cheg320 thermodynamics extended notes--2

8/2/2019 CHEG320 Thermodynamics Extended Notes--2

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Vijay Ramani, J. M. Fenton

Thermodynamics of Fuel Cells

Leibhafsky and Cairns, Hoogers, Smith and Van Ness

1. Background

Thermodynamics may be broadly defined as the study of energy conversion in a givensystem. Thermodynamic laws govern all changes of energy occurring in such a system.

These laws are based on experience, and have been found to be very sound from afundamental standpoint. However, while they may be used to predict what changes

should occur in a system, they do not explicitly tell us precisely what changes (if any)

will occur, and at what rate they will occur. Such precise predictions lie within the

purview of kinetics, which will be discussed in detail in further chapters. Initially,thermodynamics was principally concerned with the relation between mechanical work

and heat, as such relations were of fundamental importance during the days of the

industrial revolution. However, in later years, there was a growing awareness that workcould also be done by devices that had no moving parts, particularly from changes in

chemical composition. This approach led to the development of the concept of chemicalpotential, and to the development of chemical thermodynamics. The thermodynamics of acomplete fuel cell may be entirely based upon the first and second laws of

thermodynamics, and, more pertinently, upon the application of these laws to

electrochemical systems. A brief review of these concepts is provided below. Detailedinformation about any of the fundamental concepts discussed is readily available in

standard chemical engineering thermodynamics texts

2. A Quick Review

a. The first law of thermodynamics: This law (also called the law of conservation ofenergy) states energy may neither be created nor be destroyed, but may be converted

from one form to another. In other words, the total quantity of energy available in a

system does not change, but the form in which it is available may change. This maymathematically be represented as follows:

dE = Q - W (1)

where E is the system energy, and Q and W refer to the heat input into the system and the

work done by the system respectively. Q and W are path dependent functions, while E isa path independent function. A system is defined as an open system if it permits both

mass and energy transfer, a closed system if it permits only energy transfer, and an

isolated system if it permits neither mass nor energy transfer through its boundaries. A

fuel cell is an open system, which permits the flow of mass and energy through itsboundaries.

Equation 1 on integration gives:

E = Q W (2)


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For an open system, the change in energy may be expressed as follows:

E = U + KE + PE + (PV) (3)

Where U is the internal energy of the system, KE and PE are the kinetic and potentialenergies respectively, and PV refers to the pressure-volume work done on the fluid to

keep it flowing through the system. At this point, we can introduce a new thermodynamicproperty, namely the enthalpy (H). The enthalpy combines the internal energy and the PV

work terms as follows:

H = U + PV (4)

Combining equations 3 and 4, we get the following equation for the energy change in an

open system:

H = Q W (5)

Equation 5 is valid only for steady flow conditions, where KE and PE = 0. The fuelcell may be represented as a control volume as shown in Fig. 1 (Hoogers) below. It is

seen that the work obtained in this case is from the transport of electrons across apotential difference and not from rotation of turbine blades or any such mechanical

means.

Fig. 1

b. Application of the first law to fuel cells: In an operating fuel cell, the following

reaction occurs:

Fuel + oxidant = products (6)

As this reaction is accomplished electrochemically, it involves the transfer of electronsfrom one electrode to the other. The no. of electrons transferred is proportional to the no.

of equivalents of chemical change (N, obtained from stoichiometry) that occur when thefuel is oxidized. By definition, for each equivalent of chemical change, 6.023 x 10

23

(Avogardos number A) of electrons are transferred. The amount of electricityrepresented for this quantity of electrons transferred (corresponding to one equivalent ofchemical change) is given by:

F = Ae (7)


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d. Reversible processes and the concept of entropy: The concept of reversibility isfundamental to the second law of thermodynamics. A system is said to undergo a

reversible change if it remains in equilibrium as it passes from its initial state to its final

state. A reversible process is a reversible change in which the system remains in

equilibrium with its environment. The visualization of a reversible change is especiallyeasy for an electrochemical cell, and is implicit in the definition of a perfect

electrochemical apparatus (as defined by Gibbs):

