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A simple rational model is proposed for discharge of batteries with aqueous electrolytes, based on Nernst equation. Details of electrode kinetics are not taken into account. Only a few overall parameters of the battery are considered. A simple algorithm, with variable time step-length Δ t , is presented, for proposed model. The model is first applied to Daniel cell, in order to clar ify concepts and principles of battery operation. It is found that initial pinching, in time-history curve of voltage E-t, is due to initial under-concentration of product ion. Then, model is applied to a lead-acid battery. In absence of an ion product, and in order to construct nominator of Nernst ratio, such an ion, with coefficient tending to zero, is assumed, thus yielding unity in nominator. Time-history curves of voltage, for various values of internal resistance, are compared with corresponding published experimental curves. Temperature effect on voltage-time curve is examined. Proposed model can be extended to other types of batteries, which can be considered as having aqueous electrolytes, too.

For the discharge of batteries, successful empirical models have been proposed [

Computational analysis of batteries with aqueous electrolytes can be easily implemented, because only aqueous ion concentrations are introduced in Nernst equation.

Here, details of electrode kinetics [

A simple algorithm, with variable time step-length Δt, is presented, for proposed model. When exhausting of battery capacity is approached and denominator of Nernst ratio tends to become negative, Δt is halved and algorithm step is repeated, as many times are needed for denominator to remain positive.

Proposed model is first applied to Daniel cell [

Then, model is applied to a lead-acid battery [

Time-history curves of voltage E-t of lead-acid battery, for various values of internal resistance, are compared with corresponding published experimental curves [

In future research, proposed model can be extended to other types of batteries, which can be considered as having aqueous electrolytes, too, e.g. redox flow cells [

For a general form of total chemical reaction of a battery with aqueous electrolyte

a A ( s ) + b B ( l ) + c C ( aq ) + d D ( aq ) → e E ( s ) + f F ( l ) + g G ( aq ) + h H ( aq ) , (1)

the flow-chart of algorithm of proposed model, for discharge of batteries with aqueous electrolytes, is presented in

First, the physical constants are read: universal gas constant R = 8.314 J ⋅ mol − 1 ⋅ K − 1 and Faraday constant F = 96490 C ⋅ mol − 1 ⋅ e − 1 , where eelectric charge of an electron.

Then, constant input data are read: nominal voltage E_{0} of battery in V, total capacity Q_{0} in C, internal resistance r in Ω, volume v in L, temperature T in K, number n of electrons involved in chemical reaction of battery, as well as coefficients c, d of reactant ions and g, h of product ions of Equation (1).

The initial conditions are read: For step number i = 0, time t = 0. and discharged capacity Q = 0., initial concentrations [C]_{0}, [D]_{0} of reactant ions and [G]_{0}, [H]_{0} of product ions in mol∙L^{−1}, as well as initial time step-length Δt of the algorithm in sec.

Then, the initial voltage E in V is found, by Nernst equation

E = E 0 − R T n F ln ( [ G ] g [ H ] h [ C ] c [ D ] d )

and the initial current I = E / r in A.

For any i-th step of the algorithm, first step number i is increased by 1 and time t by Δt.

Then, the transferred electric charge increment Δ Q = I × Δ t in C, within present step, is determined. From this ΔQ, the corresponding increment of number of moles is found Δ N m = Δ Q / ( n F ) and the transferred concentration increment Δ c = Δ N m / v in mol∙L^{−1}.

This concentration increment Δc will be distributed to reactant and product ions, proportionally to their coefficients:

Reactants [ C ] ← [ C ] − c c + d Δ c , [ D ] ← [ D ] − d c + d Δ c

Products [ G ] ← [ G ] + g g + h Δ c , [ H ] ← [ H ] + h g + h Δ c

However, previously, the reactant which is going to be first consumed, let C here, is checked if its concentration becomes negative in present step. In such a

case, [C] is restored to its previous value ( [ C ] ← [ C ] − c c + d Δ c ), Δt is halved

Δ t ← Δ t / 2 and algorithm step is repeated as many times are needed, in order for [C] to become positive. (This procedure occurs when capacity tends to be exhausted Q ← Q 0 and abrupt drop of voltage E starts).

After updating of concentrations of reactant and product ions, the new voltage is found by Nernst equation

E = E 0 − R T n F ln ( [ G ] g [ H ] h [ C ] c [ D ] d ) ,

new current I = E / r and new discharged capacity Q ← Q + Δ Q are determined.

