Modeling and Analysis of Electrochemical Performance in Lithium-Sulfur Batteries
Explore the mathematical models, finite volume discretization, and charge conservation in the electrochemical discharge of lithium-sulfur batteries. This study delves into the concentration balance, volume balance for solids, and more to enhance battery performance and efficiency.
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Modeling and Analysis of Electrochemical Performance of Lithium-sulfur Batteries Aashutosh Mistry and Partha P. Mukherjee Energy and Transport Sciences Laboratory Texas A&M University Jul 11, 2016 Presentation at College on Multiscale Computational Modeling for Energy Applications, International Centre for Theoretical Physics (ICTP), Trieste, Italy
Contents To give a brief overview of models for electrochemical discharge of Li-sulfur cell 1. 2. 3. 4. Mathematical Model Finite Volume Discretization Lumped (0thOrder) Model MATLAB tips 2
1. Mathematical Model discharge + + Li Li e charge 1 2 1 2 3 2 charge + 2 8 S S e ( ) 8 l discharge charge + 2 6 2 8 2 S S e discharge 3 2 charge + 2 4 2 6 S S e dischar ge 1 2 1 2 charge + 2 2 2 4 S S e discharge charge + 2 2 2 S S e dischar ge precipitation S S ( ) 8 ( ) 8 l s dissolution precipitation + + 2 2 Li S Li S ( ) s 2 dissolution 5
1. Mathematical Model Concentration balance: 0 i C x D RT ( ) + = 2 8 2 6 2 4 2 2 2 = + + 0 i , , , , , , i Li S S S S S S i e C D z F C R ( ) 8 i i i i l t x x x 1 1 2 Charge conservation in electrolyte phase: charge = + 2 8 2:2 m S S e ( ) 8 l discharge + 0 i D RT C x + = 2 i 2 0 i 3 2 0 e i z F C z D F aj charge = + 2 6 2 8 3:2 m S S e i i m x x x i m discharge 3 Charge conservation in solid phase: charge = + 2 4 2 6 4:2 m S S e d ischarge 2 = c aj 1 2 1 2S c m 2 x charge = + 2 2 2 4 5: m S S e m disc harge charge = + 2 2 2 6: m S e Volume balance for solids: dischar ge d = k Q k dt = , , k S Li S Li S 8 2 2 2 6
2. Finite Volume Discretization L L L sep an cat 7
2. Finite Volume Discretization N 1 M L L L 8 sep an cat
2. Finite Volume Discretization W P E w e N 1 M L L L 9 sep an cat
2. Finite Volume Discretization Governing equation (PDE): T t T x = + C k Q p x W P E d V V T t T x = + d C k Q V w e p x V ' Leibnitz s ) ( d dt T x = d + d C T V k Q V p x V V P Gauss Divergence , area a dT d T x T x T x e = + = + Q V d C V k A Q V ka ka P p t w e w a x = , volume V x 1 dT d T x T x Q C = + k k P t C x 10 e w p p
2. Finite Volume Discretization Governing equation (PDE): T t T x = + C k Q p x W P E d V V T t T x = + d C k Q V w e p x V 1 dT d T x T x Q C = + k k P t C x P e w p p , area a ? + 1 p p 1 T T x T T T T Q C + P W k k P P E P e w t C x x a x = , volume V p p x 11
2. Finite Volume Discretization Concentration balance: 0 i C x D RT ( ) = + + 0 i i e C D z F C R i i i i t x x x W P E Charge conservation in electrolyte phase: + w e 0 i D RT C x + = 2 i 2 0 i 0 e i z F C z D F aj i i m x x x i m Charge conservation in solid phase: 2 = c aj P c m 2 x m , area a Volume balance for solids: a x = d , volume V = k Q x k dt 12
2. Finite Volume Discretization Concentration balance: 0 i C x D RT ( ) = + + 0 i i e C D z F C R i i i i t x x x W P E d V V w e + 1 p ( ) ( 2 ) ( ) ( ) ( ) ( 2 ) + 1 p p ( ) ( ) C C C C C C i i i i i i 0 i P W D P P E P t x x e w + 1 p ( ) ( ) 2 ( ) ( ) 2 P 0 i D RT C C e e + e e + + 1 p P W z F R E P i i i i x x e w , area a a x = , volume V x ( ) R C )( ) ( ) + 1 ~ p + + 1 p i R R C C ( i i i i i 13
