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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
3/17
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
4/17
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
5/17
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
6/17
Assumptions Particle sizes are distributed.
time
Small crystals
LargeMiddle
Curr
ent
Schematic interpretation of distribution model
Physical meaning of parameters
7/17
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
8/17
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
9/17
Pos. Neg. Pos.
Results and discussion
10/17
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
11/17
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.
12/17
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
13/17
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.
14/17
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
15/17
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
-18
-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
16/17
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.
17/17
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