As output of present step of algorithm, the following data are written: number i of algorithm step, time t, voltage E, current I, discharged capacity Q, concentrations of reactant ions [C], [D] and product ions [G], [H], present time step-length Δt.

At end of present step of algorithm, if voltage E is still greater than the desired cut-off voltage, e.g. 0.5E_{0} or 0.8E_{0}, we go to next step of algorithm. Otherwise, algorithm is interrupted.

Based on above simple algorithm, a short computer program has been developed, with only fifty five Fortran instructions, for discharge of batteries with aqueous electrolytes.

The proposed model is first applied to Daniel cell, in order to clarify concepts and principles of battery operation. In

A container is divided, by a porous diaphragm, into two compartments. The left compartment is filled by an aqueous electrolyte of sulfuric zinc ZnSO_{4}(aq) and the right one is filled by sulfuric copper CuSO_{4}(aq). In the left compartment, a zinc electrode Zn(s) is dipped and, in the right one, a copper electrode Cu(s). The upper ends of two electrodes, outside of the container, are connected by an external wire with resistance r. Because of potential difference, at 25˚C, E 0 = + 0.34 − ( − 0.76 ) = 1.10 V [

In the left zinc electrode, because of subtraction of electrons, zinc cations Zn^{2+} are repelled from Zn(s) to electrolyte ZnSO_{4} (aq). Whereas, in the right copper electrode, because of addition of electrons, copper cations Cu^{2+} from electrolyte CuSO_{4} (aq) are attracted to electrode Cu(s) (

The vertical porous diaphragm, in the middle of container, prevents passage of metal cations Zn^{2+}, Cu^{2+}, but allows passage of anion SO 4 2 − [^{2+} by Cu(s) electrode, free anions SO 4 2 − pass, through porous diaphragm, to left compartment, where there is an excess of Zn^{2+} cations, after their repel from electrode Zn(s), so that free SO 4 2 − anions meet free Zn^{2+} cations, in order to close the circuit and maintain electro-neutrality (

So, chemical reactions of Daniel cell are the following:

In left compartment, oxidation half-reaction

Zn ( s ) − 2 e − → Zn 2 + ( aq ) ,

with E 0 = − 0.76 V , from table of standard reduction potentials at 25˚C [

In right compartment, reduction half-reaction

Cu 2 + ( aq ) + 2 e − → Cu ( s )

with E 0 = + 0.34 V .

By adding the above two half-reactions, total chemical reaction of Daniel cell is obtained

Zn ( s ) + Cu 2 + ( aq ) → Zn 2 + ( aq ) + Cu ( s )

with E 0 = + 0.34 − ( − 0.76 ) = + 1.10 V .

If, for a Daniel cell, is chosen capacity Q 0 = 1.0 A ⋅ h = 3600 C and initial concentration of reactant ion [ Cu 2 + ( aq ) ] 0 = 1.0 mol ⋅ L − 1 , this choice implies a specific value of battery volume v, which is simply determined as follows:

[ Cu 2 + ( aq ) ] 0 × v × n × F = Q 0

1.0 mol ⋅ L − 1 × v × 2 e × 96490 C ⋅ mol − 1 ⋅ e − 1 = 3600 C

→ v = 0.01865 L = 18.65 cm 3

corresponding to a cube with side a = 2.652 cm.

If it is further assumed that, in this Daniel cell, initial concentration of product ion is very small [ Zn 2 + ( aq ) ] 0 = 10 − 5 mol ⋅ L − 1 , the initial voltage, at 25˚C, results, according to Nernst equation:

E = E 0 − R T n F ln ( [ Zn 2 + ] 0 [ Cu 2 + ] 0 ) ,

E = 1.10 V − 8.314 J ⋅ mol − 1 ⋅ K − 1 × 298 K 2 e × 96490 C ⋅ mol − 1 ⋅ e − 1 ln ( 10 − 5 1.0 ) = 1.10 V − 0.01284 J C × ( − 11.51 ) = 1.10 V + 0.1478 V = 1.248 V

That is, an initial voltage, by 0.1478 V, higher than nominal voltage E 0 = 1.10 V , is obtained. So, it is observed that initial pinching of voltage is due to initial under-concentration of product ion [ Zn 2 + ( aq ) ] 0 = 10 − 5 mol ⋅ L − 1 .

The above Daniel cell can run by the proposed algorithm of

The proposed algorithm of

It is observed, in _{0} of battery, where t_{u} duration of battery, as was expected.

