Mench M.M. Fuel Cell Engines

Подождите немного. Документ загружается.

c03 JWPR067-Mench December 18, 2007 1:59 Char Count=

90 Thermodynamics of Fuel Cell Systems

Reaction coordinate

Local thermodynamic

equilibrium can

be established

Figure 3.10 Chemical equilibrium is achieved at a minimum in Gibbs function for given tempera-

ture and pressure.

For nonreacting mixtures, the change in the Gibbs function between two states will involve

only the change in the sensible enthalpy and entropy, as the formation terms will cancel:

−

= 

g =

−

− T

(

−

)

(3.80)

For reacting mixtures where the products and reactants differ, the Gibbs formation energy

to achieve STP conditions for a given species is required. The Gibbs energy of formation

◦

is calculated based on 298 K, 1 atm pressure and is chosen to be zero for stable species

in their natural state, similar to the enthalpy of formation. For nonreacting mixtures, the

Gibbs function is calculated using molar or mass averaging the constituents. For reacting

mixtures, the methodology is the same as presented for enthalpy and entropy since the

Gibbs function is a combination of them.

One point of confusion is common. The Gibbs function of formation and enthalpy at

a reference state of 298 K, 1 atm is chosen to be zero for all stable species in their natural

state, yet the reference condition of zero entropy is 0 K. Therefore, every species has some

entropy at 298 K, and the choice of a zero Gibbs function for a stable species at 298 K

seems to conﬂict with the deﬁnition of the Gibbs function in Eq. (3.79). This is because

the reference temperatures chosen for the Gibbs function and the entropy are different.

However, since we are choosing a reference temperature for the Gibbs function of 298 K,

what we actually evaluate in Eq. (3.79) is the change in the Gibbs function from 298 K,

1 atm, to some other state. Thus, we can expand Eq. (3.79) to be

g =

◦

+ (

h −

ref

) − T (

s −

ref

) (3.81)

where the reference temperature for the enthalpy and entropy is 298 K. Since we are

actually evaluating a difference in enthalpy and entropy, the absolute value of the Gibbs

function will not be in conﬂict at reference conditions, despite the apparent discrepancy in

the reference point temperatures.

For reacting mixtures, the change in Gibbs function can be determined in an analogous

fashion to the other parameters. For the generic reaction in Eq. (3.67).



g =

−



i,P

(

)

−



j,R

(

)

(3.82)

c03 JWPR067-Mench December 18, 2007 1:59 Char Count=

3.5 Psychrometrics: Thermodynamics of Moist Air Mixtures 91

Methods for calculating for nonreacting ideal gas mixtures, reacting ideal gas mixtures,

liquids, and solids are based on determination of entropy and enthalpy changes in previous

sections. Examples of the calculation are given in Section 3.6, where the Gibbs function is

used to determine the expected open-circuit voltage (OCV) of a fuel cell.

3.5 PSYCHROMETRICS: THERMODYNAMICS

OF MOIST AIR MIXTURES

Psychrometrics is the study of nonreacting moist air mixtures and is critical to understand

the water balance in low-temperature PEFCs. Nonreacting moist mixtures can be evaluated

exactly like other nonreacting gas mixtures, but since engineering with moist air mixtures

is so common, additional parameters and special charts have been developed to aid in

calculation and analysis. In most fuel cells, water is produced as a product and must be

removed from the fuel cell as part of the efﬂuent mixture. In low-temperature PEFCs,

the water balance is critical to maintain proper electrolyte conductivity while avoiding

electrode ﬂooding.

To begin, we will ﬁrst assume a two-gas mixture of water vapor and another ideal gas

mixture, air. In this case, the total atmospheric pressure is the sum of the air and vapor

partial pressures:

P = P

+ P

= y

P + y

P (3.83)

It is important to realize there can be many other moist gas-phase mixtures besides air.

