Thermodynamically Rigorous Description of the Open Circuit Voltage of Redox Flow Batteries

Overview

AVRFB

In the case of the AVRFB, the positive electrolyte solution tank contains ${\mathrm{VO}}_{2}^{+}$ and VO²⁺ ions and the negative electrolyte solution tank contains V³⁺ and V²⁺ ions, which corresponds to vanadium in the oxidation states V, IV, III, and II. In addition, a supporting acid, which is typically sulfuric acid for the AVRFB, is added to increase the solubility of the salts and the conductivity of the electrolyte solution. The chemical reactions on the electrodes of the two half-cells and their potential vs the standard hydrogen electrode (SHE) can be expressed as Ref. 37

$$\begin{eqnarray}&&\begin{array}{r}{{\mathrm{VO}}_{2}}^{+}+2{{\rm{H}}}^{+}+{{\rm{e}}}^{-}\displaystyle \underset{\mathrm{charge}}{\overset{\mathrm{discharge}}{\rightleftharpoons }}{\mathrm{VO}}^{2+}+{{\rm{H}}}_{2}{\rm{O}}\\ \,{E}_{\mathrm{pos}}^{0}\,=\,1.000\ {\rm{V}}\ \mathrm{vs}\ \mathrm{SHE}\end{array}\end{eqnarray} \tag{ I }$$

$$\begin{eqnarray}&&\begin{array}{r}{{\rm{V}}}^{3+}+{{\rm{e}}}^{-}\displaystyle \underset{\mathrm{discharge}}{\overset{\mathrm{charge}}{\rightleftharpoons }}{{\rm{V}}}^{2+}\\ \,{E}_{\mathrm{neg}}^{0}\,=\,-0.256\ {\rm{V}}\ \mathrm{vs}\ \mathrm{SHE}.\end{array}\end{eqnarray} \tag{ II }$$

Combining the reactions (I) and (II) leads to the total cell reaction of the AVRFB:

$$\begin{eqnarray}&&\begin{array}{l}{\mathrm{VO}}_{2\ \mathrm{pos}}^{+}+2{{\rm{H}}}_{\mathrm{pos}}^{+}+{{\rm{V}}}_{\mathrm{neg}}^{2+}\displaystyle \underset{\mathrm{charge}}{\overset{\mathrm{discharge}}{\rightleftharpoons }}{\mathrm{VO}}_{\mathrm{pos}}^{2+}+{{\rm{H}}}_{2}{{\rm{O}}}_{\mathrm{pos}}+{{\rm{V}}}_{\mathrm{neg}}^{3+}\\ \,{E}_{\mathrm{cell}}^{0}\,=\,1.256\ {\rm{V}}\ \mathrm{vs}\ \mathrm{SHE}.\end{array}\end{eqnarray} \tag{ III }$$

Fe/Cd RFB

In addition to the established RFBs, the combination of different half-cell reactions and variations in the composition of the electrolyte solutions result in a very large number of other possible RFB designs. A new promising candidate is the Fe/Cd RFB developed by Zeng et al.,³⁸ which showed a small capacity reduction during the cycle test and low predicted costs for energy storage. In contrast to the AVRFB, the same initial electrolyte solution, which is an aqueous solution of FeCl₂ and CdCl₂ with HCl as the supporting acid, is typically used in both half-cells. The cell is equipped with a cation exchange membrane through which only H⁺ can diffuse. The Fe/Cd RFB is a hybrid RFB, i.e. a solid is formed during charging: at the negative electrode, cadmium metal is dissolved and redeposited. The chemical reactions occurring in the half-cells of the Fe/Cd RFB can be expressed as Ref. 38

$$\begin{eqnarray}&&\begin{array}{l}2{\mathrm{Fe}}^{3+}+2{{\rm{e}}}^{-}\displaystyle \underset{\mathrm{charge}}{\overset{\mathrm{discharge}}{\rightleftharpoons }}2{\mathrm{Fe}}^{2+}\\ \,{E}_{\mathrm{pos}}^{0}\,=\,0.77\ {\rm{V}}\ \mathrm{vs}\ \mathrm{SHE}\end{array}\end{eqnarray} \tag{ IV }$$

$$\begin{eqnarray}&&\begin{array}{l}{\mathrm{Cd}}^{2+}+2{{\rm{e}}}^{-}\displaystyle \underset{\mathrm{discharge}}{\overset{\mathrm{charge}}{\rightleftharpoons }}\mathrm{Cd}\\ \,{E}_{\mathrm{neg}}^{0}\,=\,-0.40\ {\rm{V}}\ \mathrm{vs}\ \mathrm{SHE}.\end{array}\end{eqnarray} \tag{ V }$$

