Parameter Sensitivity Analysis to Improve Material Design for Novel Zn-MnO 2 Batteries with Ionic Liquid Electrolytes

Transcription

1 Parameter Senitivity Analyi to Improve Material Deign for Novel Zn-MnO Batterie with Ionic Liquid Electrolyte Zachary T. Gima & Bernard J. Kim CE 95: Spring 06 Abtract Battery deign at the material level i often an inefficient proce that require many iteration and ignificant development cot. To improve the deign proce, parameter enitivity analyi can be performed to provide battery material cientit with better deign intiution. A parameter enitivity analyi wa developed for electrode parameter by adapting the ingle particle model (SPM) for a Zn-MnO chemitry. Parameter of interet included particle radiu and electrode thickne. The parameter enitivity analyi determined the relative enitivity of the battery output voltage to thee aforementioned parameter a well a their linear dependence on one another. A. Motivation & Background I. INTRODUCTION The Advanced Manufacturing for Energy lab at UC Berkeley i developing a printable, econdary Zn-MnO battery for Internet of Thing (IoT) and grid-cale energy torage application (Figure ). The battery utilize an ionic liquid electrolyte that enable facile recharging of zinc and could potentially lead to diplacement of lithium-baed ytem in application where afety and cot conideration are paramount. Specifically, application involving enor for food or wearable would ignificantly benefit from a afer and le hazardou energy torage ytem. Fig.. Cro ection SEM of Zn-MnO battery developed by Advanced Manufacturing for Energy lab. The ditinction between electrode and electrolyte layer i clearly viible, along with ome individual particle. Zachary T. Gima and Bernard J. Kim are with Mechanical Engineering, Univerity of Californa, Berkeley, CA 90, USA. {ztakeo, bernard.kim}@berkeley.edu Thi cla of zinc-baed econdary batterie with ionic liquid electrolyte i not yet well undertood or characterized. Becaue thi reearch i relatively recent, there exit much room for improvement beyond proof-of-concept performance []. In order to treamline experimental deign, we deire a model that can parameterize controllable battery characteritic (e.g. geometry, compoition, etc.) and predict reulting ytem performance. Specifically, thi model will inform key manufacturing deciion in order to bet improve the battery output voltage. Device for IoT demand certain voltage minimum in order to operate electronic component, and improving the battery ytem voltage will help the ytem meet thi minimum over a larger range of charge. Thi tudy utilize the ingle particle model (SPM) [], [] for it implicity over more ophiticated model (uch a Doyle-Fuller-Newman []). The model i accurate for low C- rate, which are typical of our battery. The SPM model each electrode a a ingle porou pherical particle and aume contant ion concentration within the electrolyte with repect to pace and time [], []. When applied to our battery, thi model will help optimize electrode compoition in order to further improve overall performance. B. Relevant Literature Lithium-ion batterie have emerged a the battery of choice for portable electronic, a well a in vehicular and aeropace application [5], [6]. In order to fully utilize the capability of thee batterie, a battery management ytem mut be deployed with the battery to enure afe and optimal operation for a given application [5]. More ophiticated verion of thee management ytem employ an electrochemical model decribing the battery dynamic to monitor key performance parameter uch a tate-of-charge and internal reitance []. However, if the ytem dynamic are not well undertood, then the model may provide inight into the battery behavior under certain load condition. The ingle particle model (SPM) wa derived from a implified verion of the complete Doyle-Fuller-Newman model [], [] and ha been widely utilized to model and undertand lithium-ion batterie [], [], []. While the aumption the SPM make limit the model applicability to cenario with low C-rate or negligible electrolyte dynamic, it implicity and eae of olving compared to more rigorou model [] make it an attractive tool, epecially for more fundamental analye of electrochemical dynamic. The Zn-MnO battery of interet hare propertie with previouly modeled intercalation lithium-ion batterie that

