resistance and depend on current density (Nakajima et al., 2006). Overpotentials are hence

*<sup>R</sup>*x(*I*) = *∂η*<sup>x</sup>

where x = hf, Ohm, lf. Each overpotential is then evaluated by integration of those resistances

Figure 4 shows the complex plane plots by EIS, where low frequency arc becomes large according to a decrease in hydrogen partial pressure by variation of the anode gas flow rate from H2/N2 = 40/40 cm<sup>3</sup> min−<sup>1</sup> (1atm, 25 ◦C) to 40/160 cm<sup>3</sup> min−1. Change of high

*η*<sup>x</sup> = 

(a) (b)

frequency arc is small. In the previous study, the high and low frequency arcs have been found to be attributed to the cathode and anode reactions, respectively, in the low and medium current regions, from the variation of the arcs with anode and cathode gas feeding conditions (Nakajima et al., 2010). Hence anode and cathode impedances can be separated for a cell with two electrode set-up (without the reference electrode) by frequency domain when the time constant (relaxation time) sufficiently differs between the anode and cathode as the present cell.

Fig. 4. Complex-plane plots of the SOFC at 0.51 A cm−<sup>2</sup> under different H2 partial pressures. Cathode: Dried air of 1000 cm3min−1. *U*<sup>f</sup> = 58% and *U*ox = 5.8%.

The resistances obtained by the CNLS fitting are numerically integrated according to Eq. 2. Then overpotential at each part is averagely obtained as illustrated in Fig. 5 with I-V curves. In this case, *η*hf is the cathode overpotential, *η*c, *η*Ohm is the Ohmic overpotential, and *η*lf is the anode overpotential, *η*a. The I-V curves and voltages evaluated by subtracting the sum of the overpotentials from the open circuit voltage are in good agreement. Each overpotential is presented in Figs. 6 and 7.

Fig. 5. Cell voltage and overpotential map of the SOFC. Cathode: Dried air of 1000 cm3min−1. Anode: H2/N2 = 40/40 cm3min−1. *U*<sup>f</sup> = 57% and *U*ox = 5.7% at 0.5 A cm−2.

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

(b)

(c)

H2 /N2

(a)

of the Mass Transfer in an Anode-Supported Microtubular Solid Oxide Fuel Cell

H2 /N2

H2 /N2

H2 /N2

: 40/40 cm-3 min-1

: 40/80

<sup>291</sup> Electrochemical Impedance Spectroscopy Study

: 40/120

: 40/160

0 0.2 0.4 0.6 0.8 1.0 Current density (A cm-2)

0 0.2 0.4 0.6 0.8 1.0 Current density (A cm-2)

0 0.2 0.4 0.6 0.8 1.0 Current density (A cm-2)

Fig. 7. (a) Ohmic, (b) cathode, and (c) anode overpotentials of the SOFC for different H2

composition. When hydrogen partial pressure in the fed gas is decreased, the activation overpotential also increases owing to the decrease in the exchange current density. In this

The Nernst loss due to the fuel partial pressure gradient along the axis by the fuel consumption contributes the large anode concentration overpotential as indicated from the diffusion impedance in the previous report (Nakajima et al., 2010). This Nernst loss results in

0.05 0.10 0.15 0.20

*η*c (V)

*η*a (V)

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

partial pressures. *U*<sup>f</sup> = 57% and *U*ox = 5.7% at 0.5 A cm−2.

way, EIS can be used to diagnose the cell status under operation.

current distribution along the axis as described in later section.

0

0

*η*Ohm (V)

Fig. 6. (a) Ohmic, (b) cathode, and (c) anode overpotentials of the SOFC for different anode gas flow rates. *U*ox = 5.7% at 0.5 A cm−2.

#### **3.2 Anode overpotentials**

The anode and cathode activation overpotentials, *η*aa, and *η*ca, respectively, are then separated using the Butler-Volmer (BV) equation as illustrated in Fig. 8. Thereby the concentration overpotential of the anode, *η*ac, is also separated by subtracting the anode activation overpotentials from the anode overpotentials as presented in Figs. 9 and 10. It should be noted that the overpotential described by the BV type equation is controversial in terms of the electron transfer rate limiting (the activation overpotential) or chemical reaction rate limiting process.

The separation of the overpotentials is successfully confirmed by the observation of the overpotential variation in conjunction with the variation of the anode gas flow rate and

6 Will-be-set-by-IN-TECH

/N2 : 40/40 cm-3 min-1

= 28 % at 0.5 A cm-2)

= 57 % at 0.5 A cm-2)

(a)

/N2 :120/120

/N2 : 80/80

0 0.5 1.0 1.5 Current density (A cm-2)

0 0.5 1.0 1.5 Current density (A cm-2)

0 0.5 1.0 1.5 Current density (A cm-2)

Fig. 6. (a) Ohmic, (b) cathode, and (c) anode overpotentials of the SOFC for different anode

The anode and cathode activation overpotentials, *η*aa, and *η*ca, respectively, are then separated using the Butler-Volmer (BV) equation as illustrated in Fig. 8. Thereby the concentration overpotential of the anode, *η*ac, is also separated by subtracting the anode activation overpotentials from the anode overpotentials as presented in Figs. 9 and 10. It should be noted that the overpotential described by the BV type equation is controversial in terms of the electron transfer rate limiting (the activation overpotential) or chemical reaction rate limiting

The separation of the overpotentials is successfully confirmed by the observation of the overpotential variation in conjunction with the variation of the anode gas flow rate and

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.05 0.10 0.15 0.20

*ηc* (V)

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

*η*a (V)

gas flow rates. *U*ox = 5.7% at 0.5 A cm−2.

**3.2 Anode overpotentials**

process.

0

0

*η*Ohm (V)

H2

(Uf = 19 % at 0.5 A cm-2)

(b)

(c)

(Uf

H2

H2

(Uf

Fig. 7. (a) Ohmic, (b) cathode, and (c) anode overpotentials of the SOFC for different H2 partial pressures. *U*<sup>f</sup> = 57% and *U*ox = 5.7% at 0.5 A cm−2.

composition. When hydrogen partial pressure in the fed gas is decreased, the activation overpotential also increases owing to the decrease in the exchange current density. In this way, EIS can be used to diagnose the cell status under operation.

The Nernst loss due to the fuel partial pressure gradient along the axis by the fuel consumption contributes the large anode concentration overpotential as indicated from the diffusion impedance in the previous report (Nakajima et al., 2010). This Nernst loss results in current distribution along the axis as described in later section.



Air : 1000 cm-3 min-1

of the Mass Transfer in an Anode-Supported Microtubular Solid Oxide Fuel Cell

O2 : 1000 cm-3 min-1

Fig. 11. Complex-plane plots of the SOFC for different oxygen partial pressures.