If no changes take place in the cell except during the passage of current, and all

changes which accompany the current can be reversed by reversing the current, the cell

may be called a perfect electrochemical apparatus

In a fuel cell with reactions described by equation 6 occurring, if the reactants are

identified as the initial state and the products as the final state, then it is always possibleto arrest the reaction by imposing an opposing electromotive force on the system. It is

also possible to reverse the reaction by such means. Therefore, it is not difficult tovisualize a fuel cell as a perfect electrochemical apparatus.

The second law of thermodynamics defines a property called entropy, which is a measure

of the disorder in a system. An irreversible process generates entropy via the mechanismsof frictional heat loss or heat transfer through a finite temperature difference. Such

processes (involving heat transfer) can be made reversible by reducing the finite

temperature gradient into an infinitesimal difference (thereby leading to a vastly reduced

rate of heat transfer). Entropy is based on such reversible heat transfer and is expressed asfollows:

dS = (dQ/T)rev

(12)

In other words, entropy is a system property whose differential changes are given by thequantities (dQ/T) rev. These quantities sum to zero for any series of reversible processes

that causes the system to undergo a cyclic process. For a reversible adiabatic process, dQ

is evidently zero, and hence the process is said to be isentropic (constant entropy). For anirreversible process, the entropy is calculated by assuming a series of reversible processes

that have the same initial and final points. This is valid because entropy is a state function

and is independent of the path taken. This is further seen in the integrated form ofequation 12:

S = S2 S1 = 1, 2 (dQ/T) rev (13)

If the process undergoes reversible heat transfer Q rev at a constant temperature To, then

the entropy is given by:

S = Q rev / To (14)

Mathematically, the second law may be expressed as:


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S total = S system + S surroundings 0 (15)

Where the equality applies for a reversible process.

e. The second law and the reversible fuel cell, the Gibbs free energy:

From equation 11, we have:

H = Q NFE (11)

From the discussion of entropy (eqn 14), for a reversible fuel cell, Q can be substituted asfollows:

H = TS NFE (16)

In differential form, this becomes:

dH = TdS FEdN (17)

Since the cell is operating reversibly, the work obtained from it is the maximum possibleuseful work obtainable (minimal losses). This work is expressed in a function called the

Gibbs free energy (G)

dG = -FEdN (18)

Thus, by substituting equation 18 in equation 17, we can get an expression for themaximum useful work obtainable from the system in terms of thermodynamic state

functions:

dG = dH TdS (19)

From equation 19, we arrive at yet another way of looking at G, H and S. H may beconsidered to be the total energy possessed by a system, and S may be considered to be

unavailable energy, thereby stating that G is the maximum possible useful work that

can be extracted by the system at constant temperature and pressure, and hence is knownas the free energy or energy free to be used.

f. Understanding the significance Gibbs free energy the chemical potential:

A glance at equation 19 reveals that the free energy is related to the enthalpy and

the entropy neither of which is an intrinsic chemical energy term that can be directly

related to the reaction occurring in eqn. 6. However, intuitively it is clear that the reactionis in some manner involved in the production of energy. Therefore, there is evidently

some property of the reactants and products that defines the tendency to these substances

to react, and this property most likely influences the free energy. Gibbs postulated the


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existence of an intensive (independent of mass) state function (path independent) that has

a constant value at equilibrium for a given substance through all the phases in which thesubstance is present as a component. He called this quantity the chemical potential ().