Consider a humidiﬁed anode fuel mixture of CO

,CO,H

, and H

O vapor. Then,

P = P

+ P

(3.84)

The humidity ratio is deﬁned as the mass of moisture per mass of dry mixture and can also

be deﬁned as the rate of moisture mass ﬂow per dry mass ﬂow:

ω =

dry

(3.85)

where the dry mixture includes everything but the water vapor. Considering the mixture as

an ideal gas (note the water vapor partial pressure is typically low enough that ideal gas

behavior can be assumed with little error), we can show that

ω =

dry

V · MW

dry

V · MW

dry

(3.86)

If the dry mixture is air, for instance, MW

air

= 28.85 kg/kmol, and the humidity ratio is

deﬁned as

ω =

dry

28.85

= 0.622

P − P

(3.87)

c03 JWPR067-Mench December 18, 2007 1:59 Char Count=

92 Thermodynamics of Fuel Cell Systems

where P is the total pressure, P

is the water vapor partial pressure, and P

is the air partial

pressure in the humidiﬁed mixture. The humidity ratio in a pure hydrogen humidiﬁed

stream would be

ω =

dry

= 9

P − P

(3.88)

The relative humidity is a more common parameter known to anyone who has heard or

seen a weather forecast. It is deﬁned as the ratio of actual water vapor pressure, P

,tothe

saturation water vapor pressure, P

sat

RH =

sat

(T )

(3.89)

Equations (3.88) and (3.89) can be related through the vapor pressure P

. In equilibrium,

the RH cannot exceed unity. The saturation pressure (P

sat

)isthemaximum possible vapor

pressure that can be achieved in equilibrium and is solely a function of temperature and

completely independent of other gas-phase species. Consider a closed container ﬁlled

initially with liquid water and dry air, as shown in Figure 3.11. Some of the water molecules

will have enough energy to break free of the liquid phase and enter the gas phase, and

concomitantly, some of the gas-phase molecules will collide with and rejoin the liquid

phase. Eventually, equilibrium will be established between the rate of liquid molecules

entering the gas phase and the rate returning to the liquid state, with no net rate of transfer

between phases. The vessel will thus have an equilibrium gas-phase mixture of air and vapor

at fully saturated (RH = 1 where P

= P

sat

) conditions, given enough time. Now consider

heating the vessel. The additional thermal input energy to the liquid will provide enough

energy for evaporation of more liquid into the vapor phase, and, over time, a different

equilibrium will be established.

Several concepts are often misunderstood regarding equilibrium between a liquid and

its vapor phase. The ﬁrst is that every species has a unique saturation pressure relationship

Figure 3.11 Equilibrium between liquid and gas phases in a closed container.

c03 JWPR067-Mench December 18, 2007 1:59 Char Count=

3.5 Psychrometrics: Thermodynamics of Moist Air Mixtures 93

Figure 3.12 Saturation pressure versus temperature for water.

with temperature. Water will establish phase equilibrium between the liquid and gas phases

that is a function of temperature only, and each species has a unique saturation vapor

pressure–temperature relationship. For example, methane will have a different equilibrium

vapor pressure at 30

◦

C than water. The second major misunderstanding comes from the

fact that the phase equilibrium is established between a liquid and its own vapor state. Any

other gases present have no effect on the equilibrium saturation vapor pressure. That is,

if the gas phase in the container in Figure 3.11 were hydrogen instead of air, the same

equilibrium water vapor pressure would be established. If the container were initially

evacuated, the ﬁnal total system pressure established would be the saturation pressure at

the system temperature.

A plot of the saturation pressure–temperature relationship for water is shown in Figure

3.12. Finally, the saturation pressure can be conveniently curve ﬁt, which is especially

useful for modeling:

sat

(T )(Pa) =−2846.4 +411.24 T (

◦

C) − 10.554 T (

◦

+ 0.16636 T (

◦

(3.90)

This curve ﬁt is accurate from 15

◦

C to 100

◦

C. Although a polynomial was chosen here,

other ﬁts are available [6], are easy to generate based on generally available thermodynamic

values, and may have a more accurate ﬁt in different temperature regions.

