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The document discusses a reduced-order model for a lithium metal oxide cell that simplifies a complex 1D electrochemical model comprising ten coupled partial differential equations into a more manageable form with linear ordinary differential equations. This model reduction facilitates on-board control and allows for easier parameter estimation while maintaining physical relevance. Methods such as volume averaging and profile-based approximations are employed to achieve this simplification, which is essential for predicting cell voltage and internal field profiles during various charge and discharge cycles.

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Model reduction concepts & Voltage predictions Senthil Kumar V. Chemical Reactions Modeling India Science Lab, GM Global R&D 1 Reduced order model for a single cell - with uniform reaction rate approximation
Detailed 1D Electro-chemical model for a single cell e- e- Separator Lithium Metal Oxide Graphite - + LiC6 →x Li++Li1-xC6+x e- x Li++ LiMO2 +x e- → Li1+xMO2 Li+ Discharge of a Lithium Metal Oxide cell “Current” Loss of electrons  Oxidation  Anode Gain of electrons  Reduction  Cathode Load 2 Solid Electrolyte Electrolyte Solid Electrolyte 5 phases in 3 regions (n, s & p) Isothermal 1D model: Mass and Charge balances for each phase. 5 phases x 2 balances =10 PDEs Mass balance: Diffusion equation Charge balance: Ohm’s law & Current balance Negative electrode n n n Fj a x x − =          −   1 1  ( ) 1 2 2 2 2 n n n n j t a x c D x t c + − +           =    ( ) I x c t F RT x x n n n =   − +    −    − + 2 2 2 2 1 1 ln 1 2             =   r c D r r r t c n n n 1 1 2 2 1 1 Solid phase current balance Liquid phase mass balance Total current balance Solid phase mass balance Separator           =   x c D x t c s s 2 2 2 2  ( ) I x c t F RT x s s =   − +    − + 2 2 2 2 ln 1 2  Liquid phase mass balance Total current balance Positive electrode p p p Fj a x x − =          −   1 1              =   r c D r r r t c p p p 1 1 2 2 1 1 ( ) p p p p j t a x c D x t c + − +           =   1 2 2 2 2  ( ) I x c t F RT x x p p p =   − +    −    − + 2 2 2 2 1 1 ln 1 2   Solid phase current balance Liquid phase mass balance Total current balance Solid phase mass balance
Need for Model Reduction • The single cell isothermal model has 10 coupled PDEs. Difficult to use for packs, on-board control , calendar / cycle life predictions, detailed parameters estimation, inclusion of complex degradation mechanisms etc. • The reduced order model should preferably comprise of first order ODEs. Such a mathematical structure will automatically ensure minimal computational requirement and numerical stability. If the first order ODEs turn out to be linear, optimization during parameter (re-)estimations become robust. • Equivalent circuit based single cell models are popular for on-board control due to their first order ODEs structure, and the consequent properties. Detailed electrochemical model should be reduced to a similar structure to make the physics based model attractive for on-board applications. • Need a first order ODEs based model which – Is grid / system size independent: Unlike Method of lines or PDEs discretized into ODEs. – Has physically relevant internal variables: Like Lyapunov-Schmidt model reduction (here: interfacial electrolyte fluxes and concentrations). – Has systematic mathematical approximations rather than phenomenological approximations: Unlike neglecting electrolyte potential /concentration variations - Single particle model. 3 n s p
`` `` Model reduction methodology (I): Volume averaging 4 • Volume averaging a PDE yields an ODE. Hence volume averaging is the chosen methodology for model reduction. On volume averaging information about spatial gradients or profiles are lost. This information is recovered by profile based approximation in the next step. ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n in n n n n n n in n n n n n n n n in l x n n n l x n n n n l x n n l x n n n l x n n l x n n n n n j l a t q dt c d l j a t l q dt c d j t a j t a RHS l q x c D l dx x c D x l RHS dt c d dx c l dt d dx t c l dx x f l Al dx x f A f j t a x c D x t c n n n n n n + + + + = = = = = = + − + − = − + − = − = − = − =         =           = =         =   = = = − +           =        1 1 1 1 2 1 1 1 1 LHS 1 1 2 2 2 2 2 2 2 0 2 2 2 2 0 2 2 2 0 2 0 2 2 0 0 2 2 2 2       e- e- Separator Lithium Metal Oxide Graphite - + LiC6 →x Li++Li1-xC6+x e- x Li++ LiMO2 +x e- → Li1+xMO2 Li+ Discharge of a Lithium Metal Oxide cell “Current” Load Negative electrode Accumulation of Li+ in a slice of n-electrode = Difference in diffusive out-flow and in-flow + Generation of Li+ in that slice. Accumulation of Li+ in the whole of n-electrode = - Out flux from the interface + Total generation of Li+ within the n-electrode. Negative electrode q2in Interfacial flux is a physically relevant internal variable
`` `` Model reduction methodology (II): Profile based approximations 5 • The profile approximations are constructed such that volume averages are respected. The profiles are expressed in terms of physically relevant internal variables. Eventually equations (ODEs or algebraic equations) are derived for these internal variables. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n in in n n n in n n n n n in n n n in n n in n n n n n l x n n n n n in in n in n n in n n j l a t q dt dq D l dt dc l j l a t q dt c d l t q D l t c t c l l l x dx x f l f x l D l t q t c t x c l q x c D x l q x c D x n + + = − + − = + − + − = + = = =  = − + = −            −             1 3 1 3 3 3 1 1 2 , 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 0 2 2 2 2 2 2 2 2 2 2 2 2    2. Volume average respecting lowest order profile approximation 1. Exact result from volume averaging 3. Profile is expressed in terms of interfacial flux & concentration 4. Average concentration depends only on interfacial flux & concentration 5. Volume averaged diffusion equation, exact result 6. An evolution equation for the interfacial flux & concentration n s p c2in & q2in
6 Nonlinear PDEs Linear ODEs Nonlinear algebraic expressions Approximations Consistent Reaction rate Butler – Volmer Kinetics Electrolyte concentration field Electrolyte potential field Solid potential field Solid phase concentration field Electrolyte phase Solid phase Linear ODEs Algebraic expressions Linear ODEs Algebraic expressions Algebraic expressions Uniform reaction rate from external current Diffusion approx as in Single particle model Model reduction methodology (III): Smart decomposition 1. Nonlinearity in a system is not bothersome if it can be relegated to algebraic evaluations. 2. Coupled linear ODEs can be solved by time marching without iteration, unlike non-linear coupled PDEs which require iteration to achieve a consistent solution. It is better to have a time-marchable structure for on-board algorithm. ( ) ( ) p p p Fl a t I t j − = ( ) ( ) n n n Fl a t I t j =
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 , ln 2 , 2 ln 2 , 2 , 2 1 3 3 2 ; 1 3 6 2 x l l t I x l t I t c t x c t t x l t I t c t c t l t x l D l t q t c t x c D q q l c c q q c c l l l D l D l D l l l l l D l D l D l l n n n n n in in s s mid in n in n n n in in s ip in s ip in ip ip in in ip p p s s n n p p p s s s s n s n ip p p s s n n n n n s s s s n s n in − − − +          +  =  +        =  =  − + = + + = + + = + +         − + − = + +         + + − =                    Model reduction – Bird’s eye view 7 ( ) ( ) p p p ip ip p p p ip p p in in p p n n n in ip s n s n ip n n in n n n s n n s in n n j l a t q dt dq D l l dt dq l j l a t q dt dq D l l l dt dq D l D l l l + + − + =         − + − + − =         + +         + + 1 3 1 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2             p p p R j dt c d 3 1 − = 2 2 1 1 1 2 45 30 n n n rn n rn R j R c D dt c d − = + 2 2 1 1 1 2 45 30 p p p rp p rp R j R c D dt c d − = + n n n R j dt c d 3 1 − = 1. Two coupled ODEs for the interfacial fluxes, obtained by volume averaging and profile based approximations. 2. Quartic diffusion approximations for diffusion within active material spheres. Two coupled ODEs for each electrode. 3. Rest are algebraic expressions 5. Model reduction achieved with uniform reaction rate approximation: 10 coupled PDEs to 6 linear ODEs. 4. ROM predicts the internal field profiles too, not just the cell voltage. Negative Positive 2 ODEs 4 ODEs
1C discharge prediction by uniform rate ROM 8 Fig. 1 Max abs error  9 mV Fig. 2 Fig. 3 Max % error  0.27 %
Max % error  0.2% Fig. 4 Fig. 2 Voltage recovery at the beginning of rest periods are predicted well. Fig. 3 Max abs error  10 mV 2000 s 1C discharge, Then 300 s rest, Then 2000s 1C charge, Then rest till 8000 s. 1C pulse prediction by uniform rate ROM 9 Fig. 1
7C discharge prediction by uniform rate ROM 10 Model enhancement underway to reduce errors at high current discharge & charge. Fig. 1 Max abs error  80 mV Fig. 2 Fig. 3 Max % error  3.2%
Max % error  2% Fig. 4 Model enhancement underway to reduce errors at high current discharge & charge. Fig. 2 Max abs error  80 mV Voltage recovery at the beginning of rest periods are predicted well. Fig. 3 10 s 7C discharge, Then 40 s rest, Then 10s 7C charge, Then 60 s rest. 7C pulse prediction by uniform rate ROM 11 Fig. 1
Back up: Detailed derivation 12
Electro-chemical reactions in a Lithium Metal Oxide cell e- e- Separator Lithium Metal Oxide Graphite - + LiC6 →x Li++Li1-xC6+x e- x Li++ LiMO2 +x e- → Li1+xMO2 Li+ Discharge of a Lithium Metal Oxide cell “Current” Loss of electrons  Oxidation  Anode Gain of electrons  Reduction  Cathode Load e- e- Separator Lithium Metal Oxide Graphite - + x Li++Li1-xC6+x e- → LiC6 Li1+xMO2 → x Li++ LiMO2 +x e- Li+ Charge of a Lithium Metal Oxide cell “Current” Loss of electrons  Oxidation  Anode Gain of electrons  Reduction  Cathode Charger Compound Oxidation state of M Check LiMO2 +3 +1+3-2*2 = 0 Li2MO2 +2 +1*2+2-2*2 = 0 Li1+xMO2 y = +3-x +1*(1+x)+y-2*2 = 0 M: Co, Ni, Mn Manganese can have +2 to +7 oxidation states LiMn2O4  Mn+3.5 , KMnO4  Mn+7 Basic mnemonics LEO – Loss of Electron is Oxidation GER – Gain of Electron is Reduction AnOx – Anode is where Oxidation occurs RedCat – Cathode is where Reduction occurs Process – Electrode Energy consumer Energy producer Reduction – Cathode - + Oxidation - Anode + - 13
(I) Solid phase current balances in n & p regions 14 n s p
Solid phase current balance equations e- e- Separator Lithium Metal Oxide Graphite - + Li+ “Current” x=0 x=ln x=ln+ls x=L=ln+ls+lp 15 Positive electrode Negative electrode 0 and : BCs 1 1 0 1 1 1 1 =    − =    − − =          −   = = n l x n x n n n n x I x Fj a x x    I x x Fj a x x L x p l l x p p p p s n =    − =    − − =          −   = + = 1 1 1 1 1 1 and 0 : BCs    Convention: Discharge current is positive. Subscripts Notation: n – Negative electrode p – Positive electrode s – Separator 1- solid phase 2- liquid phase ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ρ S E J E E J R V V A I D m 1 m m V/m A/m2 : Continuum : 1 1 -  =   − =   − = =   − =  =   =      • Cell reactions generate or consume electrons. Hence the gradient of current is proportional to the reaction rate.                     − =                −   s m mol j mol C F m m a m A x x 2 3 2 3 1  Separator has ionic conduction only.
