Reversible and Irreversible Potentials and an Inaccuracy in Popular Models in the Fuel Cell Literature

Abstract

Modeling is an integral part of fuel cell design and development. This paper identifies a long-standing inaccuracy in the fuel cell modeling literature. Specifically, it discusses an inexact insertion, in popular models, of cell/stack current into Nernst’s equation in the derivation of output (load) voltage. The origin of the inaccuracy is traced to the nature of reversible and irreversible potentials (equilibrium and non-equilibrium states) in the cell. The significance of the inaccuracy is explained in the context of the electrochemistry and thermodynamics of the fuel cell.

1. Introduction

Modeling the operation of fuel cells is an indispensable part of fuel cell research. In many papers in the literature involving the modeling of solid oxide fuel cells (SOFCs) and proton exchange membrane fuel cells (PEMFCs), an inaccuracy is found in the cell/stack voltage expression (see, for example, references [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]). We argue that the problem arises because of an inexact intermediate step in the derivation of the output voltage. The origin of this inaccuracy can be traced to the substitution in the Nernst equation of activity (or concentration or partial pressure) with a function of the load current. An analysis of the flaw is presented in the remainder of this paper.

2. Background: The Nernst Equation

The Nernst equation, which is the cornerstone of fuel cell thermodynamics, provides an expression for the reversible thermodynamic potential, also known as the equilibrium voltage or the open-circuit electromotive force (EMF), of the fuel cell [31]:

$$E_{Nernst}=E_0+\frac{RT}{nF}\ln \left(\frac{{\prod }_i^{}{a}_{reactan{t}_i}^{c_i}}{{\prod }_j^{}{a}_{produc{t}_j}^{c_j}}\right)$$

(1)

where $E_0$ is the reference (standard) EMF at unit activity and atmospheric pressure; i and j are the numbers of reactant and product species; a represents the activity; $c_i$ is the stoichiometric coefficient of species i; R is the universal gas constant; F is Faraday’s constant; n is the number of electrons transferred for each molecule of the fuel participating in the reaction; and T is the temperature. For a hydrogen–oxygen fuel cell (e.g., solid oxide fuel cell or proton exchange membrane fuel cell), hydrogen and oxygen are the reactants, and the product is water (or steam). The reference EMF, $E_0$, depends on the temperature, T:

$$E_0=E_0^0+(T-T_0)\frac{\Delta s}{nF},$$

(2)

where $E_0^0$ is the standard EMF at temperature $T_0$, and $\Delta s$ is the change in entropy. The activity, a, of an ideal gas is expressed in terms of its pressure (or partial pressure), p:

$$a_{H_2}=\frac{p_{H_2}}{p^0},$$

(3)

$$a_{O_2}=\frac{p_{O_2}}{p^0},$$

(4)

where $p^0$ is the standard-state pressure (1 atm). At high temperatures, such as 1000 °C (as in solid oxide fuel cells), steam can be assumed to behave as an ideal gas, and therefore,

$$a_{H_{2}O}=\frac{p_{H_{2}O}}{p^0}.$$

(5)

Using $p^0$ = 1 atm, and noting that $n=2$ for a hydrogen fuel cell, we have the following version of the Nernst equation for solid oxide fuel cells:

$$E_{Nernst}=E_0+\frac{RT}{2F}\ln \left(\frac{p_{H_2}\sqrt{p_{O_2}}}{p_{H_{2}O}}\right).$$

(6)

If the fuel cell is operated below 100 °C so that liquid water is produced (as in proton exchange membrane fuel cells), the activity of water can be taken to be unity ($a_{H_{2}O}$ = 1). In that case, the Nernst equation takes the form

$$E_{Nernst}=E_0+\frac{RT}{2F}\ln \left(p_{H_2}\sqrt{p_{O_2}}\right).$$

(7)

3. The Inaccuracy in the Model

Drawing inspiration from the pioneering work in reference [1] where an elegant dynamic model for an SOFC plant was developed, many later papers (e.g., references [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]) re-derived and/or built upon the model, expressing the partial pressures of the reactants and products as functions of the cell current (not to be confused with the exchange current [31] or the fuel crossover/internal current [31]), before inserting those partial pressure expressions into the Nernst equation.