Combining the reactions (IV) and (V) leads to the total cell reaction of the Fe/Cd RFB:

$$\begin{eqnarray}&&\begin{array}{l}2{\mathrm{Fe}}_{\mathrm{pos}}^{3+}+{\mathrm{Cd}}_{\mathrm{neg}}\displaystyle \underset{\mathrm{charge}}{\overset{\mathrm{discharge}}{\rightleftharpoons }}2{\mathrm{Fe}}_{\mathrm{pos}}^{2+}+{\mathrm{Cd}}_{\mathrm{neg}}^{2+}\\ \,{E}_{\mathrm{cell}}^{0}\,=\,1.17\ {\rm{V}}\ \mathrm{vs}\ \mathrm{SHE}.\end{array}\end{eqnarray} \tag{ VI }$$

Overview

The properties of RFBs are usually described as a function of the SOC. Throughout the present work, the SOC is defined as

$$\begin{eqnarray}&&\mathrm{SOC}=\displaystyle \frac{{b}_{\mathrm{Me},\mathrm{ox}}^{\mathrm{pos}}}{{b}_{\mathrm{Me},\mathrm{ox}}^{\mathrm{pos}}+{b}_{\mathrm{Me},\mathrm{red}}^{\mathrm{pos}}}=\displaystyle \frac{{b}_{\mathrm{Me},\mathrm{ox}}^{\mathrm{pos}}}{{b}_{\mathrm{Me},\mathrm{tot}}^{\mathrm{pos}}},\end{eqnarray} \tag{ 5 }$$

where ${b}_{\mathrm{Me},\mathrm{ox}}^{\mathrm{pos}}$ represents the molality of the active metal species (Me) in the positive half-cell (pos) in oxidized form and ${b}_{\mathrm{Me},\mathrm{red}}^{\mathrm{pos}}$ in reduced form. Hence, an SOC of 1 corresponds to a fully charged RFB, and an SOC of 0 corresponds to a fully discharged RFB. It is also possible to define the SOC with respect to the molalities in the negative half-cell, which leads to an expression equivalent to Eq. 5. In the literature, sometimes molarities are used to define the SOC instead of molalities.^18,20

Furthermore, for both RFBs considered in this work, the following mass balances, which can be derived from Eq. 5, are applicable:

$$\begin{eqnarray}&&{b}_{\mathrm{Me},\mathrm{ox}}^{\mathrm{pos}}=\mathrm{SOC}\cdot {b}_{\mathrm{Me},\mathrm{tot}}^{\mathrm{pos}}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{b}_{\mathrm{Me},\mathrm{red}}^{\mathrm{pos}}=(1-\mathrm{SOC})\cdot {b}_{\mathrm{Me},\mathrm{tot}}^{\mathrm{pos}}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{b}_{\mathrm{Me},\mathrm{ox}}^{\mathrm{neg}}=(1-\mathrm{SOC})\cdot {b}_{\mathrm{Me},\mathrm{tot}}^{\mathrm{neg}}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{b}_{\mathrm{Me},\mathrm{red}}^{\mathrm{neg}}=\mathrm{SOC}\cdot {b}_{\mathrm{Me},\mathrm{tot}}^{\mathrm{neg}}.\end{eqnarray} \tag{ 9 }$$

An additional necessary assumption for the derivation of Eqs. 8 and 9 is

$$\begin{eqnarray}&&{n}_{\mathrm{Me},\mathrm{tot}}^{\mathrm{pos}}\cdot {z}_{\mathrm{pos}}={n}_{\mathrm{Me},\mathrm{tot}}^{\mathrm{neg}}\cdot {z}_{\mathrm{neg}},\end{eqnarray} \tag{ 10 }$$

where ${n}_{\mathrm{Me},\mathrm{tot}}^{\mathrm{pos}}$ and ${n}_{\mathrm{Me},\mathrm{tot}}^{\mathrm{neg}}$ are the total number of moles of the active metal species in the positive and negative half-cells. Furthermore, ${z}_{\mathrm{pos}}$ and ${z}_{\mathrm{neg}}$ are the numbers of electrons transferred per mole of metal species in the positive and negative half-reaction, respectively. Equation 10 is valid for all the compositions of the electrolyte solutions described in this work (see Table I) and ensures that the capacity of both half-cells is the same. Therefore, at an SOC of 0 and 1, only one species of the corresponding redox couple is present in both the positive and negative half-cells. In the case of the two RFBs considered in this work, Eq. 10 leads to

$$\begin{eqnarray}&&{n}_{{\rm{V}},\mathrm{tot}}^{\mathrm{pos}}={n}_{{\rm{V}},\mathrm{tot}}^{\mathrm{neg}}\end{eqnarray} \tag{ 11 }$$

for the AVRFB and

$$\begin{eqnarray}&&{n}_{\mathrm{Fe},\mathrm{tot}}^{\mathrm{pos}}=2\cdot {n}_{\mathrm{Cd},\mathrm{tot}}^{\mathrm{neg}}\end{eqnarray} \tag{ 12 }$$

for the Fe/Cd RFB.

