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Entropy measurement of a large format lithium ion battery and its application to calculate heat generation
Accepted Manuscript Entropy measurement of a large format lithium ion battery and its application to calculate heat generation Chil-Hoon Doh, Yoon-Cheol Ha, Seung-wook Eom PII:
S0013-4686(19)30701-7
DOI:
https://doi.org/10.1016/j.electacta.2019.04.026
Reference:
EA 33974
To appear in:
Electrochimica Acta
Received Date: 5 November 2018 Revised Date:
1 April 2019
Accepted Date: 4 April 2019
Please cite this article as: C.-H. Doh, Y.-C. Ha, S.-w. Eom, Entropy measurement of a large format lithium ion battery and its application to calculate heat generation, Electrochimica Acta (2019), doi: https://doi.org/10.1016/j.electacta.2019.04.026. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Entropy Measurement of a Large Format Lithium Ion Battery and its Application to Calculate Heat Generation
Battery Research Center, Korea Electrotechnology Research Institute, and 2KERI Campus, Korea University of Science and Technology, Changwon, Gyongnam 51543, Korea [email protected];[email protected]
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*E-mail:
Abstract
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As large format lithium ion batteries are used in EV and ESS applications, the temperature control of a battery is important to achieve a safe and long cycle life operation. To control the temperature properly, we need to know the heat generation characteristics. Heat generation of a battery mainly depends on the applied current, internal resistance, temperature and entropy of the battery. However, it is not easy to measure the entropy, because they depend on SOC and temperature simultaneously. Here, we suggested a new way to measure the entropy, on the
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assumption of adiabatic condition and calculate the heat generation. Internal resistance, specific heat capacity, OCCP values are also measured experimentally. The entropy (F ∂EOC ∂T-1) at a given SOC were measured from the variation of OCCP with temperature and expressed via a polynomial equation. The temperature dependence of the internal
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resistance were also expressed in the same procedure. The battery temperature variations calculated from the above parameters showed a good agreement with those from experiments. This suggested method provides a simple and reliable temperature estimation of lithium ion batteries.
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Chil-Hoon Doh1,2,*, Yoon-Cheol Ha1,2, Seung-wook Eom1,2
Keywords
Lithium ion battery, heat generation, entropy, specific heat capacity, internal resistance, temperature estimation
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Graphical abstract
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1.
Introduction As the EV and ESS industries grow, higher energy density and larger format of lithium-ion batteries (LIBs) has been
employed widely. The energy density of LIB cells reached around 260 W h kg-1 and the size increased to accommodate
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approximately 120 A h capacity. As they are employed in a module or a pack, the temperature control becomes crucial for the safe and long cycle life operation. During charge and discharge, the temperature of a cell changes and the exact estimation of the temperature variation is important for the design of TMS (temperature management system) of the module, pack and system. Single cell in an open space may easily radiate the heat generated from charge/discharge but
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it is not easy to control the cells inside the battery. The cell located inner part of the multi-cell-packed battery package might be in a quasi-adiabatic condition [1,2], where accumulated heat leads to a faster increase in temperature at the
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inner part than the outer part of the multi-cell-packed battery pack, thus provoking a fast degradation or a thermal runaway at the worst case.
Quantifying the heat generation and the temperature variation on the basis of the C-rate of charge/discharge is important to establish the optimum operating condition for battery performance such as energy, power, cycle life, and safety. The fundamental thermochemical properties [3] related to Gibb’s free energy can be estimated under an adiabatic
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condition. The endothermic/exothermic characteristics against current amplitude and SOC range can be understood on the basis of the irreversible Joule heating and the reversible entropic heating/cooling. The investigation of electrochemical exothermic/endothermic properties, especially the entropic effect [3 - 13], is necessary for the safe 456789101112
battery operation with prolonged life cycles. Furthermore, as LIBs use an organic electrolyte, they have a low specific
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heat capacity (Cp(battery)) and the temperature of LIB could increase rapidly compared with water-based batteries. Battery heat generation and temperature variation can be expressed by Eq. (1) [3, 9-11,14- 16]. The quantity 15
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of heat generated by a battery during galvanostatic charge/discharge can be divided into two categories, namely, internal resistance and entropy (Eq. (2)). Under a galvanostatic condition, SOC has a linear relationship with elapsed time (Eq. (3)). Therefore, differential time can be directly correlated with differential SOC by using Eq. (4). Moreover, the current applied during galvanostatic charge/discharge can be directly correlated with the C-rate by using Eq. (5). total = irreversible + reversible + tab = I E − E − Qtotal =
+ + ≅ − ITΔS --------------(1),
(I2 Ri - ITΔS) dt ---------------------------------------------------------------------(2),
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SOC = (q0 - q) q0-1 = (q0 - I t) q0-1 ------------------------------------------------------------(3), dt = − q0 I-1 dSOC -----------------------------------------------------------------------(4), I
=
C-rate q0 --------------------------------------------------------------------------------(5).
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By combining Eqs. (2), (4), and (5), Qtotal can be varied as a function of the C-rate and the range of SOC (Eqs. (6) and (7)). Temperature variation can be expressed using a rearranged version of Eq. (8) as a function of initial temperature, specific heat capacity (Cp(battery)), internal resistance, C-rate, and entropy, which are intensive properties of a battery. The specific Coulomb capacity (q0 M-1) and specific internal resistance (ri) of a battery are also inherent
ITΔS
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intensive properties.
Qtotal = - ( 0 )
(I2 Ri - ITΔS) dSOC ---------------------------------------------------(6),
Q ) *+ = - q0
(C-rate q0 Ri - TΔS) dSOC = Cp(battery) M (T2 - T1) ----------------(7),
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T2 - T1 = - ( =-(
q0 ) M Cp(battery) q0
2 Cp(battery)
)
(
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q
(C-rate q0 Ri C-rate q0 2
1 TΔS) dSOC F
ri - TΔS) dSOC ------------------------------------(8).
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The following is defined in Eqs. (1)–(8):
3
, rate (or power) of heat generation (W) ; Q, heat generation (W s) ;
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EOC, open-circuit cell potential (V); Et, cell potential at given time (V);
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I, current (A), discharge (+), charge (-);
T, battery temperature (K) depending on current, initial temperature, entropy, and SOC; T1, initial temperature before charge or discharge (K); T2, final temperature after charge or discharge (K); RA, anode tab resistance (Ohm, Ω); RC, cathode tab resistance (Ohm, Ω); 4 / 34
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Ri, battery internal resistance (Ohm, Ω); ri, battery specific internal resistance (Ri M; Ohm g, Ω g); F, Faraday’s constant (A s mol-1);
t, time (s); SOC, state of charge (dimensionless); C-rate, (h−1), discharge (+), charge (-);
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q0, battery capacity (A h, 3600 A s);
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∆S, battery specific entropy (J mol-1 K-1 or W s mol-1 K-1);
Cp(battery), specific heat capacity of the battery (J g-1 K -1 or W s g-1 K-1);
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M, battery mass (g); (q0 M-1), specific coulomb capacity (A h g-1);
In this study, the useful and easy method to evaluate entropy value as function of SOC was contrived under an atmospheric surrounding that is the same as the usual battery operating condition. Molar entropy was evaluated as (F
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∆E ∆T-1) based on the Gibb’s free energy relationship and the Nernst equation of (∆G = ∆H – T ∆S = - n F E). Battery specific heat capacity (Cp(battery)) was measured by varying SOC. Open-circuit cell potential (OCCP) and internal resistance (Ri) were evaluated by varying the SOC and temperature by using the intermittent current transient method.
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The evaluated entropy, Cp(battery) and Ri were entered into the calculations by using Eq. (8), which represents an electrothermo-chemical model, to obtain temperature variation as a function of change in SOC. This calculation showed that
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adiabatic heat insulation was the worst operating condition. The calculated results showed good agreement with the experimental results. The measured results of Cp(battery), entropy and internal resistance also showed good agreement with the calculated results. The acquired intensive electro-thermo-chemical properties of the battery help determine the internal battery temperature. The measurement of internal battery temperature is impossible under normal operations. However, the limiting current for safe operation can be determined using information about the internal battery temperature. This information about the electro-thermo-chemical properties of a battery might be useful from the viewpoint of specifying the heat radiation design criteria of the battery to ensure the safe operation of electrical/electronic devices/machines.
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2.
Experiment and computational calculation A prismatic lithium ion battery (graphite/Li(Ni-Co-Mn)O2) with coulomb capacity of 104 A h (2.032 kg)
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was used herein.
2.1. OCCP with SOC and temperature
The two parallel-paired batteries were charged–discharged using the intermittent current transient technique under temperatures of 291 K, 298 K, 313 K, 328 K, and 333 K and the isothermal mode of EV-ARC (extended volume–
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accelerated rate calorimeter, THT, U.K.). A current of 9.4 A/cell was applied for 30 min. Thereafter, the battery was rested for 60 min. This periodic current transient was applied until the battery was charged to 4.15 V and discharged to
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2.7 V. The OCCP values were collected at the end of the rest time under each temperature and SOC.
2.2. Internal resistance (Ri) with SOC and temperature
Internal resistance was evaluated from a dataset (current, voltage) compiled for a previous OCCP measurement by
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using the intermittent current transient technique under temperatures of 298 K, 313 K, and 328 K. The cell potential differences between each end of galvanic current pulse to be the start of rest time and the end of rest time of 1 hour were gathered.
