Mench M.M. Fuel Cell Engines

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

70 Thermodynamics of Fuel Cell Systems

Flow in

Flow out

Figure 3.5 Flow work at the ﬂux boundaries of a control volume.

Ĺ Rotational Motion of Molecules Spinning molecules store energy in the form

of rotational momentum. Similar to the vibrational modes of storage, for a more

complex structure, the molecule has a greater number of rotational modes available.

A monatomic species such as helium has no bonds and therefore cannot store energy

in this form.

Ĺ Atomic-Level Storage At the atomic level, energy is stored in the orbital states,

intermolecular and interatomic forces, and nuclear spin states. Intermolecular forces

become important for high-pressure gases, liquids, and solid states.

Note that the stored internal energy does not contain the chemical energy that would be

released or consumed for a reaction that converts the species into other molecules. This

energy is a result of the reconﬁguration of the bond structure and can be orders of magnitude

greater in value compared to the energy stored in nonreacting species.

Enthalpy (H) Enthalpy is a unique thermodynamic parameter that is deﬁned based on

the other thermodynamic quantities of internal energy, pressure, and density:

H = U +

= U + Pv (3.8)

where v is the speciﬁc volume.

Enthalpy is a measure of the energy stored in a ﬂowing ﬂuid. It is therefore deﬁned as

the internal energy plus the ﬂow work. The ﬂow work is the energy required for ﬂow of the

ﬂuid across the control volume boundary. Consider a basic control volume of a container

with gas-phase ﬂow in and out of the control surface, as shown in Figure 3.5.

For a control volume, the ﬁrst law of thermodynamics includes not only the ﬂux of heat

and work across the control surface but also the ﬂux of energy across the control surfaces

via mass ﬂow in and out of the control volume. The ﬁrst law for a control volume is written

−





+ gz



−





+ gz



(3.9)

The reader is referred to detailed texts on thermodynamics if a deeper explanation is needed.

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3.1 Physical Nature of Thermodynamic Variables 71

Here, E is energy,

Q is the rate of heat transfer across the control surface and is positive for

heat added to the system, and

W is the rate of work across the control surface and is positive

for work done by the system on the surroundings. The ﬁnal two terms on the right-hand

side of Eq. (3.9) represent the ﬂux of energy into and out of the control volume in the form

of enthalpy, kinetic energy, and potential energy carried by mass ﬂux across the control

surfaces. In order to cross into and out of the control surface, some ﬂow work is done on the

control volume (at the entrance) and on the environment (at the exit). For a compressible

substance, this total ﬂow energy is represented as

u + P ¯v

Internal energy Flow work



(3.10)

For an ideal gas, we can also show that

h =

u + R

T (3.11)

Constant-Pressure Speciﬁc Heat (c

) The constant-pressure speciﬁc heat is deﬁned as

the rate of change of enthalpy with temperature at constant pressure:

(T , P) ≡

∂

∂T



(3.12)

Constant-Volume Speciﬁc Heat (c

) The constant-volume speciﬁc heat is deﬁned as the

rate of change of internal energy with temperature at constant volume:

(T , P) ≡

∂

∂T



(3.13)

The speciﬁc heats are a measure of the capacity for the internal energy storage, which can

change with temperature. As the temperature changes, different rotational and vibrational

modes of energy storage become active, and the speciﬁc heat values can change. For

the same energy input to two substances, the substance with the larger speciﬁc heat will

have a lower increase in temperature, since it has a relatively high capacity for thermal

energy storage in other modes besides translational velocity. Figure 3.6 shows the constant-

pressure speciﬁc heats of a variety of gas-phase species as a function of temperature. Note

the temperature independence of the monotomic species which can only store internal

energy as translational and interatomic forces. As the molecular complexity increases (i.e.,

more bonds), the capacity to store energy increases, and the speciﬁc heat is generally

higher. As the temperature is increased, more vibrational and rotational modes become

active, increasing the speciﬁc heat to a limit, where all energy storage modes are fully

occupied or the molecule disassociates into smaller fragments.