16 Volume averaging of solid phase current balances Negative electrode: 0 and : BCs 1 1 0 1 1 1 1 =    − =    − − =          −   = = n l x n x n n n n x I x Fj a x x    ( ) ( ) n n n n n n n n n l x n l x n n l x n Fl a t I t j l j F a I l j F a x dx j F a x d n n n =  = − =          −  − =          − = = =   0 1 1 0 0 1 1   Positive electrode: I x x Fj a x x L x p l l x p p p p s n =    − =    − − =          −   = + = 1 1 1 1 1 1 and 0 : BCs    ( ) ( ) p p p p p p p p p l l l l l x p l l l l l x p p l l l l l x p Fl a t I t j l j F a I l j F a x dx j F a x d p s n s n p s n s n p s n s n − =  − = − =          −  − =          − + + + = + + + = + + + =   1 1 1 1   e- e- Separator LMO Graphite - + Li+ “Current” x=0 x=ln x=ln+ls x=L=ln+ls+lp ( ) ( ) ( ) ( )     + = + = = = = = = = L l l x p p L l l x p l x n n l x n s n s n n n dx x f l Al dx x f A f dx x f l Al dx x f A f 1 1 0 0 ( ) ( ) ( ) ( ) ( ) ( ) 0 = + − = = t j l a t j l a F t I t j l a F t I t j l a p p p n n n p p p n n n Volume averaging results are exact, but information on spatial variations are lost. No accumulation of electrons. Hence average rates are coupled at any instant.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) I x x l l t I x t f l l t I x t f x l t I x l t I t j F a x x Fj a x x t j t x j Fl a t I t j x n n n n n n l x n n n n n n n n n n n n n n n n =    − − =    − + − =  =    − + − =    − − = −           −   − =          −    = = = 0 1 1 1 1 1 1 1 1 1 1 1 1 Satisfies 0 0 , : ion approximat rate reaction Uniform       17 Solid phase current in electrodes ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )   ( ) t I x l l x l t I x t f l l l t I x t f x l t I x l t I t j F a x x Fj a x x t j t x j Fl a t I t j L x p s n p p s n p l l x p p p p p p p p p p p p p p p s n =    − + − =    − + + =  =    − + =    − = −           −   − =          −    − = = + = 1 1 1 1 1 1 1 1 1 1 1 1 Satisfies 0 0 , : ion approximat rate reaction Uniform       Uniform reaction rate approximation will recover the volume average evolution equations. It decouples the potential and concentration fields, enabling a step-by-step solution of all the fields. The volume average contributions give the leading order transient solution, which contains the steady state solution within it. It is possible to derive corrections by introducing the deviations from volume average. Corrections will be attempted in future. ( ) ( ) ( ) t x j t j t x j d n n n , , + =
(II) Liquid phase mass balances in n, s, & p regions 18 n s p
Liquid phase mass balance (or electrolyte diffusion) equations ( ) ( ) ( )   ( ) 0 and , , : BCs , 0 , : IC 1 2 2 2 2 2 2 2 20 2 2 2 2 2 =   − =   − = + = − +           =   = + = + + + L x p ip l l x p ip s n p p p p x c D q x c D c t l l c c x c j t a x c D x t c s n  Positive electrode ( ) ( ) ( ) , , , 0 : BCs , 0 , : IC 1 2 2 2 2 2 0 2 2 20 2 2 2 2 2 in l x n in n x n n n n n q x c D c t l c x c D c x c j t a x c D x t c n =   − = =   − = − +           =   − = − = +  Negative electrode ( ) 20 2 2 2 2 2 0 , : IC c x c x c D x t c s s =           =    Separator e- e- Separator Lithium Metal Oxide Graphite - + Li+ “Current” x=0 x=ln x=ln+ls x=L=ln+ls+lp 19 Concentration and flux continuity conditions at N-S interface ( ) ( )   ( ) . , , , , , : BCs 2 2 2 2 2 2 2 2 2 2 ip l l x s ip s n in l x s in n q x c D c t l l c q x c D c t l c s n n =   − = + =   − = − + + = − = + Continuity conditions at N-S interface Continuity conditions at S-P interface Concentration and flux continuity conditions at S-P interface Electrolyte concentration is a continuous field through negative electrode, separator and positive electrode regions.