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2.3. Specific heat capacity (Cp(battery))
A thin flexible-film Ohmic heater with a resistance of 89 Ω and an area of 12 × 7 cm2 (KH-305/10, Omegalux,
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U.S.A.) was inserted between two batteries. This battery–(Ohmic heater)–battery assembly was installed within the EVARC cylinder. The Ohmic heater was connected to an automatic Cp measuring unit, which was constructed as a part of EV-ARC. Cp(battery) was measured using the automatic mode of the implanted software by varying the heating power from 3.2 W to 7.1 W and by varying the SOC over its entire range.
2.4. Entropy (∆S) with SOC The same set of two paired batteries was used to evaluate the entropy in atmospheric air within the EVARC cylinder. Molar entropy was evaluated as (F ∆E ∆T-1) based on the Gibb’s free energy relationship and 6 / 34
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the Nernst equation of (∆G = ∆H – T ∆S = - n F E). 2.4.1.
Isothermal galvanostatic intermittent charge–discharge
The two parallel-paired batteries were charged–discharged by using the isothermal galvanostatic intermittent transient
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technique (I-GITT) at given temperature. Five different temperature of 291 K, 298, 313 K, 328 K and 333 K were separately tested by applying the isothermal mode of EV-ARC. A current of 9.4 A/cell was applied for 30 min. Thereafter, the batteries were rested for 60 min. This periodic current transient was applied until the batteries were charged to 4.15 V and discharged to 2.7 V. Specific entropies were calculated using the gathered OCCP data at each
Cell potential evaluation with discrete stepping of temperature for various SOC
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2.4.2.
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temperature and SOC.
The SOC-regulated paired batteries were heated under the adiabatic condition by using the “Heat–Wait– Search” mode of EV-ARC. SOC was regulated with interval of 0.1 for SOC 0.0 - SOC 1.0 and SOC 0.05 - SOC 0.95 using the current of 0.1 C-rate. “Heat–Wait–Search” is a stepwise sequence to get exothermic or endothermic heat generation. Cell potential and temperature data were collected after the full stabilization of temperature (0.02 K min-1, 1
adiabatic chamber.
2.4.3.
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h). Additional heating power was applied using the EV-ARC control unit to maintain each set temperature in the
Cell potential and temperature evaluation with quasi-adiabatic slow cooling for various SOC
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The SOC-regulated paired batteries were heated to 333 K under the adiabatic condition and then slowly cooled under the quasi-adiabatic condition to room temperature (283 K-299 K) over 30 h. Cell potential
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during this cooling was used as the equilibrated cell potential of that temperature. The choice of temperature range depends on the purpose of the test and any modification for slow cooling instead of the ARC tool would be acceptable.
2.5. Computational calculation of endothermic/exothermic heat An iterative electro-thermo-chemical modeling (IETCM) calculation program was developed on the basis of Eq. (8). The temporal temperature of each SOC was calculated by iterating the mutually correlated properties of 7 / 34
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internal resistance and entropy. The IETCM program produced the patterns of variation in temperature, internal resistance, and entropy, as well as the maximum temperature and final temperature, by considering the current and its direction. Related electro-thermo-chemical properties such as voltage and accumulated heat can be calculated. The
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IETCM program can also be modified for non-galvanostatic condition. Anyway, the cases were not touched at here. The IETCM program employs the Eigen input values of entropy, internal resistance, OCCP, Coulomb capacity, weight, and specific heat capacity as the measured intensive properties. The initial parameters as boundary conditions for the iterative calculation were initial SOC, final SOC, initial temperature, and applied current. The
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current was converted into C-rate as the ratio of current against Coulomb capacity. Differential SOC was generated depending on the SOC range and the given fractionation as the input value. A convergence test was performed for
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iterative calculations with different fraction numbers of 10n (n = 1, 2, 3, 4) to confirm the iteration results. Effective and meaningful results were deduced for fraction numbers over 1000. The fractionation number of 1000 was used to get good agreement and save the calculation time. The iteration process could be considered to consist of several steps. In step 1, conditional parameters such as battery information and calculation boundary condition as initial temperature, initial SOC, final SOC, and C-rate are read. In step 2, specific entropy is evaluated for each
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corresponding SOC level. In step 3, internal resistance (Ri000), which means the initial Ri as calculated for the given SOC and temperature, is calculated as a function of initial temperature (T1000). On the basis of one instance of mutual thermo-electro-chemical calculation, T2000 is generated for (SOC000 + dSOC). T2000 would be just T1001 in the next iteration of SOC001. Furthermore, SOC001 is just (SOC000 + dSOC). The next internal resistance Ri001 can be evaluated
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in the same manner as the previous one. These iterations can be executed for the number of fractionations. From the results of the iterative calculations, the maximum temperature, final temperature (T2999), final internal resistance
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(Ri999), temperature variation (T2000 - T2999) in the objective SOC region, internal resistance variation (Ri000 - Ri999) in the objective SOC region, and average specific entropy can be deduced. The internal resistance in the temperature region outside the measured region was determined using an extrapolated linear relationship for the low-temperature region and a linear relationship between the Ri of 333 K and 0 Ω as the Ri of 2273 K.
For the conversion between molar entropy (J mol-1 K-1) and entropy (J K-1), the moles of electricity of a battery can be defined as Eq. 9. The charge/discharge capacity of the battery used herein was 103.7 Ah, which amounts to 373,320 A s battery-1 and 3.8692 mole battery-1 by applying Faraday’s constant 96,485.33289 A s mole-1. 8 / 34
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Capacity 103.7 A h battery-1 = 373, 320 A sec battery-1 = 3.8692 mole battery-1 --------------(9).
3.
Results and discussions
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3.1. OCCP behavior with SOC and temperature The OCCP of LIBs has been compared against SOC and temperature widely in the literature. The OCCP of an LIB is unique because of the differences in the cathode, anode, electrolyte, and (+/-) balancing across various batteries. In this study, OCCP information was evaluated for the iterative calculations as a function of temperature and
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charge/discharge (Fig. 1a). The obtained OCCP information against SOC was fitted as a polynomial relationship (Table S1). The OCCP information is not necessary for the galvanostatic (constant current) electro-thermo-chemical reaction
potentiostatic process in this work.
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3.2. Internal resistance (Ri)
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(Eq. 8), but it is critical for the potentiostatic (constant voltage) electro-thermo-chemical reaction. We do not study the
A plot of battery Ri versus SOC during charge/discharge is shown for the temperatures of 298 K, 313 K and 328 K with temperature on the y-axis (Fig. 1b). The polynomial functions of Ri were evaluated for different SOC and temperature during charge and discharge. Fig. 1b1 and Fig. 1b2 show a logarithmic plot of battery Ri versus SOC for
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the temperatures of 298 K, 313 K and 328 K during charging and discharging. Moreover, the average battery Ri is shown as a function of temperature at 298 K, 313 K and 328 K in Fig. 1c. The average battery Ri was evaluated in the
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SOC range of 0.05-1.0 during charge and 1.0-0.003 during discharge. Table S2 shows the relationship of Ri against temperature and SOC. The relationship between average battery Ri and temperature can be expressed using Eqs. 10 and 11. The numerical formula of Ri in terms of temperature and SOC is important in the iterative calculation of electro-thermo-chemical reactions. R
= 5.8782E-7 T2 - 3.9448E-4 T + 6.7133E-2; R² = 1.0000 --------------------------(10),
dR
= (2 x 5.8782E-7 T - 3.9448E-4) dT -------------------------------------------------(11).
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3.3. Specific heat capacity (Cp(battery)) The quantity of heat input to a battery can be determined given the heating power and heating time. The variations in temperature and heating power during Cp(battery) measurement are plotted in Fig. 1d. During this heating, the ambient
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temperature was invariably equilibrated to the battery temperature. Cp(battery) was approximately 1 (J g-1 K-1) for heating powers of 3.16 W, 6.18 W and 7.11 W and a battery of SOC 0.482 (Fig. 1e) and for different SOC levels of 0, 0.25, 0.5, 0.75, and 1 (Fig. 1f). Therefore, Cp(battery) can be considered constant at 1.0144 (J g-1 K-1). Maleki et al. [17] reported the Cp(battery) of a US18650 cell as 1.04 ± 0.02 (J g-1 K-1). Pesaran et al. [18] reported the Cp(battery) of a 6 A h lithium-ion
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battery and a 4 A h lithium-ion polymer battery as 0.795 and 1.0118 (J g-1 K-1), respectively. Related works on Cp(battery) [19 - 21] can be found in the literature. The Cp(battery) value of a given battery is a useful intensive property from the 20
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viewpoint of interpreting the electro-thermo-chemical reactions that occur during galvanostatic and potentiostatic processes.
3.4. Entropy (△ △S) 3.4.1.
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Fig. 1
Isothermal galvanostatic intermittent charge–discharge The entropy of charge and discharge were evaluated using I-GITT (Fig. 2a). Entropy could be evaluated on the
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basis of the OCCP and temperature data during I-GITT. Battery entropy can be determined using I-GITT. However, the validity of the obtained entropy might be low because of the difference between the internal battery temperature and the
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temperature of the convection oven. A battery is heated owing to the passage of current through it. At low SOCs, the entropy at the start of charging and the end of discharging were considerably different. This result could be compared with the results obtained using the two other methods described later in the paper.
3.4.2.
