Fuel Cells Lecture 2.pdf - Curtin University

Effects of Pressure 

• The change in Gibbs Free Energy with pressure of the fuel 

cell reaction H 2 + (½)O 2 H 2 O is given by 

⎡ a ⋅ a ⎤ 

1/2 

H O 

0 

2 2 

Δ g f = Δg f − RT ln ⎢ ⎥ 

⎢⎣ aH2O 

⎥⎦ 

Where R is the universal gas constant 8.314 JK-1mol-1 , T 

is the absolute temperature in K, is the change in 

Gibbs free energy at standard pressure and at 

temperature T, ‘a’ is the activity of the gas given by P/P0 where P is the pressure or partial pressure of the gas, P0 0 

Δg 

f 

is the standard pressure, 0.1 Mpa= 1 bar. 

S. Rajakaruna 2007 Renewable Energy Systems 402 1

Effects of Pressure 

• If the pressure of hydrogen and oxygen increase, Δg 

becomes more negative and hence releases more 

energy become available for conversion to electricity. 

• If the pressure of steam increases, less energy will be 

available for the conversion to electricity. 

• Note here that the effect of change of temperature is 

0 

represented by the change in the value of Δg 

with f 

changing temperature. The second term only represents 

the effect of pressure change. 

• The effect of temperature on maximum EMF can be 

−Δg 

f 

seen by substituting the above equation in 

E = 

S. Rajakaruna 2007 Renewable Energy Systems 402 2 

2F 

f

Change in EMF with Pressure -The 

Nernst Equation 

0 

1/2 1/2 

−Δg 

f RT ⎛ aH ⋅ a ⎞ O 0 RT ⎛ aH ⋅ a ⎞ O 

E = + ln = E + ln 

2F 2F a 2F 

a 

E 

0 

2 2 2 2 

⎜ ⎟ ⎜ ⎟ 

⎜ ⎟ ⎜ ⎟ 

⎝ H2O ⎠ ⎝ H2O ⎠ 

Where is the maximum EMF at standard pressure 

0 

and temperature T (K). Note E 

drops with the 

increasing temperature as seen in Table 2.2. This is 

called ‘The Nernst Equation’ and some of its other forms 

are as given below. 



E E 

0 

= + 

RT 

2F 

1 

⎛ ⎞ 

2 

⎜ PH ⎛ P 

2 O ⎞ 2 ⋅⎜ 

⎟ 

⎜ 0 0 ⎟ 

P P ⎟ 

ln ⎜ 

⎝ ⎠ 

⎟ 

P 

⎜ H2O ⎟ 

⎜ 0 ⎟ 

⎜ P ⎟ 

⎝ ⎠ 

If all the pressures are expressed in ‘bar’, then, P 0 =1. Hence above equation 

can be expressed as: 

E E 

RT 

⎛ ⎞ 

⎜ PH ⋅ PO 

1 

2 

⎟ 

( ) 

0 

2 2 

= + ln ⎜ ⎟ 

2F 

⎜ 

PH 

2O 

⎟ 

⎝ ⎠ 



• In nearly all the cases, pressures of gases in above 

equation are partial pressures as there are many gases 

in the mix both at anode and cathode. 

• Furthermore, the overall pressure in the mix of gases at 

the cathode and anode is the same. Because this 

simplifies the design of the cell. If this system operating 

pressure is ‘P’, then the partial pressures can be 

expressed as a ratio of P. 

P = α ⋅ P P = β ⋅ P P = δ ⋅ P 

H O H O 

2 2 2 

α, β and δ 

Where are constants depending on the 

concentration of gases in the mix. 



1 1 1 

⎛ ⎞ ⎛ ⎞ 

2 2 2 

0 RT α ⋅ β ⋅ P 0 RT α ⋅ β RT 

E = E + ln 

⎜ ⎟ 

= E + ln 

⎜ ⎟ 

+ ln P 

2F ⎜ δ ⎟ 2F ⎜ δ ⎟ 

⎜ ⎟ ⎜ ⎟ 4F 

⎝ ⎠ ⎝ ⎠ 

This form of Nernst equation gives the effect of system 

pressure P on the EMF as a separate term. It also shows 

how partial pressures influence the EMF. 

( ) 

Example: On the cathode, oxygen is usually supplied as 

air. The partial pressures of air at 0.1MPa are: N 2 =0.0781 

MPa, O 2 =0.02095 MPa, Argon = 0.00093 MPa etc. Thus, 

β=0.2095 and P=0.1MPa= 1 bar 

The partial pressure of H 2 depends on how fuel processor 

produces hydrogen. 


Fuel and Oxygen Utilization 

• As hydrogen passes through the anode, it is used in the 

reaction. So the partial pressure of hydrogen will 

progressively decrease as it flows from one end to the 

other end of the anode. 