Table I.  Overview of scenarios studied in the present work.

#	RFB type	Overall composition	Membrane	Pitzer parameters	Objective
1	Fe/Cd RFB	${\tilde{c}}_{{\mathrm{FeCl}}_{2}}^{0}$ = 1.0 mol l−1	catex	complete	Simulation of ${E}_{\mathrm{cell},\mathrm{real}}^{\mathrm{rev}}$ and ${E}_{\mathrm{cell},\mathrm{ideal}}^{\mathrm{rev}}$
		${\tilde{c}}_{{\mathrm{CdCl}}_{2}}^{0}$ = 0.5 mol l−1			with the developed model
		${\tilde{c}}_{\mathrm{HCl}}^{0}$ = 3.0 mol l−1
2a	AVRFB	${\tilde{c}}_{{\mathrm{VOSO}}_{4}}^{0,\mathrm{pos}}$ = 2.0 mol l−1	catex	unavailable;	Simulation of ${E}_{\mathrm{cell},\mathrm{ideal}}^{\mathrm{rev}}$ and
		${\tilde{c}}_{{{\rm{H}}}_{2}{\mathrm{SO}}_{4}}^{0,\mathrm{pos}}$ = 3.0 mol l−1		no activity based	comparison with measured ${E}_{\mathrm{cell},\mathrm{real}}^{\mathrm{rev}}$ data37
		${\tilde{c}}_{{{\rm{V}}}_{2}{({\mathrm{SO}}_{4})}_{3}}^{0,\mathrm{neg}}$ = 1.0 mol l−1		model used
		${\tilde{c}}_{{{\rm{H}}}_{2}{\mathrm{SO}}_{4}}^{0,\mathrm{neg}}$ = 2.0 mol l−1
2b	AVRFB	${\tilde{c}}_{{\mathrm{VOSO}}_{4}}^{0,\mathrm{pos}}$ = 1.6 mol l−1	anex	unavailable;	Simulation of ${E}_{\mathrm{cell},\mathrm{ideal}}^{\mathrm{rev}}$ and
		${\tilde{c}}_{{{\rm{H}}}_{2}{\mathrm{SO}}_{4}}^{0,\mathrm{pos}}$ = 3.2 mol l−1		no activity based	comparison with measured ${E}_{\mathrm{cell},\mathrm{real}}^{\mathrm{rev}}$ data37
		${\tilde{c}}_{{{\rm{V}}}_{2}{({\mathrm{SO}}_{4})}_{3}}^{0,\mathrm{neg}}$ = 0.8 mol l−1		model used
		${\tilde{c}}_{{{\rm{H}}}_{2}{\mathrm{SO}}_{4}}^{0,\mathrm{neg}}$ = 0.8 mol l−1
3	Fe/Cd RFB	varied	catex	complete	Sensitivity analysis
4	Fe/Cd RFB	${\tilde{c}}_{{\mathrm{FeCl}}_{2}}^{0}$ = 1.0 mol l−1	catex	complete	Calculation of voltage losses using
		${\tilde{c}}_{{\mathrm{CdSO}}_{4}}^{0}$ = 0.5 mol l−1		(except $\mathrm{Cd}{({\mathrm{HSO}}_{4})}_{2}$)	measured Ecell data38
		${\tilde{c}}_{\mathrm{HCl}}^{0}$ = 3.0 mol l−1

All salts are assumed to dissociate completely. Furthermore, it is assumed that hydrochloric acid dissociates completely according to

$$\begin{eqnarray}&&\mathrm{HCl}\to {{\rm{H}}}^{+}+{\mathrm{Cl}}^{-}.\end{eqnarray} \tag{ VII }$$

In the case of sulfuric acid, only the first dissociation step is considered to be complete, while the incomplete dissociation of bisulfate is captured via its dissociation equilibrium constant:

$$\begin{eqnarray}&&{{\rm{H}}}_{2}{\mathrm{SO}}_{4}\to {{\rm{H}}}^{+}\ +\ {{\mathrm{HSO}}_{4}}^{-}\end{eqnarray} \tag{ VIII }$$

$$\begin{eqnarray}&&{{\mathrm{HSO}}_{4}}^{-}\rightleftharpoons {{\rm{H}}}^{+}+{{\mathrm{SO}}_{4}}^{2-}\end{eqnarray} \tag{ IX }$$

The thermodynamic equilibrium constant of reaction (IX) (K_a = 1. 05 · 10⁻² at 298.15 K and 1 bar) is taken from the literature.³⁹ More details on the calculation of this dissociation equilibrium are provided in the supplementary material.