Cell potential evaluation with discrete stepping of temperature for various SOC
The typical cell potential and temperature behavior can be obtained (Fig. 2b1), and specific entropyies can be obtained for various SOCs and temperature range of 303K and 333K (Fig. 2b2). The entropy obtained herein gave some limited data sets depending on the size of temperature step in spite of the need for an extended period of measuring time. 10 / 34
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3.4.3. Cell potential and temperature evaluation with quasi-adiabatic slow cooling for various SOCs The typical cell potential and temperature behavior can be obtained (Fig. 2c1 and Fig. 2c2). Fig. 2c1
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shows the polynomial fitting of the cell potential and temperature of a battery (SOC 0.482) against time. A new polynomial fitting of cell potential versus temperature of a battery could be obtained as Fig. 2c2 based on the results of Fig. 2c1. These stepwise polynomial fittings generate easy matches between the datasets of cell potential and temperature over different time intervals. The polynomial fitting of temperature and cell
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potential against measurement time was performed by increasing the SOC in steps of 0.05. As a result, the relationships given in Table S3 and Table S4 were obtained. The polynomial fitting of cell potential versus
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temperature was given in Fig. 2c3 and Table S5. This polynomial relation could be used in multi-physics interpretation. The relationship between cell potential and temperature of Table S5 could be considered as linear relationship. The linear fitting data of cell potential (V) against temperature (K) for each SOC level are compiled as Table S6 and plotted in Fig. 2c3. The derivative results of (dV dT-1), △S, △H were obtained
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using the Gibb’s free energy relationship. The specific entropy for any given SOC range could be obtained for the full temperature range and various SOC levels used in the iterative electro-thermo-chemical
3.4.4.
Fig. 2
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simulation.
Comparison of the entropy results obtained using three different methods
The tendencies of entropy against SOC obtained using the I-GITT (method 1), temperature stepping (method 2), and quasi-adiabatic slow cooling methods (method 3) were similar. Nevertheless, I-GITT (method 1) is incorrect compared with temperature stepping (method 2) and adiabatic slow cooling (method 3) because the cell potential data used in IGITT were gathered under the condition of current flow. Temperature stepping (method 2) was employed to measure the equilibrated cell potential for discrete temperatures. This measurement required a long waiting time and yielded a small dataset, thus reducing representativeness. Adiabatic slow cooling (method 3) was employed to continuously 11 / 34
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measure the data of temperature and cell potential during the slow cooling of a heated battery in an adiabatic chamber. Therefore, quasi-adiabatic slow cooling method could generate a representative entropy value. Table S5 summarizes the entropy values measured using temperature stepping (method 2) and quasi-adiabatic slow cooling (method 3). In case of
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quasi-adiabatic slow cooling (method 3), polynomial fitted entropy values are listed together with the measured values. Fig. 2d shows the entropy values obtained using adiabatic slow cooling between 333 K and 293 K, and first-order fitting of adiabatic slow cooling between 333 K and 293 K.
The entropy value at a given SOC and the average entropy over a given SOC range for the iterative calculation can
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be extracted from Table S5. The polynomial relationship is necessary for the flexible and accurate iteration of the electro-thermo-chemical reaction in a battery. However, the obtained results exhibit complex patterns, and it would be
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impossible to express them as one polynomial. To overcome this difficulty, regionally polynomial-fitted entropy was obtained with temperature stepping between 333 K and 303 K and adiabatic slow cooling between 333 K and 293 K (Fig. 2e). Table 1 summarizes the regional polynomial expressions of entropy for four different SOC ranges. The entropy of region 1 for an SOC of 0–0.1 has a negative value, and its linear relationship with SOC is linear. The entropy of region 2 for an SOC of 0.1–0.28 is negative, and its relationship with SOC is linear. The entropy of region 3 for an
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SOC of 0.28–0.57(5) is negative according to temperature stepping (method 2) but is positive according to adiabatic slow cooling (method 3). The temperature of the adiabatic discharge of the battery with a low C-rate of 0.03 C-rate indicated that the reaction was initially exothermic, but it later changed to endothermic and finally changed again to exothermic (Fig. 3a). To verify the temperature-following characteristics of the adiabatic chamber against the battery
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temperature, the temperatures of the top plate, bottom plate, and cylinder plate of the adiabatic chamber are plotted in Fig. 3a. It was possible to measure these clear exothermic/endothermic reactions owing to the low C-rate of 0.03 C-rate.
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The low C-rate illuminated the effect of entropy on the thermochemical reaction by minimizing the contribution of Ohmic Joule heating. In line with this result, the entropy of region 3 was positive in contrast to the entropies of regions 1 and 2. Therefore, the result of adiabatic slow cooling (method 3) corresponds to the experimental result and approximates well the real phenomenon. The entropy of region 4 for SOC of 0.57(5)–1 was negative. Region 4 had five breakpoints with a small significance, and these were combined into a six-order polynomial function. Adiabatic slow cooling (method 3) showed the highest multiplicity and the best agreement with the experimental results among the abovementioned three methods for measuring entropy. The entropy values obtained using adiabatic slow cooling (method 3) was used in the iterative simulation. 12 / 34
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Table 1. Polynomial expression of entropy for four different SOC ranges. ∆S (cooling, fit, 333 K–303 Classification
∆S (temp. step, 333 K–303 K)
∆S (cooling, 333 K–303 K)
Region 1
SOC range: 0–0.0983375
Function: ∆S = b1 SOC + a1
b1
2.50E+02
528.26471
a1
−4.89E+01
−70.55344
R 1²
1
0.995
−72.2888035
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SOC range: 0.0983375−0.28 Function: ∆S = c2 SOC2 + b2 SOC + a2
c2
1.89E+02
b2
−5.23E+01
a2
−2.14E+01
R 2²
0.789
Region 3
539.3344063
0.998
SOC range: 0.28−0.57459
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Region 2
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K)
80.09035889
82.55883466
−26.00540646
−26.80023123
0.909
0.906
Function: ∆S = d3 SOC3 + c3 SOC2 + b3 SOC + a3
9.73E+02
c3
−.43E+03
b3
6.88E+02
19.16202512
19.64044168
a3
−1.08E+02
−1.34453314
−1.21573825
0.967
0.925
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R 3²
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d3
0.8821
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SOC range: 0.57459−1 Region 4
Function: ∆S = g4 SOC6 + f4 SOC5 + e4 SOC4 + d4 SOC3 + c4 SOC2 + b4 SOC + a4 2.68E+05
−36115.46404
−15837.61063
−1.26E+06
160191.9635
65925.20666
e4
2.43E+06
−294133.1811
−111847.7554
d4
−2.49E+06
286448.9981
98654.33399
c4
1.43E+06
−156228.7381
−47461.39921
b4
−4.31E+05
45281.48644
11691.30821
g4 f4
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5.39E+04
−5453.422546
−1132.079173
R 4²
0.983
0.984
0.994
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a4
3.5. Battery temperature behavior during adiabatic charge–discharge and comparison with computational calculation results for C-rates of 0.05 C, 0.3 C, and 0.5 C
The trends of temperature and cell potential during adiabatic charge and discharge are plotted in Fig. 3 against
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SOC with different initial temperatures and C-rates. A thermocouple was set at the center of the two sandwiched lithium ion batteries within the adiabatic chamber. Computational iterative calculation generated a full dataset of
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given SOC span as shown at Fig. 3a and Table S8. Calculated results were plotted as sporadic emphatic marks at Fig. 3a together with experimental results. Entropy of each SOC levels were calculated based on regionally polynomial fitted values following Table 1 as generated by method 3.
During charge/discharge at the rate of 0.05 C, the cell potential pattern was maintained nevertheless the increase of temperature by the advance of cycling. However, the temperature of battery was increased gradually by the
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passage of current. The temperature was more rapidly increased at the initial charge/discharge to be from 301.5 K to 302.1 K for charge and from 302.1K to 306.3 K for discharge than at the next charge/discharge to be from 306.3 K to 305.2 K for charge and from 305.2 K to 308.4 K for discharge. The endothermic and exothermic reactions were
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measured during each charge and discharge stage. The breakpoints between exothermic and endothermic reaction are lined vertically and highlighted with red color. The classification of these breakpoints corresponds to the four-step
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polynomial classification of entropy. A few meaningful breakpoints at SOC 0.0482 were not separated because of the very narrow SOC span and the presence of a region with very high internal resistance. Appropriately colored dots at Fig. 3b shows calculated temperature obtained by the iterative calculation of electro-thermo-chemical reaction according to Eq. (8), Cp(battery), △S, Ri, battery capacity, and battery weight. The better correspondence between experiment and calculation in the SOC region of 0.1−1.0 could be ascribed to the better stability of entropy and internal resistance than that in the SOC region of 0–0.1. The low correspondence in the SOC region of 0.0–0.1 could be ascribed to the instability of entropy and internal resistance. Cell potential at Fig. 3b was replotted at Fig. 3b1 as an expanded y-axis scale. 14 / 34
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Fig. 3c and Fig. 3d show the experimental and iterative calculation results for charge/discharge with high C-rates of 0.3 and 0.5 C. The agreement between the calculated and experimental results is as good as that in the case of the C-rate of 0.05 C. The good agreement between the measured results and the IETCM results shows that the measured
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intensive parameters of entropy, specific heat capacity, and internal resistance are effective.
Fig. 3
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3.6. Battery temperature behavior during room temperature and adiabatic charge– discharge
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The trends of temperature and cell potential under charge/discharge at a rate of 0.1 C-rate at room temperature are plotted in Fig. 4a against SOC. A thermocouple was set at the center of the two sandwiched lithium ion batteries within the adiabatic chamber. Room temperature was varied as 282 K–293 K during the daytime and nighttime. The temperature of the chamber containing the batteries was set to 286.5 K–289.3 K. The battery temperature followed the chamber temperature with a few perturbations owing to the influence of Ohmic Joule heating and entropic heating. In
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any case, the battery external temperature was consistent with the chamber temperature. Battery temperature over three continuous charge/discharge cycles remained between 286.5 K and 291.8 K without any serious heating. The same charge/discharge cycle was executed under the adiabatic condition instead of room temperature, and the results are
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plotted in Fig. 4b.