• The same is true for oxygen. As oxygen flows from one 

end to the other on the cathode, its partial pressure will 

also gradually decrease. 

• Since α and β both decrease, the EMF generated 

continues to drop as the gases flow from the entry (inlet) 

to the exit (outlet). It will be worst near the outlet. 


Fuel and Oxygen Utilization 

• Since the bipolar plates on the electrode are good 

conductors, there cannot be different voltages in the 

same cell. Therefore, it is the current density that drops 

as the gases flow towards outlet. 

⎛ ⎞ 

1 

2 

0 RT α ⋅ β 

E = E + ln 

⎜ ⎟ RT 

+ ln P 

( ) 

2F ⎜ δ ⎟ 

⎜ ⎟ 4F 

⎝ ⎠ 

• Due to the RT term, it is also clear that the effects of 

decreasing partial pressures is more at high 

temperature fuel cells such as ‘Solid Oxide Fuel Cell’. 


Fuel Utilization 

• For high fuel efficiency, it is required that all the fuel 

supplied to the cell used in the reaction. But at the same 

time it can be seen from the above discussion that the 

pressure of H 2 at outlet should not be reduced to very 

low levels. As a result, fuel and oxygen utilization need 

careful optimising, especially in high temperature fuel 

cells. 

• Fuel Utilization Ratio (u) is defined as, 

N N − N 

u = = 

N N 

used in out 

H H H 

2 2 2 

in in 

H H 

2 2 


Fuel Utilization Ratio 

where N is the flow rate in moles/s. For a solidoxide 

fuel cell the desired range of u is from 0.7 

to 0.9. 

• Overused-fuel condition: i.e. u> 0.9, will lead 

to fuel starvation near the outlet and cause 

permanent damage to cells. 

• Underused-fuel condition: i.e. u

Terminal Voltage of Practical Cells 

−Δg 

• The maximum EMF given by the equation E = 

2F 

is an ideal value assuming there are no losses. 

The practical terminal voltage of a fuel cell is 

considerably less than the above value due to 

various types of losses. 

• At temperatures below 100°C, the maximum EMF 

E is about 1.2 V. Figure 3.1 shows the variation of 

terminal voltage with current density of a practical 

fuel cell operating at 40°C and standard pressure. 


f

Terminal voltage of a low- 

temperature fuel cell 


Terminal voltage of a low- 


Note the following in the figure. 

• Even the open circuit voltage is less than the 

theoretical value. 

• There is a rapid initial fall in voltage with 

increasing current density. 

• Then there is a range of current density where 

drop of voltage is almost linear. In this linear 

region, the slope is less than in the initial region. 

• When current density approaches some high 

values, the voltage starts to fall rapidly. 


Terminal voltage of a high- 


• As discussed earlier, the maximum EMF (E) falls with 

the increasing temperature. At about 800°C, E is ab out 

1.0 V. Figure 3.2 shows how the terminal voltage of a 

practical cell at 800°C varies with the current den sity. 

Note the following in the figure. 

• Open circuit voltage is almost equal to the theoretical 

value. 

• There is no rapid drop in voltage at low current densities. 

The graph is much more linear. 

• At about the same high values of current densities, the 

voltage starts to fall rapidly. 


Terminal voltage of a high- 



Fuel Cell Irreversibilities – Causes 

of Voltage Drop 

• There are four main types of irreversibilities that cause 

the terminal voltage to drop below the Maximum EMF 

(E). E is also called the ‘ideal emf’, ‘no-loss emf’ or the 

‘reversible emf’. 

1. Activation Loss 

2. Fuel Crossover and Internal Currents 

3. Ohmic Losses 

4. Mass Transport or Concentration Loss 


Activation Loss 

• Activation Loss is the main reason for rapid drop in 

terminal voltage of low-temperature fuel cells at low 

current densities. It is caused by the slowness of the 

reactions taking place on the surface of the electrode. A 

portion of the available energy is lost in driving the 

chemical reaction or to supply the ‘activation energy’. 

• The activation loss of voltage is described by “Tafel 

Equation’ as: 

⎛ i ⎞ 

Δ Va = Aln 

⎜ ⎟ 

i 

⎝ 0 ⎠ 

where A is a constant which is higher for slower 

reactions and i 0 is a constant higher for faster reactions. 



• For the slow 

reaction A is higher, 

i 0 is smaller. 

• For the fast 

reaction A is smaller 

and i 0 is higher. 

• Due to the 

logarithmic function 

it is i 0 which affects 

the voltage drop 

more. 



• The above equation describes the voltage drop only for i 

> i 0 . For i 

the value of i 0 , the lower is the voltage drop. 

• The constant i 0 is called the ‘exchange current density’. 

This is the current corresponding to the electrode 

reaction in equilibrium under open-circuit condition. 