AVRFB

In case of the AVRFB, the SOC (see Eq. 5) is defined as

$$\begin{eqnarray}&&\mathrm{SOC}=\displaystyle \frac{{b}_{{\mathrm{VO}}_{2}^{+}}^{\mathrm{pos}}}{{b}_{{\mathrm{VO}}_{2}^{+}}^{\mathrm{pos}}+{b}_{{\mathrm{VO}}^{2+}}^{\mathrm{pos}}}=\displaystyle \frac{{b}_{{\mathrm{VO}}_{2}^{+}}^{\mathrm{pos}}}{{b}_{{\rm{V}},\mathrm{tot}}^{\mathrm{pos}}}.\end{eqnarray} \tag{ 13 }$$

By applying Eqs. 6–9 to an AVRFB with cation exchange membrane, in which protons diffuse freely between the two half-cells, Eqs. 14–17 are obtained.

$$\begin{eqnarray}&&{b}_{{\mathrm{VO}}_{2}^{+}}^{\mathrm{pos}}=\mathrm{SOC}\cdot {b}_{{\rm{V}},\mathrm{tot}}^{\mathrm{pos}}\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{b}_{{\mathrm{VO}}^{2+}}^{\mathrm{pos}}=(1-\mathrm{SOC})\cdot {b}_{{\rm{V}},\mathrm{tot}}^{\mathrm{pos}}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{b}_{{{\rm{V}}}^{3+}}^{\mathrm{neg}}=(1-\mathrm{SOC})\cdot {b}_{{\rm{V}},\mathrm{tot}}^{\mathrm{neg}}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{b}_{{{\rm{V}}}^{2+}}^{\mathrm{neg}}=\mathrm{SOC}\cdot {b}_{{\rm{V}},\mathrm{tot}}^{\mathrm{neg}}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{b}_{{{\rm{H}}}^{+},\mathrm{add}}^{\mathrm{pos}}=+{b}_{{\mathrm{VO}}_{2}^{+}}^{\mathrm{pos}}\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{b}_{{{\rm{H}}}^{+},\mathrm{add}}^{\mathrm{neg}}=+{b}_{{{\rm{V}}}^{2+}}^{\mathrm{neg}}.\end{eqnarray} \tag{ 19 }$$

In addition to the metal ions, the molality of the protons in the two half-cells is crucial for the calculations. During operation of the RFB, this molality varies and is influenced by ${b}_{{{\rm{H}}}^{+},\mathrm{add}}$, which represents the amount of additional protons due to redox reactions and diffusion through the membrane. In the positive half-cell, two moles of protons are formed per mole of ${\mathrm{VO}}_{2}^{+}$ during charging (see reaction (I)). At the same time, one mole of protons diffuses through the cation exchange membrane into the negative half-cell. Here, ${b}_{{{\rm{H}}}^{+},\mathrm{add}}$ is defined relative to a fully discharged RFB (SOC = 0). Of course, the actual proton concentration also depends on the overall acid concentration and the dissociation equilibrium of the corresponding acid (see reactions (VII)–(IX)).

Water is formed in the positive half-cell of the AVRFB during discharge. The mass balance of water in this case can be expressed as

$$\begin{eqnarray}&&{m}_{{\rm{W}}}^{\mathrm{pos}}={m}_{{\rm{W}},\mathrm{SOC}=0}^{\mathrm{pos}}-{b}_{{\mathrm{VO}}_{2}^{+}}^{\mathrm{pos}}\cdot {m}_{{\rm{W}}}^{\mathrm{pos}}\cdot {M}_{{\rm{W}}},\end{eqnarray} \tag{ 20 }$$

where ${m}_{{\rm{W}},\mathrm{SOC}=0}^{\mathrm{pos}}$ is the mass of water in the positive half-cell at an SOC of 0 and M_W is the molar mass of water. Inserting Eq. 14 into Eq. 20 and rearranging yields

$$\begin{eqnarray}&&{m}_{{\rm{W}}}^{\mathrm{pos}}=\displaystyle \frac{{m}_{{\rm{W}},\mathrm{SOC}=0}^{\mathrm{pos}}}{1+\mathrm{SOC}\cdot {b}_{{\rm{V}},\mathrm{tot}}^{\mathrm{pos}}\cdot {M}_{{\rm{W}}}}.\end{eqnarray} \tag{ 21 }$$

In the case of an AVRFB setup with an anion exchange membrane, ${\mathrm{HSO}}_{4}^{-}$ ions instead of protons can diffuse freely between the two half-cells. Therefore, in this case, Eqs. 18 and 19 have to be replaced by the following relationships:

$$\begin{eqnarray}&&{b}_{{{\rm{H}}}^{+},\mathrm{add}}^{\mathrm{pos}}=+2\ {b}_{{\mathrm{VO}}_{2}^{+}}^{\mathrm{pos}}\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{b}_{{\mathrm{HSO}}_{4}^{-},\mathrm{add}}^{\mathrm{pos}}=+{b}_{{\mathrm{VO}}_{2}^{+}}^{\mathrm{pos}}\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{b}_{{\mathrm{HSO}}_{4}^{-},\mathrm{add}}^{\mathrm{neg}}=-{b}_{{{\rm{V}}}^{2+}}^{\mathrm{neg}}.\end{eqnarray} \tag{ 24 }$$