When the number of charge/discharge cycles increased, battery temperature increased gradually owing to Joule
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heating and entropic heating. The temperature of the adiabatic chamber was consistent with the battery temperature through the automatic synchronization of the temperature of adiabatic chamber. The battery temperature increased from 298.8 K to 316.2 K over three continuous charge/discharge cycles. This means that the heat generated during the charge/discharge cycles accumulated continuously, thus leading to an increase in temperature. Moreover, it is important to understand the significance of heat radiation for a serial and parallelintegrated large-scale battery system.
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Fig. 4c shows the plots of the cell potential and battery temperature under adiabatic discharge at a few different C-rates and the same initial temperature of 298 K. The alteration of cell potential behavior by C-rate range of 0.03 C to 1.0 C was not so serious. The calculated temperature is plotted together with the experimental result. In all
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cases, the C-rate of 0.2 was used for charging. The calculated temperature shows good agreement with the experimental result for the low C-rates of 0.03 and 0.1. As the C-rate increased to 0.3, the calculated temperature became somewhat higher than the experimental temperature. Table 2 summarizes the initial voltage, final voltage, initial temperature, final experiment temperature, and final calculation temperature. The electro-thermo-chemical reaction results calculated
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using the proposed model are consistent for C-rates of up to 0.5 C-rate for an SOC of 0.1. The temperature calculated using the proposed model for the 1.0 C-rate was somewhat higher than the experimental temperature. This deviation can
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be ascribed to the property of heat flux due to the spatial inhomogeneity of temperature in the battery [1,2]. Heat flux was not considered in this study.
Fig. 4
Initial cell potential (V) 4.130
Final cell potential (V) 2.7
0.0039
4.128
2.7
0.0000
4.128
2.7
4.109 4.128
C-rate (h-1)
Initial temperature (experiment/calculation)
Final temperature (experiment)
Final temperature (calculation)
0.03
24.771
28.489
29.733
0.1
25.489
31.889
33.459
0.0159
0.3
24.578
35.446
38.586
2.7
0.0159
0.4
25.233
37.616
40.802
2.7
0.0150
0.5
24.975
(37.475)
43.211
2.7
0.0205
1.0
25.697
44.987
60.768
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Final SOC
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Table 2. Temperature increase during adiabatic discharge
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A possible iterative simulation result is plotted in Fig. 5 as a function of the C-rate for the full range of charge and discharge under the initial temperatures of 291 K, 308 K and 333 K under the adiabatic condition. The final temperatures showed considerably different behaviors between charge and discharge but similar behaviors for the initial
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temperatures of 291 K and 308 K. For adiabatic/pseudo-adiabatic charge and discharge, a C-rate lower than 2 must be applied to ensure that the temperature remains below 373 K. The heat generation during charge was lower than that during discharge owing to the contribution of the entropic term. Inclusion of entropy pattern to electro-thermo-chemical
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Fig. 5
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multi-physics modeling could surely improve accuracy of temperature anticipation.
Conclusions
Entropy of a large format lithium ion batteries was evaluated efficiently using the method of adiabatic slow cooling to measure the variation of OCCP with temperature. Specific heat capacity (Cp(battery)), internal resistance, and OCCP were also measured experimentally by using an adiabatic chamber. The battery temperature variations calculated
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from the acquired parameters of entropy, specific heat capacity, internal resistance showed a good agreement with those from experiments. The good agreements between experimental and simulation indicates that the parameters of entropy, specific heat capacity and internal resistance were evaluated accurately. The acquired results would be used in multi-
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Acknowledgments
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physics interpretation on heat generation and dissipation.
This study was supported by the KERI R&D program of MSIP/NST (18-11-N0103-01 and 19-12-N0102-01) and by MOTIE (18-02-N0108-02 and 19-02-N0108-01) of Korea. The authors express their gratitude to Hanwha Land Systems Co., Ltd., and Daewoo Shipbuilding & Marine Engineering Co., Ltd. (DSME).
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Figure caption Thermochemical properties of OCCP, internal resistance and Cp(battery). (a) OCCP against SOC and temperature; (b) internal resistance against SOC for temperature of 298 K, 313 K, and 328 K during charge and discharge; (b1) logarithmic plot of internal resistance during charge; (b2) logarithmic plot of internal resistance during discharge; (c) average internal resistance against temperature together with charge and discharge; (d) Temperature and heating power during Cp(battery) measuring; (e) Specific heat capacity of battery (Cp(battery)) measured for some different heating power; (f) Specific heat capacity for different SOC levels.
Fig. 2.
(a) Typical plot of entropy during charge and discharge with I-GITT; (b1) cell potential and temperature behavior against time for SOC 0.482 for temperature stepping; (b2) specific entropy as obtained from temperature stepping (method 2) with temperature range of 303K and 333K; (c1) cell potential and temperature against time for slow cooling (cell SOC 0.482); (c2) cell potential against temperature for slow cooling; (c3) polynomial fitting of cell potential against temperature for various SOC range for slow cooling (cell SOC 0.482); (d) specific entropy of experimental results from slow cooling (method 3) with temperature range of 293K and 333K; (e) polynomial-fitted entropy originated from experimental results from temperature stepping (method 2) and slow cooling (method 3).
Fig. 3.
Temperature and cell potential during adiabatic charge and discharge (cell potential range: 2.7–4.15 V) plotted together with calculated temperature on the basis of heat relationship by using the measured properties of Cp(battery), Ri, and △S. (a) 0.03, (b) 0.05, (b1) 0.05 C; cell potential, (c) 0.3 and (d) 0.5 C.
Fig. 4.
Temperature and cell potential patterns of (a, b) continuous charge and discharge sequences with C-rate of 0.1 for a cell potential range of 2.7–4.15 V (a) at room temperature and (b) under adiabatic condition through ARC. (c) Adiabatic discharge at various C-rates together with temperature calculated using the proposed model. C-rates: 0.03, 0.1, 0.3, 0.4, 0.5, and 1.0 C.
Fig. 5.
Calculated final temperature with C-rate for the full charge/discharge of the battery.
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Fig. 1.
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(a) OCCP against SOC and temperature
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(b) Internal resistance against SOC and temperature
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(b1) During charge
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(b2) During discharge
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(c) Average internal resistance against temperature
(d) Temperature and heating power during specific heat capacity of battery (Cp(battery)) measuring 21 / 34
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(e) Specific heat capacity of battery (Cp(battery)) measured for some different heating power
(f) Specific heat capacity for different SOC levels Fig. 1. Thermochemical properties of OCCP, internal resistance and Cp(battery). (a) OCCP against SOC and temperature; (b) internal resistance against SOC for temperature of 298 K, 313 K, and 328 K during charge and discharge; (b1) logarithmic plot of internal resistance during charge; (b2) logarithmic plot of internal resistance during discharge; (c) average internal resistance against temperature together with charge and discharge; (d) Temperature and heating power during Cp(battery) measuring; (e) Specific heat capacity of 22 / 34
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battery (Cp(battery)) measured for some different heating power; (f) Specific heat capacity for different SOC levels.
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(a) Typical plot of entropy during charge and discharge with I-GITT.
(b1) Cell potential and temperature behavior against time for SOC 0.482 for temperature stepping. 23 / 34
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(b2) Specific entropy as obtained from several temperature range for temperature stepping
(c1) Cell potential and temperature against time for slow cooling (cell SOC 0.482)
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(c2) Cell potential against temperature for slow cooling (cell SOC 0.482)
(c3) Polynomial fitting of cell potential against temperature for various SOC range for slow cooling
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(d) Entropy of experimental results from temperature stepping (method 2) and slow cooling (method 3).
(e) Polynomial-fitted entropy originated from experimental results from temperature stepping (method 2) and 26 / 34
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slow cooling (method 3). Fig. 2. (a) Typical plot of entropy during charge and discharge with I-GITT; (b1) cell potential and temperature behavior against time for SOC 0.482 for temperature stepping; (b2) specific entropy as obtained from temperature stepping (method 2) with temperature range of 303K and 333K; (c1) cell potential and temperature
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against time for slow cooling (cell SOC 0.482); (c2) cell potential against temperature for slow cooling; (c3) polynomial fitting of cell potential against temperature for various SOC range for slow cooling (cell SOC 0.482); (d) specific entropy of experimental results from slow cooling (method 3) with temperature range of 293K and 333K; (e) polynomial-fitted entropy originated from experimental results from temperature stepping
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(method 2) and slow cooling (method 3).
(a) 0.03 C-rate
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(b) 0.05 C-rate
(b1) 0.05 C-rate; cell potential
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(c) 0.3 C-rate
(d) 0.5 C-rate Fig. 3. Temperature and cell potential during adiabatic charge and discharge (cell potential range: 2.7–4.15 V) plotted together with calculated temperature on the basis of heat relationship by using the measured properties of Cp(battery), Ri, and △S. (a) 0.03, (b) 0.05, (b1) 0.05 C; cell potential, (c) 0.3 and (d) 0.5 C. 29 / 34
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Room temperature continuous charge/discharge condition
(b)
Adiabatic continuous charge/discharge condition
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(c)
Adiabatic discharge for different C-rates
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Fig. 4. Temperature and cell potential patterns of (a, b) continuous charge and discharge sequences with C-rate of 0.1 for a cell potential range of 2.7–4.15 V (a) at room temperature and (b) under adiabatic condition through ARC. (c) Adiabatic discharge at various C-rates together with temperature calculated using the proposed model. C-rates: 0.03,
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0.1, 0.3, 0.4, 0.5, and 1.