A couple of points are worth noting from Figure 3.12. First, the slope of the change

in P

sat

with temperature is very shallow at typical ambient conditions. Around 20

◦

there is very little difference in the saturation vapor pressure for a few degrees change in

temperature. In terms of fuel cell operation, this means the water present in the ambient air

commonly used as the oxidizer actually has very little variation with ambient temperature

or humidity conditions, compared to the water exchanged through various modes in an fuel

cell operating at elevated temperatures >50

◦

C. Although the change in saturation pressure

with temperature is modest at room temperature (the typical operating temperature of

c03 JWPR067-Mench December 18, 2007 1:59 Char Count=

94 Thermodynamics of Fuel Cell Systems

PEFCs is 80

◦

C), the saturation pressure is extremely sensitive to temperature. At 80

◦

C and

1 atm total pressure, a 1

◦

C change is about a 5% change in the saturation pressure. Therefore,

for low-temperature PEFCs, heat and water management are major design considerations.

For higher temperature fuel cells, the water generated is generally in the vapor state or part

of the electrolyte.

For operating temperatures above 100

◦

C, water is not always in a vapor state. According

to Eq. (3.89), if the total system pressure is increased, the RH can still be less than 1 for

temperatures above the atmospheric pressure boiling point. This is the physical principle

behind a pressure cooker. Since the container can be sealed and pressurized, the water

inside will boil, and concomitantly cook the food, at a higher temperature.

Example 3.8 Calculation of Maximum Water Uptake in a Flow Given a ﬂow inlet to

a5-cm

active area fuel cell at 3 atm, 50% RH at 80

◦

C, and an anode stoichiometry of

3.0. Determine the maximum possible molar rate of water uptake from the incoming anode

ﬂow if the fuel cell is operating at a current density of 0.8 A/cm

. You can assume the ﬂow

rate, pressure, and temperature are constant in the fuel cell.

SOLUTION This problem integrates some of our understanding from the previous chap-

ters as well. To achieve water balance, the vapor uptake into the anode and cathode streams

plus any liquid water droplets removed from the system must equal the water produced by

the reaction.

First, we must solve for the molar ﬂow rate of gas into the fuel cell anode. The ﬂow

rate of dry gas is given as

= λ

= 3 ×

(0.8A/cm

)(5 cm

)

(2 e

−

eq/mol H

)(96,485 C/eq)

= 6.22 × 10

−5

mol H

The water mole fraction of the incoming water vapor is

RH =

sat

(T )

⇒ y

RH · P

sat

(T )

0.5 × 47,684 Pa

303,975 Pa

= 0.0784

where the saturation pressure is found from Eq. (3.90). The maximum possible water uptake

in this ﬂow, at RH = 1, would be double the actual input value at RH = 50% (0.157).

From the deﬁnition of mole fraction

total

dry

Solving for the vapor ﬂow rate input to the fuel cell anode gives

dry

1 − y

(6.22 × 10

−5

mol/s) × 0.0784

1 − 0.0784

= 5.291 × 10

−6

mol/s

For the fully humidiﬁed maximum, this value is

dry

1 − y

(6.22 × 10

−5

mol/s) × 0.157

1 − 0.157

= 1.158 × 10

−5

mol/s

c03 JWPR067-Mench December 18, 2007 1:59 Char Count=

3.5 Psychrometrics: Thermodynamics of Moist Air Mixtures 95

The maximum uptake is simply the difference in the maximum value in the ﬂow at 100%

RH and the actual value input:

v,uptake,max

= 5.291 × 10

−6

mol/s − 1.158 × 10

−5

mol/s = 6.289 × 10

−6

mol/s

COMMENTS: What we have solved for is the maximum possible uptake of water vapor

into the anode ﬂow stream, which may not actually be achieved. Depending on the design,

it is quite possible that the ﬂow residence time in the fuel cell is not long enough to achieve

the equilibrium condition. We have also assumed constant pressure, temperature, and ﬂow

rate to simplify the problem. In reality, there are often variations in these values through

the fuel cell, which can even be used to help control the water uptake in advanced designs.