Fe/Cd RFB

In case of the Fe/Cd RFB, the SOC is defined as

$$\begin{eqnarray}&&\mathrm{SOC}=\displaystyle \frac{{b}_{{\mathrm{Fe}}^{3+}}^{\mathrm{pos}}}{{b}_{{\mathrm{Fe}}^{3+}}^{\mathrm{pos}}+{b}_{{\mathrm{Fe}}^{2+}}^{\mathrm{pos}}}=\displaystyle \frac{{b}_{{\mathrm{Fe}}^{3+}}^{\mathrm{pos}}}{{b}_{\mathrm{Fe},\mathrm{tot}}^{\mathrm{pos}}}.\end{eqnarray} \tag{ 25 }$$

By applying Eqs. 6–9 to an Fe/Cd RFB with cation exchange membrane, the following mass balances are obtained:

$$\begin{eqnarray}&&{b}_{{\mathrm{Fe}}^{3+}}^{\mathrm{pos}}=\mathrm{SOC}\cdot {b}_{\mathrm{Fe},\mathrm{tot}}^{\mathrm{pos}}\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{b}_{{\mathrm{Fe}}^{2+}}^{\mathrm{pos}}=(1-\mathrm{SOC})\cdot {b}_{\mathrm{Fe},\mathrm{tot}}^{\mathrm{pos}}\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&{b}_{{\mathrm{Cd}}^{2+}}^{\mathrm{neg}}=(1-\mathrm{SOC})\cdot {b}_{\mathrm{Cd},\mathrm{tot}}^{\mathrm{neg}}\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&{b}_{\mathrm{Cd}}^{\mathrm{neg}}=\mathrm{SOC}\cdot {b}_{\mathrm{Cd},\mathrm{tot}}^{\mathrm{neg}}\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&{b}_{{{\rm{H}}}^{+},\mathrm{add}}^{\mathrm{pos}}=-{b}_{{\mathrm{Fe}}^{3+}}^{\mathrm{pos}}\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&{b}_{{{\rm{H}}}^{+},\mathrm{add}}^{\mathrm{neg}}=2\ {b}_{\mathrm{Cd}}^{\mathrm{neg}}.\end{eqnarray} \tag{ 31 }$$

Since diffusion of water across the membrane is neglected, a constant amount of solvent can be assumed for the Fe/Cd RFB with the result that ${b}_{\mathrm{Fe},\mathrm{tot}}^{\mathrm{pos}}$ and ${b}_{\mathrm{Cd},\mathrm{tot}}^{\mathrm{neg}}$ always equal their initial values. This assumption is not valid for the AVRFB due to the formation of water at the positive half-cell during discharge.

For the Fe/Cd RFB, the Nernst equation was derived in the present work (see Appendix) and is given by

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\mathrm{cell},\mathrm{real}}^{\mathrm{rev}} & = & {E}_{\mathrm{cell}}^{0}+\displaystyle \frac{{RT}}{2F}\mathrm{ln}\left(\displaystyle \frac{{\left({a}_{{\mathrm{Fe}}^{3+}}^{\mathrm{pos}}\right)}^{2}{\left({a}_{{{\rm{H}}}^{+}}^{\mathrm{neg}}\right)}^{2}{a}_{\mathrm{Cd}}^{\mathrm{neg}}}{{\left({a}_{{\mathrm{Fe}}^{2+}}^{\mathrm{pos}}\right)}^{2}{\left({a}_{{{\rm{H}}}^{+}}^{\mathrm{pos}}\right)}^{2}{a}_{{\mathrm{Cd}}^{2+}}^{\mathrm{neg}}}\right)\ \\ \ \mathrm{with}\ {E}_{\mathrm{cell}}^{0} & = & 1.17\ {\rm{V}}.\end{array}\end{eqnarray} \tag{ 32 }$$

For the AVRFB, the Nernst equation was derived by Pavelka et al.³⁷ In case of a cation exchange membrane, it is given by

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\mathrm{cell},\mathrm{real}}^{\mathrm{rev}} & = & {E}_{\mathrm{cell}}^{0}+\displaystyle \frac{{RT}}{F}\mathrm{ln}\left(\displaystyle \frac{{a}_{{\mathrm{VO}}_{2}^{+}}^{\mathrm{pos}}\ {a}_{{{\rm{V}}}^{2+}}^{\mathrm{neg}}\ {a}_{{{\rm{H}}}^{+}}^{\mathrm{pos}}\ {a}_{{{\rm{H}}}^{+}}^{\mathrm{neg}}}{{a}_{{\mathrm{VO}}^{2+}}^{\mathrm{pos}}\ {a}_{{{\rm{V}}}^{3+}}^{\mathrm{neg}}\ {a}_{{{\rm{H}}}_{2}{\rm{O}}}^{\mathrm{pos}}}\right)\ \\ \ \mathrm{with}\ {E}_{\mathrm{cell}}^{0} & = & 1.256\ {\rm{V}}.\end{array}\end{eqnarray} \tag{ 33 }$$