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Fig. 5. Calculated final temperature with C-rate for the full charge/discharge of the battery.
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S0013-4686(19)30701-7
DOI:
https://doi.org/10.1016/j.electacta.2019.04.026
Reference:
EA 33974
To appear in:
Electrochimica Acta
Received Date: 5 November 2018 Revised Date:
1 April 2019
Accepted Date: 4 April 2019
Please cite this article as: C.-H. Doh, Y.-C. Ha, S.-w. Eom, Entropy measurement of a large format lithium ion battery and its application to calculate heat generation, Electrochimica Acta (2019), doi: https://doi.org/10.1016/j.electacta.2019.04.026. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Entropy Measurement of a Large Format Lithium Ion Battery and its Application to Calculate Heat Generation
Battery Research Center, Korea Electrotechnology Research Institute, and 2KERI Campus, Korea University of Science and Technology, Changwon, Gyongnam 51543, Korea [email protected];[email protected]
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*E-mail:
Abstract
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As large format lithium ion batteries are used in EV and ESS applications, the temperature control of a battery is important to achieve a safe and long cycle life operation. To control the temperature properly, we need to know the heat generation characteristics. Heat generation of a battery mainly depends on the applied current, internal resistance, temperature and entropy of the battery. However, it is not easy to measure the entropy, because they depend on SOC and temperature simultaneously. Here, we suggested a new way to measure the entropy, on the
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assumption of adiabatic condition and calculate the heat generation. Internal resistance, specific heat capacity, OCCP values are also measured experimentally. The entropy (F ∂EOC ∂T-1) at a given SOC were measured from the variation of OCCP with temperature and expressed via a polynomial equation. The temperature dependence of the internal
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resistance were also expressed in the same procedure. The battery temperature variations calculated from the above parameters showed a good agreement with those from experiments. This suggested method provides a simple and reliable temperature estimation of lithium ion batteries.
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Chil-Hoon Doh1,2,*, Yoon-Cheol Ha1,2, Seung-wook Eom1,2
Keywords
Lithium ion battery, heat generation, entropy, specific heat capacity, internal resistance, temperature estimation
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Graphical abstract
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1.
Introduction As the EV and ESS industries grow, higher energy density and larger format of lithium-ion batteries (LIBs) has been
employed widely. The energy density of LIB cells reached around 260 W h kg-1 and the size increased to accommodate
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approximately 120 A h capacity. As they are employed in a module or a pack, the temperature control becomes crucial for the safe and long cycle life operation. During charge and discharge, the temperature of a cell changes and the exact estimation of the temperature variation is important for the design of TMS (temperature management system) of the module, pack and system. Single cell in an open space may easily radiate the heat generated from charge/discharge but
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it is not easy to control the cells inside the battery. The cell located inner part of the multi-cell-packed battery package might be in a quasi-adiabatic condition [1,2], where accumulated heat leads to a faster increase in temperature at the
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inner part than the outer part of the multi-cell-packed battery pack, thus provoking a fast degradation or a thermal runaway at the worst case.
Quantifying the heat generation and the temperature variation on the basis of the C-rate of charge/discharge is important to establish the optimum operating condition for battery performance such as energy, power, cycle life, and safety. The fundamental thermochemical properties [3] related to Gibb’s free energy can be estimated under an adiabatic
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condition. The endothermic/exothermic characteristics against current amplitude and SOC range can be understood on the basis of the irreversible Joule heating and the reversible entropic heating/cooling. The investigation of electrochemical exothermic/endothermic properties, especially the entropic effect [3 - 13], is necessary for the safe 456789101112
battery operation with prolonged life cycles. Furthermore, as LIBs use an organic electrolyte, they have a low specific
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heat capacity (Cp(battery)) and the temperature of LIB could increase rapidly compared with water-based batteries. Battery heat generation and temperature variation can be expressed by Eq. (1) [3, 9-11,14- 16]. The quantity 15
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of heat generated by a battery during galvanostatic charge/discharge can be divided into two categories, namely, internal resistance and entropy (Eq. (2)). Under a galvanostatic condition, SOC has a linear relationship with elapsed time (Eq. (3)). Therefore, differential time can be directly correlated with differential SOC by using Eq. (4). Moreover, the current applied during galvanostatic charge/discharge can be directly correlated with the C-rate by using Eq. (5). total = irreversible + reversible + tab = I E − E − Qtotal =
+ + ≅ − ITΔS --------------(1),
(I2 Ri - ITΔS) dt ---------------------------------------------------------------------(2),
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SOC = (q0 - q) q0-1 = (q0 - I t) q0-1 ------------------------------------------------------------(3), dt = − q0 I-1 dSOC -----------------------------------------------------------------------(4), I
=
C-rate q0 --------------------------------------------------------------------------------(5).
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By combining Eqs. (2), (4), and (5), Qtotal can be varied as a function of the C-rate and the range of SOC (Eqs. (6) and (7)). Temperature variation can be expressed using a rearranged version of Eq. (8) as a function of initial temperature, specific heat capacity (Cp(battery)), internal resistance, C-rate, and entropy, which are intensive properties of a battery. The specific Coulomb capacity (q0 M-1) and specific internal resistance (ri) of a battery are also inherent
ITΔS
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intensive properties.
Qtotal = - ( 0 )
(I2 Ri - ITΔS) dSOC ---------------------------------------------------(6),
Q ) *+ = - q0
(C-rate q0 Ri - TΔS) dSOC = Cp(battery) M (T2 - T1) ----------------(7),
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T2 - T1 = - ( =-(
q0 ) M Cp(battery) q0
2 Cp(battery)
)
(
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q
(C-rate q0 Ri C-rate q0 2
1 TΔS) dSOC F
ri - TΔS) dSOC ------------------------------------(8).
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The following is defined in Eqs. (1)–(8):
3
, rate (or power) of heat generation (W) ; Q, heat generation (W s) ;
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EOC, open-circuit cell potential (V); Et, cell potential at given time (V);
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I, current (A), discharge (+), charge (-);
T, battery temperature (K) depending on current, initial temperature, entropy, and SOC; T1, initial temperature before charge or discharge (K); T2, final temperature after charge or discharge (K); RA, anode tab resistance (Ohm, Ω); RC, cathode tab resistance (Ohm, Ω); 4 / 34
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Ri, battery internal resistance (Ohm, Ω); ri, battery specific internal resistance (Ri M; Ohm g, Ω g); F, Faraday’s constant (A s mol-1);
t, time (s); SOC, state of charge (dimensionless); C-rate, (h−1), discharge (+), charge (-);
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q0, battery capacity (A h, 3600 A s);
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∆S, battery specific entropy (J mol-1 K-1 or W s mol-1 K-1);
Cp(battery), specific heat capacity of the battery (J g-1 K -1 or W s g-1 K-1);
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M, battery mass (g); (q0 M-1), specific coulomb capacity (A h g-1);
In this study, the useful and easy method to evaluate entropy value as function of SOC was contrived under an atmospheric surrounding that is the same as the usual battery operating condition. Molar entropy was evaluated as (F
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∆E ∆T-1) based on the Gibb’s free energy relationship and the Nernst equation of (∆G = ∆H – T ∆S = - n F E). Battery specific heat capacity (Cp(battery)) was measured by varying SOC. Open-circuit cell potential (OCCP) and internal resistance (Ri) were evaluated by varying the SOC and temperature by using the intermittent current transient method.
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The evaluated entropy, Cp(battery) and Ri were entered into the calculations by using Eq. (8), which represents an electrothermo-chemical model, to obtain temperature variation as a function of change in SOC. This calculation showed that
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adiabatic heat insulation was the worst operating condition. The calculated results showed good agreement with the experimental results. The measured results of Cp(battery), entropy and internal resistance also showed good agreement with the calculated results. The acquired intensive electro-thermo-chemical properties of the battery help determine the internal battery temperature. The measurement of internal battery temperature is impossible under normal operations. However, the limiting current for safe operation can be determined using information about the internal battery temperature. This information about the electro-thermo-chemical properties of a battery might be useful from the viewpoint of specifying the heat radiation design criteria of the battery to ensure the safe operation of electrical/electronic devices/machines.
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2.
Experiment and computational calculation A prismatic lithium ion battery (graphite/Li(Ni-Co-Mn)O2) with coulomb capacity of 104 A h (2.032 kg)
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was used herein.
2.1. OCCP with SOC and temperature
The two parallel-paired batteries were charged–discharged using the intermittent current transient technique under temperatures of 291 K, 298 K, 313 K, 328 K, and 333 K and the isothermal mode of EV-ARC (extended volume–
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accelerated rate calorimeter, THT, U.K.). A current of 9.4 A/cell was applied for 30 min. Thereafter, the battery was rested for 60 min. This periodic current transient was applied until the battery was charged to 4.15 V and discharged to
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2.7 V. The OCCP values were collected at the end of the rest time under each temperature and SOC.
2.2. Internal resistance (Ri) with SOC and temperature
Internal resistance was evaluated from a dataset (current, voltage) compiled for a previous OCCP measurement by
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using the intermittent current transient technique under temperatures of 298 K, 313 K, and 328 K. The cell potential differences between each end of galvanic current pulse to be the start of rest time and the end of rest time of 1 hour were gathered.
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2.3. Specific heat capacity (Cp(battery))
A thin flexible-film Ohmic heater with a resistance of 89 Ω and an area of 12 × 7 cm2 (KH-305/10, Omegalux,
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U.S.A.) was inserted between two batteries. This battery–(Ohmic heater)–battery assembly was installed within the EVARC cylinder. The Ohmic heater was connected to an automatic Cp measuring unit, which was constructed as a part of EV-ARC. Cp(battery) was measured using the automatic mode of the implanted software by varying the heating power from 3.2 W to 7.1 W and by varying the SOC over its entire range.