2. Finite Volume Discretization Charge conservation in electrolyte phase: + 0 i D RT C x + = 2 i 2 0 i 0 e i z F C z D F aj W P E i i m x x x i m d V V w e + 1 p ( ) ( ) 2 ( ) ( ) 2 D RT C C e e e e 2 i 2 i P W z F E P i i x x e w ( ) ( 2 ) ( ) ( 2 ) ( ) ( ) C C C C i i i i i + 0 i P W z F D E P P i x x e w + 1 p , area a + = 0 aj m m a x = , volume V x ( )( ) e ~ ( ) ( ) + + 1 ~ p 1 ~ p = + e e 14
2. Finite Volume Discretization Charge conservation in solid phase: 2 = c aj c m 2 x W P E m d V V w e + ( ) ( ) x ( ) 1 p + 1 p + 2 c c c = E P W aj c m 2 m P ( )( ) c ~ ( ) ( ) + + 1 ~ p 1 ~ p = + , area a c c a x = , volume V x 15
3. Lumped (0th Order) Model Concentration balance: 0 i C x D RT ( ) = + + 0 i i e C D z F C R i i i i t x x x 0 i C x D RT = 0 i i e J D z F C C i i x Charge conservation in electrolyte phase: + 0 i D RT C x + = 2 i 2 0 i 0 e i z F C z D F aj i i m x x x i m = J J , ionic C i Charge conservation in solid phase: i 2 = c aj c m 2 x m = c J electronic c x Volume balance for solids: d = k Q = = J J J k dt app ionic electronic / / anode separator cathode current collector 16
3. Lumped (0th Order) Model Concentration balance: 0 i C x D RT ( ) = + + 0 i i e C D z F C R i i i i t x x x Charge conservation in electrolyte phase: + 0 i D RT C x + = 2 i 2 0 i 0 e i z F C z D F aj i i m x x x i m Charge conservation in solid phase: 2 = c aj c m 2 x m Volume balance for solids: d = k Q k dt 17
3. Lumped (0th Order) Model Concentration balance: = + x L cat L se p 0 i C x D RT ( ) = + + 0 i d i e C D z F C R x i i i i t x x x = 0 x ( ) d dt ( ) + = + L L C R L i sep R L , , sep sep cat cat i i cat cat sep Charge conservation in electrolyte phase: x L = + cat L sep 0 i D RT C x + + = 2 i 2 0 i 0 d e i z F C z D F aj x i i m x x x i m = 0 x = J aL j Charge conservation in solid phase: app cat m x L = + L m se p cat 2 = d c a j x c m 2 x m x L = sep Volume balance for solids: x L = + L sep c at d d = = k Q d k Q x k k dt dt = 0 x 18
3. Lumped (0th Order) Model + = 2 8 2 6 2 4 2 2 2 , , , , , , i Li S S S S S S ( ) 8 Concentration balance: ( sep dt l ( ) d ) + = + = L L C R L i sep R L R , , sep cat cat i i cat cat sep i 1 1 2 charge = + 2 8 2:2 m S S e ( ) 8 l discharge 3 2 charge = + 2 6 2 8 3:2 Charge conservation m S S e discharge 3 = J aL j charge = + 2 4 2 6 4:2 m S S e app cat m d ischarge m 1 2 1 2S charge = + 2 2 2 4 5: m S S e disc harge Volume balance for solids: charge = + 2 2 2 6: m S e dischar ge d = k Q k dt = , , k S Li S Li S 8 2 2 2 19
4. MATAB tips input() & disp() Syntax: varName = input( inputstring ); Syntax: disp( outputstring ); Syntax: disp(varName); clear, clc Example: write a script to compute sine of a given number do not modify script file to enter different input value 20
4. MATAB tips linspace() & plot() Syntax: linspace(lower_bound, upper_bound); Syntax: plot(x,y,format_specification); title(), xlabel(), ylabel(), axis() Example: write a script to compute sinh of numbers between and , and plot the function y = sin(x) 21
4. MATAB tips save() & load() Syntax: save(filename,variables); Syntax: load(filename,variables); Example: write a script to compute sin(1/x) of numbers between 0.01 and 0.01, plot the function y = sin(1/x), save data as a .mat file, clear workspace, load data and replot 22
5. Team Projects 1. Effect of porosity on cell performance 2. Effect of mean pore size on cell performance 3. Effect of sulfur loading on cell performance 4. Effect of S8 dissolution rate 5. Effect of Li2S precipitation rate Perform computations at C/10, C/5, C/4, C/3 and C/2 23
5. Team Projects Vcell: cell voltage in V j2 to j6: rates of electrochemical reactions r7 & r8: rates of chemical reactions time: time of discharge in hr cap: discharge capacity in mAh/g of S8 ES8s, ELi2Ss, Esep, Ecat: volume fractions a: active area C s: concentrations 24
Thank You! 25