Under zero current supplied to the external circuit, the 

electrode reaction is still happening, but in both 

directions. 

At the cathode: 

At the anode: 2 

O + 4e + 4H ↔ 2H 

O 

− + 

2 2 

2H ↔ 4H + 

4e 

+ − 



• When the external circuit is switched on, the exchange 

current density is readily available so drawing a current 

up to i 0 will cause no voltage drop. 

• The anode and cathode have two different i 0 values as 

i 0a and i 0c respectively. 

• For a hydrogen fuel cell, i 0c

Effect of i 0 


Methods to increase i 0 

1. Increasing the cell temperature: Increase of 

temperature increases the reaction under open-circuit 

condition. Therefore i 0 is higher and the voltage drop is 

less. By considering low and high values for i 0, the 

shapes of V-I characteristics of Figure 3.1 and 3.2 at 

low current levels can be described. For a low 

temperature cell i 0 is about 0.1 mA and that for a 

800°C cell is about 10 mA, which is 100 times large r. 

2. Using more effective catalysts. 

3. Increasing the contact area of the electrode. 

4. Increasing reactant concentration. Ex. Use pure 

oxygen instead of air. 

5. Increasing the pressure of reactants. 


Fuel Crossover and Internal 

Currents 

• Although the electrolyte is not supposed to allow the 

passing of the fuel (H 2 ) and electrons between the two 

electrodes, a very small “leakage” does happen in all fuel 

cells. Both hydrogen and electron crossovers have the 

same effect that they reduce the charges available to the 

external circuit. 

• Although, this leakage current is not large enough to 

affect the energy efficiency, it does cause a very 

considerable drop in open-circuit voltage. 

• Because of the leakage, the current produced under 

open-circuit condition is not zero. 

• If i n is the internal current density, it can be considered 

in the Tafel equation as: 


Fuel Crossover and Internal 

Currents 

⎛ i + i ⎞ n 

V = E − Aln 

⎜ ⎟ 

i 

⎝ 0 ⎠ 

• Thus, the open-circuit terminal voltage is given by, 

ln n ⎛ i ⎞ 

VOC = E − A ⎜ ⎟ 

i 

⎝ 0 ⎠ 

Using typical values for a low-temperature cell, E= 

1.2v, A=0.06 V, i 0 =0.04mA.cm -2 and i n =3 mA.cm-2, V-I 

characteristics can be drawn as below. 


Activation and Internal Current 

Losses 

Compare this with Figure 3.1 and see how closely it matches the V-I 

characteristic of a practical cell at low current densities. 


Ohmic Losses 

• Ohmic losses are due to the resistance to the flow of (a) 

electrons through the electrodes, bipolar plates and 

other stack connections and (b) ions through the 

electrolyte. 

• In most fuel cells, ohmic loss is mainly due to the 

electrolyte. 

• The voltage drop is directly proportional to the current 

density. 

Δ V = i ⋅r 

o 

Where i is the current density and r is the resistance 

per unit area. 


Ohmic Losses 

• Ohmic losses are important in all types of fuel 

cells. To reduce the ohmic losses, following 

can be done. 

1. Use of electrodes with highest possible 

conductivity. 

2. Good design and use of suitable material for 

bipolar plates. 

3. Making the electrolyte as thin as practically 

possible while making sure no direct contact 

between electrodes is possible and enough 

physical strength is available in the stack. 


Mass Transport or Concentration 

Loss 

• As current is drawn to the external circuit, gases are 

consumed, thus causing a reduction in pressure of both 

hydrogen and oxygen at the electrodes if the gas supply 

rates are constant. This reduction in pressure at 

electrodes due to consumption is the reason for the 

concentration loss. 

i 

• If is the limiting current corresponding to zero 

l 

pressure at the electrodes, the concentration loss can be 

expressed using the Nernst equation as, 

⎛ i ⎞ 

Δ V = −B ln ⎜1− ⎟ for i ≤ i 

⎝ il 

⎠ 

c l 


Concentration Loss 

⎛ i ⎞ 

V = E + B ln ⎜1− ⎟ 

⎝ il 

⎠ 

E = 1.2 V , i = 

1000mA 


l

Combined Effect of All Losses 

• The effects on terminal voltage by all the 

irreversibilities that were discussed can be 

represented by the following equation. 

⎛ i + i ⎞ ⎛ n i + i ⎞ n 

V = E − ( i + in ) r − Aln ⎜ ⎟ + B ln ⎜1− ⎟ 

i i 

⎝ 0 ⎠ ⎝ l ⎠ 

• Where E is the maximum EMF (reversible opencircuit 

voltage), V is the terminal voltage, i is the 

current density in the external circuit. 


Typical Values of Parameters

Fuel Cells Lecture 2.pdf - Curtin University

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