Gas-Phase Speciﬁc Heat The speciﬁc heat of an ideal gas is solely a function of tem-

perature, and even nonideal gases have an insigniﬁcant dependence on pressure. For most

cases involving fuel cells except hydrogen storage, the conditions are such that an ideal gas

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

72 Thermodynamics of Fuel Cell Systems

Figure 3.6 Speciﬁc heat as function of temperature for several species.

model is valid over the operating range, and the partial derivatives of Eqs. (3.12) and (3.13)

can be replaced with ordinary differentials:

(T ) =



(3.14)

(T ) =



(3.15)

To determine the enthalpy or internal energy change for a temperature change for a non-

reacting species, we integrate Eqs. (3.14) and (3.15):



h =



(T ) dT (3.16)



u =



(T ) dT (3.17)

The enthalpy change between two temperature states of a nonreacting substance is known

as the sensible energy. Determination of the sensible enthalpy at a given temperature can

be done in several ways with varying accuracy. If we can assume constant speciﬁc heats,

for a nonreacting gas we can simplify Eqs. (3.16) and (3.17) as

−

(

− T

)

(3.18)

−

(

− T

)

(3.19)

To use constant speciﬁc heats, a suitable average temperature should be chosen to minimize

inaccuracies in the result, since any nonmonatomic specie will have a functional dependence

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3.1 Physical Nature of Thermodynamic Variables 73

with temperature, as shown in Figure 3.6. This is generally acceptable for fuel cells since

they tend to operate over a narrow temperature range. If we do not assume constant speciﬁc

heats, we must integrate the speciﬁc heat curve-ﬁt expression with respect to temperature.

This is still a simple matter and more acurrate:

−



(T ) dT (3.20)

−



(T ) dT (3.21)

where the proper polynomial expression for

(T )or

(T ) is inserted and integrated. The

absolute speciﬁc internal energy and enthalpy at a given temperature T

is given by

u(T

) =



T,ref

(T ) dT (3.22)

h(T

) =



T,ref

(T ) dT (3.23)

where 298 K is typically chosen as the standard reference temperature.

Speciﬁc heat values have been measured for a wide variety of gases and can be used

to determine the internal energy and enthalpy. Speciﬁc heat relationships are generally

modeled with a high-order polynomial, such as those listed below for common fuel cell

gases, valid in the range of 300–1000 K [1]:

(T )

= 3.057 + 2.677 ×10

−3

T − 5.810 × 10

−6

+ 5.521 × 10

−9

− 1.812 × 10

−12

(3.24)

(T )

= 3.626 − 1.878 ×10

−3

T + 7.055 × 10

−6

− 6.764 × 10

−9

+ 2.156 × 10

−12

(3.25)

(T )

= 3.675 − 1.208 ×10

−3

T + 2.324 × 10

−6

− 0.632 × 10

−9

− 0.226 × 10

−12

(3.26)

(T )

Air

= 3.653 − 1.337 ×10

−3

T + 3.294 × 10

−6

− 1.913 × 10

−9

+ 0.2763 × 10

−12

(3.27)

(T )

= 4.070 − 1.108 ×10

−3

T + 4.152 × 10

−6

− 2.964 × 10

−9

+ 0.807 × 10

−12

(3.28)

(T )

monatomic gas

= 2.5 (3.29)

where T is in kelvins.

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74 Thermodynamics of Fuel Cell Systems

Liquid- and Solid-Phase Speciﬁc Heat For an incompressible liquid or solid, the density

is constant, so that

(T ) =



dP/ ¯ρ



(T ) =

c(T ) (3.30)

Solids and liquid have no distinction between constant-pressure or constant-volume speciﬁc

heats, and a single speciﬁc heat value is appropriate. The speciﬁc heat of most incompress-

ible substances varies only slightly with temperature and can generally be considered

constant over a limited temperature range of interest to fuel cell studies. For example, the

mass speciﬁc heat of liquid water at 25

◦

C is 4.179 kJ/kg · K, while at 100

◦

C, the speciﬁc

heat of the liquid phase is 4.218 kJ/kg · K, only about 1% higher.