Volume averaged Li mass balance in electrolyte phase 20 ( ) ( )   = = = = n n l x n n l x n dx x f l Al dx x f A f 0 0 1 ( ) ( ) ( ) ( ) ( ) ( ) p p p ip p p p ip in s s s n n n in n n n n n n in n n l x n n n n n l x n n n l x n n l x n n l x n n n n n j l a t q dt c d l q q dt c d l j l a t q dt c d l j t a l q j t a x c D l j t a dx x c D x l RHS dt c d dx c l dt d dx t c l dx j t a x c D x t c l n n n n n + + + + = + = = = = + − + = − = − + − = − + − = − +         = − +           = =         =   =       − +           =       1 1 1 1 1 1 1 1 LHS 1 1 2 2 2 2 2 2 2 2 2 2 2 0 2 2 2 2 0 2 2 2 0 2 0 2 2 0 2 2 2 2        `` Negative electrode Positive electrode Separator e- e- Separator Lithium Metal Oxide Graphite - + Li+ “Current” x=0 x=ln x=ln+ls x=ln+ls+lp ( ) 0 : balance e electrolyt Overall 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 = + + + + = + + dt c d l dt c d l dt c d l c l l l A c Al c Al c Al p p p s s s n n n p p s s n n p p p s s s n n n          Interfacial fluxes appear in the volume averaged equations, hence they are identified as physically relevant variables. 0 0 0 0 0 : discharge For 0 2 0 2    − =    =  = = t p p p p t n n n n dt c d Fl a I j dt c d Fl a I j I 0 = + p p p n n n j l a j l a
21 Lowest order approximation in negative electrode ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t f l D l q c l x t f x D l q t x c x D l q x c x c x l q x c D t f t f l l q q l x t f x l q x c D l q x c D x l q x c D x j l a t q dt c d l j t a x c D x t c n n n in in n n n in n n in x n in n n n in in n n in n n in n n in n n n n in n n n n n n n + − = = + − = − =   =    − =   =  + − = − = + − =   −            −            − + − = − +           =   = + + 2 : @ 2 , 0 Satisfies 0 : @ 1 1 2 2 2 2 2 2 2 2 2 2 2 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2   ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n in in n n n in n n in n n in n n in n in n n n n in in n n n n l x n n n n n in in j l a t q dt dq D l dt dc l dt dq D l dt dc dt c d D q l c c l D l q c t x c l l l x dx x f l f x l D l t q t c t x c n + = − + − = + + = + = + = = =  = − + =  1 3 3 balance. e electrolyt overall the in later used be Will 3 3 2 2 , 3 3 1 1 2 , 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 0 2 2 2 2 2 2   n s p
Lowest order approximation in positive electrode ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t f l D l q c l l x t f x L D l q c x L D l q x c q x c D x L l q x c D t f L l q L x t f x l q x c D l q x c D x l q x c D x j l a t q dt c d l j t a x c D x t c p p p ip ip s n p p ip p p ip ip l l x p p ip p p ip p ip p p ip p p ip p p p p ip p p p p p p p s n + = + = + − = − − =   =   −  − − =   + = = + =                         − + = − +           =   + = + + 2 : @ 2 Satisfies 0 : @ 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2   ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( ) p p p ip ip p p p ip p p ip p p ip p p ip p ip p p p p ip ip p p L l l x p p p p p ip ip j l a t q dt dq D l dt dc l dt dq D l dt dc dt c d D q l c c l D l q c t x c l x L dx x f l f x L l D l t q t c t x c s n + + = − + = − − = − = − = = −  = − − − =  1 3 3 3 3 2 2 , 3 1 2 , 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2   n s p 22
Electrolyte concentration profile in the separator 23 ( ) ( ) ( ) ( ) ( ) ( ) ( ) t f l D q c l x t f l x D l q q x D q c l x l D q q D q x c q x c D l x l q q q x c D t f l l q q q l x t f x l q q x c D l q q x c D x l q q x c D x q q dt c d l x c D x t c n s in in n n s s ip in s in n s s ip in s in ip l l x s n s ip in in s n s ip in in n s ip in s s ip in s s ip in s ip in s s s s s s n + − = = + −         − + − = −         − + − =   =   −  −         − + − =   +         − = − = +         − =   −             −            − =           =   + = 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 : @ 2 Satsifies : @   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ip in ip s s s in s s s in s s ip s s in s s in s s ip s s in s in s s ip in s s in s in s s s s s n s s s s n l l l x s s s ip in s ip in s ip s s in s in ip s ip in s s in s in ip s n n s s ip in n s in in q q dt dq D l dt dq D l dt dc l dt dq D l dt dq D l dt dc dt c d D q l D q l c c D q q l D q l c c l l l l x l l l l x dx x f l f D q q l c c D q l D q l c c D q q l D q l c c l l x l x D l q q l x D q c t x c s n n 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 3 6 3 6 3 6 2 3 3 1 , 2 2 1 1 2 2 2 2 : @ 2 , − = − − − − = − − = − + − = = = − = = −  = + = − − − = − + − = + = −         − + − − =  + =    n s p An inter-relationship between the four internal variables, at any time.
Overall electrolyte balance – Lowest order approximation 24 ( ) ( ) ( ) 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 6 3 3 3 3 ; 6 3 ; 3 c l l l q D l D l c l q D l D l c l l D q l c c D q l D q l c c D q l c c c l l l c l c l c l p p s s n n ip s s s p p p ip p p in s s s n n n in s s n n p ip p ip p s ip s s in s in s n in n in n p p s s n n p p p s s s n n n                 + + =         + − +         − + + − = − − = + = + + = + + `` ( ) ( ) ( ) ( ) ( ) 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 3 3 3 2 1 3 1 3 c l l l q D l D l c l q D l D l c l l D q q l c c j l a t q dt dq D l dt dc l j l a t q dt dq D l dt dc l p p s s n n ip s s s p p p ip p p in s s s n n n in s s n n s ip in s ip in p p p ip ip p p p ip p p n n n in in n n n in n n               + + =         + − +         − + + + = − − + = − − + − = + + + Four unknowns & Four equations ( ) ( ) ( ) ( ) t q t c t q t c ip ip in in 2 2 2 2 , , ,
(4 unknowns x 4 equations) → (2 unknowns x 2 equations) 25 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) p p p ip ip p p p ip p p in in p p n n n in ip s n s n ip n n in n n n s n n s in n n ip s s ip in s s in in ip s s in s s ip in ip ip in in ip p p s s n n p p p s s s s n s n ip p p s s n n n n n s s s s n s n in ip ip in in ip p p s s n n ip s s s p p p ip p p in s s s n n n in s s n n s ip in s ip in s ip in s ip in p p p ip ip p p p ip p p n n n in in n n n in n n j l a t q dt dq D l l dt dq l j l a t q dt dq D l l l dt dq D l D l l l dt dq D l dt dq D l dt dc dt dq D l dt dq D l dt dc dt dc dt dq dt dq dt dc l l l D l D l D l l l l l D l D l D l l q q c c c l l l q D l D l c l q D l D l c l l D q q l c c D q q l c c j l a t q dt dq D l dt dc l j l a t q dt dq D l dt dc l + + + + − + =         − + − + − =         + +         + +         + +         + =  + + = + = + +         − + − = + +         + + − = + + = + + =         + − +         − + + + + =  + = − − + = − − + − = + 1 3 1 2 3 2 2 2 2 2 1 3 3 2 ; 1 3 6 2 6 3 3 3 2 2 1 3 1 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2                                               `` ( ) ( ) ( ) ( ) t q t c t q t c ip ip in in 2 2 2 2 , , , ( ) ( ) t q t q ip in 2 2 ,
(III) Total current balances in n, s, & p regions 26 n s p
Total current balance Positive electrode Negative electrode Separator e- e- Separator Lithium Metal Oxide Graphite - + Li+ “Current” x=0 x=ln x=ln+ls x=L=ln+ls+lp ( ) ( ) ( ) ( ) t I x t t l x I x c t F RT x x in l x n in n x n n n n n 2 2 2 2 2 0 2 2 2 2 2 2 1 1 and , , 0 : BCs ln 1 2 =    −  =  =    − =   − +    −    − = = +      ( ) ( ) ( )+ − + − + = + = = = +    − =    −    − =    − =   − +    − s n s n n n l l x p l l x s l x s l x n s s x x x x I x c t F RT x 2 2 2 2 2 2 2 2 2 2 2 2 and : BCs ln 1 2       ( ) ( ) ( ) 0 and : BCs ln 1 2 2 2 2 2 2 2 2 2 2 2 1 1 =    −    − =    − =   − +    −    − = + = + = + + − L x p l l x p l l x s p p p x x x I x c t F RT x x s n s n       27 0 : Newman Reference, 2 / 2 =  + = s n l l x Ohmic current Migrational current Diffusional current 0 : Venkat 0 : Comsol : styles Reference Other 2 0 1 =  =  = = L x x Electrolyte potential is a continuous field through the three regions. Hence, liquid potential and migrational current continuities apply at the interfaces. ( ) + − =  t F RT 1
Liquid potential in the separator 28 ( ) ( ) ( )   ( ) ( ) ( ) ( )             + − −        =  +       + −  =       +    + −  =  −    =    =    +    − + = 2 , ln 2 , 2 ln 2 0 , 2 0 : Reference ln 2 , ln 2 ln 2 2 2 2 2 2 2 2 2 2 / 2 2 2 2 2 2 2 2 2 2 2 s n s mid s n s mid s n mid l l x s s s s l l x I t c t x c t x t f l l I t c t l l c c t f x I c t x I x c x I x c x s n       ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t c t q D t I x t I t c t q D t I x t I x c c I x ip ip s s l l x s ip in in s s l x s in s s s n n 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2  + =    − =  + =    − =    − =    − + = =       Liquid current & potential continuities ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) s s mid ip s n ip s s mid in n in l t I t c t c t l l t l t I t c t c t l t 2 2 2 2 2 2 2 2 2 2 2 ln 2 , 2 ln 2 ,   −        = +  =  +        =  =  ( ) ( ) ( ) ( ) ip s s in s s in s n mid n s s ip in n s in in q D l q D l c t l l c t c l x D l q q l x D q c t x c 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 8 8 3 , 2 2 , − − =       + = −         − + − − =