Reference [1], in its “dynamic behaviour algorithm” (Figure 2 on p. 497 of that paper), expressed (i) the partial pressures of the reactants and products in terms of the corresponding molar flow rates, and (ii) the molar flow rates taking part in the reaction in terms of cell (load) current, and inserted the resulting expressions of $p_{H_2}$, $p_{O_2}$ and $p_{H_{2}O}$ into Equation (6).

Building upon reference [1], other papers (e.g., references [6,7,12,14,15,16,19,20]) obtained essentially the same core model with the following fallacious form of the Nernst equation:

$$E_{Nernst}=E_0+\frac{RT}{2F}\left\{\ln \left(\frac{K_{H_{2}O}}{K_{H_2}}\sqrt{\frac{K_r}{r_{HO}{K}_{O_2}}}\right)+\frac{1}{2}\ln \left(I_{FC}{\left(\frac{1}{u}-1\right)}^2\left(\frac{2}{u}-r_{HO}\right)\right)\right\},$$

(8)

where $K_{H_2}$, $K_{O_2}$, and $K_{H_{2}O}$ are valve molar constants [1]; $K_r=1/\left(4F\right)$ [1,12,13]; $r_{HO}$ is the ratio of the input molar flow rates of hydrogen and oxygen [2,6]; u is the fuel utilization ratio [2,6]; and $I_{FC}$ is the load current [6]. A detailed derivation of Equation (8) is given in the following sub-section.

Derivation of the Fallacious Expression

A derivation of the fallacious form of the Nernst voltage expression (Equation (8)) is presented here. This derivation follows the treatment in references [1,6,12]. A single cell is considered here; the extension of the derivation to a stack is trivially easy.

Assuming perfect gas properties for hydrogen at the anode channel, we have

$$p_{H_2}^{anode}{V}_{anode}=n_{H_2}^{anode}RT,$$

(9)

where $p_{H_2}^{anode}=p_{H_2}^{anode}\left(t\right)$ is the partial pressure of hydrogen in the anode channel; $n_{H_2}^{anode}=n_{H_2}^{anode}\left(t\right)$ is the number of moles of hydrogen in the anode channel; and $V_{anode}$ is the anode volume. Taking the derivative with respect to time, we can express the rate of change of partial pressure in terms of the molar flow rate:

$$\frac{d}{dt}{p}_{H_2}^{anode}\left(t\right)=\frac{RT}{V_{anode}}\left(q_{H_2}^{anode}\left(t\right)\right),$$

(10)

with $q_{H_2}^{anode}\left(t\right)$ defined as

$$q_{H_2}^{anode}\left(t\right)=\frac{d}{dt}{n}_{H_2}^{anode}\left(t\right).$$

(11)

Splitting $q_{H_2}^{anode}\left(t\right)$ into its three components, namely, the input flow rate, the flow rate that takes part in the reaction, and the output flow rate, we have

$$q_{H_2}^{anode}\left(t\right)=q_{H_2}^{in}\left(t\right)-q_{H_2}^{react}\left(t\right)-q_{H_2}^{out}\left(t\right),$$

(12)

and thus (omitting the t for simplicity)

$$\frac{d}{dt}{p}_{H_2}^{anode}=\frac{RT}{V_{anode}}\left(q_{H_2}^{in}-q_{H_2}^{react}-q_{H_2}^{out}\right).$$

(13)

The process of oxygen inflow–reaction–outflow at the cathode channel can be described by

$$\frac{d}{dt}{p}_{O_2}^{cathode}=\frac{RT}{V_{cathode}}\left(q_{O_2}^{in}-q_{O_2}^{react}-q_{O_2}^{out}\right).$$

(14)