2.4. Entropy (∆S) with SOC The same set of two paired batteries was used to evaluate the entropy in atmospheric air within the EVARC cylinder. Molar entropy was evaluated as (F ∆E ∆T-1) based on the Gibb’s free energy relationship and 6 / 34
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the Nernst equation of (∆G = ∆H – T ∆S = - n F E). 2.4.1.
Isothermal galvanostatic intermittent charge–discharge
The two parallel-paired batteries were charged–discharged by using the isothermal galvanostatic intermittent transient
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technique (I-GITT) at given temperature. Five different temperature of 291 K, 298, 313 K, 328 K and 333 K were separately tested by applying the isothermal mode of EV-ARC. A current of 9.4 A/cell was applied for 30 min. Thereafter, the batteries were rested for 60 min. This periodic current transient was applied until the batteries were charged to 4.15 V and discharged to 2.7 V. Specific entropies were calculated using the gathered OCCP data at each
Cell potential evaluation with discrete stepping of temperature for various SOC
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2.4.2.
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temperature and SOC.
The SOC-regulated paired batteries were heated under the adiabatic condition by using the “Heat–Wait– Search” mode of EV-ARC. SOC was regulated with interval of 0.1 for SOC 0.0 - SOC 1.0 and SOC 0.05 - SOC 0.95 using the current of 0.1 C-rate. “Heat–Wait–Search” is a stepwise sequence to get exothermic or endothermic heat generation. Cell potential and temperature data were collected after the full stabilization of temperature (0.02 K min-1, 1
adiabatic chamber.
2.4.3.
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h). Additional heating power was applied using the EV-ARC control unit to maintain each set temperature in the
Cell potential and temperature evaluation with quasi-adiabatic slow cooling for various SOC
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The SOC-regulated paired batteries were heated to 333 K under the adiabatic condition and then slowly cooled under the quasi-adiabatic condition to room temperature (283 K-299 K) over 30 h. Cell potential
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during this cooling was used as the equilibrated cell potential of that temperature. The choice of temperature range depends on the purpose of the test and any modification for slow cooling instead of the ARC tool would be acceptable.
2.5. Computational calculation of endothermic/exothermic heat An iterative electro-thermo-chemical modeling (IETCM) calculation program was developed on the basis of Eq. (8). The temporal temperature of each SOC was calculated by iterating the mutually correlated properties of 7 / 34
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internal resistance and entropy. The IETCM program produced the patterns of variation in temperature, internal resistance, and entropy, as well as the maximum temperature and final temperature, by considering the current and its direction. Related electro-thermo-chemical properties such as voltage and accumulated heat can be calculated. The
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IETCM program can also be modified for non-galvanostatic condition. Anyway, the cases were not touched at here. The IETCM program employs the Eigen input values of entropy, internal resistance, OCCP, Coulomb capacity, weight, and specific heat capacity as the measured intensive properties. The initial parameters as boundary conditions for the iterative calculation were initial SOC, final SOC, initial temperature, and applied current. The
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current was converted into C-rate as the ratio of current against Coulomb capacity. Differential SOC was generated depending on the SOC range and the given fractionation as the input value. A convergence test was performed for
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iterative calculations with different fraction numbers of 10n (n = 1, 2, 3, 4) to confirm the iteration results. Effective and meaningful results were deduced for fraction numbers over 1000. The fractionation number of 1000 was used to get good agreement and save the calculation time. The iteration process could be considered to consist of several steps. In step 1, conditional parameters such as battery information and calculation boundary condition as initial temperature, initial SOC, final SOC, and C-rate are read. In step 2, specific entropy is evaluated for each
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corresponding SOC level. In step 3, internal resistance (Ri000), which means the initial Ri as calculated for the given SOC and temperature, is calculated as a function of initial temperature (T1000). On the basis of one instance of mutual thermo-electro-chemical calculation, T2000 is generated for (SOC000 + dSOC). T2000 would be just T1001 in the next iteration of SOC001. Furthermore, SOC001 is just (SOC000 + dSOC). The next internal resistance Ri001 can be evaluated
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in the same manner as the previous one. These iterations can be executed for the number of fractionations. From the results of the iterative calculations, the maximum temperature, final temperature (T2999), final internal resistance
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(Ri999), temperature variation (T2000 - T2999) in the objective SOC region, internal resistance variation (Ri000 - Ri999) in the objective SOC region, and average specific entropy can be deduced. The internal resistance in the temperature region outside the measured region was determined using an extrapolated linear relationship for the low-temperature region and a linear relationship between the Ri of 333 K and 0 Ω as the Ri of 2273 K.
For the conversion between molar entropy (J mol-1 K-1) and entropy (J K-1), the moles of electricity of a battery can be defined as Eq. 9. The charge/discharge capacity of the battery used herein was 103.7 Ah, which amounts to 373,320 A s battery-1 and 3.8692 mole battery-1 by applying Faraday’s constant 96,485.33289 A s mole-1. 8 / 34
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Capacity 103.7 A h battery-1 = 373, 320 A sec battery-1 = 3.8692 mole battery-1 --------------(9).
3.
Results and discussions
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3.1. OCCP behavior with SOC and temperature The OCCP of LIBs has been compared against SOC and temperature widely in the literature. The OCCP of an LIB is unique because of the differences in the cathode, anode, electrolyte, and (+/-) balancing across various batteries. In this study, OCCP information was evaluated for the iterative calculations as a function of temperature and
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charge/discharge (Fig. 1a). The obtained OCCP information against SOC was fitted as a polynomial relationship (Table S1). The OCCP information is not necessary for the galvanostatic (constant current) electro-thermo-chemical reaction
potentiostatic process in this work.
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3.2. Internal resistance (Ri)
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(Eq. 8), but it is critical for the potentiostatic (constant voltage) electro-thermo-chemical reaction. We do not study the
A plot of battery Ri versus SOC during charge/discharge is shown for the temperatures of 298 K, 313 K and 328 K with temperature on the y-axis (Fig. 1b). The polynomial functions of Ri were evaluated for different SOC and temperature during charge and discharge. Fig. 1b1 and Fig. 1b2 show a logarithmic plot of battery Ri versus SOC for
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the temperatures of 298 K, 313 K and 328 K during charging and discharging. Moreover, the average battery Ri is shown as a function of temperature at 298 K, 313 K and 328 K in Fig. 1c. The average battery Ri was evaluated in the
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SOC range of 0.05-1.0 during charge and 1.0-0.003 during discharge. Table S2 shows the relationship of Ri against temperature and SOC. The relationship between average battery Ri and temperature can be expressed using Eqs. 10 and 11. The numerical formula of Ri in terms of temperature and SOC is important in the iterative calculation of electro-thermo-chemical reactions. R
= 5.8782E-7 T2 - 3.9448E-4 T + 6.7133E-2; R² = 1.0000 --------------------------(10),
dR
= (2 x 5.8782E-7 T - 3.9448E-4) dT -------------------------------------------------(11).
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3.3. Specific heat capacity (Cp(battery)) The quantity of heat input to a battery can be determined given the heating power and heating time. The variations in temperature and heating power during Cp(battery) measurement are plotted in Fig. 1d. During this heating, the ambient
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temperature was invariably equilibrated to the battery temperature. Cp(battery) was approximately 1 (J g-1 K-1) for heating powers of 3.16 W, 6.18 W and 7.11 W and a battery of SOC 0.482 (Fig. 1e) and for different SOC levels of 0, 0.25, 0.5, 0.75, and 1 (Fig. 1f). Therefore, Cp(battery) can be considered constant at 1.0144 (J g-1 K-1). Maleki et al. [17] reported the Cp(battery) of a US18650 cell as 1.04 ± 0.02 (J g-1 K-1). Pesaran et al. [18] reported the Cp(battery) of a 6 A h lithium-ion
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battery and a 4 A h lithium-ion polymer battery as 0.795 and 1.0118 (J g-1 K-1), respectively. Related works on Cp(battery) [19 - 21] can be found in the literature. The Cp(battery) value of a given battery is a useful intensive property from the 20
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viewpoint of interpreting the electro-thermo-chemical reactions that occur during galvanostatic and potentiostatic processes.
3.4. Entropy (△ △S) 3.4.1.
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Fig. 1
Isothermal galvanostatic intermittent charge–discharge The entropy of charge and discharge were evaluated using I-GITT (Fig. 2a). Entropy could be evaluated on the
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basis of the OCCP and temperature data during I-GITT. Battery entropy can be determined using I-GITT. However, the validity of the obtained entropy might be low because of the difference between the internal battery temperature and the
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temperature of the convection oven. A battery is heated owing to the passage of current through it. At low SOCs, the entropy at the start of charging and the end of discharging were considerably different. This result could be compared with the results obtained using the two other methods described later in the paper.
3.4.2.