Liquid potential in the negative electrode 29 ( ) ( )   ( ) ( )   ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n in in n n n n n in in n in n in n n n n n n n n n n n n n n n n n n n n brug n n n brug n n l t I t c t x c t t x x l l t I x l t I t c t x c t t x t f l t I c l x t f Ix c x l l t I I c x l l t I x I x c x x l l t I x x l l t I x I x c x x t c t t x c t 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 2 1 1 2 2 2 2 2 2 2 2 2 , 0 ln 2 , 0 2 , ln 2 , ln 2 : At ln 2 2 ln 2 2 ln 2 2 ln 2 ,                      +         =  +  = =  − − − +          +  =  + =  +  − = + =  +  − − − =        +  − − −   =    +    −       − −   − =    − =    +    −    −    =
Liquid potential in the positive electrode 30 ( ) ( )   ( ) ( )   ( ) ( ) ( )   ( ) ( )   ( ) ( ) ( )   ( ) ( ) ( )   ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )   ( ) ( )   ( ) ( ) ( ) ( ) ( ) p p ip ip s n p p s n p ip ip s n ip p ip p s n p p s n p p p s n p p p s n p s n p p p p p brug p p p brug p p l t I t c t L x c t t L x l l x l t I l l x t I t c t x c t t x t f l l t I c l l x t f x t I c l l x l t I t I c l l x l t I x t I x c x l l x l t I x l l x l t I x t I x c x x t c t t x c t 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 2 1 1 2 2 2 2 2 2 2 2 2 , ln 2 , 2 , ln 2 , ln 2 : At ln 2 2 ln 2 2 ln 2 2 ln 2 ,                      −         =  +  = =  + − + + − −          +  =  + + =  +  − + = + =  +  − + − =            +  − + −   =    +    −           + −   + − =    − =    +    −    −    =
(IV) Diffusion approximations & Solid phase potential in n & p regions 31 n s p
Solid state diffusion within the electrode particles ( ) k r r k k r k k k k k k k j r c D r c D c r c p n k r c D r r r t c k =   − =   − = =           =   = = 1 1 0 1 1 0 1 1 1 1 2 2 1 and 0 : BCs , 0 , : IC . , ; 1 32 e- e- Separator Lithium Metal Oxide Graphite - + Li+ “Current” x=0 x=ln x=ln+ls x=ln+ls+lp Negative and Positive electrodes Volume averaged diffusion eqns ( ) ( )   = = = = k k r r k k r r dr r f r r r dr r f r f 0 2 3 3 0 2 3 3 4 4   flux. averaged volume electrode with particle electrode an in ion concentrat average of Evolution 3 : averaging volume electrode with However, . of functions are and general In 3 3 3 1 3 3 3 1 1 1 0 1 1 2 3 0 2 1 1 2 2 3 1 0 1 2 3 0 2 1 3 k k k k k k k k k k r r k k k r r k k k k r r k k r r k k r j dt c d x j c r j dt c d r j r c D r r dr r r c D r r r r RHS dt c d dr c r r t dr r t c r LHS k k k k − = − = − =         =           = =   =   = = = = =   
Solid diffusion – Parabolic approx 33 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 5 3 0 2 2 3 2 0 2 3 2 2 1 1 2 2 1 1 2 1 2 1 1 1 1 1 1 3 1 1 2 3 1 1 2 2 1 1 2 1 1 2 2 1 1 2 2 5 3 5 3 3 3 2 2 2 : @ 2 0 0 : 0 @ 3 3 3 3 1 3 1 k k k r r k r r k k k k k sk k k k k k sk k k k k k sk k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k r r r dr r r r r dr r f r r f r r r D j c c r r r D j c c t f r r D j c r r t f r r D j c r r D j r c r r D j r c r r j r c D r t f r t f r r j r c D r r r j r c D r r r j r c D r r r r j r c D r r r k k = = =  = − + = − = − + − = = + − =  − =   − =    − =   + − = = + − =   − =           −            − =             = = ( ) ( ) k k k k k k k k k sk k k k sk k k k k k sk k k k k k k sk k r j dt c d t j t x j D r j c c D r j c c r r D j c c r r r D j c c 3 , 5 5 5 2 2 5 3 2 1 1 1 1 1 2 1 1 2 2 1 1 − =  − = + =       + =       − + =
Solid diffusion – Quartic approx (I) 34 ( ) ( ) earlier. as Same . 5 3 3 5 3 : @ 5 3 5 3 0 0 : 0 @ 5 3 3 5 3 1 5 3 5 3 3 1 3 1 2 3 2 3 1 1 2 5 3 1 1 2 2 5 3 1 1 2 2 4 2 1 1 2 1 1 2 2 2 5 3 0 2 2 3 2 2 2 1 1 2 2 1 1 2 2 B A r j r r B r A j r r r r B r A r c D r r B r A r c D r t f r t f r r B r A r c D r r r B Ar r c D r r r j B A r c D r r r r r r dr r r r r r r B A r c D r r r r j r c D r r r k k k k k k k k k k k k k k k k k k k k k k k k k k r r k k k k k k k k k + = − + = − = + =   + =   + − = = + + =   + =           − = + =           = = = + =           − =           = kr k k k k kr k k k k kr k k k k k k kr k k k k k k k k k k k kr k kr k k k r r k k k k r r k k k k k k k k c r D j r B c r D j r A B c r D j r B A j r B A c r D D Br D Ar r r D B r D A c r c c r r r dr r r r r r r r dr rr r r r r D B r D A r c r r B r A r c D k k 1 1 1 1 1 1 1 1 1 1 3 2 1 1 1 1 1 3 6 3 0 2 3 3 3 4 3 0 2 3 3 2 1 1 1 2 3 1 1 20 15 12 6 5 1 4 3 5 3 3 5 2 4 10 4 2 1 5 4 3 3 2 1 6 3 3 4 3 4 3 3 5 3 5 3 − − = + = = − − + = − + = + = + =    = = = = = = + =   + =     = =
Solid diffusion – Quartic approx (II) 35 ( ) ( ) ( ) ( ) ( ) ( ) B D r A D r c c r r r D B r r D A c c r r r dr r r r r r r r dr r r r r r r r D B r r D A c c r r r D B r r D A c c t f r r D B r D A c r r t f r r D B r D A c r r D B r D A r c r r B r A r c D k k k k sk k k k k k k k k sk k k k k r r k k k k r r k k k k k k sk k k k k k k sk k k k k k k sk k k k k k k k k k k k k k k 1 2 1 2 1 4 4 2 1 2 2 1 1 4 7 3 0 2 4 3 4 2 5 3 0 2 2 3 2 4 4 2 1 2 2 1 1 4 4 2 1 2 2 1 1 4 2 1 2 1 4 2 1 2 1 1 3 2 1 1 1 2 3 1 1 35 15 7 3 20 5 3 6 7 3 7 3 3 5 3 5 3 3 20 6 20 6 20 6 : @ 20 6 5 3 5 3 − − =       − −       − − = = = = = = = − − − − = − − − − = − + + = = + + = + =   + =     = = r c r j D r c c r c r j D r c c r c r D j r D r c c r c r D j r B A r c r D j r B r c r D j r B r c r D j r A B A D r c c k k k k k k sk k k k k k sk k k k k k k k k sk k k k k k k k k k k k k k k k k k k k k k k k sk k   + − =   − + =           + − − =   + − = +   − − =   − − =   + =       + − = 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 1 35 8 35 35 8 35 7 24 7 3 15 7 24 7 3 7 3 7 60 7 45 7 3 20 15 12 6 7 3 15
Solid diffusion – Quartic approx (III) 36 k kr k k k k k k k k r r k k k k k k k k k k k k k k k k k k k r B dt c d r r B r c dt d r r B r c t r r r dr rr r r r r B r c t r r B r c D r r r r r r B A r c D r r r r c D r r r r r c t t c r r c D r r r t c r r c D r r r t c k 2 3 4 3 2 2 4 3 4 3 3 2 2 1 1 1 1 1 1 2 1 2 1 4 3 0 2 3 2 1 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 1 1 1 1 2 2 1 1 1 2 2 1 = =           =           = = = =           =                   + =                             =           =                           =               =   = k k kr k k kr k k k k k k k k k k k k k k k k k k k k j r c r D dt c d r c r D j r r c dt d r c r D j r B r r c dt d r B r c dt d r c r D j r B 2 1 2 1 1 1 2 1 2 1 1 2 1 2 1 1 1 1 2 45 30 30 2 45 30 2 45 2 3 2 3 20 15 − = +   − − =             − − = =           =             − − =
Uniform rate ROM – Model overview 37 ( ) ( ) ( ) n n n n Fl a t I t j t j =  ( ) ( ) ( ) p p p p Fl a t I t j t j − =  ( ) ( ) ( ) ( )         + =  + = =          = −  −        −  −  = − = − − 0 1 2 1 0 1 2 1 2 1 0 5 . 0 2 5 . 0 5 . 0 max 0 2 sinh 2 , 0 , 0 2 sinh 2 2 sinh 2 n n n n n n n n n sn sn n s n n j j F RT t x U t x j j F RT U U RT F j j c c c c k j 2 2 1 1 1 2 45 30 n n n rn n rn r j r c D dt c d − = + n n n rn n n sn D r j c r c c 1 1 1 35 35 8 − + = p p p r j dt c d 3 1 − = 2 2 1 1 1 2 45 30 p p p rp p rp r j r c D dt c d − = + p p p rp p p sp D r j c r c c 1 1 1 35 35 8 − + = ( ) ( ) ( ) ( )         + =  + = =          = −  −        −  −  = − = − − 0 1 2 1 0 1 2 1 2 1 0 5 . 0 2 5 . 0 5 . 0 max 0 2 sinh 2 , , 2 sinh 2 2 sinh 2 ; p p p p p p p p p sp sp p s p p j j F RT t L x U t L x j j F RT U U RT F j j c c c c k j Positive electrode Negative electrode Diffusion approximations ( ) ( ) ( ) n n n n sn D r t j t c t c 1 1 5 − = ( ) ( ) ( ) p p p p sp D r t j t c t c 1 1 5 − = Parabolic Quartic Parabolic Quartic n n n r j dt c d 3 1 − =

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Uniform_Reaction_Rate_ROM_slides_2013.pdf

  • 1. Model reduction concepts & Voltage predictions Senthil Kumar V. Chemical Reactions Modeling India Science Lab, GM Global R&D 1 Reduced order model for a single cell - with uniform reaction rate approximation
  • 2. Detailed 1D Electro-chemical model for a single cell e- e- Separator Lithium Metal Oxide Graphite - + LiC6 →x Li++Li1-xC6+x e- x Li++ LiMO2 +x e- → Li1+xMO2 Li+ Discharge of a Lithium Metal Oxide cell “Current” Loss of electrons  Oxidation  Anode Gain of electrons  Reduction  Cathode Load 2 Solid Electrolyte Electrolyte Solid Electrolyte 5 phases in 3 regions (n, s & p) Isothermal 1D model: Mass and Charge balances for each phase. 5 phases x 2 balances =10 PDEs Mass balance: Diffusion equation Charge balance: Ohm’s law & Current balance Negative electrode n n n Fj a x x − =          −   1 1  ( ) 1 2 2 2 2 n n n n j t a x c D x t c + − +           =    ( ) I x c t F RT x x n n n =   − +    −    − + 2 2 2 2 1 1 ln 1 2             =   r c D r r r t c n n n 1 1 2 2 1 1 Solid phase current balance Liquid phase mass balance Total current balance Solid phase mass balance Separator           =   x c D x t c s s 2 2 2 2  ( ) I x c t F RT x s s =   − +    − + 2 2 2 2 ln 1 2  Liquid phase mass balance Total current balance Positive electrode p p p Fj a x x − =          −   1 1              =   r c D r r r t c p p p 1 1 2 2 1 1 ( ) p p p p j t a x c D x t c + − +           =   1 2 2 2 2  ( ) I x c t F RT x x p p p =   − +    −    − + 2 2 2 2 1 1 ln 1 2   Solid phase current balance Liquid phase mass balance Total current balance Solid phase mass balance