Mathematically, it was introduced as follows:

dU = dQ dW (20)

for a reversible process, Q = TdS. Therefore:

dU = TdS PdV + idni (21)

where i is the chemical potential (intensity factor) for each species present and n i is theamount of each species present (capacity factor). The product of the two represents the

useful work done by each species, and the summation is performed to include

contributions of all species. Since H = U + PV, 21 becomes:

dH = TdS + idni (22)

Combining equations 17, 18 and 22, we arrive at a relation between the Gibbs free

energy and the chemical potential, which is an intrinsic quantity and bears a relation to

the reactants and products in equation 6:

dG = idni (23)

g. The chemical potential and the electromotive force the Nernst equation: Thechemical potential is an intensive property, and does not define the amount of a particular

species present in a given volume. This is obtained from gas pressures or concentrations.A direct link between chemical potential and readily obtainable measures such as

concentration is not possible. Links have been established by using invented functions

such as activity. Conveniently, the activity of an ideal gas is equal to its partial pressure The following treatment is limited to ideal gases as a detailed discussion of the activity,

fugacity and related concepts are more suited for a thermodynamics class.

We have the following relation for G from

G = H TS (24)

dG = dH TdS - SdT (25)

Since H = U + PV, dH = dU + PdV + VdP. Therefore:

dG = dU + PdV + VdP TdS SdT (26)

Substituting for dU from equation 21,

dG = TdS PdV + idni + PdV + VdP TdS SdT (27)


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Which simplifies to:

dG = VdP SdT + idni (28)

Now, by partially differentiating equation 28 with respect to P:

(G/P) T, n = V (29)

However, since we are considering ideal gases, we also have:

V = nRT / P (30)

Combining equations 29 and 30, we have:

(G/P) T, n = nRT / P (31)

Since only differences in G (and not absolute values of G) are significant from a practicalviewpoint, a standard state is generally chosen, and is represented by the superscript

o.

Integrating equation 31, we have:

G = G - G o = nRT ln (P/P o) (32)

Equation 32 can be written for a species in solution by replacing the pressure for the

component partial pressure:

Gi = Gi - Gio

= niRT ln (pi/pio) (33)

Now applying the definition of the chemical potential (equation 23) we can write:

i = io

+ RT ln (pi/pio) (34)

In equation 34, we have a relationship between the chemical potential, and a measurablequantity such as pressure. It may be recalled that this is precisely the link we have been

seeking.

Now, for a given reaction, the change in Gibbs free energy may be written as follows:

G = pp - rr(35)

Where is the stoichiometric number, and the subscripts p and r refer to products andreactants respectively (in some ways, equation 35 may also be considered as a statement

of the first law). We also know from equation 18 that :

G = - NFE (36)


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Now, substituting equation 34 into equation 35, and combining equations 35 and 36 we

have:

G = pp0

- rr0

+ RT ln{(ppp

/prr

)} = - NFE (37)

We can now consider two states that merit further discussion. We can define a standardstate, where all the gases (reactants and products) are at unit pressure. In this case,

ln{(ppp

/prr

)} is equal to 0, and equation 37 reduces to the form:

Go = pp0

- rr0

= - NFEo

(38)

The second state is the equilibrium case, where G = 0, and (ppp

/prr

) equals theequilibrium constant K. In this case equation 37 reduces to:

0 = pp0

- rr0

+ RT lnK (39)

Substituting equation 38 into equation 39, we obtain the following equilibriumrelationship:

Go = -RT lnK = - NFEo (40)

Equation 40 is valid only at equilibrium. A more general representation of equation 37 is

given below:

G = Go + RT ln{(ppp

/prr

)} = - NFE (41)

Now if we take the latter equality in equation 41, and divide throughout by NF, we get:

Go/ NF + (RT/NF) ln{(ppp

/prr

)} = -E (42)

Substituting forGo/ NF from equation 40, we have:

- Eo + (RT/NF) ln{(ppp/prr)}= -E (43)On rearrangement, equation 43 gives:

E = Eo

- (RT/NF) ln{(ppp/pr

r)}(44)

Equation 44 is called the Nernst equation, and is widely used in electrochemistry toevaluate the effect of simple changes in reactant or product activity, and temperature on

the cell voltage. It must be noted that the free energy route (using tabulated values of G)

is still preferred for complex changes.