3.2 HEAT OF FORMATION, SENSIBLE ENTHALPY,

AND LATENT HEAT

Enthalpy of Formation The enthalpy of a given component is an important parameter, as

it is a measure of the internal energy and ﬂow energy potentially available for conversion into

other forms of energy and work. In order to determine the relative thermal energy exchange

between states, we need to deﬁne an arbitrary baseline from which the differences are

measured. Consider any species at standard temperature and pressure (STP; 298 K, 1 atm).

This is our chosen baseline state. The enthalpy of the substance at this state is termed the

enthalpy of formation or the heat of formation:

◦

= heat of formation (3.31)

Heats of formation for some fuel cell species are given in Table 3.2. Others can be found

in thermodynamics references or online. For an element in its stable state, for example

diatomic hydrogen or monotonic helium, no energy is required or released to achieve a

stable state at STP. Therefore, the heat of formation of atomic species in their stable state

at STP is deﬁned as zero. For a compound like water, which is a product of an exothermic

Table 3.2 Heats of Formation of Common Fuel Cell Species at 1 atm, 298 K

Species Formula

◦

(

kJ/kmol

)

Water vapor H

−241,820

Liquid water H

−285,830

Carbon dioxide CO

−393,520

Carbon monoxide CO −110,530

Methanol vapor CH

−200,890

Liquid methanol CH

−238,810

Methane CH

4,g

−74,850

Nitrogen N

Oxygen O

Hydrogen H

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3.2 Heat of Formation, Sensible Enthalpy, and Latent Heat 75

25°C

1 atm

Combustion

out

= h°

25°C

1 atm

25°C

1 atm

Figure 3.7 Heat of formation.

reaction between O

and H

, however, thermal energy must be exchanged to achieve a

stable compound at STP, resulting in a nonzero heat of formation.

Consider Figure 3.7, in which a steady ﬂow of oxygen and hydrogen at 1 atm, 25

◦

is mixed and completely burned in a ﬂow reactor to form a water vapor stream that exits

the reactor at 25

◦

C, 1 atm. We know from experience with combustion that the reaction

is exothermic and heat would be released by the reaction. In this example, heat must be

removed from the reactor to cool the product water vapor to its exit condition of 1 atm,

◦

C. The heat removed to achieve this is equivalent to the heat of formation of the water

vapor. From this example, we see that for a water molecule to exist at STP some thermal

energy conversion (and therefore a nonzero heat of formation) is required. In this case,

thermal energy is released, so that the chemical energy state of the product water is lower

than the initial reactants. Therefore, the heat of formation for the water vapor is negative,

as is the case for any exothermic reaction. Note from Figure 3.7 that there is a difference

between the heat of formation of water in a liquid or vapor state. This is due to the latent

energy required for vaporization and is discussed in the following section. If we were to

take the reactor in Figure 3.7 and produce liquid-phase exhaust, additional heat would have

to be removed to condense the vapor.

Sensible Enthalpy Consider a water vapor stream exiting the reaction chamber in Figure

3.7 at 25

◦

C, 1 atm. In order to change this stream from 25

◦

C to some other temperature T

without additional reaction, some heat transfer is required. This energy required is called

sensible enthalpy:

= sensible enthalpy =

h(T ) −

(

ref

)

(3.32)

where the reference temperature is 25

◦

Latent Heat When a substance is undergoing a phase change, the temperature is generally

constant, but there is still a thermal energy exchange when the molecular structure is

reordered to a different phase. This thermal energy difference between the molecular

structures of the two phases is termed latent heat:

LH = latent heat (3.33)

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76 Thermodynamics of Fuel Cell Systems

vap

t = t

vap

Figure 3.8 Schematic of evaporating droplet.

Consider a droplet evaporating at ambient pressure, as shown in Figure 3.8. At this pressure,

the boiling point (where the liquid will undergo phase change) is 100

◦

C. The energy

required to boil off the water is termed the latent heat of vaporization, h

. The latent heat

of vaporization for water at 100

◦

C is 2257 kJ/kg. Besides vaporization, there is also a

latent heat released when vapor condenses and reforms into a liquid at a lower molecular

energy state. The enthalpy of condensation is simply the additive inverse, −2257 kJ/kg.