  • 3. Need for Model Reduction • The single cell isothermal model has 10 coupled PDEs. Difficult to use for packs, on-board control , calendar / cycle life predictions, detailed parameters estimation, inclusion of complex degradation mechanisms etc. • The reduced order model should preferably comprise of first order ODEs. Such a mathematical structure will automatically ensure minimal computational requirement and numerical stability. If the first order ODEs turn out to be linear, optimization during parameter (re-)estimations become robust. • Equivalent circuit based single cell models are popular for on-board control due to their first order ODEs structure, and the consequent properties. Detailed electrochemical model should be reduced to a similar structure to make the physics based model attractive for on-board applications. • Need a first order ODEs based model which – Is grid / system size independent: Unlike Method of lines or PDEs discretized into ODEs. – Has physically relevant internal variables: Like Lyapunov-Schmidt model reduction (here: interfacial electrolyte fluxes and concentrations). – Has systematic mathematical approximations rather than phenomenological approximations: Unlike neglecting electrolyte potential /concentration variations - Single particle model. 3 n s p
  • 4. `` `` Model reduction methodology (I): Volume averaging 4 • Volume averaging a PDE yields an ODE. Hence volume averaging is the chosen methodology for model reduction. On volume averaging information about spatial gradients or profiles are lost. This information is recovered by profile based approximation in the next step. ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n in n n n n n n in n n n n n n n n in l x n n n l x n n n n l x n n l x n n n l x n n l x n n n n n j l a t q dt c d l j a t l q dt c d j t a j t a RHS l q x c D l dx x c D x l RHS dt c d dx c l dt d dx t c l dx x f l Al dx x f A f j t a x c D x t c n n n n n n + + + + = = = = = = + − + − = − + − = − = − = − =         =           = =         =   = = = − +           =        1 1 1 1 2 1 1 1 1 LHS 1 1 2 2 2 2 2 2 2 0 2 2 2 2 0 2 2 2 0 2 0 2 2 0 0 2 2 2 2       e- e- Separator Lithium Metal Oxide Graphite - + LiC6 →x Li++Li1-xC6+x e- x Li++ LiMO2 +x e- → Li1+xMO2 Li+ Discharge of a Lithium Metal Oxide cell “Current” Load Negative electrode Accumulation of Li+ in a slice of n-electrode = Difference in diffusive out-flow and in-flow + Generation of Li+ in that slice. Accumulation of Li+ in the whole of n-electrode = - Out flux from the interface + Total generation of Li+ within the n-electrode. Negative electrode q2in Interfacial flux is a physically relevant internal variable
  • 5. `` `` Model reduction methodology (II): Profile based approximations 5 • The profile approximations are constructed such that volume averages are respected. The profiles are expressed in terms of physically relevant internal variables. Eventually equations (ODEs or algebraic equations) are derived for these internal variables. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n in in n n n in n n n n n in n n n in n n in n n n n n l x n n n n n in in n in n n in n n j l a t q dt dq D l dt dc l j l a t q dt c d l t q D l t c t c l l l x dx x f l f x l D l t q t c t x c l q x c D x l q x c D x n + + = − + − = + − + − = + = = =  = − + = −            −             1 3 1 3 3 3 1 1 2 , 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 0 2 2 2 2 2 2 2 2 2 2 2 2    2. Volume average respecting lowest order profile approximation 1. Exact result from volume averaging 3. Profile is expressed in terms of interfacial flux & concentration 4. Average concentration depends only on interfacial flux & concentration 5. Volume averaged diffusion equation, exact result 6. An evolution equation for the interfacial flux & concentration n s p c2in & q2in
  • 6. 6 Nonlinear PDEs Linear ODEs Nonlinear algebraic expressions Approximations Consistent Reaction rate Butler – Volmer Kinetics Electrolyte concentration field Electrolyte potential field Solid potential field Solid phase concentration field Electrolyte phase Solid phase Linear ODEs Algebraic expressions Linear ODEs Algebraic expressions Algebraic expressions Uniform reaction rate from external current Diffusion approx as in Single particle model Model reduction methodology (III): Smart decomposition 1. Nonlinearity in a system is not bothersome if it can be relegated to algebraic evaluations. 2. Coupled linear ODEs can be solved by time marching without iteration, unlike non-linear coupled PDEs which require iteration to achieve a consistent solution. It is better to have a time-marchable structure for on-board algorithm. ( ) ( ) p p p Fl a t I t j − = ( ) ( ) n n n Fl a t I t j =
  • 7. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 , ln 2 , 2 ln 2 , 2 , 2 1 3 3 2 ; 1 3 6 2 x l l t I x l t I t c t x c t t x l t I t c t c t l t x l D l t q t c t x c D q q l c c q q c c l l l D l D l D l l l l l D l D l D l l n n n n n in in s s mid in n in n n n in in s ip in s ip in ip ip in in ip p p s s n n p p p s s s s n s n ip p p s s n n n n n s s s s n s n in − − − +          +  =  +        =  =  − + = + + = + + = + +         − + − = + +         + + − =                    Model reduction – Bird’s eye view 7 ( ) ( ) p p p ip ip p p p ip p p in in p p n n n in ip s n s n ip n n in n n n s n n s in n n j l a t q dt dq D l l dt dq l j l a t q dt dq D l l l dt dq D l D l l l + + − + =         − + − + − =         + +         + + 1 3 1 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2             p p p R j dt c d 3 1 − = 2 2 1 1 1 2 45 30 n n n rn n rn R j R c D dt c d − = + 2 2 1 1 1 2 45 30 p p p rp p rp R j R c D dt c d − = + n n n R j dt c d 3 1 − = 1. Two coupled ODEs for the interfacial fluxes, obtained by volume averaging and profile based approximations. 2. Quartic diffusion approximations for diffusion within active material spheres. Two coupled ODEs for each electrode. 3. Rest are algebraic expressions 5. Model reduction achieved with uniform reaction rate approximation: 10 coupled PDEs to 6 linear ODEs. 4. ROM predicts the internal field profiles too, not just the cell voltage. Negative Positive 2 ODEs 4 ODEs
  • 8. 1C discharge prediction by uniform rate ROM 8 Fig. 1 Max abs error  9 mV Fig. 2 Fig. 3 Max % error  0.27 %
  • 9. Max % error  0.2% Fig. 4 Fig. 2 Voltage recovery at the beginning of rest periods are predicted well. Fig. 3 Max abs error  10 mV 2000 s 1C discharge, Then 300 s rest, Then 2000s 1C charge, Then rest till 8000 s. 1C pulse prediction by uniform rate ROM 9 Fig. 1
  • 10. 7C discharge prediction by uniform rate ROM 10 Model enhancement underway to reduce errors at high current discharge & charge. Fig. 1 Max abs error  80 mV Fig. 2 Fig. 3 Max % error  3.2%
  • 11. Max % error  2% Fig. 4 Model enhancement underway to reduce errors at high current discharge & charge. Fig. 2 Max abs error  80 mV Voltage recovery at the beginning of rest periods are predicted well. Fig. 3 10 s 7C discharge, Then 40 s rest, Then 10s 7C charge, Then 60 s rest. 7C pulse prediction by uniform rate ROM 11 Fig. 1
  • 12. Back up: Detailed derivation 12
  • 13. Electro-chemical reactions in a Lithium Metal Oxide cell e- e- Separator Lithium Metal Oxide Graphite - + LiC6 →x Li++Li1-xC6+x e- x Li++ LiMO2 +x e- → Li1+xMO2 Li+ Discharge of a Lithium Metal Oxide cell “Current” Loss of electrons  Oxidation  Anode Gain of electrons  Reduction  Cathode Load e- e- Separator Lithium Metal Oxide Graphite - + x Li++Li1-xC6+x e- → LiC6 Li1+xMO2 → x Li++ LiMO2 +x e- Li+ Charge of a Lithium Metal Oxide cell “Current” Loss of electrons  Oxidation  Anode Gain of electrons  Reduction  Cathode Charger Compound Oxidation state of M Check LiMO2 +3 +1+3-2*2 = 0 Li2MO2 +2 +1*2+2-2*2 = 0 Li1+xMO2 y = +3-x +1*(1+x)+y-2*2 = 0 M: Co, Ni, Mn Manganese can have +2 to +7 oxidation states LiMn2O4  Mn+3.5 , KMnO4  Mn+7 Basic mnemonics LEO – Loss of Electron is Oxidation GER – Gain of Electron is Reduction AnOx – Anode is where Oxidation occurs RedCat – Cathode is where Reduction occurs Process – Electrode Energy consumer Energy producer Reduction – Cathode - + Oxidation - Anode + - 13
  • 14. (I) Solid phase current balances in n & p regions 14 n s p
  • 15. Solid phase current balance equations e- e- Separator Lithium Metal Oxide Graphite - + Li+ “Current” x=0 x=ln x=ln+ls x=L=ln+ls+lp 15 Positive electrode Negative electrode 0 and : BCs 1 1 0 1 1 1 1 =    − =    − − =          −   = = n l x n x n n n n x I x Fj a x x    I x x Fj a x x L x p l l x p p p p s n =    − =    − − =          −   = + = 1 1 1 1 1 1 and 0 : BCs    Convention: Discharge current is positive. Subscripts Notation: n – Negative electrode p – Positive electrode s – Separator 1- solid phase 2- liquid phase ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ρ S E J E E J R V V A I D m 1 m m V/m A/m2 : Continuum : 1 1 -  =   − =   − = =   − =  =   =      • Cell reactions generate or consume electrons. Hence the gradient of current is proportional to the reaction rate.                     − =                −   s m mol j mol C F m m a m A x x 2 3 2 3 1  Separator has ionic conduction only.
