Electroreduction Kinetics of Lead Sulfate in Lead-acid Battery Negative Electrode

Yasuyuki Hamano, Ikumi Ban, Kenji Hirakawa, Yoshiaki Yamaguchi GS Yuasa International Ltd.

Technical Development Division, Global Technical Headquarters

10th International Conference on Lead-acid Batteries

目次

1 Background

2 Theory and Experimental Theory: Mathematical model for cathodic reduction of PbSO4

Experimental: Potentiostatic reduction from PSOC

3 Results and Discussion Validity check

Temperature dependence of cathodic reduction current

4 Conclusion

Table of contents

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Charge acceptance from regenerative braking energy

Incomplete charging causes the formation of large and highly crystalline PbSO4.

Focus on charge reaction of negative plates PbSO4 + H+ + 2e− → Pb + HSO4

−

Cycling endurance

SEM image of negative active material with progressive sulfation

GS Yuasa Technical Report, 13 (2), 15 (2016).

Very short period / High current

Requirements for Enhanced Flooded Batteries

Previously reported mathematical model for i-t curve

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Instantaneous nucleation and 3D diffusion controlled growth model “Microscopic reaction site model”

PbSO4

Pb active sites

PbSO4

Pb

𝑖𝑖(𝑡𝑡)/𝐴𝐴 = 𝑃𝑃1𝑡𝑡−1/2[1− exp(−𝑃𝑃2𝑡𝑡)]+𝑃𝑃3𝑡𝑡−1/2 𝑖𝑖(𝑡𝑡) =2𝐹𝐹𝑁𝑁𝐷𝐷𝑐𝑐Pb 2+

𝑑𝑑[2(𝑎𝑎 + 𝑏𝑏) + 𝑎𝑎𝑏𝑏](ℎ − 𝐵𝐵𝑡𝑡)2

F. E. Varela, L. M. Gassa, J. R. Vilche, Electrochim. Acta, 37, 1119 (1992). etc…

K. Kanamura, Z. Takehara, J. Electrochem. Soc ., 139, 345 (1992)

Solid-state mechanism Rate determining step: Pb crystal growth

Dissolution-precipitation mechanism Rate determining step: PbSO4 dissolution

These models were validated by using planar electrodes. The kinetics model for porous lead electrodes cannot be found in the literature.

Flux of Pb2+

Theory 1/2

PbSO4 Crystal

0l

tMkρ

0l

)(tl

charging for t s l0

Dissolution occurs on the surface of PbSO4 at a constant rate.

Assumptions Dissolution-precipitation mechanism 3D dissolution of PbSO4

Cubic crystal Almost the same as “microscopic reaction site model”

l0 initial side length of PbSO4 [cm] M molecular mass of PbSO4 [g mol-1] ρ density of PbSO4 [g cm-3] k mass transfer coefficient of Pb2+ [mol cm-2 s-1]

𝑖𝑖𝑙𝑙0 𝑡𝑡 =6𝑧𝑧𝐹𝐹𝑧𝑧 𝑙𝑙0 − 2𝑧𝑧

𝑀𝑀𝜌𝜌 𝑡𝑡

2

, 𝑡𝑡 ≤𝑙𝑙0𝜌𝜌2𝑧𝑧𝑀𝑀

0, 𝑡𝑡 >𝑙𝑙0𝜌𝜌2𝑧𝑧𝑀𝑀

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Equation of current

Theory 2/2

Pareto type distribution ⇨ Long tailed shape

10

m0 )( += α

ααl

llP

10

mtotal0total0 )()( +== α

ααl

lNlPNlN

Probability density function P(l0)

Total current

02

010

m

00total

)2(6

)()()(0

dltMklzFkl

lN

dltilNti

total

l

∫

∫−⋅=

=

+ ραα

α

　　　

PbSO4 crystal number density N(l0)

Current within 50 s was omitted from the curve fitting to avoid the effect of the dissolved Pb2+ ion at the initial, the non-Faradaic current and the detachment of PbSO4 from the surface of lead electrode.

α shape factor lm scale factor [µm]

Ntotal total number

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Assumptions Particle sizes are distributed.

time

Small crystals

LargeMiddle

Curr

ent

Schematic interpretation of distribution model

Physical meaning of parameters

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10

mtotal0total0 )()( +== α

ααl

lNlPNlN α α

α 1

0

mtotal0total0 )()( +== α

ααl

lNlPNlN Ntotal

α Ntotal Sharpness of the distribution

Number density distribution

Total number of lead sulfate crystals

α is high crystals are small

Number density distribution

Ntotal is linear with N(l0)

Physical meaning of parameters

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k

02

010

mtotaltotal )2(6)( dltMklzFkk

llNti ∫ −⋅= + ρ

αα

α

　

Mass transfer coefficient [mol s-1 cm-2]

k k

Step (1) Dissolution reaction rate: PbSO4 + H+ ⇒ Pb2+ + HSO4

-

Step (2) Diffusion rate of Pb2+

k includes

Pb2+

Pb

PbSO

4

(1)

(2)

Electrochemical potential

Experimental

Cell configuration: one negative plate, two positive plates Negative plate: 0.85 Ah(theoretical) Reference electrode: Pb|PbSO4 s.g. 1.30 Electrolyte: s.g. 1.280

Test procedure 1) DOD adjustment

Full charge ⇨ 0.097 A discharge, 30 min, 25 °C

2) Rest for 12 h

3) Potentiostatic reductioin

precharge: 1.46 A, 6 s

potential step chronoamperometry: -300 mV

Potentiostatic reduction of negative plates The potential was kept at sufficiently negative to achieve the mass transfer limitation

Test cell

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Pos. Neg. Pos.