In SOFCs, water vapor (steam) forms at the anode:

$$\frac{d}{dt}{p}_{H_{2}O}^{anode}=\frac{RT}{V_{anode}}\left(0+q_{H_{2}O}^{react}-q_{H_{2}O}^{out}\right).$$

(15)

In PEMFCs, water forms at the cathode. Of course, when liquid water is produced in a fuel cell (as in PEMFCs), we have

$$p_{H_{2}O}=1.$$

(16)

Assuming that “the molar flow of any gas through the valve is proportional to its partial pressure inside the channel” [1], we have for an SOFC

$$\frac{q_{H_2}^{out}}{p_{H_2}}=K_{H_2}$$

(17)

and

$$\frac{q_{H_{2}O}^{out}}{p_{H_{2}O}}=K_{H_{2}O}$$

(18)

at the anode and

$$\frac{q_{O_2}^{out}}{p_{O_2}}=K_{O_2}$$

(19)

at the cathode, with $K_{H_2}$, $K_{H_{2}O}$, and $K_{O_2}$ being constants.

From electrochemistry we know that

$$q_{H_2}^{react}=q_{H_{2}O}^{react}=2q_{O_2}^{react}=\frac{I_{FC}}{2F},$$

(20)

where $I_{FC}=I_{FC}\left(t\right)$ is the cell current. Upon the introduction of a constant [1,13],

$$K_r=\frac{1}{4F},$$

(21)

Equation (20) yields

$$q_{H_2}^{react}=q_{H_{2}O}^{react}=2K_r{I}_{FC}$$

(22)

and

$$q_{O_2}^{react}=K_r{I}_{FC}.$$

(23)

Now, inserting Equations (17) and (22) into Equation (13) and taking the Laplace transform (we use the notation that the transform of $x\left(t\right)$ is $\mathcal{X}\left(s\right)$), we get [12]

$$s{\mathcal{P}}_{H_2}\left(s\right)-p_{H_2}\left(0\right)=\frac{RT}{V_{anode}}\left({\mathcal{Q}}_{H_2}^{in}\left(s\right)-2K_r{\mathcal{I}}_{FC}\left(s\right)-K_{H_2}{\mathcal{P}}_{H_2}\left(s\right)\right)$$

(24)

which, upon the substitution of $p_{H_2}\left(0\right)=0$ and

$${\tau }_{H_2}=\frac{V_{anode}}{K_{H_2}RT},$$

(25)

becomes

$${\mathcal{P}}_{H_2}\left(s\right)=\frac{1/K_{H_2}}{1+{\tau }_{H_2}s}\left({\mathcal{Q}}_{H_2}^{in}\left(s\right)-2K_r{\mathcal{I}}_{FC}\left(s\right)\right).$$

(26)

Similarly, for oxygen, Equations (14), (19) and (23) yield

$$s{\mathcal{P}}_{O_2}\left(s\right)-p_{O_2}\left(0\right)=\frac{RT}{V_{cathode}}\left({\mathcal{Q}}_{O_2}^{in}\left(s\right)-K_r{\mathcal{I}}_{FC}\left(s\right)-K_{O_2}{\mathcal{P}}_{O_2}\left(s\right)\right)$$

(27)

which, with the substitution of $p_{O_2}\left(0\right)=0$ and

$${\tau }_{O_2}=\frac{V_{cathode}}{K_{O_2}RT},$$

(28)

gives

$${\mathcal{P}}_{O_2}\left(s\right)=\frac{1/K_{O_2}}{1+{\tau }_{O_2}s}\left({\mathcal{Q}}_{O_2}^{in}\left(s\right)-K_r{\mathcal{I}}_{FC}\left(s\right)\right).$$

(29)

Finally, for steam, we have from Equations (15), (18), and (22)

$${\mathcal{P}}_{H_{2}O}\left(s\right)=\frac{1/K_{H_{2}O}}{1+{\tau }_{H_{2}O}s}2K_r{\mathcal{I}}_{FC}\left(s\right)$$

(30)

where

$${\tau }_{H_{2}O}=\frac{V_{anode}}{K_{H_{2}O}RT}.$$

(31)

(Equations (26), (29) and (30) were used in Figure 2 of reference [1].)