  • 16. 16 Volume averaging of solid phase current balances Negative electrode: 0 and : BCs 1 1 0 1 1 1 1 =    − =    − − =          −   = = n l x n x n n n n x I x Fj a x x    ( ) ( ) n n n n n n n n n l x n l x n n l x n Fl a t I t j l j F a I l j F a x dx j F a x d n n n =  = − =          −  − =          − = = =   0 1 1 0 0 1 1   Positive electrode: I x x Fj a x x L x p l l x p p p p s n =    − =    − − =          −   = + = 1 1 1 1 1 1 and 0 : BCs    ( ) ( ) p p p p p p p p p l l l l l x p l l l l l x p p l l l l l x p Fl a t I t j l j F a I l j F a x dx j F a x d p s n s n p s n s n p s n s n − =  − = − =          −  − =          − + + + = + + + = + + + =   1 1 1 1   e- e- Separator LMO Graphite - + Li+ “Current” x=0 x=ln x=ln+ls x=L=ln+ls+lp ( ) ( ) ( ) ( )     + = + = = = = = = = L l l x p p L l l x p l x n n l x n s n s n n n dx x f l Al dx x f A f dx x f l Al dx x f A f 1 1 0 0 ( ) ( ) ( ) ( ) ( ) ( ) 0 = + − = = t j l a t j l a F t I t j l a F t I t j l a p p p n n n p p p n n n Volume averaging results are exact, but information on spatial variations are lost. No accumulation of electrons. Hence average rates are coupled at any instant.
  • 17. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) I x x l l t I x t f l l t I x t f x l t I x l t I t j F a x x Fj a x x t j t x j Fl a t I t j x n n n n n n l x n n n n n n n n n n n n n n n n =    − − =    − + − =  =    − + − =    − − = −           −   − =          −    = = = 0 1 1 1 1 1 1 1 1 1 1 1 1 Satisfies 0 0 , : ion approximat rate reaction Uniform       17 Solid phase current in electrodes ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )   ( ) t I x l l x l t I x t f l l l t I x t f x l t I x l t I t j F a x x Fj a x x t j t x j Fl a t I t j L x p s n p p s n p l l x p p p p p p p p p p p p p p p s n =    − + − =    − + + =  =    − + =    − = −           −   − =          −    − = = + = 1 1 1 1 1 1 1 1 1 1 1 1 Satisfies 0 0 , : ion approximat rate reaction Uniform       Uniform reaction rate approximation will recover the volume average evolution equations. It decouples the potential and concentration fields, enabling a step-by-step solution of all the fields. The volume average contributions give the leading order transient solution, which contains the steady state solution within it. It is possible to derive corrections by introducing the deviations from volume average. Corrections will be attempted in future. ( ) ( ) ( ) t x j t j t x j d n n n , , + =
  • 18. (II) Liquid phase mass balances in n, s, & p regions 18 n s p
  • 19. Liquid phase mass balance (or electrolyte diffusion) equations ( ) ( ) ( )   ( ) 0 and , , : BCs , 0 , : IC 1 2 2 2 2 2 2 2 20 2 2 2 2 2 =   − =   − = + = − +           =   = + = + + + L x p ip l l x p ip s n p p p p x c D q x c D c t l l c c x c j t a x c D x t c s n  Positive electrode ( ) ( ) ( ) , , , 0 : BCs , 0 , : IC 1 2 2 2 2 2 0 2 2 20 2 2 2 2 2 in l x n in n x n n n n n q x c D c t l c x c D c x c j t a x c D x t c n =   − = =   − = − +           =   − = − = +  Negative electrode ( ) 20 2 2 2 2 2 0 , : IC c x c x c D x t c s s =           =    Separator e- e- Separator Lithium Metal Oxide Graphite - + Li+ “Current” x=0 x=ln x=ln+ls x=L=ln+ls+lp 19 Concentration and flux continuity conditions at N-S interface ( ) ( )   ( ) . , , , , , : BCs 2 2 2 2 2 2 2 2 2 2 ip l l x s ip s n in l x s in n q x c D c t l l c q x c D c t l c s n n =   − = + =   − = − + + = − = + Continuity conditions at N-S interface Continuity conditions at S-P interface Concentration and flux continuity conditions at S-P interface Electrolyte concentration is a continuous field through negative electrode, separator and positive electrode regions.