2 make the SPM uitable to modeling thi ytem. It ha been determined that the main mechanim for charge torage in thi ytem i with intercalation of Zn + ion into the MnO cathode [8]. Furthermore, the electrolyte-agnotic aumption of the SPM alo prove to be beneficial a thi battery utilize an ionic liquid electrolyte whoe olvent behavior differ ignificantly from traditional aqueou electrolyte [9]. The Zn- MnO battery i alo being cycled at low C-rate, further agreeing with thi aumption [0]. By adapting the SPM to thi Zn-MnO ytem, we hope to leverage parameter enitivity analyi on the reulting model parameter to optimize controllable geometric and compoitional factor to better undertand and improve the ytem. C. Focu of thi Study Thi tudy will utilize the SPM to model the zinc-baed battery in order to perform parameter enitivity analyi of the battery output voltage to particle radiu and electrode thickne. The reult will be ued to directly modify electrode geometrie and compoition in phyical experiment to better tailor performance output for IoT demand. II. TECHNICAL DESCRIPTION Thi tudy utilize a reduced electrochemical battery model, the SPM. Thi model approximate each electrode of the battery a collection of uniform particle. The dynamic of one particle at each electrode are tudied and then caled up to reflect the volume of the entire electrode. Thi approximation reult from auming that the concentration of inertion ion in the electrolyte phae remain contant in pace and time and alo that the entirety of the electrode evenly contribute to the total cell dynamic. The firt aumption i reaonably valid for batterie charged and dicharged at low C-rate. See Figure for a graphical repreentation of the SPM. For clarity, Table I provide definition for all parameter related to the SPM that follow. A. Single Particle Model Specifically, the SPM reult from applying the above implifying aumption to the more complex DFN model. The partial differential equation which therefore define the SPM come directly out of Fick Second Law of Diffuion in pherical coordinate for ymmetric, -D diffuion at each electrode ()-(). Thee PDE decribe the dynamic of the inertion ion in the olid phae. (r, t) = D (r, c + (r, t) + D+ (r, c + (0,t) = (R I(t),t)= (0,t) = 0, (R+,t)= D + Fa + AL + () Fig.. Graphic repreentation of the SPM []. The electrode are each repreented a ingle porou pherical particle. Thi reult from auming electrolyte concentration remain contant in pace and time. TABLE I SINGLE PARTICLE MODEL PARAMETER DEFINITIONS Symbol Decription SI Unit A Cell cro ectional area m a j Specific interfacial urface area m /m c j Concentration in olid phae mol/m c j Concentration at particle urface mol/m c j,max Max concentration in olid phae mol/m D j Diffuion coefficient in olid phae m /m F Faraday contant C/mol I Input current A i j 0 Exchange current denity V j Poitive (+) or negative (-) electrode - L j Electrode thickne m R Univeral ga contant J/mol-K R f Lumped current collector reitance R j Particle radiu m r Radial coordinate m or m/m T Cell temperature T t Time ec or ec/ec U j Equilibrium potential V V Output voltage V j Anodic/cathodic tranfer coefficient - The Neumann boundary condition () and () at the urface of the electrode (r = R ± ) proportionally relate the flux entering or exiting the electrode to the input current, I(t). The boundary condition at the center of the electrode (r = 0) are required for well-poedne and pherical ymmetry. The ytem output i the cell voltage and i governed by a combination of Butler-Volmer kinetic, electrode thermodynamic, electrode OCP, and internal reitance (5).