: 500/500 cm-3 min-1

<sup>293</sup> Electrochemical Impedance Spectroscopy Study

Air / N2

0 0.5 1.0 1.5

Air / N2

Air : 1000 cm-3 min-1

O2 : 1000 cm-3 min-1

overpotentials of the SOFC for different oxygen partial pressures.

: 500/500 cm-3 min-1

Fig. 12. (a) Overpotential for the low frequency arc and (b) cathode concentration

(a) (b)

Current density (A cm ) -2

0

*Z* " (Ω cm2

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35

the Sherwood number, *Sh*, is

1985) is used as *Sh*.

*η* lf (V)

)

0.1

㻜 㻜㻚㻝 㻜㻚㻞 㻜㻚㻟 㻜㻚㻠 㻜㻚㻡 㻜㻚㻢

*Z*' (Ω cm2 )

> 0 0.02 0.04 0.06 0.08 0.10

Air / N2

Air : 1000 cm-3 min-1

*η*cc (V)

Then the concentration overpotential at the cathode is analyzed in the light of the oxygen transfer at the cathode surface boundary layer. The author calculates the concentration overpotentials from the convective mass transfer using the Sherwood numbers for forced and natural convections for the cells having two different diameters in the cylindrical quartz tube. Then the calculated overpotential is compared with that derived from the EIS measurements. In analogy with the Nusselt number, *Nu*, for the circular-tube annulus (Kays & Perkins, 1985),

> *Sh* <sup>=</sup> *<sup>h</sup>*O2*D*<sup>e</sup> *D*O2

where *h*O2 and *D*O2 represent the mass transfer coefficient and the binary molecular diffusivity of O2, respectively. *D*O2 can be calculated from reported values at low temperatures on the basis of the Chapman-Enskog model (Bird et al., 2007). *D*<sup>e</sup> is the hydraulic diameter, *D*<sup>o</sup> − *D*i, difference between the outer and inner diameters of the circular-tube annulus. In the present

In the case of forced convection, known *Nu* for fully developed laminar flow (Kays & Perkins,

case, *D*<sup>o</sup> and *D*<sup>i</sup> are the diameters of the quartz tube and the cell, respectively.

0 0.5 1.0 1.5

Current density (A cm ) -2

(3)

: 500/500 cm-3 min-1

Assuming that the cathode concentration overpotential is zero when oxygen gas is fed, it can be separated by subtracting *η*lf for oxygen gas from *η*lf for other oxygen gas partial pressures as presented in Fig. 12(b). This is so-called O2 gain with reference to the case of oxygen gas.

Fig. 8. (a) Activation and (b) concentration overpotentials at the anode of the SOFC for different anode gas flow rates.

Fig. 9. (a) Activation and (b) concentration overpotentials at the anode of the SOFC for different anode gas flow rates.

Fig. 10. (a) Activation and (b) concentration overpotentials at the anode of the SOFC for different anode gas flow rates.

#### **3.3 Cathode overpotentials**

At the cathode, the oxygen utilization was rather smaller than fuel utilization according to practical operation conditions. So, the Nernst loss from the oxygen concentration gradient along the axis is small in contrast to the anode. In the low and medium current regions, the concentration overpotential at the cathode is almost negligible and the activation overpotential is observed as shown in Figs. 6 and 7. However, the large current region in the case of large hydrogen flow rate, decrease in the oxygen partial pressure of the fed gas in the cathode side leads to an increase in the diameter of the low frequency arc of the complex plane plot, that is *R*lf, as seen in Fig. 11. Thus the oxygen mass transfer impedance is included in the low frequency arc. Hence *η*lf increases with a decrease in the oxygen partial pressure in the fed gas as shown in Fig. 12(a).

8 Will-be-set-by-IN-TECH

Overpotential (V)

Fig. 8. (a) Activation and (b) concentration overpotentials at the anode of the SOFC for

Fig. 9. (a) Activation and (b) concentration overpotentials at the anode of the SOFC for

*η*ac (V)

Fig. 10. (a) Activation and (b) concentration overpotentials at the anode of the SOFC for

At the cathode, the oxygen utilization was rather smaller than fuel utilization according to practical operation conditions. So, the Nernst loss from the oxygen concentration gradient along the axis is small in contrast to the anode. In the low and medium current regions, the concentration overpotential at the cathode is almost negligible and the activation overpotential is observed as shown in Figs. 6 and 7. However, the large current region in the case of large hydrogen flow rate, decrease in the oxygen partial pressure of the fed gas in the cathode side leads to an increase in the diameter of the low frequency arc of the complex plane plot, that is *R*lf, as seen in Fig. 11. Thus the oxygen mass transfer impedance is included in the low frequency arc. Hence *η*lf increases with a decrease in the oxygen partial pressure in

0 0.02 0.04 0.06 0.08 0.10 0.12

(a) (b)

Experimental

0 0.05 0.10 0.15 0.20

0 0.05 0.10 0.15 0.20 0.25 0.30

0 0.05 0.10 0.15 0.20 0.25 (a)

Overpotential (V)

different anode gas flow rates.

*η*aa (V)

different anode gas flow rates.

*η*aa (V)

different anode gas flow rates.

the fed gas as shown in Fig. 12(a).

**3.3 Cathode overpotentials**

Fit curve with the BV equation

0 0.2 0.4 0.6 0.8

0 0.5 1.0 1.5

/N2 : 120/120

/N2 : 40/40 cm-3 min-1

H2

H2 /N2 : 80/80

H2

Current density (A cm-2)

0 0.2 0.4 0.6 0.8 1.0 Current density (A cm-2)

: 40/40 cm-3 min-1

(a) (b)

H2 /N2

H2 /N2 : 40/80

H2 /N2 : 40/120

H2 /N2 : 40/160

Current density (A cm-2)

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

> 0 0.02 0.04 0.06 0.08 0.10 0.12

*η*ac (V)

(b)

0 0.2 0.4 0.6 0.8

Current density (A cm-2)

0 0.5 1.0 1.5 Current density (A cm-2)

0 0.2 0.4 0.6 0.8 1.0 Current density (A cm-2)

Fig. 11. Complex-plane plots of the SOFC for different oxygen partial pressures.

Assuming that the cathode concentration overpotential is zero when oxygen gas is fed, it can be separated by subtracting *η*lf for oxygen gas from *η*lf for other oxygen gas partial pressures as presented in Fig. 12(b). This is so-called O2 gain with reference to the case of oxygen gas.

Fig. 12. (a) Overpotential for the low frequency arc and (b) cathode concentration overpotentials of the SOFC for different oxygen partial pressures.

