Vijay Ramani, J. M. Fenton Thermodynamics of Fuel Cells

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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 given
system. 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 a
fundamental 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 work
could 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 chemical
potential, and to the development of chemical thermodynamics. The thermodynamics of a
complete 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. Detailed
information 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 of
energy) 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 may
mathematically be represented as follows:
dE = δQ - δW (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 is
a 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 its
Equation 1 on integration gives:
∆E = Q – W (2)
Vijay Ramani, J. M. Fenton
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 potential
energies 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 thermodynamic
property, 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 fuel
cell 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 a
potential difference and not from rotation of turbine blades or any such mechanical
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 electrons
from 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 the
fuel is oxidized. By definition, for each equivalent of chemical change, 6.023 x 1023
(Avogardo’s number A) of electrons are transferred. The amount of electricity
represented for this quantity of electrons transferred (corresponding to one equivalent of
chemical change) is given by:
F = Ae (7)
Vijay Ramani, J. M. Fenton
Where e is the unit electronic charge in coloumbs (1.6 x 10 –19). This quantity comes out
to be equal to 96493 coloumbs and is referred to as Faraday’s constant (F). From this one
can state that the amount of electricity transferred when reaction (eqn. 6) occurs is given
Electricity = NF (8)
This amount of electricity transferred corresponds to a certain amount of electrical work
being done by the cell, given by:
- We = NFE (9)
Where E is the cell voltage (the potential difference between the two electrode terminals
of the operating cell). The negative sign indicates that work is being done by the system.
This work can also be described in terms of voltage, current and time as:
We = ∫0,t EIdt (10)
Where I is the current and t is the time. This is identical to stating that the current is the
rate of change of charge (I = q / t). Substituting equation 9 in equation 5, we get a
mathematical statement of the first law for a fuel cell:
∆H = Q – NFE (11)
c. The second law of thermodynamics: While the first law of thermodynamics states that
the total energy of a system is conserved, it does not impose any restrictions on the
direction of energy transfer (but we know that heat cannot flow from a cooler body to a
hotter body). Furthermore, it considers the energy content of work and heat to be
equivalent. However, from day to day experience, this is evidently not true. While work
can readily be transferred from one form to another with conversion efficiency limited
only by frictional losses, it is very difficult to convert heat efficiently to work. Indeed,
efficiencies of 40% are considered to be excellent for these processes. Therefore, heat is a
less useful form of energy than an equivalent quantity of work. These shortcomings of
the first law are addressed in the second law. The second law can be stated in several
ways. One such statement is (Smith, Van Ness and Abbot):
No apparatus can operate in such a way that its only effect on the system
and surroundings is to convert heat absorbed by a system completely to
work done by the system (or) it is impossible by a cyclic process to
convert the heat absorbed by a system completely into work done by the
No process is possible that consists solely in the transfer of heat from one
temperature level to a higher one.
Thus, while the second law does not proscribe the conversion of heat to work, it places a
limit on the fraction of heat that may be converted to work.
Vijay Ramani, J. M. Fenton
d. Reversible processes and the concept of entropy: The concept of reversibility is
fundamental 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 especially
easy 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 possible
to 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 to
visualize 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 mechanisms
of 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 as
dS = (dQ/T) rev (12)
In other words, entropy is a system property whose differential changes are given by the
quantities (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 an
irreversible 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 of
equation 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:
Vijay Ramani, J. M. Fenton
∆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 as
∆H = T∆S – 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 possible
useful 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 the
maximum useful work obtainable from the system in terms of thermodynamic state
dG = dH – TdS (19)
From equation 19, we arrive at yet another way of looking at G, H and S. H may be
considered 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 known
as 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 reaction
is 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
Vijay Ramani, J. M. Fenton
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 the
substance is present as a component. He called this quantity the chemical potential (µ).
Mathematically, it was introduced as follows:
dU = dQ – dW (20)
for a reversible process, Q = TdS. Therefore:
dU = TdS – PdV +Σ µidni (21)
where µi is the chemical potential (intensity factor) for each species present and ni is the
amount 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 + Σ µidni (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 = Σ µidni (23)
g. The chemical potential and the electromotive force – the Nernst equation: The
chemical 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 +Σ µidni + PdV + VdP – TdS – SdT (27)
Vijay Ramani, J. M. Fenton
Which simplifies to:
dG = VdP – SdT + Σ µidni (28)
Now, by partially differentiating equation 28 with respect to P:
(δG/δP) 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:
(δG/δP) T, n = nRT / P (31)
Since only differences in G (and not absolute values of G) are significant from a practical
viewpoint, 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 - Gi o = niRT ln (pi/pi o) (33)
Now applying the definition of the chemical potential (equation 23) we can write:
µi = µio + RT ln (pi/pi o) (34)
In equation 34, we have a relationship between the chemical potential, and a measurable
quantity such as pressure. It may be recalled that this is precisely the link we have been
Now, for a given reaction, the change in Gibbs free energy may be written as follows:
∆G = Σνpµp - Σνrµr (35)
Where ν is the stoichiometric number, and the subscripts p and r refer to products and
reactants 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)
Vijay Ramani, J. M. Fenton
Now, substituting equation 34 into equation 35, and combining equations 35 and 36 we
∆G = Σνpµp0 - Σνrµr0 + RT ln{Π(ppνp/prνr)} = - NFE (37)
We can now consider two states that merit further discussion. We can define a standard
state, where all the gases (reactants and products) are at unit pressure. In this case,
ln{Π(ppνp/prνr)} is equal to 0, and equation 37 reduces to the form:
∆Go = Σνpµp0 - Σνrµr0 = - NFEo (38)
The second state is the equilibrium case, where ∆G = 0, and Π(ppνp/prνr) equals the
equilibrium constant K. In this case equation 37 reduces to:
0 = Σνpµp0 - Σνrµr0 + RT lnK (39)
Substituting equation 38 into equation 39, we obtain the following equilibrium
∆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{Π(ppνp/prνr)} = - NFE (41)
Now if we take the latter equality in equation 41, and divide throughout by NF, we get:
∆Go/ NF + (RT/NF) ln{Π(ppνp/prνr)} = -E (42)
Substituting for ∆Go/ NF from equation 40, we have:
Eo + (RT/NF) ln{Π(ppνp/prνr)}= -E (43)
On rearrangement, equation 43 gives:
E = Eo - (RT/NF) ln{Π(ppνp/prνr)}(44)
Equation 44 is called the Nernst equation, and is widely used in electrochemistry to
evaluate 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
Vijay Ramani, J. M. Fenton
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)
ηcarnot = 1 – T2/T1 (45)
From equation 45, it is seen that the efficiency is 1 only at an infinitely hot source. More
realistically, 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 in
maintaining 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 in
maintaining 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 efficiency
of 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 Carnot
efficiency”. In reality, the fuel cell is far from 100% efficient. The efficiency based upon
the first law of thermodynamics is readily expressed as:
η = 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:
η = ∆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:
ηmax = ∆Go/HHV = NFEo /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 this
efficiency. 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 (and
hence E) decreases with temperature (remember the Nernst equation). Thus there exists a
temperature beyond which the fuel cell efficiency is actually lower than 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 ….. temperatures
above 950K! – not a fair comparison as the mechanisms are different).
Vijay Ramani, J. M. Fenton
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 efficiencies
predicted 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 a
ratio of equations 47 and 48:
η 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 due
to 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 current
produced to the current expected (based on Faraday’s law) for the amount of fuel fed to
the cell:
ηI = 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 more
comprehensive definition of cell efficiency (this is also called the Gibbs free energy
Vijay Ramani, J. M. Fenton

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