If an anion exchange membrane is used in an AVRFB instead of the cation exchange membrane, Eq. 33 is replaced by

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\mathrm{cell},\mathrm{real}}^{\mathrm{rev}} & = & {E}_{\mathrm{cell}}^{0}+\displaystyle \frac{{RT}}{F}\mathrm{ln}\left(\displaystyle \frac{{a}_{{\mathrm{VO}}_{2}^{+}}^{\mathrm{pos}}\ {a}_{{{\rm{V}}}^{2+}}^{\mathrm{neg}}\ {\left({a}_{{{\rm{H}}}^{+}}^{\mathrm{pos}}\right)}^{2}\ {a}_{{\mathrm{HSO}}_{4}^{-}}^{\mathrm{pos}}}{{a}_{{\mathrm{VO}}^{2+}}^{\mathrm{pos}}\ {a}_{{{\rm{V}}}^{3+}}^{\mathrm{neg}}\ {a}_{{{\rm{H}}}_{2}{\rm{O}}}^{\mathrm{pos}}\ {a}_{{\mathrm{HSO}}_{4}^{-}}^{\mathrm{neg}}}\right)\ \\ \ \mathrm{with}\ {E}_{\mathrm{cell}}^{0} & = & 1.256\ {\rm{V}}.\end{array}\end{eqnarray} \tag{ 34 }$$

The investigated RFBs employ electrolyte solutions with high concentrations of salts and acids. A suitable thermodynamic model must be chosen for describing these solutions appropriately. In the present work, the Pitzer model is applied, which was implemented as suggested by Bea et al.⁴⁰ In the literature, there is a plethora of models for activity coefficients in electrolyte solutions, which differ in their complexity and the number of adjustable parameters. In this work, the Pitzer model^41–45 is used to describe the activity coefficients, as it was developed especially for aqueous multicomponent solutions and has proven to yield reliable predictions at high ionic strengths.⁴⁶ All necessary model parameters for the simulations conducted in this work are taken from the literature and listed in the supplementary material. When selecting the parameters, care was taken to ensure that the concentrations used in the RFBs do not exceed the maximum molality of the parameterization data. Typically, Pitzer parameters are only available for room temperature (25 °C), which must be taken into account.

The composition of the electrolyte solution is frequently reported in molarities. However, since the Pitzer model is formulated in molalities, a conversion of the concentrations is necessary. For this purpose, overall molarities were converted into molalities using the formulae and tabulated data given in the supplementary material.

In scenario #1, a Fe/Cd RFB with chloride salts only was studied, see Table I. Figure 1a shows the OCV curves for the ideal (${\gamma }_{i}^{b* }=1$) and real case (${\gamma }_{i}^{b* }$ calculated with the Pitzer model) as functions of the SOC.

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As could be expected from the high concentrations of ionic species, a significant difference between the two curves corresponding to the ideal and the real case is found. The observed difference in the OCVs is solely caused by the influence of the activity coefficients, i.e. by the excess OCV E_ex, which can be calculated using Eq. 3 and which is plotted in Fig. 1b. Therefore, activity coefficients should not be neglected in the Nernst equation.

When modeling the OCV of RFBs, it is often assumed that the ratio of activity coefficients and thus also E_ex remains constant over the SOC, so that this contribution may simply be viewed as an additional additive contribution to the OCV, the so-called "formal potential term".^33–35 The present results show that this is not the case. By contrast, the excess OCV depends linearly on the SOC, with a coefficient of determination of about R² = 0.99998. In the supplementary material, a possible reason for the linear trend of the excess OCV vs the SOC in the case of the Fe/Cd RFB is presented. There, the natural logarithm of all species' activity coefficients is shown as a function of the SOC. For most species, that property depends approximately linearly on the SOC. Consequently, when summing up these terms in the Nernst equation that linearity is preserved. However, we can only investigate this thoroughly for the Fe/Cd RFB, so that we cannot make any predictions whether this is only a peculiarity of the Fe/Cd RFB or a generic feature of RFBs.

Moreover, E_ex varies between 0.038 V (SOC $\to$ 0) and −0.014 V (SOC $\to$ 1). Thus, it is in the same order of magnitude as the voltage losses during operation, which are discussed in more detail below.