Then the concentration overpotential at the cathode is analyzed in the light of the oxygen transfer at the cathode surface boundary layer. The author calculates the concentration overpotentials from the convective mass transfer using the Sherwood numbers for forced and natural convections for the cells having two different diameters in the cylindrical quartz tube. Then the calculated overpotential is compared with that derived from the EIS measurements. In analogy with the Nusselt number, *Nu*, for the circular-tube annulus (Kays & Perkins, 1985), the Sherwood number, *Sh*, is

$$Sh = \frac{h\_{\rm O\_2} D\_{\rm e}}{D\_{\rm O\_2}} \tag{3}$$

where *h*O2 and *D*O2 represent the mass transfer coefficient and the binary molecular diffusivity of O2, respectively. *D*O2 can be calculated from reported values at low temperatures on the basis of the Chapman-Enskog model (Bird et al., 2007). *D*<sup>e</sup> is the hydraulic diameter, *D*<sup>o</sup> − *D*i, difference between the outer and inner diameters of the circular-tube annulus. In the present case, *D*<sup>o</sup> and *D*<sup>i</sup> are the diameters of the quartz tube and the cell, respectively.

In the case of forced convection, known *Nu* for fully developed laminar flow (Kays & Perkins, 1985) is used as *Sh*.

Current density (A cm-2)

0 0.5 1.0 1.5

0 0.5 1.0 1.5 Current density (A cm-2)

*q*op,x = *Iη*<sup>x</sup> (9)

(b)

*η*cc (V)

<sup>295</sup> Electrochemical Impedance Spectroscopy Study

Fig. 13. Comparison of the cathode concentration overpotentials between those from EIS and

Current density (A cm-2)

0 0.5 1.0 1.5

those calculated for (a) forced convection and (b) natural convection.

: 1000 / 1000 cm-3 min-1

: 1000 / 1000 cm-3 min-1

Air : 2000 cm-3 min-1

Air : 2000 cm-3 min-1

Air / N2

**3.4.1 Heat production rates at the anode and cathode**

product of the overpotentials and current as follows.

Air / N2

0 0.5 1.0 1.5 Current density (A cm-2)

Air : 1000 cm-3 min-1

Air : 1000 cm-3 min-1

of the Mass Transfer in an Anode-Supported Microtubular Solid Oxide Fuel Cell

: 500 / 500 cm-3 min-1

: 500 / 500 cm-3 min-1

Air / N2

Air / N2

(a)

(a)

㻯㼍㼘㼏㼡㼘㼍㼠㼑㼐㻌 㻯㼍㼘㼏㼡㼘㼍㼠㼑㼐

㻹㼑㼍㼟㼡㼞㼑㼐 㻹㼑㼍㼟㼡㼞㼑㼐

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

*η*cc (V)

natural convection.

previous section.

originating in the electrolyte.

calculated from the overpotentials.

*η*cc (V)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

㻯㼍㼘㼏㼡㼘㼍㼠㼑㼐㻌 㻯㼍㼘㼏㼡㼘㼍㼠㼑㼐

㻹㼑㼍㼟㼡㼞㼑㼐 㻹㼑㼍㼟㼡㼞㼑㼐

> 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

> > (b)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

*η*cc(V)

Fig. 14. Comparison of the cathode concentration overpotentials for the cell with twice the diameter between those from EIS and those calculated for (a) forced convection and (b)

In this section, current distribution in the axial direction of the cell by the fuel consumption is estimated by comparing the measured and calculated cell surface temperatures on the basis of the derivation of the relation between current (heat production rate) and cell temperature. This relation is derived using the anode, Ohmic, and cathode overpotentials evaluated in the

Rates of irreversible heat production ascribed to the overpotentials are calculated by the

The author regards the heat production associated with the Ohmic overpotential as that

Figure 15(a) presents the irreversible heat production rates at the respective parts of the cell

**3.4 Current distribution in the cell derived from surface temperature measurements**

In the case of natural convection with laminar flow, in analogy with *Nu* for vertical fluid layer (Churchill, 1983),

$$Sh = \frac{h\_{\rm O\_2}(r\_{\rm o} - r\_{\rm i})}{D\_{\rm O\_2}} = 0.28 Ra^{1/4} \left(\frac{l}{r\_{\rm o} - r\_{\rm i}}\right)^{-1/4} \tag{4}$$

where

$$Ra = Gr \text{Sc} \tag{5}$$

$$=\frac{g(\rho^\*-\rho^{\rm el})(r\_0-r\_i)^3}{\rho^\*\upsilon D\_{\rm O\_2}}\tag{6}$$

*l* is the cathode axial length. *ρ*<sup>∗</sup> and *ρ*el are the densities of the fed and cathode surface gas, respectively, *r*<sup>o</sup> and *r*<sup>i</sup> are the radii of the quartz tube and the cell, respectively. *ν* is the kinematic viscosity of air. In the present chapter, *<sup>h</sup>*O2 at *<sup>ρ</sup>*el of fed nitrogen density is used for simplification.

Oxygen flux, *J*O2 is expressed as

$$J\_{\rm O\_2} = h\_{\rm O\_2} (\mathsf{C}\_{\rm O\_2}^{\rm b} - \mathsf{C}\_{\rm O\_2}^{\rm el}) \tag{7}$$

where C<sup>b</sup> and Cel O2 are oxygen concentrations in the fed gas and at the cathode surface, respectively. In the case of natural convection, the mass transfer coefficient is compensated with the average radius, *r*m, by multiplying *r*m/*r*<sup>i</sup> = (*r*<sup>o</sup> − *r*i)/*r*i*ln*(*r*o/*r*i) (Churchill, 1983) to correlate the current density. Since *<sup>J</sup>*O2 can be calculated from current, and *<sup>h</sup>*O2 and *<sup>C</sup>*<sup>b</sup> O2 are known, *C*el O2 is yielded.

The cathode concentration overpotential, *η*cc, is given by substituting *C*el O2 into the following equation.

$$\eta\_{\rm cc} = \frac{RT}{anF} \left( \ln \frac{\rm C\_{O\_2}^{b}}{C\_{O\_2}^{el}} - \ln \frac{C\_{O\_2}^{\ast b}}{C\_{O\_2}^{\ast el}} \right) \tag{8}$$

Here, *η*cc is derived as the *O*<sup>2</sup> gain with the second term in the right-hand side for *O*<sup>2</sup> gas having negligibly small value.

As presented in Fig. 13, the concentration overpotentials measured from EIS are larger than that calculated for forced convection and smaller than that for natural convection. Despite that *h*O2 for natural convection is overestimated for simplification, the concentration overpotentials measured are smaller than those for natural convection. The cathode concentration overpotentials in the present experimental set-up are thus determined by the mass transfer in the transition region between forced and natural convections.

Figure 14 shows the measured and calculated cathode concentration overpotentials for the cell having twice the diameter (Ishihara et al., 2009; Watanabe et al., 2010). The concentration overpotential is larger as predicted from the Sherwood numbers of Eq. 3. The concentration overpotentials measured from the EIS are also between those calculated for forced and natural convections.

In this way, the cathode concentration overpotential can be related with the Sherwood number. This is useful for actual SOFC systems because the Sherwood number can be determined for the structure of a cell, alignment of the cells in a stack, and gas feed conditions in the actual systems.