3 ! V (t) = RT F inh I(t) a + AL + i + 0 c + (t)! RT F inh I(t) a AL i 0 c (t) + U + c + (t) U c (t) + R f I(t) (5) The exchange current denity i j 0 and olid electrolyte urface concentration c j are decribed by (6) and (). It hould be noted that c j imply decribe the olid concentration of each electrode at the boundary of the patial dimenion, i.e. c j (t) = c j (R j,t). i j 0 cj = k rc j 0 ec j (t) c j,max c j (t) (6) c j (t) = c j R j,t, j {+, } () The full ytem plant i preented graphically in Figure. # Input Current Dynamic Sytem Σ Parameter " State Variable % && $ Output Voltage Fig.. Block diagram of ytem plant with input I(t) and output V(t). B. SPM Normalization The SPM equation ()-() can be normalized in both time and pace in order to facilitate the enitivity analyi, a will be clarified later. Thi normalization i carried out by caling the radial r and time t coordinate by the particle radiu and characteritic diffuion time repectively (8). r = r, t = D t (8) Applying thi normalization to the ytem equation ()-() yield the following modified PDE and boundary condition t ( r, t) = r ( r, r t ( r, t) = r ( r, c r r (0, t) = r (0, t) = r (, R t) = I( t) r (, R t) = I( t) D + Fa + AL + () The notable change i that the particle radiu, R ± now appear explicitly in the boundary condition after normalization. Henceforth, the bar above r and t will be dropped for clarity and brevity. C. Senitivity Equation Next, we derive the enitivity equation. We want to tudy the influence of the parameter with repect to the ytem output, not the tate. That i, for a given change in our parameter, we want to tudy how the ytem output voltage However, in order to calculate the enitivity of the output with repect to the parameter, we mut firt find the enitivity of the tate with repect to the parameter. We can then combine thi with the derivative of the output with repect to the tate by uing chain @ The parameter of interet to be tudied are the particle radiu R ± and the electrode thickne L ±. Thi reult in four eparate parameter for both the anode and cathode (5). = = 6L R + L + 5 (5) The firt tep i to take the derivative of the output equation with repect to the tate. Thi equation i omitted for brevity. A can be noted from (5), the only tate that the output voltage i dependent on i c j, which i imply a boundary value of c j. The econd tep i to derive the individual enitivity equation for each parameter. Thi i done by firt evaluating the nominal tate equation to determine the nominal olution x(t, 0), evaluating the Jacobian matrice given by (6), and latly olving the enitivity equation () for S(t) x, ) A(t, 0) x=x(t, 0), = x, ) B(t, 0) x=x(t, 0), = 0 (6) Ṡ(t) =A(t, 0)S(t)+B(t, 0), S(t 0 )=0 () Therefore, the next tep i to derive the enitivity equation with repect to the tate. Thi i done by taking the derivative of (9)-() with repect to i for the appropriate electrode (8). S i (t, x, x, ) The enitivity equation for each parameter with repect to the tate can then be computed. The enitivity equation w.r.t. = are computed a in (9), (0), and ().

4 S t (r, t) =S rr (r, t; 0 ) (9) S r (0,t)=0 (0) S r (,t)= I(t) () The enitivity equation w.r.t. = L are computed a in (), (), and (). S t (r, t) =S rr (r, t; 0 ) () S r (0,t)=0 () S r (,t)= I(t) () The enitivity equation w.r.t. + = R+ are computed a in (5), (6), and (). S + t (r, t) =S+ rr (r, t; 0) (5) S r + (0,t)=0 (6) S r + (,t)= D + I(t) Fa + AL + () The enitivity equation w.r.t. + = L+ are computed a in (8), (9), and (0). S + t (r, t) =S+ rr (r, t; 0) (8) S r + (0,t)=0 (9) S r + (,t)= D + I(t) (0) Fa + AL With the enitivity equation derived, we can now olve thee four et of PDE to compute S (r,t), S (r,t), S + (r,t), and S + (r,t). We are now poitioned to apply the chain rule to get the enitivity of the output with repect to the tate. That i, for every parameter, we can now multiply it enitivity equation to obtain our deired reult (). Note that the enitivity equation with repect to the tate provide the enitivity evolution with time for all patial coordinate. The appropriate value of S i (r,t) to ue occur at the boundary, i.e. S j i R j S i (,t) () The final tep i to normalize () with repect to each parameter. Without thi tep, the value provided by the enitivity vector become kewed by the magnitude of each parameter, making it difficult to perform an objective analyi of each parameter influence on the output. The normalization i performed by multiplying the final enitivity vector by the parameter divided by the @ j i j i V = S i () The reult of () can then be gathered in a enitivity matrix S (). S = S S S + S + The enitivity derivation i ummarized in Figure. I(t) Diffuion PDE c ±(r,t) c ±(t) Senitivity PDE S i ±(r,t) Output V(c ±) S i ±(,t) V c ± V S i ±(,t) c ± V θ i ± θ i ± V () Fig.. Block diagram of enitivity equation derivation. Only the boundary value of c ± (r,t) and S ± i (r,t) (c± (r,t)) and S ± i (,t) are ued in the final enitivity matrix. D. Senitivity Analyi Let S T S = D T CD. The enitivity analyi can then be performed by decompoing the