Lead-acid battery is an old type of battery, which is still used in many applications [

The oxidation half-reaction of a lead-acid cell is

Pb ( s ) + HSO 4 − ( aq ) − 2 e − → PbSO 4 ( s ) + H + ( aq ) ,

with E 0 = − 0.36 V .

and the reduction half-reaction

PbO 2 ( s ) + HSO 4 − ( aq ) + 3 H + ( aq ) + 2 e − → PbSO 4 ( s ) + 2 H 2 O ( l )

with E 0 = + 1.69 V .

So, the total chemical reaction of the cell is

Pb ( s ) + PbO 2 ( s ) + 2 H + ( aq ) + 2 HSO 4 − ( aq ) → 2 PbSO 4 ( s ) + 2 H 2 O ( l )

with E 0 = + 1.69 − ( − 0.36 ) = 2.05 V .

In the absence of an aqueous ion among products, and in order to construct nominator of Nernst ratio, such an ion P(aq), with coefficient tending to zero, let 10^{−5}, is assumed, thus yielding unity in nominator [P(aq)]^{0} = 1.0.

So, total reaction becomes

Pb ( s ) + PbO 2 ( s ) + 2 H + ( aq ) + 2 HSO 4 − ( aq ) → 2 PbSO 4 ( s ) + 2 H 2 O ( l ) + 10 − 5 P ( aq )

and the Nernst equation becomes

E = E 0 − R T n F ln ( 1.0 [ H + ] 2 [ HSO 4 − ] 2 )

All concentrations of reactant ions are gradually transformed to concentration of product ion [P(aq)], which, however, has meaningless participation in chemical reaction.

Here, a battery, consisting of three lead-acid cells, is considered, so nominal voltage is E 0 = 3 × 2.05 V = 6.15 V ≈ 6.20 V .

If a total initial capacity Q 0 = 1.0 A ⋅ h = 3600 C of the battery is chosen and initial concentrations of reactant ions [ H + ] 0 = [ HSO 4 − ( aq ) ] 0 = 1.0 mol ⋅ L − 1 , this choice implies a specific value of battery volume v, which is simply determined as follows:

( [ H + ] 0 + [ HSO 4 − ] 0 ) × v × n × F = Q 0

2.0 mol ⋅ L − 1 × v × 2 e × 96490 C ⋅ mol − 1 ⋅ e − 1 = 3600 C

→ v = 9.3274 × 10 − 3 L = 9.3274 cm 3

corresponding to a cube with side a = 2.105 cm

The above lead-acid battery can run with the algorithm of

The proposed algorithm of

In

Then, for the same lead-acid battery, temperature effect on time-history of voltage E-t is examined. For internal resistance r = 31.0 Ω and three values of

temperature T: 10˚C = 283 K, 50˚C = 323 K, 90˚C = 363 K, the three corresponding curves for time-history of voltage E-t have been drawn in _{u} of battery slightly increases.

For the temperature T values tried here, effect of temperature on voltage E is not significant; variations up to 0.035 V only are observed.

1) A simple rational (not empirical) model is proposed, for discharge of batteries with aqueous electrolytes, based on Nernst equation.

2) Details of electrode kinetics are not taken into account. Only a few overall parameters of the battery are considered.

3) A simple algorithm is presented, for proposed model, with variable time step-length Δt, describing both regions of the E-t curve, that with slow reduction of voltage, as well as that with abrupt reduction.

4) Proposed model is first applied to Daniel cell, in order to clarify concepts and principles of battery operation.

5) It is found that initial pinching, in time-history curve of voltage, is due to initial under-concentration of product ions.

6) As in the chemical reaction of a lead-acid battery, no product aqueous ion exists, such an ion, with coefficient tending to zero, is considered, yielding unity in the nominator of Nernst ratio.

7) Time-history curves of voltage of a lead-acid battery, for five values of resistance, obtained by proposed model, are compared with corresponding published experimental curves.

8) Effect of temperature, on time-history of voltage, is examined.

9) Proposed model can be extended to other types of batteries, which can be considered as having aqueous electrolytes, too.

The authors declare no conflicts of interest regarding the publication of this paper.

Papadopoulos, P.G., Koutitas, C.G., Karayannis, C.G., Kiousis, P.D. and Dimitropoulos, Y.N. (2021) Simple Rational Model for Discharge of Batteries with Aqueous Electrolytes, Based on Nernst Equation. Open Journal of Physical Chemistry, 11, 1-11. https://doi.org/10.4236/ojpc.2021.111001