10 Will-be-set-by-IN-TECH

In the case of natural convection with laminar flow, in analogy with *Nu* for vertical fluid layer

= 0.28*Ra*1/4

<sup>=</sup> *<sup>g</sup>*(*ρ*<sup>∗</sup> <sup>−</sup> *<sup>ρ</sup>*el)(*r*<sup>o</sup> <sup>−</sup> *<sup>r</sup>*i)<sup>3</sup> *ρ*∗*νD*O2

*l* is the cathode axial length. *ρ*<sup>∗</sup> and *ρ*el are the densities of the fed and cathode surface gas, respectively, *r*<sup>o</sup> and *r*<sup>i</sup> are the radii of the quartz tube and the cell, respectively. *ν* is the kinematic viscosity of air. In the present chapter, *<sup>h</sup>*O2 at *<sup>ρ</sup>*el of fed nitrogen density is used

respectively. In the case of natural convection, the mass transfer coefficient is compensated with the average radius, *r*m, by multiplying *r*m/*r*<sup>i</sup> = (*r*<sup>o</sup> − *r*i)/*r*i*ln*(*r*o/*r*i) (Churchill, 1983) to correlate the current density. Since *<sup>J</sup>*O2 can be calculated from current, and *<sup>h</sup>*O2 and *<sup>C</sup>*<sup>b</sup>

O2 <sup>−</sup> *<sup>C</sup>*el O2

O2 are oxygen concentrations in the fed gas and at the cathode surface,

− ln

*C*∗<sup>b</sup> O2 *C*∗el O2

*<sup>J</sup>*O2 = *<sup>h</sup>*O2 (*C*<sup>b</sup>

 ln *C*b O2 *C*el O2

Here, *η*cc is derived as the *O*<sup>2</sup> gain with the second term in the right-hand side for *O*<sup>2</sup> gas

As presented in Fig. 13, the concentration overpotentials measured from EIS are larger than that calculated for forced convection and smaller than that for natural convection. Despite that *h*O2 for natural convection is overestimated for simplification, the concentration overpotentials measured are smaller than those for natural convection. The cathode concentration overpotentials in the present experimental set-up are thus determined by the

Figure 14 shows the measured and calculated cathode concentration overpotentials for the cell having twice the diameter (Ishihara et al., 2009; Watanabe et al., 2010). The concentration overpotential is larger as predicted from the Sherwood numbers of Eq. 3. The concentration overpotentials measured from the EIS are also between those calculated for forced and natural

In this way, the cathode concentration overpotential can be related with the Sherwood number. This is useful for actual SOFC systems because the Sherwood number can be determined for the structure of a cell, alignment of the cells in a stack, and gas feed conditions

The cathode concentration overpotential, *η*cc, is given by substituting *C*el

*<sup>η</sup>*cc <sup>=</sup> *RT αnF*

mass transfer in the transition region between forced and natural convections.

 *l r*<sup>o</sup> − *r*<sup>i</sup> −1/4

) (7)

*Ra* = *GrSc* (5)

(4)

(6)

O2 are

(8)

O2 into the following

*Sh* <sup>=</sup> *<sup>h</sup>*O2 (*r*<sup>o</sup> <sup>−</sup> *<sup>r</sup>*i) *D*O2

(Churchill, 1983),

for simplification.

where C<sup>b</sup> and Cel

known, *C*el

equation.

convections.

in the actual systems.

Oxygen flux, *J*O2 is expressed as

O2 is yielded.

having negligibly small value.

where

Fig. 13. Comparison of the cathode concentration overpotentials between those from EIS and those calculated for (a) forced convection and (b) natural convection.

Fig. 14. Comparison of the cathode concentration overpotentials for the cell with twice the diameter between those from EIS and those calculated for (a) forced convection and (b) natural convection.

#### **3.4 Current distribution in the cell derived from surface temperature measurements**

In this section, current distribution in the axial direction of the cell by the fuel consumption is estimated by comparing the measured and calculated cell surface temperatures on the basis of the derivation of the relation between current (heat production rate) and cell temperature. This relation is derived using the anode, Ohmic, and cathode overpotentials evaluated in the previous section.

#### **3.4.1 Heat production rates at the anode and cathode**

Rates of irreversible heat production ascribed to the overpotentials are calculated by the product of the overpotentials and current as follows.

$$q\_{\rm op,x} = I\eta\_{\rm x} \tag{9}$$

The author regards the heat production associated with the Ohmic overpotential as that originating in the electrolyte.

Figure 15(a) presents the irreversible heat production rates at the respective parts of the cell calculated from the overpotentials.

to that of fed gases. For this calculation, 50 - 75 % of total current is assumed between the inlet and midpoint of the anode tube according to hydrogen utilization to estimate the amounts of the product water and remaining hydrogen at the midpoint, considering current distribution along the axial direction owing to the Nernst loss. Strictly speaking, this partial pressure terms should be excluded for the calculation of total heat production since those terms are included

<sup>297</sup> Electrochemical Impedance Spectroscopy Study

The entropy received by the cathode is also derived from the entropy balance between the

The relation between current and reversible heat production rates of the anode and cathode

Since the total heat production rates at the anode, *q*a, and cathode, *q*c, are expressed as the sum of the heat production rates associated with the overpotentials and the single electrode

*<sup>q</sup>*<sup>a</sup> <sup>=</sup> *<sup>q</sup>*op,a <sup>−</sup> *<sup>I</sup>π*<sup>a</sup>

the total heat production rates at the respective parts of the cell are shown in Fig. 15(c). These heat production rates seem to represent those in the middle part of the cell (Nakajima et al., 2009). The heat production rate at the cathode is significantly large compared with those at the anode and electrolyte. The heat absorption by the single electrode Peltier heat associated

By substituting the total heat production rates per unit volume into the heat conduction (energy balance) equations for the constitutive layers and integrating the differential equations with boundary conditions between the layers and at the surfaces, temperatures

Assuming uniform heat flux in the axial direction and temperature in the circumferential direction, the energy balance (the heat conduction) equation at steady state of the anode,

Here, *r*, *T*, *λ*, and *V* are the radial coordinate, temperature, thermal conductivity, and volume of each layer, respectively. The subscripts, x = a, el (Ohm), c represent the anode, electrolyte and cathode layers. The radiation heat transfer is also assumed to be negligible owing to metallic cathode current collector layer (Nakajima et al., 2009). Integration of this simplified

*dT*x(*r*) *dr* <sup>+</sup> *<sup>q</sup>*<sup>x</sup> *V*x

*Iπ*c

*q*<sup>c</sup> = *q*op,c +

are given from the product of the Peltier heats and current as presented in Fig. 15(b).

1 2

*S*H2 = *S*H2O (12)

*<sup>F</sup>* (13)

*<sup>F</sup>* (14)

(15)

in the measured overpotential as the Nernst loss.