Finally, in this problem we only considered the anode, while the complete water balance

must include the cathode as well.

Evaporation and Condensation Equilibrium thermodynamics cannot predict the rate of

phase change and can only be applied to the beginning and ending quasi-equilibrium states.

The difference between the actual vapor pressure of the liquid in the gas phase and the

maximum saturation pressure is the driving force for evaporation. This is similar to the

temperature and voltage potential gradients being the driving forces for heat and ion

transport, respectively. The higher the temperature and the dryer the gas phase, the faster

the evaporation of the liquid into the gas phase. Conversely, if a moist mixture is suddenly

cooled so that the vapor pressure exceeds the saturation pressure at the new temperature

conditions, water will condense into liquid until the vapor pressure is equal to the maximum

saturation pressure at the new temperature. The rapid cooling and droplet wise condensation

of the moist air from our lungs are how we can see our breath on a cold day.

It should be emphasized that evaporation and condensation are very complex nonequi-

librium topics that are the source of many specialized textbooks. The processes of conden-

sation and evaporation are related to physicochemical parameters, including temperature,

vapor pressure, surface tension, surface energy and contact angle, surface impurities, ho-

mogeneity and roughness. Phase change is a local phenomenon, and some degree of local

supersaturation is required to initiate condensation and desublimation.

Note that there is a difference between boiling and evaporation. In boiling, the entire

volume of liquid is brought to a temperature where the saturation pressure is at least equal

to the atmospheric pressure. The appearance of vapor bubbles occurs volumetrically within

the ﬂuid. However, we also know from common experience that, if we leave a pan of water

out, it will slowly evaporate into the air with no volumetric vapor bubbles formed, even

though the temperature is far below the boiling point. The end result is the same, a dry

pan, but the processes of evaporation and boiling are clearly different. The water in the pan

will evaporate over time because the air around us is rarely fully humidiﬁed. As discussed,

evaporation is a surface phenomenon related to the imbalance between the saturation vapor

pressure and actual vapor pressure just above the liquid surface. If there is less than full

humidiﬁcation in the ambient gas, the imbalance between the saturation pressure and the

actual vapor pressure acts as a driving force for evaporation. The imbalance will drive the

vapor pressure above the liquid toward a thermodynamic equilibrium of full saturation.

Some molecules in the liquid state at the gas–liquid interface will have enough stored

energy to escape the surface and go into the gas phase, cooling the resulting liquid in a

process known as evaporative cooling (this is a reason why you feel cold when you get out

c03 JWPR067-Mench December 18, 2007 1:59 Char Count=

96 Thermodynamics of Fuel Cell Systems

of a swimming pool on a hot day). The cooling liquid must then absorb heat from the

surroundings in an amount equivalent to the latent heat of evaporation for the lost mass.

Clearly, the volatility of a liquid is related to the intermolecular forces near the surface

of the liquid. This can be affected by temperature, impurities, surface morphology, and

other factors. If the gas in the room becomes fully saturated and the air is static, a dynamic

equilibrium will be established and net evaporation will return to zero.

If a saturated gas mixture is cooled, a net ﬂux of gas-phase vapor molecules will

condense into the liquid phase upon collision with the liquid surface until a new equilibrium

is established at the cooler temperature. Upon condensation to a lower thermal energy state,

the new liquid will release thermal energy and heat the droplet in an amount equivalent to

the latent heat of fusion for the condensed mass.