Now, applying the inverse Laplace transform to Equation (26) allows us go from the s-domain back to the time domain:

$$p_{H_2}\left(t\right)=\frac{1}{K_{H_2}{\tau }_{H_2}}exp\left(\frac{-t}{{\tau }_{H_2}}\right){\int }_{\theta =0}^t\left(q_{H_2}^{in}\left(\theta \right)-2K_r{I}_{FC}\left(\theta \right)\right)exp\left(\frac{\theta }{{\tau }_{H_2}}\right)d\theta .$$

(32)

The steady-state partial pressure, then, is obtained from the above equation as

$$p_{H_2,ss}=\frac{1}{K_{H_2}}\left(q_{H_2,ss}^{in}-2K_r{I}_{FC,ss}\right).$$

(33)

(A simpler derivation of Equation (33) is given in Appendix A.1) The steady-state partial pressures of oxygen and steam are obtained similarly:

$$p_{O_2,ss}=\frac{1}{K_{O_2}}\left(q_{O_2,ss}^{in}-K_r{I}_{FC,ss}\right)$$

(34)

and

$$p_{H_{2}O,ss}=\frac{1}{K_{H_{2}O}}2K_r{I}_{FC,ss}.$$

(35)

Now, plugging the above three Equations (33)–(35), as well as the definitions of u and $r_{HO}$ (Equations (A4) and (A5), respectively, from Appendix A.2) into the Nernst EMF expression (Equation (6) of Section 2), we obtain, after some algebra (see Appendix A.2), our familiar Equation (8).

4. Analysis of the Inaccuracy

The terminal (load) voltage is generally obtained by subtracting the following types of losses (or “irreversibilities”) from $E_{Nernst}$:

• activation loss;
• concentration loss;
• ohmic loss; and
• losses due to fuel crossover and internal current.

Equation (8), or one of its many (implicit or explicit) variants, has been the mainstay of a line of fuel cell modeling research for about two decades now (examples abound; a few representative articles are listed in the References section). Not all of the models in references [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] show the explicit form of Equation (8), but they all use some form of this equation, by expressing—directly or indirectly—the Nernst voltage as a function of, among other variables, current.

The problem with Equation (8) is that it mixes equilibrium and non-equilibrium expressions. The Nernst voltage (EMF) is the reversible thermodynamic potential that applies only to the equilibrium condition of the cell; the equilibrium is lost when current is drawn from the cell. In other words, the Nernst voltage is, by definition, the open-circuit EMF and cannot therefore be expressed in terms of the cell (load) current or current density.

That the use of $I_{FC}$ in Equation (8) is questionable can also be seen from the fact that setting

$$I_{FC}=0$$

causes the right side to be mathematically undefined.

The effect of Equation (8) on the output voltage can be seen in the SOFC polarization curve (Figure 1) obtained from the following relationship:

$$V=E_{Nernst}-r{I}_{FC},$$

(36)

where V is the output (load) voltage; $E_{Nernst}$ is given by Equation (8); and r is the ohmic resistance of the cell (cell parameter values, taken from references [1,6,12], are given in

Table 1. Numerical values of parameters and constants.

Parameter	Value
T	1273 K
$E_0$	1.18 V
u	0.8
$K_r$	1/(4F) mol/(s.A)
$K_{H_2}$	0.843 mol/(s.atm)
$K_{H_{2}O}$	0.281 mol/(s.atm)
$K_{O_2}$	2.52 mol/(s.atm)
r	3.28125 × 10${}^{-4}$ $\mathsf{\Omega }$
$r_{HO}$	1.145
n	2
Constants
F	96,485 Coulombs/mol
R	8.31 J/(mol K)

). The upper curve (red) in Figure 1 shows the Nernst voltage, $E_{Nernst}$, computed from Equation (8), while the lower one plots the output voltage, V. Equation (36), like Equation (12) of reference [1] and Equation (6) of reference [6], considers only the ohmic loss out of the four types of losses mentioned earlier.