Cell potential evaluation with discrete stepping of temperature for various SOC
The typical cell potential and temperature behavior can be obtained (Fig. 2b1), and specific entropyies can be obtained for various SOCs and temperature range of 303K and 333K (Fig. 2b2). The entropy obtained herein gave some limited data sets depending on the size of temperature step in spite of the need for an extended period of measuring time. 10 / 34
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3.4.3. Cell potential and temperature evaluation with quasi-adiabatic slow cooling for various SOCs The typical cell potential and temperature behavior can be obtained (Fig. 2c1 and Fig. 2c2). Fig. 2c1
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shows the polynomial fitting of the cell potential and temperature of a battery (SOC 0.482) against time. A new polynomial fitting of cell potential versus temperature of a battery could be obtained as Fig. 2c2 based on the results of Fig. 2c1. These stepwise polynomial fittings generate easy matches between the datasets of cell potential and temperature over different time intervals. The polynomial fitting of temperature and cell
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potential against measurement time was performed by increasing the SOC in steps of 0.05. As a result, the relationships given in Table S3 and Table S4 were obtained. The polynomial fitting of cell potential versus
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temperature was given in Fig. 2c3 and Table S5. This polynomial relation could be used in multi-physics interpretation. The relationship between cell potential and temperature of Table S5 could be considered as linear relationship. The linear fitting data of cell potential (V) against temperature (K) for each SOC level are compiled as Table S6 and plotted in Fig. 2c3. The derivative results of (dV dT-1), △S, △H were obtained
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using the Gibb’s free energy relationship. The specific entropy for any given SOC range could be obtained for the full temperature range and various SOC levels used in the iterative electro-thermo-chemical
3.4.4.
Fig. 2
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simulation.
Comparison of the entropy results obtained using three different methods
The tendencies of entropy against SOC obtained using the I-GITT (method 1), temperature stepping (method 2), and quasi-adiabatic slow cooling methods (method 3) were similar. Nevertheless, I-GITT (method 1) is incorrect compared with temperature stepping (method 2) and adiabatic slow cooling (method 3) because the cell potential data used in IGITT were gathered under the condition of current flow. Temperature stepping (method 2) was employed to measure the equilibrated cell potential for discrete temperatures. This measurement required a long waiting time and yielded a small dataset, thus reducing representativeness. Adiabatic slow cooling (method 3) was employed to continuously 11 / 34
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measure the data of temperature and cell potential during the slow cooling of a heated battery in an adiabatic chamber. Therefore, quasi-adiabatic slow cooling method could generate a representative entropy value. Table S5 summarizes the entropy values measured using temperature stepping (method 2) and quasi-adiabatic slow cooling (method 3). In case of
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quasi-adiabatic slow cooling (method 3), polynomial fitted entropy values are listed together with the measured values. Fig. 2d shows the entropy values obtained using adiabatic slow cooling between 333 K and 293 K, and first-order fitting of adiabatic slow cooling between 333 K and 293 K.
The entropy value at a given SOC and the average entropy over a given SOC range for the iterative calculation can
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be extracted from Table S5. The polynomial relationship is necessary for the flexible and accurate iteration of the electro-thermo-chemical reaction in a battery. However, the obtained results exhibit complex patterns, and it would be
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impossible to express them as one polynomial. To overcome this difficulty, regionally polynomial-fitted entropy was obtained with temperature stepping between 333 K and 303 K and adiabatic slow cooling between 333 K and 293 K (Fig. 2e). Table 1 summarizes the regional polynomial expressions of entropy for four different SOC ranges. The entropy of region 1 for an SOC of 0–0.1 has a negative value, and its linear relationship with SOC is linear. The entropy of region 2 for an SOC of 0.1–0.28 is negative, and its relationship with SOC is linear. The entropy of region 3 for an
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SOC of 0.28–0.57(5) is negative according to temperature stepping (method 2) but is positive according to adiabatic slow cooling (method 3). The temperature of the adiabatic discharge of the battery with a low C-rate of 0.03 C-rate indicated that the reaction was initially exothermic, but it later changed to endothermic and finally changed again to exothermic (Fig. 3a). To verify the temperature-following characteristics of the adiabatic chamber against the battery
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temperature, the temperatures of the top plate, bottom plate, and cylinder plate of the adiabatic chamber are plotted in Fig. 3a. It was possible to measure these clear exothermic/endothermic reactions owing to the low C-rate of 0.03 C-rate.
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The low C-rate illuminated the effect of entropy on the thermochemical reaction by minimizing the contribution of Ohmic Joule heating. In line with this result, the entropy of region 3 was positive in contrast to the entropies of regions 1 and 2. Therefore, the result of adiabatic slow cooling (method 3) corresponds to the experimental result and approximates well the real phenomenon. The entropy of region 4 for SOC of 0.57(5)–1 was negative. Region 4 had five breakpoints with a small significance, and these were combined into a six-order polynomial function. Adiabatic slow cooling (method 3) showed the highest multiplicity and the best agreement with the experimental results among the abovementioned three methods for measuring entropy. The entropy values obtained using adiabatic slow cooling (method 3) was used in the iterative simulation. 12 / 34
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Table 1. Polynomial expression of entropy for four different SOC ranges. ∆S (cooling, fit, 333 K–303 Classification
∆S (temp. step, 333 K–303 K)
∆S (cooling, 333 K–303 K)
Region 1
SOC range: 0–0.0983375
Function: ∆S = b1 SOC + a1
b1
2.50E+02
528.26471
a1
−4.89E+01
−70.55344
R 1²
1
0.995
−72.2888035
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SOC range: 0.0983375−0.28 Function: ∆S = c2 SOC2 + b2 SOC + a2
c2
1.89E+02
b2
−5.23E+01
a2
−2.14E+01
R 2²
0.789
Region 3
539.3344063
0.998
SOC range: 0.28−0.57459
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Region 2
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K)
80.09035889
82.55883466
−26.00540646
−26.80023123
0.909
0.906
Function: ∆S = d3 SOC3 + c3 SOC2 + b3 SOC + a3
9.73E+02
c3
−.43E+03
b3
6.88E+02
19.16202512
19.64044168
a3
−1.08E+02
−1.34453314
−1.21573825
0.967
0.925
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R 3²
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d3
0.8821
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SOC range: 0.57459−1 Region 4
Function: ∆S = g4 SOC6 + f4 SOC5 + e4 SOC4 + d4 SOC3 + c4 SOC2 + b4 SOC + a4 2.68E+05
−36115.46404
−15837.61063
−1.26E+06
160191.9635
65925.20666
e4
2.43E+06
−294133.1811
−111847.7554
d4
−2.49E+06
286448.9981
98654.33399
c4
1.43E+06
−156228.7381
−47461.39921
b4
−4.31E+05
45281.48644
11691.30821
g4 f4
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5.39E+04
−5453.422546
−1132.079173
R 4²
0.983
0.984
0.994
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a4
3.5. Battery temperature behavior during adiabatic charge–discharge and comparison with computational calculation results for C-rates of 0.05 C, 0.3 C, and 0.5 C
The trends of temperature and cell potential during adiabatic charge and discharge are plotted in Fig. 3 against
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SOC with different initial temperatures and C-rates. A thermocouple was set at the center of the two sandwiched lithium ion batteries within the adiabatic chamber. Computational iterative calculation generated a full dataset of
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given SOC span as shown at Fig. 3a and Table S8. Calculated results were plotted as sporadic emphatic marks at Fig. 3a together with experimental results. Entropy of each SOC levels were calculated based on regionally polynomial fitted values following Table 1 as generated by method 3.
During charge/discharge at the rate of 0.05 C, the cell potential pattern was maintained nevertheless the increase of temperature by the advance of cycling. However, the temperature of battery was increased gradually by the
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passage of current. The temperature was more rapidly increased at the initial charge/discharge to be from 301.5 K to 302.1 K for charge and from 302.1K to 306.3 K for discharge than at the next charge/discharge to be from 306.3 K to 305.2 K for charge and from 305.2 K to 308.4 K for discharge. The endothermic and exothermic reactions were
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measured during each charge and discharge stage. The breakpoints between exothermic and endothermic reaction are lined vertically and highlighted with red color. The classification of these breakpoints corresponds to the four-step
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polynomial classification of entropy. A few meaningful breakpoints at SOC 0.0482 were not separated because of the very narrow SOC span and the presence of a region with very high internal resistance. Appropriately colored dots at Fig. 3b shows calculated temperature obtained by the iterative calculation of electro-thermo-chemical reaction according to Eq. (8), Cp(battery), △S, Ri, battery capacity, and battery weight. The better correspondence between experiment and calculation in the SOC region of 0.1−1.0 could be ascribed to the better stability of entropy and internal resistance than that in the SOC region of 0–0.1. The low correspondence in the SOC region of 0.0–0.1 could be ascribed to the instability of entropy and internal resistance. Cell potential at Fig. 3b was replotted at Fig. 3b1 as an expanded y-axis scale. 14 / 34
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Fig. 3c and Fig. 3d show the experimental and iterative calculation results for charge/discharge with high C-rates of 0.3 and 0.5 C. The agreement between the calculated and experimental results is as good as that in the case of the C-rate of 0.05 C. The good agreement between the measured results and the IETCM results shows that the measured
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intensive parameters of entropy, specific heat capacity, and internal resistance are effective.
Fig. 3
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3.6. Battery temperature behavior during room temperature and adiabatic charge– discharge
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The trends of temperature and cell potential under charge/discharge at a rate of 0.1 C-rate at room temperature are plotted in Fig. 4a against SOC. A thermocouple was set at the center of the two sandwiched lithium ion batteries within the adiabatic chamber. Room temperature was varied as 282 K–293 K during the daytime and nighttime. The temperature of the chamber containing the batteries was set to 286.5 K–289.3 K. The battery temperature followed the chamber temperature with a few perturbations owing to the influence of Ohmic Joule heating and entropic heating. In
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any case, the battery external temperature was consistent with the chamber temperature. Battery temperature over three continuous charge/discharge cycles remained between 286.5 K and 291.8 K without any serious heating. The same charge/discharge cycle was executed under the adiabatic condition instead of room temperature, and the results are
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plotted in Fig. 4b.