Results and discussion

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Significant decrease at the beginning and then gradual decrease.

Fig. 1 Potentiostatic current transient. Potential of negative electrode was kept at −300 mV vs. Pb|PbSO4|H2SO4 s.g. 1.30.

Results and discussion

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Previously reported mathematical models cannot explain the experimental curve.

Instantaneous nucleation and 3D diffusion controlled growth model

Microscopic reaction site model

𝑖𝑖(𝑡𝑡)/𝐴𝐴 = 𝑃𝑃1𝑡𝑡−1/2[1− exp(−𝑃𝑃2𝑡𝑡)]+𝑃𝑃3𝑡𝑡−1/2

𝑖𝑖(𝑡𝑡) =2𝐹𝐹𝑁𝑁𝐷𝐷𝑐𝑐Pb 2+

𝑑𝑑[2(𝑎𝑎 + 𝑏𝑏) + 𝑎𝑎𝑏𝑏](ℎ − 𝐵𝐵𝑡𝑡)2

Fig. 1 Potentiostatic current transient. Potential of negative electrode was kept at −300 mV vs. Pb|PbSO4|H2SO4 s.g. 1.30.

Results and discussion

α 1.64 PbSO4 size parameter

Ntotal 2.54×108 Total number of PbSO4

k [mol s−1 cm−2] 3.42×10−8 Mass transfer coefficient

Table Parameters used in the calculation

The calculated potentiostatic recharge curve is consistent with the experimental curve in the range t ≥ 50 s.

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Fig. 1 Potentiostatic current transient. Potential of negative electrode was kept at −300 mV vs. Pb|PbSO4|H2SO4 s.g. 1.30.

Results and discussion

Fig. 2 Calculated PbSO4 size distribution at the start of the potential step chronoamperometry.

Why does the long tailed distribution work well ?

Discharge for DOD adjustment Nucleation continues progressively

Nucleation

Discharge for 30 min

Growth

Nucleation

Crystal size distribution of PbSO4

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PbSO

4nu

mbe

rden

sity

/µm

-1

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+09

0 20 40 60 80 100100101

102103

104105

106107108109

l0 / µm

α = 1.64Ntotal = 2.54×108

Temperature dependence of mass transfer coefficient k

Fig. 3 Temperature dependence of potentiostatic current transient.

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Experimental conditions DOD adjustment Discharge 0.097 A, 60 min, 25 °C Rest for 15 min Potentiostatic reduction Precharge 1.46 A, 6 s Potential step -300 mV

Experimental

0

1

2

3

4

5

0 200 400 600 800 1000 1200

i/ A

t / s

25 ℃（実験値）40 ℃（実験値）52 ℃（実験値）

25 ºC40 ºC52 ºC

Experimental

0

1

2

3

4

5

0 200 400 600 800 1000 1200

i/ A

t / s

25 ℃（実験値）40 ℃（実験値）52 ℃（実験値）

25 ºC40 ºC52 ºC

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Experimental conditions DOD adjustment Discharge 0.097 A, 60 min, 25 °C Rest for 15 min Potentiostatic reduction Precharge 1.46 A, 6 s Potential step -300 mV

Fig. 3 Temperature dependence of potentiostatic current transient.

Fitting conditions α and Ntotal are fixed. α 1.64 Ntotal 4.38×108

Temperature dependence of mass transfer coefficient k

0

1

2

3

4

5

0 200 400 600 800 1000 1200

i/ A

t / s

25 ℃（実験値）40 ℃（実験値）52 ℃（実験値）25 ℃（計算値）

25 ºC40 ºC52 ºC25 ºC k = 3.59 × 10-8 mol s-1 cm-2

0

1

2

3

4

5

0 200 400 600 800 1000 1200

i/ A

t / s

25 ℃（実験値）40 ℃（実験値）52 ℃（実験値）25 ℃（計算値）40 ℃（計算値）

25 ºC40 ºC52 ºC25 ºC40 ºC

k = 3.59 × 10-8 mol s-1 cm-2

k = 8.45 × 10-8 mol s-1 cm-2

0

1

2

3

4

5

0 200 400 600 800 1000 1200

i/ A

t / s

25 ℃（実験値）40 ℃（実験値）52 ℃（実験値）25 ℃（計算値）40 ℃（計算値）52 ℃（計算値）

25 ºC40 ºC52 ºC25 ºC40 ºC52 ºC

k = 3.59 × 10-8 mol s-1 cm-2

k = 8.45 × 10-8 mol s-1 cm-2

k = 15.9 × 10-8 mol s-1 cm-2

Experimental

Results and discussion

Fig. 4 Arrhenius plot of mass transfer coefficient

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-17

-16

-15

3 3,1 3,2 3,3 3,4

lnk

1000×T-1 / K-1

Arrhenius equation

𝑧𝑧 = 𝐴𝐴exp −𝐸𝐸𝑎𝑎𝑅𝑅𝑅𝑅

A Pre-exponential factor Ea Activation energy

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A 2.2 mol s-1 cm-2

Ea 44.4 kJ mol-1

Thermodynamic parameters of the mass transfer of Pb2+ were determined!

Further understanding of the chemistry of the lead sulfate reduction

Conclusion

Mathematical model for the potentiostatic reduction current of PbSO4 considering the crystal size distribution of PbSO4 in negative plates

By using this model, the crystal size and the mass transfer coefficient of Pb2+ (k) have been determined.

The activation energy of the mass transfer of Pb2+ have been determined from the temperature dependence of k.

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