*π*c *<sup>T</sup>* <sup>+</sup> 1 2 *S*O2<sup>−</sup> +

of the Mass Transfer in an Anode-Supported Microtubular Solid Oxide Fuel Cell

formation of O2 and transport of O2−.

with current can be seen at the anode.

electrolyte and cathode layers reduces to

at the anode and cathode surfaces can be evaluated.

**3.4.2 Relation between the surface temperature and local current**

<sup>0</sup> <sup>=</sup> <sup>1</sup> *r d dr λ*x*r*

Peltier heats as follows,

Fig. 15. Heat production rates of the SOFC with (a) the overpotentials, (b) the single electrode Peltier heats, and (c) the sum of them. H2/N2 = 40/40 cm−<sup>3</sup> min−1. Cathode: Dried air of 1000 cm−<sup>3</sup> min−1. *U*<sup>f</sup> = 57% and *U*ox = 5.7% at 3.0 A.

Reversible heat production with the entropy received reversibly by the anode, the electrochemical Peltier heat at the anode, is then derived from the entropy balance at the anode. The entropy balance for the anode reaction

$$
\frac{1}{2}\text{H}\_2 + \text{O}^{2-} \rightarrow \frac{1}{2}\text{H}\_2\text{O} + \text{e}^-\tag{10}
$$

is expressed as follows on the basis of the local equilibrium hypothesis(Kjelstrup & Bedeaux, 1997; Nakajima et al., 2004).

$$\frac{\pi\_\text{a}}{T} + \frac{1}{2}\mathcal{S}\_{\text{O}^{2-}} + \frac{1}{2}\mathcal{S}\_{\text{H}\_2} = \mathcal{S}\_{\text{H}\_2\text{O}}\tag{11}$$

where *π*a/*T* and *S*O2<sup>−</sup> denote the entropies reversibly received by the anode and transported by O2<sup>−</sup> in the electrolyte, respectively. *π*<sup>a</sup> corresponds to the single electrode Peltier heat. *S*H2 , *S*H2O are the entropies consumed by the formations of H2 and H2O, respectively.

Here the entropy transported by electrons at the electrode is neglected since its value in metal is negligibly small compared with the other terms in general (Moore & Graves, 1973; Vedernikov, 1969). In this section, *T* and *F* have their common meanings. Because *S*O2<sup>−</sup> in the LSGM has not been reported, that in (ZrO2)0.92(Y2O3)0.08(YSZ) obtained by thermoelectric power measurement (Ahlgren & Willy Poulsen, 1994) is applied.

Then *S*H2 and *S*H2O are obtained from those at 1 atm in thermodynamic data. Here, partial pressures of H2 and H2O are calculated from the ratio of molar flow rate of the product water

12 Will-be-set-by-IN-TECH

 

&XUUHQW\$

Fig. 15. Heat production rates of the SOFC with (a) the overpotentials, (b) the single electrode Peltier heats, and (c) the sum of them. H2/N2 = 40/40 cm−<sup>3</sup> min−1. Cathode: Dried air of

Reversible heat production with the entropy received reversibly by the anode, the electrochemical Peltier heat at the anode, is then derived from the entropy balance at the

2

1 2

where *π*a/*T* and *S*O2<sup>−</sup> denote the entropies reversibly received by the anode and transported by O2<sup>−</sup> in the electrolyte, respectively. *π*<sup>a</sup> corresponds to the single electrode Peltier heat. *S*H2 ,

Here the entropy transported by electrons at the electrode is neglected since its value in metal is negligibly small compared with the other terms in general (Moore & Graves, 1973; Vedernikov, 1969). In this section, *T* and *F* have their common meanings. Because *S*O2<sup>−</sup> in the LSGM has not been reported, that in (ZrO2)0.92(Y2O3)0.08(YSZ) obtained by thermoelectric

Then *S*H2 and *S*H2O are obtained from those at 1 atm in thermodynamic data. Here, partial pressures of H2 and H2O are calculated from the ratio of molar flow rate of the product water

is expressed as follows on the basis of the local equilibrium hypothesis(Kjelstrup & Bedeaux,

H2 <sup>+</sup> <sup>O</sup>2<sup>−</sup> <sup>1</sup>

*S*H2O are the entropies consumed by the formations of H2 and H2O, respectively.

Heat production rate (Js-1)

E

&XUUHQW\$

H2O + e<sup>−</sup> (10)

*S*H2 = *S*H2O (11)

&XUUHQW\$

Heat production rate (Js-1)

1000 cm−<sup>3</sup> min−1. *U*<sup>f</sup> = 57% and *U*ox = 5.7% at 3.0 A.

anode. The entropy balance for the anode reaction

1997; Nakajima et al., 2004).

 

> 1 2

*π*a *<sup>T</sup>* <sup>+</sup> 1 2 *S*O2<sup>−</sup> +

power measurement (Ahlgren & Willy Poulsen, 1994) is applied.

F

Heat production rate (Js-1)

&DWKRGH \$QRGH (OHFWURO\WH

D

to that of fed gases. For this calculation, 50 - 75 % of total current is assumed between the inlet and midpoint of the anode tube according to hydrogen utilization to estimate the amounts of the product water and remaining hydrogen at the midpoint, considering current distribution along the axial direction owing to the Nernst loss. Strictly speaking, this partial pressure terms should be excluded for the calculation of total heat production since those terms are included in the measured overpotential as the Nernst loss.

The entropy received by the cathode is also derived from the entropy balance between the formation of O2 and transport of O2−.

$$\frac{\pi\_{\text{C}}}{T} + \frac{1}{2} \text{S}\_{\text{O}^{2-}} + \frac{1}{2} \text{S}\_{\text{H}\_{2}} = \text{S}\_{\text{H}\_{2}\text{O}} \tag{12}$$

The relation between current and reversible heat production rates of the anode and cathode are given from the product of the Peltier heats and current as presented in Fig. 15(b).

Since the total heat production rates at the anode, *q*a, and cathode, *q*c, are expressed as the sum of the heat production rates associated with the overpotentials and the single electrode Peltier heats as follows,

$$q\_{\mathbf{a}} = q\_{\text{op}, \mathbf{a}} - \frac{I \pi\_{\mathbf{a}}}{F} \tag{13}$$

$$q\_{\rm c} = q\_{\rm op,c} + \frac{I\pi\_{\rm c}}{F} \tag{14}$$

the total heat production rates at the respective parts of the cell are shown in Fig. 15(c). These heat production rates seem to represent those in the middle part of the cell (Nakajima et al., 2009). The heat production rate at the cathode is significantly large compared with those at the anode and electrolyte. The heat absorption by the single electrode Peltier heat associated with current can be seen at the anode.

By substituting the total heat production rates per unit volume into the heat conduction (energy balance) equations for the constitutive layers and integrating the differential equations with boundary conditions between the layers and at the surfaces, temperatures at the anode and cathode surfaces can be evaluated.