matrix S T S into it contituent C and D. Thee two matrice are defined a follow (). D = C = S S , S + S,S i k +,S i + +,S i + k,s i k +,S i k + +,S i k + k +,S i +,S+ i k + +,S+ i + k +,S+ i + k,s+ i k + k +,S+ i + + k () Matrix D provide a direct quantification of parameter enitivity with larger number indicating higher enitivity to that parameter. Matrix C provide a meaure of linear dependence between parameter. Value cloe to and - indicate that thoe parameter are very trongly linearly dependent. Value equal to and - indicate that the parameter are completely proportional or inverely proportional. A. Simulation III. DISCUSSION The PDE were dicretized for imulation uing a combination of econd order central finite difference method and firt order forward and backward finite difference method for the boundary condition. The input current for the ytem wa a puled dicharge at 0.5C with a 50% duty cycle. All parameter value ued were for a LiCoO battery due to a lack of comprehenive data for our Zn-MnO ytem. Depite the mimatch of parameter value for thi imulation, the reult are till applicable for any battery ytem a thi enitivity analyi relate the relative effect of different parameter to each other. While the abolute magnitude 5

5 5 preent in the D matrix may not be repreentative, their relative value are relevant in decribing the relationhip between choen parameter. B. Reult The D and C matrice for the imulation ran are provided below (5) D = , C = 6 5 (5) Again, for clarification, the parameter vector a in (5) i reproduced below. = = 6L + L + 5 The value in D indicate that the particle radiu i more influential on the output voltage than the electrode thickne for both electrode. For thi particular imulation, thi i ubject to the parameter ued and may change baed on different value for D j, j, etc. The more ignificant reult come from the value preent in C. The entrie correponding to the linear dependence of particle radiu and electrode thickne for each electrode are equal to -. Thee value are expected becaue both parameter only appear in the boundary condition within the enitivity equation (), (), (), (0). Thi enitivity analyi ugget that the particle radiu and electrode thickne are inverely proportional to one another. In reality, thi invere proportionality i not true becaue the SPM itelf aume that the dynamic occurring at a ingle particle can be caled up to repreent the entire volume of each electrode. For real batterie, the fraction of each electrode participating in the reaction i limited by electrolyte dynamic, which are aumed away in the SPM, o change in particle radiu cannot be compenated for by change in electrode thickne and vice vera. However, thi invere proportionality doe reveal that the particle radiu and electrode thickne are trongly linearly dependent, even if they are not completely proportional. A a practical point, both parameter hould not be changed imultaneouly to avoid obfucating the effect of either on the output voltage. Ultimately, thi analyi yield inight into which parameter hould be prioritized for proce control during manufacturing. Baed on thee reult for the parameter value ued, control of particle ize hould be prioritized over electrode thickne when eeking to tightly control output voltage. C. Future Work Future work hould eek to add more inightful parameter that till remain controllable during the manufacturing proce. Thee can include the mole of cycleable zinc n Zn,, the volume fraction of active material " j, and the pecific interfacial urface area j. Furthermore, the enitivitie of additional metric (uch a tate of charge and cycle life) to thee parameter hould alo be invetigated. The SPM itelf may need to be augmented by adding back in electrolyte dynamic to more accurately model the true ytem dynamic []. IV. SUMMARY A parameter enitivity analyi wa performed for battery output voltage with repect to electrode parameter by adapting the ingle particle model (SPM). The parameter of interet were the particle radiu and electrode thickne. Analyi concluded that output voltage i more enitive to particle radiu than electrode thickne and that they are inverely proportional to each other, which i influenced by the implifying aumption inherent in the SPM. The reult rank the importance of each parameter for ue in manufacturing proce control. REFERENCES [] C. C. Ho, J. W. Evan, and P. K. Wright, Direct write dipener printing of a zinc microbattery with an ionic liquid gel electrolyte, Journal of Micromechanic and Microengineering, vol. 0, no. 0, p. 0009, Sep 00. [] S. J. Moura, M. Krtic, and N. A. Chaturvedi, Adaptive pde oberver for battery oc/oh etimation, ASME 0 5th Annual Dynamic Sytem and Control Conference joint with the JSME 0 th Motion and Vibration Conference, pp. 0 0, Oct 0, american Society of Mechanical Engineer. [] H. E. Perez and S. J. Moura, Senitivity-baed interval pde oberver for battery oc etimation, American Control Conference (ACC), pp. 8, Jul 05, ieee. [] M. Doyle, T. Fuller, and J. 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