For the AVRFB, no Pitzer parameters are available. Thus, the calculation presented in the previous subsection cannot be simply repeated for the AVRFB. However, measurement data of the OCV are available,³⁷ which can be used to estimate the excess OCV.

In the experiments conducted by Pavelka et al.,³⁷ two AVRFB setups were investigated: one with a catex and one with an anex membrane. Both setups are considered here, see scenarios #2a and #2b in Table I, using the reported overall compositions by Pavelka et al.³⁷

Figures 2a and 3a show the measured OCV data by Pavelka et al.³⁷ in comparison to the ideal OCV calculated with the present model. Additionally, the simulation results of the ideal OCV of Pavelka et al. are shown. Their calculations differ from the ones presented here in terms of the mass balances and the dissociation equilibrium of sulfuric acid, for which Pavelka et al. made additional assumptions. In the present work, the dissociation equilibrium is treated in a thermodynamically rigorous way. Furthermore, Figs. 2b and 3b show the calculated excess OCV (see Eq. 3), which is the difference between the measured (real) OCV and the simulated (ideal) OCV illustrated in Figs. 2a and 3a.

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As illustrated in Figs. 2 and 3, the simulated ideal OCVs for the setup with a catex membrane deviate significantly from the measured data, while in the case of the setup with an anex membrane, the differences are moderate, but still important. The maximum excess OCV is 0.28 V (SOC $\to$ 0) for the setup with the catex membrane and 0.08 V (SOC $\to$ 0) for the setup with the anex membrane, which is significantly higher than for the Fe/Cd RFB discussed in the previous subsection. Furthermore, the changes of the excess OCV over the SOC shown in Figs. 2b and 3b are larger than in the simulation results of the Fe/Cd RFB.

In this work, the dissociation equilibrium is treated in a thermodynamically rigorous way, but since the activity coefficients cannot be calculated due to a lack of Pitzer parameters, some uncertainties as to the reliability of the calculations remain and it is not possible to estimate the accuracy of the simulation results. It can only be noted that the developed model yields lower ideal OCVs for both simulated setups and thus larger excess OCVs than the model of Pavelka et al.³⁷ Thus, the treatment of the dissociation equilibrium of sulfuric acid has a crucial influence on the OCV. Nevertheless, both sets of calculations, i.e. those of Pavelka et al. and those of the present work, indicate that the assumption of a constant excess OCV E_ex^32,34 is not valid.

Evaluation of the regression lines obtained with the present model support the assumption that the dependence of E_ex on the OCV could be described more accurately by a straight line, which validates the findings for the Fe/Cd RFB in the previous subsection.

In this subsection, an optimization of the composition of the electrolyte solution toward high OCVs as described in scenario #3 in Table I is discussed. In order to investigate the influence of activity coefficients on the results, the ideal and real OCV plots are compared. The contour plots shown in Fig. 4 summarize the results of the variation of the overall salt molarities and the variation of the overall HCl molarity. There, the OCV at an SOC of 0.5 is shown as an example, which is, however, representative for most of the entire SOC range (except for very high SOC).

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For both the ideal and the real case, the model predicts that lower overall salt molarities lead to a higher OCV. In contrast, the variation of the HCl molarity shows different trends for the ideal and real cases: while the OCV rises with increasing HCl molarity in the ideal case, the real case predicts the opposite (except for very low acid molarities at high SOC). Hence, neglecting the activity coefficients is misleading when optimizing the OCV by varying the electrolyte composition: looking at the ideal case suggests lowering the acid concentration is beneficial, while in reality, the opposite is true.

In Fig. 5, the results already shown in Fig. 4 are depicted in a different way to show the difference of the OCV between the typical operating point (cross) and other operating conditions: there, the same color scale is used for both the ideal and real cases. That color scale is set to zero for the typical operating point of the Fe/Cd RFB, see scenario #1 in Table I. Figure 5 illustrates which overall compositions of the electrolyte solutions could enhance the OCV and shows the absolute difference to typical concentrations used in Fe/Cd RFBs.³⁸ The influence of the variation is assumed to be much smaller in the ideal case than it is in the real case. Hence, the full optimization potential of the RFB is underestimated if only the ideal part of the Nernst equation is taken into account.

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RFBs during operation are considered in the following as described in scenario #4 in Table I, in contrast to the currentless state of equilibrium discussed above. Measurement data provided by Zeng et al.³⁸ are compared with calculated OCVs to estimate the occurring voltage losses. The measurement data of Zeng et al.³⁸ were carried out with a Fe/Cd RFB with chloride and sulfate salts. For this system, one set of Pitzer parameters, namely those of the Cd²⁺-${\mathrm{HSO}}_{4}^{-}$ interaction, are not available. In principle, this can lead to inaccuracies in modeling the activity coefficients. However, as the concentration of Cl⁻ ions in the investigated RFB is typically 10 times larger than that of the ${\mathrm{SO}}_{4}^{2-}$ and ${\mathrm{HSO}}_{4}^{-}$ ions, the impact of the missing parameters is presumably rather small. Furthermore, the binary parameters of all other combinations of anions and cations of the described electrolyte solution are available.