  • 20. Volume averaged Li mass balance in electrolyte phase 20 ( ) ( )   = = = = n n l x n n l x n dx x f l Al dx x f A f 0 0 1 ( ) ( ) ( ) ( ) ( ) ( ) p p p ip p p p ip in s s s n n n in n n n n n n in n n l x n n n n n l x n n n l x n n l x n n l x n n n n n j l a t q dt c d l q q dt c d l j l a t q dt c d l j t a l q j t a x c D l j t a dx x c D x l RHS dt c d dx c l dt d dx t c l dx j t a x c D x t c l n n n n n + + + + = + = = = = + − + = − = − + − = − + − = − +         = − +           = =         =   =       − +           =       1 1 1 1 1 1 1 1 LHS 1 1 2 2 2 2 2 2 2 2 2 2 2 0 2 2 2 2 0 2 2 2 0 2 0 2 2 0 2 2 2 2        `` Negative electrode Positive electrode Separator e- e- Separator Lithium Metal Oxide Graphite - + Li+ “Current” x=0 x=ln x=ln+ls x=ln+ls+lp ( ) 0 : balance e electrolyt Overall 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 = + + + + = + + dt c d l dt c d l dt c d l c l l l A c Al c Al c Al p p p s s s n n n p p s s n n p p p s s s n n n          Interfacial fluxes appear in the volume averaged equations, hence they are identified as physically relevant variables. 0 0 0 0 0 : discharge For 0 2 0 2    − =    =  = = t p p p p t n n n n dt c d Fl a I j dt c d Fl a I j I 0 = + p p p n n n j l a j l a
  • 21. 21 Lowest order approximation in negative electrode ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t f l D l q c l x t f x D l q t x c x D l q x c x c x l q x c D t f t f l l q q l x t f x l q x c D l q x c D x l q x c D x j l a t q dt c d l j t a x c D x t c n n n in in n n n in n n in x n in n n n in in n n in n n in n n in n n n n in n n n n n n n + − = = + − = − =   =    − =   =  + − = − = + − =   −            −            − + − = − +           =   = + + 2 : @ 2 , 0 Satisfies 0 : @ 1 1 2 2 2 2 2 2 2 2 2 2 2 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2   ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n in in n n n in n n in n n in n n in n in n n n n in in n n n n l x n n n n n in in j l a t q dt dq D l dt dc l dt dq D l dt dc dt c d D q l c c l D l q c t x c l l l x dx x f l f x l D l t q t c t x c n + = − + − = + + = + = + = = =  = − + =  1 3 3 balance. e electrolyt overall the in later used be Will 3 3 2 2 , 3 3 1 1 2 , 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 0 2 2 2 2 2 2   n s p
  • 22. Lowest order approximation in positive electrode ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t f l D l q c l l x t f x L D l q c x L D l q x c q x c D x L l q x c D t f L l q L x t f x l q x c D l q x c D x l q x c D x j l a t q dt c d l j t a x c D x t c p p p ip ip s n p p ip p p ip ip l l x p p ip p p ip p ip p p ip p p ip p p p p ip p p p p p p p s n + = + = + − = − − =   =   −  − − =   + = = + =                         − + = − +           =   + = + + 2 : @ 2 Satisfies 0 : @ 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2   ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( ) p p p ip ip p p p ip p p ip p p ip p p ip p ip p p p p ip ip p p L l l x p p p p p ip ip j l a t q dt dq D l dt dc l dt dq D l dt dc dt c d D q l c c l D l q c t x c l x L dx x f l f x L l D l t q t c t x c s n + + = − + = − − = − = − = = −  = − − − =  1 3 3 3 3 2 2 , 3 1 2 , 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2   n s p 22
  • 23. Electrolyte concentration profile in the separator 23 ( ) ( ) ( ) ( ) ( ) ( ) ( ) t f l D q c l x t f l x D l q q x D q c l x l D q q D q x c q x c D l x l q q q x c D t f l l q q q l x t f x l q q x c D l q q x c D x l q q x c D x q q dt c d l x c D x t c n s in in n n s s ip in s in n s s ip in s in ip l l x s n s ip in in s n s ip in in n s ip in s s ip in s s ip in s ip in s s s s s s n + − = = + −         − + − = −         − + − =   =   −  −         − + − =   +         − = − = +         − =   −             −            − =           =   + = 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 : @ 2 Satsifies : @   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ip in ip s s s in s s s in s s ip s s in s s in s s ip s s in s in s s ip in s s in s in s s s s s n s s s s n l l l x s s s ip in s ip in s ip s s in s in ip s ip in s s in s in ip s n n s s ip in n s in in q q dt dq D l dt dq D l dt dc l dt dq D l dt dq D l dt dc dt c d D q l D q l c c D q q l D q l c c l l l l x l l l l x dx x f l f D q q l c c D q l D q l c c D q q l D q l c c l l x l x D l q q l x D q c t x c s n n 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 3 6 3 6 3 6 2 3 3 1 , 2 2 1 1 2 2 2 2 : @ 2 , − = − − − − = − − = − + − = = = − = = −  = + = − − − = − + − = + = −         − + − − =  + =    n s p An inter-relationship between the four internal variables, at any time.
  • 24. Overall electrolyte balance – Lowest order approximation 24 ( ) ( ) ( ) 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 2 6 3 3 3 3 ; 6 3 ; 3 c l l l q D l D l c l q D l D l c l l D q l c c D q l D q l c c D q l c c c l l l c l c l c l p p s s n n ip s s s p p p ip p p in s s s n n n in s s n n p ip p ip p s ip s s in s in s n in n in n p p s s n n p p p s s s n n n                 + + =         + − +         − + + − = − − = + = + + = + + `` ( ) ( ) ( ) ( ) ( ) 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 3 3 3 2 1 3 1 3 c l l l q D l D l c l q D l D l c l l D q q l c c j l a t q dt dq D l dt dc l j l a t q dt dq D l dt dc l p p s s n n ip s s s p p p ip p p in s s s n n n in s s n n s ip in s ip in p p p ip ip p p p ip p p n n n in in n n n in n n               + + =         + − +         − + + + = − − + = − − + − = + + + Four unknowns & Four equations ( ) ( ) ( ) ( ) t q t c t q t c ip ip in in 2 2 2 2 , , ,
  • 25. (4 unknowns x 4 equations) → (2 unknowns x 2 equations) 25 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) p p p ip ip p p p ip p p in in p p n n n in ip s n s n ip n n in n n n s n n s in n n ip s s ip in s s in in ip s s in s s ip in ip ip in in ip p p s s n n p p p s s s s n s n ip p p s s n n n n n s s s s n s n in ip ip in in ip p p s s n n ip s s s p p p ip p p in s s s n n n in s s n n s ip in s ip in s ip in s ip in p p p ip ip p p p ip p p n n n in in n n n in n n j l a t q dt dq D l l dt dq l j l a t q dt dq D l l l dt dq D l D l l l dt dq D l dt dq D l dt dc dt dq D l dt dq D l dt dc dt dc dt dq dt dq dt dc l l l D l D l D l l l l l D l D l D l l q q c c c l l l q D l D l c l q D l D l c l l D q q l c c D q q l c c j l a t q dt dq D l dt dc l j l a t q dt dq D l dt dc l + + + + − + =         − + − + − =         + +         + +         + +         + =  + + = + = + +         − + − = + +         + + − = + + = + + =         + − +         − + + + + =  + = − − + = − − + − = + 1 3 1 2 3 2 2 2 2 2 1 3 3 2 ; 1 3 6 2 6 3 3 3 2 2 1 3 1 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 20 2 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2                                               `` ( ) ( ) ( ) ( ) t q t c t q t c ip ip in in 2 2 2 2 , , , ( ) ( ) t q t q ip in 2 2 ,
  • 26. (III) Total current balances in n, s, & p regions 26 n s p
  • 27. Total current balance Positive electrode Negative electrode Separator e- e- Separator Lithium Metal Oxide Graphite - + Li+ “Current” x=0 x=ln x=ln+ls x=L=ln+ls+lp ( ) ( ) ( ) ( ) t I x t t l x I x c t F RT x x in l x n in n x n n n n n 2 2 2 2 2 0 2 2 2 2 2 2 1 1 and , , 0 : BCs ln 1 2 =    −  =  =    − =   − +    −    − = = +      ( ) ( ) ( )+ − + − + = + = = = +    − =    −    − =    − =   − +    − s n s n n n l l x p l l x s l x s l x n s s x x x x I x c t F RT x 2 2 2 2 2 2 2 2 2 2 2 2 and : BCs ln 1 2       ( ) ( ) ( ) 0 and : BCs ln 1 2 2 2 2 2 2 2 2 2 2 2 1 1 =    −    − =    − =   − +    −    − = + = + = + + − L x p l l x p l l x s p p p x x x I x c t F RT x x s n s n       27 0 : Newman Reference, 2 / 2 =  + = s n l l x Ohmic current Migrational current Diffusional current 0 : Venkat 0 : Comsol : styles Reference Other 2 0 1 =  =  = = L x x Electrolyte potential is a continuous field through the three regions. Hence, liquid potential and migrational current continuities apply at the interfaces. ( ) + − =  t F RT 1