This concept is a little more difﬁcult to conceptualize, but a droplet condensing on a surface

will release thermal energy and actually heat the surface. This latent heat release is what can

cause severe burns when people are exposed to steam. Thermal energy is also released when

a substance freezes. To take advantage of this effect, some citrus fruit farmers spray crops

with water when freezing temperatures are expected, a technique known as irrigational frost

protection [4]. When the water freezes, it releases 334 kJ/kg and forms a thermal barrier

for heat loss, protecting the fruit from damage. Overall, latent heat is required for melting,

boiling, and sublimation and is released for condensation, freezing, and desublimation.

Example 3.2 Latent Heat of Vaporization Consider a 1-mg droplet of liquid water at

l atm pressure. Determine the sensible enthalpy required to heat the droplet from 25 to

100

◦

C, and the total energy required for complete vaporization.

SOLUTION The latent heat required for vaporization is:

E = mh

= (1 mg)(2257 kJ/kg)



1kg

1,000,000 mg



(1000 J/kJ) = 2.257 J

The liquid droplet enthalpy values can be determined using a constant liquid speciﬁc heat

of water of 4.182 kJ/kg ·K. The sensible enthalpy required to get the same droplet from

◦

C to the boiling point of 100

◦

Cis:

− H

= m[h

) − h

)] = m[h(100) − h(25)] = mc(T

− T

(1 mg)c(100 − 20) = (4.182 kJ/kg ·K)(80) = 0.335 J

COMMENTS: The latent heat of vaporization is actually quite large compared to the

sensible enthalpy required to increase the droplet temperature. This is why it takes a

relatively long time to actually boil water into vapor, compared to the time required to heat

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3.2 Heat of Formation, Sensible Enthalpy, and Latent Heat 77

the water to the boiling temperature on the stove. The energy difference between phases

can be an important part of the overall energy in a fuel cell, especially for low-temperature

PEFCs where phase change processes are common.

Total Enthalpy Putting it all together, for a substance that deviates from the reference

state at some state A, the mass speciﬁc enthalpy can be written as

= h

+ h

+ LH

for anyheatLatentSensibleofEnthalpyatEnthalpy

phase chan

eenthalp

formationAstatesome

(3.34)

At any temperature, the enthalpy is the heat of formation required to arrive at STP plus the

sensible enthalpy required to heat (or cool) the substance to depart from STP plus whatever

latent heat is required to achieve the phase state at the given temperature. If there is no

phase change from the formation state, the LH term is obviously zero.

Example 3.3 Determination of Enthalpy of a Single Species Determine the enthalpy

per unit mass of water vapor at 600 K, 1 atm.

SOLUTION In order to get water vapor at 600 K, there is an enthalpy of formation

required to obtain liquid water at 298 K. Next, there is a sensible enthalpy required to heat

from 298 K to the boiling point in liquid form, at 373 K. Then, there is a latent heat of

vaporization required at 373 K to boil the water at 1 atm. Finally, there is a sensible enthalpy

input required to raise the water vapor temperature from 373 to 600 K. We can write all of

this as

Increase to boiling temperature

h = h

+ h

s,l

+ LH + h

s,v

Get to Boil Go from 393 K

298 K to 600 K

From Table 3.2, at 298 K, 1 atm, the liquid water

◦

=−285,830 kJ/kmol, which equals

−15,879 kJ/kg.

The sensible enthalpy to heat the water to its boiling point at 373 K is

s,l

= h(373) − h(293) ≈ c

O,l

(

)

≈ (4.182 kJ/kg · K)(80 K) = 336 kJ/kg

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78 Thermodynamics of Fuel Cell Systems

The latent heat of vaporization of water is 2257 kJ/kg. Finally, the vapor-phase water must

be heated from 373 to 600 K via another sensible enthalpy change:

s,v

= h(600) − h(373) ≈ c

p,ave,H

O,v

× 227 = 18 kg/kmol/35.06 kJ/kmol ·K × 227 K

= 442 kJ/kg

where an average vapor-phase water speciﬁc heat has been used for convenience, although

a more precise value can be obtained from integration of Eq. (3.28). Since the total amount

of sensible enthalpy contribution is quite small compared to the latent heat required for

vaporization, the error with using an average value of speciﬁc heat is small.