3.6 THERMODYNAMIC EFFICIENCY OF A FUEL CELL

The process of energy conversion in a fuel cell must satisfy the ﬁrst law of thermodynamics

and conserve energy. The initial chemical bond energy available as the difference between

the enthalphy of the products and reactants in a galvanic process is conserved, but it is

converted into electrical energy (i.e., current and voltage) and thermal energy (i.e., heat),

as discussed in Chapter 2. Since the purpose of a fuel cell is to convert chemical energy

into electrical energy, the thermodynamic efﬁciency of a fuel cell can be written as

actual electrical work

maximum available work

(3.91)

The question to answer now is: What are the actual electrical work and maximum available

work for a given process?

Maximum Electrical Work for a Reversible Process Consider a generic reversible system

with mechanical and electrical work and heat transfer at constant temperature. From the

ﬁrst law of thermodynamics for a simple compressible system

dU = δ Q − δW (3.92)

The work can be divided into mechanical expansion work and electrical work:

δW = δW

+ δW

= PdV+δW

(3.93)

For a reversible system, the second law of thermodynamics can be written as

δ Q = TdS (3.94)

The differential change in the Gibbs free energy for a constant-temperature reaction is

dG = dH − TdS− SdT

(3.95)

c03 JWPR067-Mench December 18, 2007 1:59 Char Count=

3.6 Thermodynamic Efﬁciency of a Fuel Cell 97

The differential change in enthalpy for a given reaction is

dH = dU + PdV+ VdP (3.96)

We can then show that

dU = TdS− δW

− PdV (3.97)

By substitution, we can show that

−dG = δW

(3.98)

Since this derivation was done for a reversible system, it is an expression of the maximum

electrical work possible from a system. Therefore the change in Gibbs free energy is related

to the maximum conversion of chemical to electrical energy for a given reaction. From Eq.

(3.78) the direction of spontaneous reaction is that of decreasing free energy.

Maximum Expected Voltage (E

◦

) Next, we will consider electrical work. It is generally

easier for most people to envision the concept of classical mechanical work, that is, moving

a weight through some distance. The concept of electrical work is similar if one considers

electrical work to be the moving of an electron through a distance. Consider the energy

(work) required to move a given charge:

¯w

= nFE =



eq electrons

mol reactant



eq electrons





 

Charge to be passed in coulombs









Work to move

a coulomb

(3.99)

Note that here E is not energy; it is voltage. The symbol E is commonly used to designate

a voltage based on the concept of electromotive force (EMF). Combining Eqs. (3.98) and

(3.99), an expression for the maximum possible reversible voltage of an electrochemical

cell can be deduced and is shown as E

◦

−

= E

◦

−G

for 1 mol oxidized (3.100)

where E

◦

is also called the reversible voltage, because it is the maximum possible voltage

without any irreversible polarization losses. This is the maximum possible voltage of an

electrochemical cell, since it is derived assuming a reversible process. If we are looking at

the redox reaction on a per-mole-of-fuel basis, the absolute Gibbs function is equivalent

to the molar speciﬁc value. All fuel cell losses are associated with departure from this

maximum. Since F is a constant and n is constant for a particular global redox reaction,

the functional dependence of the maximum possible voltage of an electrochemical cell

is related strictly to the dependencies of the Gibbs free energy, namely temperature and

pressure of the reactants and products.

Thermal Voltage What if all the potential chemical energy for a reaction went into

electrical work? If there were no heat transfer, there would be no entropy change; from Eq.

(3.95), dG = dH. In this case, we can show that

−H

= E

◦◦

(3.101)

c03 JWPR067-Mench December 18, 2007 1:59 Char Count=

98 Thermodynamics of Fuel Cell Systems

◦◦

(also shown as E

) is known as the thermal voltage for a reaction and is the maximum

voltage for a reversible, adiabatic system. Since there is an entropy change associated

with every real reaction process, this voltage is merely a limit representing the case of all

chemical energy converted into electrochemical work, with no heat transfer or change in

available microstates (entropy). The thermal voltage can be simply calculated using the

concepts of this chapter. Since enthalpy is only a function of temperature for an ideal gas

or liquid, calculation is a straightforward matter.