Zeng et al.³⁸ reported measured cell voltage data during charging of a Fe/Cd RFB as a function of time for four different current densities. Furthermore, the electrolyte utilization at the cutoff voltage (1.35 V) is given for each case. Explanations for the relatively low electrolyte utilization are already provided by Zeng et al.³⁸ and are not discussed here. Based on the reported data, the cell voltage curves as a function of the SOC (see Eq. 25) were derived. In terms of the SOC definition used in the present work, the charging experiments by Zeng et al. start at SOC = 0. The SOC at the end of the experiment can be determined from the electrolyte utilization provided by Zeng et al.,³⁸ in the following we discuss only the current densities 40 mA cm⁻² and 160 mA cm⁻².

The data of the charge process provided by Zeng et al.³⁸ show that the measured cell voltage rises with increasing current density, which is due to current-dependent voltage losses during charging of the RFB. As outlined in the introduction, the voltage losses are calculated as

$$\begin{eqnarray}&&{E}_{\mathrm{losses},\mathrm{real}}={E}_{\mathrm{cell}}-{E}_{\mathrm{cell},\mathrm{real}}^{\mathrm{rev}},\end{eqnarray} \tag{ 35 }$$

where E_losses,real is the true sum of all voltage losses, i.e. considering the activity coefficients when calculating the OCV. Hence, inaccuracies in the calculation of the OCV, such as the neglect of activity coefficients, lead to errors in the calculation of voltage losses. To distinguish the two approaches, the term E_losses,ideal is introduced in the following as

$$\begin{eqnarray}&&{E}_{\mathrm{losses},\mathrm{ideal}}={E}_{\mathrm{cell}}-{E}_{\mathrm{cell},\mathrm{ideal}}^{\mathrm{rev}}\end{eqnarray} \tag{ 36 }$$

to indicate the losses that one would expect when calculating the OCV without activity coefficients. Combining Eqs. 4, 35 and 36 yields

$$\begin{eqnarray}&&{E}_{\mathrm{losses},\mathrm{ideal}}={E}_{\mathrm{losses},\mathrm{real}}+{E}_{\mathrm{ex}}.\end{eqnarray} \tag{ 37 }$$

Equation 37 clearly shows that the neglect of the excess OCV leads to erroneous estimations of the voltage losses.

Figures 6a and 7a illustrate the comparison of the simulated ideal and real OCV with the measurement data of Zeng et al.³⁸ for the two applied current densities. Furthermore, Figs. 6b and 7b show the ideal and real voltage losses calculated from the data in Figs. 6a and 7a, as well as the excess OCV.

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The excess OCV describes the equilibrium state and is therefore only a function of the SOC, but does not depend on the applied current density. At an applied current density of 40 mA cm⁻², the calculated voltage losses are small and in the order of magnitude of the excess OCV. Hence, when calculating voltage losses with respect to ${E}_{\mathrm{cell},\mathrm{ideal}}^{\mathrm{rev}}$, an appreciable error can be made: the voltage losses can be under- or overestimated to a large extent. Higher applied current densities lead to increasing voltage losses caused by overpotentials and ohmic losses, so that the relative difference between ideal and real voltage losses decreases. Therefore, the thermodynamically rigorous calculation of the OCV is especially important when studying voltage losses in RFB setups at low current densities.

In the literature, the experimental data of ideal voltage losses of lab-scale RFB setups are often used to adjust model parameters, e.g. the kinetic parameters in the Butler-Volmer equations for the calculation of activation overpotentials.^{17,22,23,28,49,50} However, the present results indicate that the ideal voltage losses under- or overestimate the real voltage losses. As a result, the obtained fit parameters have to effectively capture phenomena that should rather be included in the description of the equilibrium state. Both the voltage losses and the excess OCV depend on the SOC, but not necessarily in the same functional form. Even if they did, that functional form would not be known. The excess OCV and the voltage losses (see Eq. 1) should therefore be treated independently. Hence, a precise calculation of the OCV can help to evaluate the required fit parameters more accurately for a wide range of operating regimes in order to improve existing models.

For the AVRFB, the calculations presented in this subsection cannot be simply repeated due to a lack of necessary Pitzer parameters or data from which these parameters could be obtained. Instead, we implemented the zero-dimensional model by Sharma et al.¹⁷ to get an estimate of the magnitude of the different types of voltages losses. The predicted types of voltage losses (activation overpotentials and ohmic losses of the current collector, electrolyte, and membrane) are smaller or of the same magnitude as the estimated excess OCV of a typical AVRFB examined above. This supports the hypothesis that activity coefficients are essential for an accurate modeling of RFBs and should not be neglected.