h. Fuel cell vs. Carnot efficiencies: Briefly, any heat engine that absorbs heat at a high

temperature (T1) must reject energy at a lower temperature (T2) to do useful work. The


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efficiency (called Carnot efficiency after the scientist who developed this postulation) of

an ideal engine was found to be second law limited, and can be written (sans derivation)as:

carnot = 1 T2/T1 (45)

From equation 45, it is seen that the efficiency is 1 only at an infinitely hot source. Morerealistically, the higher the hot source, the higher the efficiency. However, this is a

specious argument, because the amount of energy lost due to irreversible processes inmaintaining the temperature of the hot source far exceeds the work output of the cell,

thereby leading to reduced efficiencies.

In a fuel cell, the operation is essentially isothermal, and therefore less energy is lost inmaintaining the temperature of the hot source. Therefore, the fuel cell is inherently less

irreversible. Also, due to the absence of temperature cycling, it does not follow the

Carnot cycle, and therefore not limited by the Carnot efficiency. Therefore, the efficiencyof the fuel cell is expressed by other means. It is often reported (erroneously) that a fuel

cell is 100% efficient. This is better stated as The fuel cell is not limited by the Carnotefficiency. In reality, the fuel cell is far from 100% efficient. The efficiency based uponthe first law of thermodynamics is readily expressed as:

= W out / Q in (46)

We know that for a fuel cell, the work done is given by the G (or equivalently by NFE),and that the heat input may be calculated based upon the higher heating value (HHV) of

the fuel. Thus:

= G/HHV = NFE / HHV (47)

In a similar manner, the maximum efficiency is obtained at the highest cell voltage

(namely the open circuit potential), and is given by:

max = Go/HHV = NFE

o/HHV (48)

For a typical H2/O2 fuel cell, the maximum efficiency calculated from eqn. 48 is 83%

(with the rest being entropic losses), and a Carnot engine would have to have a high

temperature of 1753 K (with a corresponding low temperature of 298K) to achieve thisefficiency. However, it must be noted that while the work done by a Carnot cycle engine

(and hence the Carnot efficiency) increases with increasing temperature (of the hot

source), the reverse is true for the G based fuel cell efficiency. This is because G (andhence E) decreases with temperature (remember the Nernst equation). Thus there exists atemperature beyond which the fuel cell efficiency is actually lowerthan the Carnot

efficiency. This is illustrated in Fig. 2 (Hoogers). The temperature at which this occurs is

approximately 950 K for a H2/O2 system (and they run SOFCs at .. temperaturesabove 950K! not a fair comparison as the mechanisms are different).


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Fig. 2

Now, the maximum efficiency predicted by equation 48 is quite meaningless from a

practical viewpoint (no current is drawn!). therein lies the limitation of the efficienciespredicted by the first law. This can be redressed by defining efficiency from a second law

viewpoint, which indicates the actual work obtained to the maximum possible work

obtainable. This is also referred to as a voltage efficiency. In some ways, it is merely aratio of equations 47 and 48:

2nd law = NFE / NFEo

= E / Eo

(49)

The value of the actual voltage is always less than the value of the theoretical open

circuit voltage (even at open circuit!) due to inefficiencies within the fuel cell also

called overpotentials (or should we say underpotentials?). These overpotentials arise dueto several reasons, the most important of which will be discussed in detail in later

chapters. A current efficiency may also be defined based upon the ratio of the currentproduced to the current expected (based on Faradays law) for the amount of fuel fed tothe cell:

I = I / NF (dn/dt) (50)

where n is the number of moles of fuel. This efficiency can indeed be close to 100%, but

is misleading as it does not refer to the efficiency with which work is being done by the

cell. The product of the current and voltage efficiencies is sometimes used as a morecomprehensive definition of cell efficiency (this is also called the Gibbs free energy

efficiency)

cheg320 thermodynamics extended notes--2

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