  • 28. Liquid potential in the separator 28 ( ) ( ) ( )   ( ) ( ) ( ) ( )             + − −        =  +       + −  =       +    + −  =  −    =    =    +    − + = 2 , ln 2 , 2 ln 2 0 , 2 0 : Reference ln 2 , ln 2 ln 2 2 2 2 2 2 2 2 2 2 / 2 2 2 2 2 2 2 2 2 2 2 s n s mid s n s mid s n mid l l x s s s s l l x I t c t x c t x t f l l I t c t l l c c t f x I c t x I x c x I x c x s n       ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t c t q D t I x t I t c t q D t I x t I x c c I x ip ip s s l l x s ip in in s s l x s in s s s n n 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2  + =    − =  + =    − =    − =    − + = =       Liquid current & potential continuities ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) s s mid ip s n ip s s mid in n in l t I t c t c t l l t l t I t c t c t l t 2 2 2 2 2 2 2 2 2 2 2 ln 2 , 2 ln 2 ,   −        = +  =  +        =  =  ( ) ( ) ( ) ( ) ip s s in s s in s n mid n s s ip in n s in in q D l q D l c t l l c t c l x D l q q l x D q c t x c 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 8 8 3 , 2 2 , − − =       + = −         − + − − =
  • 29. Liquid potential in the negative electrode 29 ( ) ( )   ( ) ( )   ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n in in n n n n n in in n in n in n n n n n n n n n n n n n n n n n n n n brug n n n brug n n l t I t c t x c t t x x l l t I x l t I t c t x c t t x t f l t I c l x t f Ix c x l l t I I c x l l t I x I x c x x l l t I x x l l t I x I x c x x t c t t x c t 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 2 1 1 2 2 2 2 2 2 2 2 2 , 0 ln 2 , 0 2 , ln 2 , ln 2 : At ln 2 2 ln 2 2 ln 2 2 ln 2 ,                      +         =  +  = =  − − − +          +  =  + =  +  − = + =  +  − − − =        +  − − −   =    +    −       − −   − =    − =    +    −    −    =
  • 30. Liquid potential in the positive electrode 30 ( ) ( )   ( ) ( )   ( ) ( ) ( )   ( ) ( )   ( ) ( ) ( )   ( ) ( ) ( )   ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )   ( ) ( )   ( ) ( ) ( ) ( ) ( ) p p ip ip s n p p s n p ip ip s n ip p ip p s n p p s n p p p s n p p p s n p s n p p p p p brug p p p brug p p l t I t c t L x c t t L x l l x l t I l l x t I t c t x c t t x t f l l t I c l l x t f x t I c l l x l t I t I c l l x l t I x t I x c x l l x l t I x l l x l t I x t I x c x x t c t t x c t 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 2 1 1 2 2 2 2 2 2 2 2 2 , ln 2 , 2 , ln 2 , ln 2 : At ln 2 2 ln 2 2 ln 2 2 ln 2 ,                      −         =  +  = =  + − + + − −          +  =  + + =  +  − + = + =  +  − + − =            +  − + −   =    +    −           + −   + − =    − =    +    −    −    =
  • 31. (IV) Diffusion approximations & Solid phase potential in n & p regions 31 n s p
  • 32. Solid state diffusion within the electrode particles ( ) k r r k k r k k k k k k k j r c D r c D c r c p n k r c D r r r t c k =   − =   − = =           =   = = 1 1 0 1 1 0 1 1 1 1 2 2 1 and 0 : BCs , 0 , : IC . , ; 1 32 e- e- Separator Lithium Metal Oxide Graphite - + Li+ “Current” x=0 x=ln x=ln+ls x=ln+ls+lp Negative and Positive electrodes Volume averaged diffusion eqns ( ) ( )   = = = = k k r r k k r r dr r f r r r dr r f r f 0 2 3 3 0 2 3 3 4 4   flux. averaged volume electrode with particle electrode an in ion concentrat average of Evolution 3 : averaging volume electrode with However, . of functions are and general In 3 3 3 1 3 3 3 1 1 1 0 1 1 2 3 0 2 1 1 2 2 3 1 0 1 2 3 0 2 1 3 k k k k k k k k k k r r k k k r r k k k k r r k k r r k k r j dt c d x j c r j dt c d r j r c D r r dr r r c D r r r r RHS dt c d dr c r r t dr r t c r LHS k k k k − = − = − =         =           = =   =   = = = = =   
  • 33. Solid diffusion – Parabolic approx 33 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 5 3 0 2 2 3 2 0 2 3 2 2 1 1 2 2 1 1 2 1 2 1 1 1 1 1 1 3 1 1 2 3 1 1 2 2 1 1 2 1 1 2 2 1 1 2 2 5 3 5 3 3 3 2 2 2 : @ 2 0 0 : 0 @ 3 3 3 3 1 3 1 k k k r r k r r k k k k k sk k k k k k sk k k k k k sk k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k r r r dr r r r r dr r f r r f r r r D j c c r r r D j c c t f r r D j c r r t f r r D j c r r D j r c r r D j r c r r j r c D r t f r t f r r j r c D r r r j r c D r r r j r c D r r r r j r c D r r r k k = = =  = − + = − = − + − = = + − =  − =   − =    − =   + − = = + − =   − =           −            − =             = = ( ) ( ) k k k k k k k k k sk k k k sk k k k k k sk k k k k k k sk k r j dt c d t j t x j D r j c c D r j c c r r D j c c r r r D j c c 3 , 5 5 5 2 2 5 3 2 1 1 1 1 1 2 1 1 2 2 1 1 − =  − = + =       + =       − + =
  • 34. Solid diffusion – Quartic approx (I) 34 ( ) ( ) earlier. as Same . 5 3 3 5 3 : @ 5 3 5 3 0 0 : 0 @ 5 3 3 5 3 1 5 3 5 3 3 1 3 1 2 3 2 3 1 1 2 5 3 1 1 2 2 5 3 1 1 2 2 4 2 1 1 2 1 1 2 2 2 5 3 0 2 2 3 2 2 2 1 1 2 2 1 1 2 2 B A r j r r B r A j r r r r B r A r c D r r B r A r c D r t f r t f r r B r A r c D r r r B Ar r c D r r r j B A r c D r r r r r r dr r r r r r r B A r c D r r r r j r c D r r r k k k k k k k k k k k k k k k k k k k k k k k k k k r r k k k k k k k k k + = − + = − = + =   + =   + − = = + + =   + =           − = + =           = = = + =           − =           = kr k k k k kr k k k k kr k k k k k k kr k k k k k k k k k k k kr k kr k k k r r k k k k r r k k k k k k k k c r D j r B c r D j r A B c r D j r B A j r B A c r D D Br D Ar r r D B r D A c r c c r r r dr r r r r r r r dr rr r r r r D B r D A r c r r B r A r c D k k 1 1 1 1 1 1 1 1 1 1 3 2 1 1 1 1 1 3 6 3 0 2 3 3 3 4 3 0 2 3 3 2 1 1 1 2 3 1 1 20 15 12 6 5 1 4 3 5 3 3 5 2 4 10 4 2 1 5 4 3 3 2 1 6 3 3 4 3 4 3 3 5 3 5 3 − − = + = = − − + = − + = + = + =    = = = = = = + =   + =     = =
  • 35. Solid diffusion – Quartic approx (II) 35 ( ) ( ) ( ) ( ) ( ) ( ) B D r A D r c c r r r D B r r D A c c r r r dr r r r r r r r dr r r r r r r r D B r r D A c c r r r D B r r D A c c t f r r D B r D A c r r t f r r D B r D A c r r D B r D A r c r r B r A r c D k k k k sk k k k k k k k k sk k k k k r r k k k k r r k k k k k k sk k k k k k k sk k k k k k k sk k k k k k k k k k k k k k k 1 2 1 2 1 4 4 2 1 2 2 1 1 4 7 3 0 2 4 3 4 2 5 3 0 2 2 3 2 4 4 2 1 2 2 1 1 4 4 2 1 2 2 1 1 4 2 1 2 1 4 2 1 2 1 1 3 2 1 1 1 2 3 1 1 35 15 7 3 20 5 3 6 7 3 7 3 3 5 3 5 3 3 20 6 20 6 20 6 : @ 20 6 5 3 5 3 − − =       − −       − − = = = = = = = − − − − = − − − − = − + + = = + + = + =   + =     = = r c r j D r c c r c r j D r c c r c r D j r D r c c r c r D j r B A r c r D j r B r c r D j r B r c r D j r A B A D r c c k k k k k k sk k k k k k sk k k k k k k k k sk k k k k k k k k k k k k k k k k k k k k k k k sk k   + − =   − + =           + − − =   + − = +   − − =   − − =   + =       + − = 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 1 35 8 35 35 8 35 7 24 7 3 15 7 24 7 3 7 3 7 60 7 45 7 3 20 15 12 6 7 3 15
  • 36. Solid diffusion – Quartic approx (III) 36 k kr k k k k k k k k r r k k k k k k k k k k k k k k k k k k k r B dt c d r r B r c dt d r r B r c t r r r dr rr r r r r B r c t r r B r c D r r r r r r B A r c D r r r r c D r r r r r c t t c r r c D r r r t c r r c D r r r t c k 2 3 4 3 2 2 4 3 4 3 3 2 2 1 1 1 1 1 1 2 1 2 1 4 3 0 2 3 2 1 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 1 1 1 1 2 2 1 1 1 2 2 1 = =           =           = = = =           =                   + =                             =           =                           =               =   = k k kr k k kr k k k k k k k k k k k k k k k k k k k k j r c r D dt c d r c r D j r r c dt d r c r D j r B r r c dt d r B r c dt d r c r D j r B 2 1 2 1 1 1 2 1 2 1 1 2 1 2 1 1 1 1 2 45 30 30 2 45 30 2 45 2 3 2 3 20 15 − = +   − − =             − − = =           =             − − =
  • 37. Uniform rate ROM – Model overview 37 ( ) ( ) ( ) n n n n Fl a t I t j t j =  ( ) ( ) ( ) p p p p Fl a t I t j t j − =  ( ) ( ) ( ) ( )         + =  + = =          = −  −        −  −  = − = − − 0 1 2 1 0 1 2 1 2 1 0 5 . 0 2 5 . 0 5 . 0 max 0 2 sinh 2 , 0 , 0 2 sinh 2 2 sinh 2 n n n n n n n n n sn sn n s n n j j F RT t x U t x j j F RT U U RT F j j c c c c k j 2 2 1 1 1 2 45 30 n n n rn n rn r j r c D dt c d − = + n n n rn n n sn D r j c r c c 1 1 1 35 35 8 − + = p p p r j dt c d 3 1 − = 2 2 1 1 1 2 45 30 p p p rp p rp r j r c D dt c d − = + p p p rp p p sp D r j c r c c 1 1 1 35 35 8 − + = ( ) ( ) ( ) ( )         + =  + = =          = −  −        −  −  = − = − − 0 1 2 1 0 1 2 1 2 1 0 5 . 0 2 5 . 0 5 . 0 max 0 2 sinh 2 , , 2 sinh 2 2 sinh 2 ; p p p p p p p p p sp sp p s p p j j F RT t L x U t L x j j F RT U U RT F j j c c c c k j Positive electrode Negative electrode Diffusion approximations ( ) ( ) ( ) n n n n sn D r t j t c t c 1 1 5 − = ( ) ( ) ( ) p p p p sp D r t j t c t c 1 1 5 − = Parabolic Quartic Parabolic Quartic n n n r j dt c d 3 1 − =