The ratio of maximum expected voltage (E

◦

) to thermal voltage (E

◦◦

) represents the

maximum electrical work to the total available potential electrical work, the maximum

thermodynamic efﬁciency possible:

t,max

maximum electrical work

maximum available work

−G/nF

−H/nF

◦

◦◦

H − T S

H

(3.102)

Therefore, an expression of the maximum possible thermodynamic efﬁciency of a fuel cell

can be written as

t,max

= 1 −

T S

H

(3.103)

Table 3.3 shows some enthalpic, entropic, and Gibbs formation values for many fuel-cell-

related reactants and products at 25

◦

C, 1 atm conditions to aid in calculation.

Table 3.3 Enthalpy of Formation, Gibbs Energy of Formation, and Entropy Values at 298 K, 1 atm

Species Molecular Formula

◦

(kJ/kmol)

◦

(kJ/kmol)

◦

(kJ/kmol ·K)

Carbon C

0 0 5.74

Hydrogen H

2,g

0 0 130.57

Nitrogen N

2,g

0 0 191.50

Oxygen O

0 0 205.03

Carbon monoxide CO

−110,530 −137,150 197.54

Carbon dioxide CO

2,g

−393,520 −394,380 213.69

Water vapor H

−241,820 −228,590 188.72

Liquid water H

−285,830 −237,180 69.95

Hydrogen peroxide H

2,g

−136,310 −105,600 232.63

Ammonia NH

3,g

−46,190 −16,590 192.33

Hydroxyl OH

39,460 34,280 183.75

Methane CH

4,g

−74,850 −50,790 186.16

Ethane C

6,g

−84,680 −32,890 229.49

Propane C

8,g

−103,850 −23,490 269.91

Octane vapor C

18,g

−208,450 17,320 463.67

Octane liquid C

18,l

−249,910 6,610 360.79

Benzene C

6,g

82,930 129,660 269.20

Methanol vapor CH

−200,890 −162,140 239.70

Methanol liquid CH

−238,810 −166,290 126.80

Ethanol vapor C

−235,310 −168,570 282.59

Ethanol liquid C

−277,690 −174,890 160.70

Source: From [1].

c03 JWPR067-Mench December 18, 2007 1:59 Char Count=

3.6 Thermodynamic Efﬁciency of a Fuel Cell 99

Figure 3.13 Comparison of maximum thermodynamic efﬁciency for heat engine (Carnot cycle)

and fuel cell engine (hydrogen fuel cell, HHV assumed).

Chapter 1 discussed the advantages of the fuel cell in comparison with conventional

heat engine technologies and noted the fuel cell (or battery) was not constrained to a Carnot

efﬁciency. At the time it was noted that this does not mean a fuel cell can have unlimited

efﬁciency. Recall from Chapter 2 that the natural limitation on the efﬁciency for an ideal

Carnot cycle heat engine can be shown as

t,max

= 1 −

(3.104)

where the T

and T

are the temperatures of heat rejection and heat addition, respectively.

Figure 3.13 shows a comparison of the maximum thermodynamic efﬁciency of an ideal

heat engine and an ideal hydrogen fuel cell with liquid water vapor as the exhaust. Several

points concerning this plot are useful to keep in mind:

1. The plot shows that the maximum thermodynamic efﬁciency is not always greater

for a fuel cell. At high temperatures, a heat engine can theoretically be more efﬁcient.

2. The plot shows only the maximum possible efﬁciency, which will not be obtained

in practice for the heat engine or fuel cell. For a fuel cell, the efﬁciency decreases

with increasing electrical power, so that it only approaches the theoretical value at

open-circuit conditions, where no useful electrical work is produced.

3. The hydrogen fuel cell shows a decreasing efﬁciency with temperature, but this

is not a universal result and depends on the fuel used. Some types of fuels show

almost no relationship to temperature, and some show an increasing trend with

temperature. This is related to the entropy of reaction, as shown in Figure 3.13 and

discussed in greater detail in the following section.