#### **3.4.2 Relation between the surface temperature and local current**

Assuming uniform heat flux in the axial direction and temperature in the circumferential direction, the energy balance (the heat conduction) equation at steady state of the anode, electrolyte and cathode layers reduces to

$$0 = \frac{1}{r}\frac{d}{dr}\left(\lambda\_\chi r \frac{dT\_\chi(r)}{dr}\right) + \frac{q\_\chi}{V\_\chi} \tag{15}$$

Here, *r*, *T*, *λ*, and *V* are the radial coordinate, temperature, thermal conductivity, and volume of each layer, respectively. The subscripts, x = a, el (Ohm), c represent the anode, electrolyte and cathode layers. The radiation heat transfer is also assumed to be negligible owing to metallic cathode current collector layer (Nakajima et al., 2009). Integration of this simplified

In the present study, overpotentials are regarded as uniform in the axis direction because the

<sup>299</sup> Electrochemical Impedance Spectroscopy Study

In the above analysis, the sensitivity of the derived temperatures to the heat transfer coefficient at the anode surface is rather larger than that to the heat transfer coefficient at the cathode surface and the other thermal conductivities. That is, the contributions of hydrogen partial pressure due to the large thermal conductivity and of anode-supported tube design are

Figure 16 shows the surface temperatures of the cell measured at anode gas flow rates of H2/N2 = 40/40, 40/160 cm−<sup>3</sup> min−<sup>1</sup> and a cathode flow rate of dried air of 1000 cm−<sup>3</sup> min−1. In this case, the cathode concentration overpotential is not significant. The temperatures increase with an increase in the cell average current density. The increase in the temperature

Temperature(C°)

Current density(A cm-2) Current density(A cm -2)

Fig. 16. Surface temperatures against the average current density of the SOFC for (a) H2/N2 = 40/40 cm−<sup>3</sup> min−<sup>1</sup> and (b) H2/N2 = 40/160 cm−<sup>3</sup> min−1. Cathode: Dried air of 1000 cm−<sup>3</sup>

The temperature distributions represent the current density distributions. With an increase in the cell current, the surface temperature of the anode becomes higher than that of the cathode, especially at the upper part (downstream). This is probably ascribed to the temperature rise of the anode gas from the upstream to the downstream along the axis. Thus the cathode surface temperatures calculated and measured are compared to determine the local current densities at the upper part. In the author's previous report, this anode gas temperature effect was

Anode Cathode

Temperature(C°)

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

Current density(A cm -2)

Cathode

voltage between the upper and lower ends of the cathode was smaller than 50 mV.

of the Mass Transfer in an Anode-Supported Microtubular Solid Oxide Fuel Cell

at the lower parts (upstream) of the anode and cathode are largest.

Upper Middle Lower

Upper Middle Lower

E

D

0 0.2 0.4 0.6 0.8 1

Current density(A cm-2)

0 0.2 0.4 0.6 0.8 1

Anode

significant.

**3.4.3 Local current densities**

min−1. *U*<sup>f</sup> = 57% and *U*ox = 5.7% at 0.5 A cm−2.

Temperature(C°)

Temperature(C°)

equation yields the relation between the local current density and temperature in the anode and cathode (Nakajima et al., 2009).

The heat fluxes at the boundaries give the following boundary conditions.

$$-\lambda\_{\rm el} \left( \frac{dT\_{\rm el}(r)}{dr} \right)\_{r=r\_{\rm a-el}} = -\lambda\_{\rm a} \left( \frac{dT\_{\rm a}(r)}{dr} \right)\_{r=r\_{\rm a-el}} \tag{16}$$

$$-\lambda\_{\rm el} \left(\frac{dT\_{\rm el}(r)}{dr}\right)\_{r=r\_{\rm c-ol}} = -\lambda\_{\rm c} \left(\frac{dT\_{\rm c}(r)}{dr}\right)\_{r=r\_{\rm c-ol}}\tag{17}$$

$$-\lambda\_{\rm a} \left( \frac{dT\_{\rm a}(r)}{dr} \right)\_{r=r\_{\rm a,s}} = h\_{\rm fu} (T\_{\rm fu} - T\_{\rm a,s}) \tag{18}$$

$$\left(-\lambda\_{\mathbb{C}}\left(\frac{dT\_{\mathbb{C}}(r)}{dr}\right)\_{r=r\_{\mathbb{C}s}}\right)\_{r=r\_{\mathbb{C}s}} = \hbar\_{\text{air}}(T\_{\mathbb{C},s} - T\_{\text{air}})\tag{19}$$

where the subscripts, "c,s" and "c-el" represent the cathode surface and cathode-electrolyte boundary, respectively. *h*fu and *h*air denote the heat transfer coefficients at the anode and cathode surfaces, respectively. *T*fu and *T*air are temperatures of fed gases in the anode and cathode sides, respectively, which equal to the cell surface temperature at zero current. The thermal conductivities of the anode and cathode layers are effective values obtained from those of gas phase and materials with the common mixture law in porous media.

The above heat transfer coefficients are calculated by the Nusselt numbers for fully developed laminar flow of forced convection in a circular tube annulus (Kays & Perkins, 1985). The Nusselt numbers on the anode and cathode are 3.66 and 11.3, respectively for the present experimental set-up. Here, the thermal conductivity of the anode gas, *λ*fu, is estimated from a partial-pressure-weighted average of individual thermal conductivities of fed gases and the product water. Thus *λ*fu varies with current according to the amounts of the product water and remaining hydrogen.


Substituting these heat production rates into Eq. 15, the author derives the relation between the local current density and temperatures at the surfaces of the anode and cathode using thermal conductivities presented in Table 1.

Table 1. Thermal conductivities of the materials and gases in the SOFC.

Thereby the current density at each part can be determined so that the surface temperature calculated at each part of the anode and cathode is identical with the measured temperature. In the present study, overpotentials are regarded as uniform in the axis direction because the voltage between the upper and lower ends of the cathode was smaller than 50 mV.

In the above analysis, the sensitivity of the derived temperatures to the heat transfer coefficient at the anode surface is rather larger than that to the heat transfer coefficient at the cathode surface and the other thermal conductivities. That is, the contributions of hydrogen partial pressure due to the large thermal conductivity and of anode-supported tube design are significant.

#### **3.4.3 Local current densities**

14 Will-be-set-by-IN-TECH

equation yields the relation between the local current density and temperature in the anode

= −*λ*<sup>a</sup>

= −*λ*<sup>c</sup>

*dT*a(*r*) *dr*

 *dT*c(*r*) *dr*

*r*=*r*a−el

*r*=*r*c−el

= *h*fu(*T*fu − *T*a,s) (18)

= *h*air(*T*c,s − *T*air) (19)

(16)

(17)

The heat fluxes at the boundaries give the following boundary conditions.

those of gas phase and materials with the common mixture law in porous media.