When the number of charge/discharge cycles increased, battery temperature increased gradually owing to Joule
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heating and entropic heating. The temperature of the adiabatic chamber was consistent with the battery temperature through the automatic synchronization of the temperature of adiabatic chamber. The battery temperature increased from 298.8 K to 316.2 K over three continuous charge/discharge cycles. This means that the heat generated during the charge/discharge cycles accumulated continuously, thus leading to an increase in temperature. Moreover, it is important to understand the significance of heat radiation for a serial and parallelintegrated large-scale battery system.
3.7. Discharge C-rate effect on the battery temperature behavior 15 / 34
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Fig. 4c shows the plots of the cell potential and battery temperature under adiabatic discharge at a few different C-rates and the same initial temperature of 298 K. The alteration of cell potential behavior by C-rate range of 0.03 C to 1.0 C was not so serious. The calculated temperature is plotted together with the experimental result. In all
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cases, the C-rate of 0.2 was used for charging. The calculated temperature shows good agreement with the experimental result for the low C-rates of 0.03 and 0.1. As the C-rate increased to 0.3, the calculated temperature became somewhat higher than the experimental temperature. Table 2 summarizes the initial voltage, final voltage, initial temperature, final experiment temperature, and final calculation temperature. The electro-thermo-chemical reaction results calculated
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using the proposed model are consistent for C-rates of up to 0.5 C-rate for an SOC of 0.1. The temperature calculated using the proposed model for the 1.0 C-rate was somewhat higher than the experimental temperature. This deviation can
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be ascribed to the property of heat flux due to the spatial inhomogeneity of temperature in the battery [1,2]. Heat flux was not considered in this study.
Fig. 4
Initial cell potential (V) 4.130
Final cell potential (V) 2.7
0.0039
4.128
2.7
0.0000
4.128
2.7
4.109 4.128
C-rate (h-1)
Initial temperature (experiment/calculation)
Final temperature (experiment)
Final temperature (calculation)
0.03
24.771
28.489
29.733
0.1
25.489
31.889
33.459
0.0159
0.3
24.578
35.446
38.586
2.7
0.0159
0.4
25.233
37.616
40.802
2.7
0.0150
0.5
24.975
(37.475)
43.211
2.7
0.0205
1.0
25.697
44.987
60.768
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Final SOC
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4.115
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Table 2. Temperature increase during adiabatic discharge
3.8. Simulation of the Final Temperature by the C-rate 16 / 34
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A possible iterative simulation result is plotted in Fig. 5 as a function of the C-rate for the full range of charge and discharge under the initial temperatures of 291 K, 308 K and 333 K under the adiabatic condition. The final temperatures showed considerably different behaviors between charge and discharge but similar behaviors for the initial
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temperatures of 291 K and 308 K. For adiabatic/pseudo-adiabatic charge and discharge, a C-rate lower than 2 must be applied to ensure that the temperature remains below 373 K. The heat generation during charge was lower than that during discharge owing to the contribution of the entropic term. Inclusion of entropy pattern to electro-thermo-chemical
4.
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Fig. 5
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multi-physics modeling could surely improve accuracy of temperature anticipation.
Conclusions
Entropy of a large format lithium ion batteries was evaluated efficiently using the method of adiabatic slow cooling to measure the variation of OCCP with temperature. Specific heat capacity (Cp(battery)), internal resistance, and OCCP were also measured experimentally by using an adiabatic chamber. The battery temperature variations calculated
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from the acquired parameters of entropy, specific heat capacity, internal resistance showed a good agreement with those from experiments. The good agreements between experimental and simulation indicates that the parameters of entropy, specific heat capacity and internal resistance were evaluated accurately. The acquired results would be used in multi-
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Acknowledgments
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physics interpretation on heat generation and dissipation.
This study was supported by the KERI R&D program of MSIP/NST (18-11-N0103-01 and 19-12-N0102-01) and by MOTIE (18-02-N0108-02 and 19-02-N0108-01) of Korea. The authors express their gratitude to Hanwha Land Systems Co., Ltd., and Daewoo Shipbuilding & Marine Engineering Co., Ltd. (DSME).
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Figure caption Thermochemical properties of OCCP, internal resistance and Cp(battery). (a) OCCP against SOC and temperature; (b) internal resistance against SOC for temperature of 298 K, 313 K, and 328 K during charge and discharge; (b1) logarithmic plot of internal resistance during charge; (b2) logarithmic plot of internal resistance during discharge; (c) average internal resistance against temperature together with charge and discharge; (d) Temperature and heating power during Cp(battery) measuring; (e) Specific heat capacity of battery (Cp(battery)) measured for some different heating power; (f) Specific heat capacity for different SOC levels.
Fig. 2.
(a) Typical plot of entropy during charge and discharge with I-GITT; (b1) cell potential and temperature behavior against time for SOC 0.482 for temperature stepping; (b2) specific entropy as obtained from temperature stepping (method 2) with temperature range of 303K and 333K; (c1) cell potential and temperature against time for slow cooling (cell SOC 0.482); (c2) cell potential against temperature for slow cooling; (c3) polynomial fitting of cell potential against temperature for various SOC range for slow cooling (cell SOC 0.482); (d) specific entropy of experimental results from slow cooling (method 3) with temperature range of 293K and 333K; (e) polynomial-fitted entropy originated from experimental results from temperature stepping (method 2) and slow cooling (method 3).
Fig. 3.
Temperature and cell potential during adiabatic charge and discharge (cell potential range: 2.7–4.15 V) plotted together with calculated temperature on the basis of heat relationship by using the measured properties of Cp(battery), Ri, and △S. (a) 0.03, (b) 0.05, (b1) 0.05 C; cell potential, (c) 0.3 and (d) 0.5 C.
Fig. 4.
Temperature and cell potential patterns of (a, b) continuous charge and discharge sequences with C-rate of 0.1 for a cell potential range of 2.7–4.15 V (a) at room temperature and (b) under adiabatic condition through ARC. (c) Adiabatic discharge at various C-rates together with temperature calculated using the proposed model. C-rates: 0.03, 0.1, 0.3, 0.4, 0.5, and 1.0 C.
Fig. 5.
Calculated final temperature with C-rate for the full charge/discharge of the battery.
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Fig. 1.
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(a) OCCP against SOC and temperature
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(b) Internal resistance against SOC and temperature
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(b1) During charge
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(b2) During discharge
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(c) Average internal resistance against temperature
(d) Temperature and heating power during specific heat capacity of battery (Cp(battery)) measuring 21 / 34
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(e) Specific heat capacity of battery (Cp(battery)) measured for some different heating power
(f) Specific heat capacity for different SOC levels Fig. 1. Thermochemical properties of OCCP, internal resistance and Cp(battery). (a) OCCP against SOC and temperature; (b) internal resistance against SOC for temperature of 298 K, 313 K, and 328 K during charge and discharge; (b1) logarithmic plot of internal resistance during charge; (b2) logarithmic plot of internal resistance during discharge; (c) average internal resistance against temperature together with charge and discharge; (d) Temperature and heating power during Cp(battery) measuring; (e) Specific heat capacity of 22 / 34
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battery (Cp(battery)) measured for some different heating power; (f) Specific heat capacity for different SOC levels.
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(a) Typical plot of entropy during charge and discharge with I-GITT.
(b1) Cell potential and temperature behavior against time for SOC 0.482 for temperature stepping. 23 / 34
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(b2) Specific entropy as obtained from several temperature range for temperature stepping
(c1) Cell potential and temperature against time for slow cooling (cell SOC 0.482)
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(c2) Cell potential against temperature for slow cooling (cell SOC 0.482)
(c3) Polynomial fitting of cell potential against temperature for various SOC range for slow cooling
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(d) Entropy of experimental results from temperature stepping (method 2) and slow cooling (method 3).
(e) Polynomial-fitted entropy originated from experimental results from temperature stepping (method 2) and 26 / 34
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slow cooling (method 3). Fig. 2. (a) Typical plot of entropy during charge and discharge with I-GITT; (b1) cell potential and temperature behavior against time for SOC 0.482 for temperature stepping; (b2) specific entropy as obtained from temperature stepping (method 2) with temperature range of 303K and 333K; (c1) cell potential and temperature
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against time for slow cooling (cell SOC 0.482); (c2) cell potential against temperature for slow cooling; (c3) polynomial fitting of cell potential against temperature for various SOC range for slow cooling (cell SOC 0.482); (d) specific entropy of experimental results from slow cooling (method 3) with temperature range of 293K and 333K; (e) polynomial-fitted entropy originated from experimental results from temperature stepping
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(b1) 0.05 C-rate; cell potential
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(c) 0.3 C-rate
(d) 0.5 C-rate Fig. 3. Temperature and cell potential during adiabatic charge and discharge (cell potential range: 2.7–4.15 V) plotted together with calculated temperature on the basis of heat relationship by using the measured properties of Cp(battery), Ri, and △S. (a) 0.03, (b) 0.05, (b1) 0.05 C; cell potential, (c) 0.3 and (d) 0.5 C. 29 / 34
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Adiabatic continuous charge/discharge condition
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(c)
Adiabatic discharge for different C-rates
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Fig. 4. Temperature and cell potential patterns of (a, b) continuous charge and discharge sequences with C-rate of 0.1 for a cell potential range of 2.7–4.15 V (a) at room temperature and (b) under adiabatic condition through ARC. (c) Adiabatic discharge at various C-rates together with temperature calculated using the proposed model. C-rates: 0.03,
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Fig. 5. Calculated final temperature with C-rate for the full charge/discharge of the battery.
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