Figure 1 has the horizontal (current) axis starting at 1 A, not zero, because in Equation (8), the Nernst voltage is undefined at $I_{FC}=0$.

Figure 1 shows an increase in the Nernst voltage with an increasing load current. At first sight, it may not be immediately clear how the model represented by Equation (8) produces the V-values in this figure (it may not be impossible for the polarization characteristics in Figure 1 to be obtained from actual measurements of a physical fuel cell stack under specific operating conditions; much depends on fuel flow, fuel utilization, thermal effects, diffusion, back-diffusion, load current and voltage requirements, and not all complex interactions are fully understood). It can, however, be argued that Equation (8) produces this polarization curve by holding both u and $r_{HO}$ constant to mimic a constant fuel utilization ratio and a constant hydrogen–oxygen ratio of input flow rates. We need to account for the changes in the V-values without having to make $E_{Nernst}$ a function of current.

For a head-to-head comparison of the polarization produced by Equation (8) with the “correct” polarization, we need the “correct” Nernst voltage (that is, the Nernst voltage at open circuit), which, unfortunately, cannot be obtained from Equation (8). (We could, of course, obtain the correct Nernst voltage by assuming reasonable values for $p_{H_2}$, $p_{O_2}$ and $p_{H_{2}O}$ and plugging those values, along with the relevant parameter values from Table 1, into Equation (6), but the resulting Nernst voltage would have no connection to Equation (8).) We circumvent this difficulty by extrapolating the open-circuit Nernst voltage from values produced by Equation (8). Specifically, we use a trivial extrapolation where we obtain the point where the line joining the first two points—(1 A, 0.670472 V) and (2 A, 0.689471 V)—meets the $I_{FC}$ = 0 line. This gives us an extrapolated

$$E_{Nernst}=0.651473\mathrm{V},$$

which is used in Figure 2 to show the “correct” polarization behavior. The upper plot (red) in Figure 2 represents the (constant) Nernst voltage, and the lower one shows the output values from Equation (36) (obviously, both the plots in this figure are straight lines). For ease of comparison with Figure 1, the current axis in Figure 2 starts from 1 A.

5. Discussion

If a non-zero current is to be considered, the proper equation to use is the Butler–Volmer equation, not the Nernst equation. The current–voltage relationship at a non-zero current (away from equilibrium) is perhaps best investigated using reaction kinetic expressions with microscopic reversibility. When modeling specific polarization behaviors, it is important to distinguish between the “reversible” potential and the “irreversibilities”.

The difference between reversible and irreversible potentials and its connection to the Nernst equation are important in the study of solid oxide electrolysis cells (e.g., [32]), too.

In reference [33], the partial pressures of hydrogen and oxygen of a proton exchange membrane fuel cell are expressed (in Equations (2) and (3) of that paper [33]) in terms of current, but before these partial pressures are plugged into the Nernst equation Equation (1) of that paper [33]) to obtain the reversible potential, the current, correctly, is set to zero. Reference [33] also reported using non-zero values of current in the Nernst equation.

Reference [12] used Equation (8) to generate both training and test data, and therefore, none of the conclusions in that paper are invalidated by the inaccuracy discussed here.

6. Conclusions

An inaccuracy in the fuel cell modeling literature involving the nature of the reversible thermodynamic potential was identified in this paper. An analysis of the inaccuracy was provided and its significance explained. To summarize, there is nothing wrong with the Nernst equation that provides the reversible (equilibrium) voltage of a fuel cell; what is wrong is the mixing up of equilibrium and non-equilibrium conditions by inserting cell (load) current into the Nernst voltage expression. It can be argued that expressing the Nernst voltage as a function of current is a “quick-and-dirty” trick that makes modeling easy for us. However, by taking such a shortcut, we compromise scientific rigor for the sake of practical convenience.