Now, in total, we have

h = h

◦

+ h

s,L

+ LH + h

s,v

=−15,879 + 336 + 2257 + 442 =−12,844 kJ/kg

COMMENTS: Note the small contribution from sensible enthalpy relative to the heat of

formation or heat of vaporization. Also note that we could have also started with vapor at

293 K by using the heat of formation of vapor-phase water. This would have eliminated

the need to calculate the latent heat term since it is included in the heat of formation, and

we would only have to calculate the sensible enthalpy of the gas phase to go from 293 to

600 K.

3.3 DETERMINATION OF CHANGE IN ENTHALPY FOR

NONREACTING SPECIES AND MIXTURES

Nonreacting Species Calculation Consider a single nonreacting ideal gas with a temper-

ature change. For example, consider preheating hydrogen in a SOFC cathode intake from

300 to 600 K. To determine the change in enthalpy from a state at T

to a state at T

with

no phase change, we can follow Eq. (3.34):

f,1

= h

f,2

= LH

− h

= h

+ h

+ LH

− h

+ h

+ LH

(3.35)

For a nonreacting species with no phase change, the heat of formation and LH terms cancel

out, and what is left is the change in sensible enthalpy. So, for a nonreacting gas we can

show that

(

)

− h

(

)

= h

s,2

− h

s,1

[

(

)

− h

(

ref

)

]

s, 2

−

[

(

)

− h

(

ref

)

]

s, 1

(3.36)

For an enthalpy change between two temperature states of a nonreacting species, the

reference temperature enthalpy terms cancel out. This will not be the case for reacting

mixtures but simpliﬁes things here. There are several ways to solve Eq. (3.36) for the

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3.3 Determination of Change in Enthalpy for Nonreacting Species and Mixtures 79

sensible enthalpy change:

1. For many species there are tables available to look up the enthalpy values.

2. We can integrate the speciﬁc heat expressions in Eqs. (3.24)–(3.29).

3. We can assume constant speciﬁc heats at an appropriate average temperature and

solve Eq. (3.18).

The assumption of constant speciﬁc heats is generally the easiest to implement and is also

justiﬁed in most cases for fuel cells, as we shall see in Example 3.4.

Example 3.4 Calculation of Enthalpy for Nonreacting Species Consider water vapor

entering a SOFC. Determine the molar and mass speciﬁc enthalpy change required to heat

the water vapor from a 298 K inlet temperature to 1073 K using

(a) constant speciﬁc heats at an average temperature and

(b) evaluation of the integration of the proper polynomial expression for the speciﬁc

heat.

SOLUTION (a) For water vapor the proper polynomial expression is Eq. (3.28):

(T )

= R

(4.070 − 1.108 × 10

−3

T + 4.152 × 10

−6

− 2.964 × 10

−9

+ 0.807 × 10

−12

)

At an average temperature of 685 K, the above expression reduces to

p,ave,H

= 37.3kJ/(kmol · K)

−

p,ave,H

(

1073 − 298

)

= 28,894 kJ/kmol

(b) The evaluation of the integral with Eq. (3.28) yields

−

1073



298

(T )

dT = 29,029 kJ/kmol

COMMENTS: Although the integrated solution is shown quite compactly, integration of

a fourth-order polynomial is a very tedious exercise if done by hand. Note that for even a

large temperature change of around 700 K there is little difference (0.5%) in the result; it

is often not worth the additional effort for simple calculations.

Nonreacting Ideal Gas Mixture Calculation In many cases relevant to fuel cells, we

must deal with mixtures rather than pure gases. Thermodynamic properties of mixtures

can be easily calculated based on the mole or mass fractions of the constituents. A good

example of this is air, which is a nonreacting mixture of mostly nitrogen and oxygen. For

these mixtures, we can assume each species in the mixture is occupying the total volume

but at a partial pressure in the mixture, where the partial pressure is deﬁned as

= y

P (3.37)