*r*=*r*a−el

*r*=*r*c−el

*r*=*r*a,s

*r*=*r*c,s

where the subscripts, "c,s" and "c-el" represent the cathode surface and cathode-electrolyte boundary, respectively. *h*fu and *h*air denote the heat transfer coefficients at the anode and cathode surfaces, respectively. *T*fu and *T*air are temperatures of fed gases in the anode and cathode sides, respectively, which equal to the cell surface temperature at zero current. The thermal conductivities of the anode and cathode layers are effective values obtained from

The above heat transfer coefficients are calculated by the Nusselt numbers for fully developed laminar flow of forced convection in a circular tube annulus (Kays & Perkins, 1985). The Nusselt numbers on the anode and cathode are 3.66 and 11.3, respectively for the present experimental set-up. Here, the thermal conductivity of the anode gas, *λ*fu, is estimated from a partial-pressure-weighted average of individual thermal conductivities of fed gases and the product water. Thus *λ*fu varies with current according to the amounts of the product water

Substituting these heat production rates into Eq. 15, the author derives the relation between the local current density and temperatures at the surfaces of the anode and cathode using

> Anode (Effective value)(Wang et al., 2009) 1.4 Electrolyte (Yasuda et al., 2000) 2.08

Thereby the current density at each part can be determined so that the surface temperature calculated at each part of the anode and cathode is identical with the measured temperature.

Cathode (Effective value)(Campanari & Iora, 2004) 2.0 Hydrogen (1 atm, 700◦C)(PROPATH-group, 2008) 0.438 Nitrogen (1 atm, 700◦C)(PROPATH-group, 2008) 0.064 Water vapor (1 atm, 700◦C)(PROPATH-group, 2008) 0.094 Air (1 atm, 700◦C)(PROPATH-group, 2008) 0.066

Table 1. Thermal conductivities of the materials and gases in the SOFC.

Materials and gases *λ* (W m−<sup>1</sup> K−1)

 *dT*el(*r*) *dr*

 *dT*el(*r*) *dr*

> *dT*a(*r*) *dr*

 *dT*c(*r*) *dr*

and cathode (Nakajima et al., 2009).

and remaining hydrogen.

thermal conductivities presented in Table 1.

−*λ*el

−*λ*el

−*λ*<sup>a</sup>

−*λ*<sup>c</sup>

Figure 16 shows the surface temperatures of the cell measured at anode gas flow rates of H2/N2 = 40/40, 40/160 cm−<sup>3</sup> min−<sup>1</sup> and a cathode flow rate of dried air of 1000 cm−<sup>3</sup> min−1. In this case, the cathode concentration overpotential is not significant. The temperatures increase with an increase in the cell average current density. The increase in the temperature at the lower parts (upstream) of the anode and cathode are largest.

Fig. 16. Surface temperatures against the average current density of the SOFC for (a) H2/N2 = 40/40 cm−<sup>3</sup> min−<sup>1</sup> and (b) H2/N2 = 40/160 cm−<sup>3</sup> min−1. Cathode: Dried air of 1000 cm−<sup>3</sup> min−1. *U*<sup>f</sup> = 57% and *U*ox = 5.7% at 0.5 A cm−2.

The temperature distributions represent the current density distributions. With an increase in the cell current, the surface temperature of the anode becomes higher than that of the cathode, especially at the upper part (downstream). This is probably ascribed to the temperature rise of the anode gas from the upstream to the downstream along the axis. Thus the cathode surface temperatures calculated and measured are compared to determine the local current densities at the upper part. In the author's previous report, this anode gas temperature effect was

0 0.2 0.4 0.6 0.8 1.0 1.2

Cell voltage (V)

1000 cm−<sup>3</sup> min−1.

**4. Conclusion**

**5. Acknowledgments**

**6. References**

0123456

for assistance with the measurements and calculations.

70-71(PART 1): 528–532.

& Sons, New York.

Measured Calculated

Current (A)

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Cell voltage (V)

<sup>301</sup> Electrochemical Impedance Spectroscopy Study

of the Mass Transfer in an Anode-Supported Microtubular Solid Oxide Fuel Cell

Fig. 19. Comparison of the calculated and measured I-V characteristics of the SOFC for (a) H2/N2 = 40/40 cm−<sup>3</sup> min−<sup>1</sup> and (b) H2/N2 = 40/160 cm−<sup>3</sup> min−1. Cathode: Dried air of

In this chapter, EIS analysis clarifies the concentration overpotentials determined by the fuel and oxygen transfers in an intermediate temperature anode-supported microtubular SOFC. Current distribution in the cell by the Nernst loss due to the fuel partial pressure gradient along the axis of the cell is also described from surface temperature measurements. These results give the information for the optimization of the cell structure, cell alignment in the stack and operation conditions to decrease the anode concentration overpotential including the Nernst loss, for effective use of the whole electrodes in the cell, and to improve the

This work was supported by a grant of the Fukuoka Strategy Conference for Hydrogen Energy in the Fukuoka prefectural government, Japan. The author is also grateful to graduate students, Satoshi IGAUE, Atsushi OKAZAKI, Ryota MATSUMOTO, and Ken-ichi KIYAMA

Ahlgren, E. & Willy Poulsen, F. (1994). Thermoelectric power of YSZ, *Solid State Ionics*

Barfod, R., Mogensen, M., Klemensø, T., Hagen, A., Liu, Y. L. & Hendriksen, P. V. (2007).

Barsoukov, E. & Macdonald, J. R. (eds) (2005). *Impedance Spectroscopy: Theory, Experiment, and*

Bird, R. B., Stewart, W. E. & Lightfoot, E. N. (2007). *Transport Phenomenon*, 2nd edn, John Wiley

Campanari, S. & Iora, P. (2004). Definition and sensitivity analysis of a finite volume SOFC model for a tubular cell geometry, *Journal of Power Sources* 132(1-2): 113–126. Churchill, S. W. (1983). *in* E. U. Schlunder (ed.), *Heat exchanger design handbook*, Hemisphere.

Detailed characterization of anode-supported SOFCs by impedance spectroscopy,

durability of the cell by more uniform current and temperature distributions.

*Journal of the Electrochemical Society* 154(4): B371–B378.

*Applications*, 2nd edn, John Wiley & Sons, New York.

012345

Current (A)

Measured Calculated

(a) (b)

overestimated, so that the current densities derived at the upper part were extremely small (Nakajima & Kitahara, 2011).

The local current densities determined are plotted against the total current in Fig. 17. The current density decreases with the decrease in the hydrogen partial pressure in the fed fuel in accordance with an increase in the average anode activation and concentration overpotentials in the previous section. Hence the current distributions exhibited in the cell are probably ascribed to the hydrogen consumption in upstream. In particular, the current density at the upper part is small in both cases, which shows that the upper part does not effectively take part in the power generation. The higher hydrogen partial pressure results in significantly larger current in the lower part with increasing the temperature and decreasing the overpotentials there. Thus the current distribution is enhanced although the cell power output is increased.

Figure 18 shows the local I-V characteristics. The current distribution seems to be attributed mainly to the concentration overpotential, which also indicates the Nernst loss by the hydrogen consumption in the upstream. The higher temperatures at the lower part also would decrease the Ohmic and activation overpotentials. Total currents calculated by the integration of linearly interpolated local current densities along the axis agree well with the measured current as shown in Fig. 19. Hence the local current densities obtained in the present study are reasonable.

Fig. 17. Local current densities against the total cell current of the SOFC for (a) H2/N2 = 40/40 cm−<sup>3</sup> min−<sup>1</sup> and (b) H2/N2 = 40/160 cm−<sup>3</sup> min−1. Cathode: Dried air of 1000 cm−<sup>3</sup> min−1. *U*<sup>f</sup> = 57% and *U*ox = 5.7% at 3.0 A.

Fig. 18. Local I-V characteristics of the SOFC for (a) H2/N2 = 40/40 cm−<sup>3</sup> min−<sup>1</sup> and (b) H2/N2 = 40/160 cm−<sup>3</sup> min−1. Cathode: Dried air of 1000 cm−<sup>3</sup> min−1.

Fig. 19. Comparison of the calculated and measured I-V characteristics of the SOFC for (a) H2/N2 = 40/40 cm−<sup>3</sup> min−<sup>1</sup> and (b) H2/N2 = 40/160 cm−<sup>3</sup> min−1. Cathode: Dried air of 1000 cm−<sup>3</sup> min−1.

### **4. Conclusion**

16 Will-be-set-by-IN-TECH

overestimated, so that the current densities derived at the upper part were extremely small

The local current densities determined are plotted against the total current in Fig. 17. The current density decreases with the decrease in the hydrogen partial pressure in the fed fuel in accordance with an increase in the average anode activation and concentration overpotentials in the previous section. Hence the current distributions exhibited in the cell are probably ascribed to the hydrogen consumption in upstream. In particular, the current density at the upper part is small in both cases, which shows that the upper part does not effectively take part in the power generation. The higher hydrogen partial pressure results in significantly larger current in the lower part with increasing the temperature and decreasing the overpotentials there. Thus the current distribution is enhanced although the cell power

Figure 18 shows the local I-V characteristics. The current distribution seems to be attributed mainly to the concentration overpotential, which also indicates the Nernst loss by the hydrogen consumption in the upstream. The higher temperatures at the lower part also would decrease the Ohmic and activation overpotentials. Total currents calculated by the integration of linearly interpolated local current densities along the axis agree well with the measured current as shown in Fig. 19. Hence the local current densities obtained in the present study

> 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

 

(a) (b)

&HOOYROWDJH9

(a) (b)

Current density (A cm-2)

Fig. 17. Local current densities against the total cell current of the SOFC for (a) H2/N2 = 40/40 cm−<sup>3</sup> min−<sup>1</sup> and (b) H2/N2 = 40/160 cm−<sup>3</sup> min−1. Cathode: Dried air of 1000 cm−<sup>3</sup>

Fig. 18. Local I-V characteristics of the SOFC for (a) H2/N2 = 40/40 cm−<sup>3</sup> min−<sup>1</sup> and (b)

0LGGOH /RZHU 8SSHU

012345 Total cell current (A)

/RFDOFXUUHQWGHQVLW\\$FP

(Nakajima & Kitahara, 2011).

output is increased.

are reasonable.

Current density (A cm-2)

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0 0.2 0.4 0.6 0.8 1.0 1.2

&HOOYROWDJH9

min−1. *U*<sup>f</sup> = 57% and *U*ox = 5.7% at 3.0 A.

012345 Total cell current (A)

0 0.5 1.0 1.5

/RFDOFXUUHQWGHQVLW\\$FP

H2/N2 = 40/160 cm−<sup>3</sup> min−1. Cathode: Dried air of 1000 cm−<sup>3</sup> min−1.

Lower Middle Upper

In this chapter, EIS analysis clarifies the concentration overpotentials determined by the fuel and oxygen transfers in an intermediate temperature anode-supported microtubular SOFC. Current distribution in the cell by the Nernst loss due to the fuel partial pressure gradient along the axis of the cell is also described from surface temperature measurements. These results give the information for the optimization of the cell structure, cell alignment in the stack and operation conditions to decrease the anode concentration overpotential including the Nernst loss, for effective use of the whole electrodes in the cell, and to improve the durability of the cell by more uniform current and temperature distributions.

#### **5. Acknowledgments**

This work was supported by a grant of the Fukuoka Strategy Conference for Hydrogen Energy in the Fukuoka prefectural government, Japan. The author is also grateful to graduate students, Satoshi IGAUE, Atsushi OKAZAKI, Ryota MATSUMOTO, and Ken-ichi KIYAMA for assistance with the measurements and calculations.

#### **6. References**


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**13** 

*USA* 

**Mass Transport Limitations in** 

**Proton Exchange Membrane** 

**Fuel Cells and Electrolyzers** 

Elise B. Fox and Héctor R. Colón-Mercado *Savannah River National Laboratory; Aiken, SC 29808* 

E= -1.229V (5)

The performance of Proton Exchange Membrane Fuel Cells (PEMFC) and Electrolyzers (PEME) is subject to mass transport limitations. Within this chapter we will discuss the origination of those limitations and the current research efforts for mitigation. Hydrogen powered fuel cells operate based on the reaction of hydrogen and oxygen, (Figure 1) where the anode reaction is found in Eq. 1, the cathode reaction in Eq. 2 and the overall reaction in Eq. 3. The reverse of this reaction (Eq. 4) is electrolysis. Where, in the electrolyzer the anode

H2 Æ 2H+ + 2e E= 0V (1)

½ O2 + 2H+ +2e- Æ H2O E= 1.229V (2)

H2 + ½O2 Æ H2O E= 1.229V (3)

H2O Æ H2 + ½ O2 E= -1.229V (4)

 2H+ + 2e- Æ H2 E= 0V (6) Basic cell construction is very similar for both PEMFC and PEME. During electrolysis a voltage is applied to the cell while an ion conductor with electrocatalyst layers, such as Pt black on Nafion®, is used to split water into hydrogen and oxygen, as in Figure 2. As water is split into hydrogen and oxygen ions at the anode, the hydrogen ions travel across the PEM and oxygen is collected and exhausted at the bipolar plate. At the cathode, hydrogen ions recombine to create diatomic hydrogen, which can be then be stored for later use. The cell components are similar to those used in a PEM fuel cell, but different bipolar plates must be used due to the corrosive environment. PEMFCs typically use graphite bipolar plates that will degrade under the conditions used in a PEME. Corrosion resistant bipolar plates are substituted for graphite. Titanium plates are typically used, but are very expensive. Stainless steel bipolar plates have also been used, but there is a risk of leaching iron into the water, which would affect the performance of the catalysts and the membrane.

**1. Introduction** 

reaction is Eq. 5 and the cathode reaction is Eq. 6.

H2O Æ ½ O2 + 2H+ + 2e-

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