This study is preceded by another that developed a model to predict LIC performance as a function of constituent components [15]. The earlier model was the first known physics-based model to predict a LIC’s energy storage as a function of its charge current and constituent materials. This initial model was written in Simulink using a Randles ECM framework and relied upon charge current and constituent materials’ effects upon the LIC’s internal temperature in accordance with the Butler-Volmer equation

i d = i o ( e α a F c n e R u T q η − e − α c F c n e R u T a η ) (1)

where i_d is current density, i_o is exchange current density, α a is the specific surface area of the anode electrode, α c is the specific surface area of the cathode electrode, n_e is the number of electrons per ion (1 for lithium), η is the activation overpotential, F_c is Faraday’s constant, R_u is the universal gas constant, and T_a is the absolute temperature of the electrochemical cell. Because in a LIC the energy storage reaction at the anode is orders of magnitude greater than at the cathode [3], one can assume

e − α c F c n e R u T a η = 1 (2)

Therefore, the Butler-Volmer equation may be rearranged to compute the temperature of the LIC as follows

T a = α a n F c R u ln ( i d i o − 1 ) η (3)

The Butler-Volmer temperature increase and the LIC’s exchange current, in turn, affected the capacitance of the LIC as follows

C W = c i R W 2 F 2 A s l 2 R T (4)

where C_W is the Warburg capacitance, c_i is the ionic concentration of the electrolyte in moles per kilogram of electrolyte, A_s is the surface area, and l is the thickness of the electrode’s active layer [3]. Once these relationships were understood, the model was able to predict energy storage as a function of charge power. Additionally, the model was able to estimate LIC voltage and therefore state of charge.

This model computed energy storage within 4% accuracy over a wide range of charge powers, as shown in Figure 3.

Of note, energy storage may range as much as 300%, depending upon charge power, but performance is not linear. For example, energy storage increases slightly at very high power (≈120 W or 6 kW∙kg⁻¹). Other studies indicate electrolyte breaks down at high temperature [28]. Electrolyte breakdown momentarily adds additional ions to the solution that have the effect of increasing energy storage [4] [10]. Because

i d = i A (5)

where i is current and A is the surface area, Equation (3) implies

i ∝ 1 T a (6)

Many efforts have been applied to model known energy storage devices as a function of temperature and other variables (e.g. [16] [29]), but this is the study and its predecessor are the first attempts to predict a LIC’s performance as a function of its constituent components. Another potential concern is hysteresis, as temperature effects on a device may not be observed immediately [30]. A cornerstone of this study is Uno and Tanaka’s previous work demonstrating that elevated temperature in LICs degrades them in a similar manner as elevated current does [5]. This relationship and its implications are a cornerstone of this study.

Voltage in the model should be computed as a sum of all voltage drops across major RC elements in the Randles ECM depicted in Figure 2. Voltage drops across all series (V_S), double layer (V_dl//CT), and Warburg (V_W) elements are given by Equations (7)-(9), respectively.

V S = i R S (7)

V d l = i R c t ( 1 − e − t R c t C d l ) (8)

V W = i R W ( 1 − e − t R W C W ) (9)

And Equation (9) is rewritten for every branch, n, of the equivalent R_W elements as follows

V n = i R n ( 1 − e − t R n C n ) (10)

So that

V W = V 1 + V 2 + ⋯ + V n (11)

The total voltage increase, V_T, can be expressed by

V T = V s + V d l + V W (12)

This model could not accurately compute the voltage. Error was low at very high charge power but averaged 30% at low charge power. This is important because voltage is often used to indicate the state of charge in a capacitor.

Reinvestigation identified some discrepancies in temperature computations. The temperature was computed from the model’s computed capacitance (T_mod) and from experimental energy storage (T_exp), using Equation (4) and a modification of the relationship between capacitance and energy

E = 1 2 C Δ V 2 (13)

where E is energy stored, C is capacitance, and ΔV is the change in voltage. In this situation C will be replaced by C_T, a total capacitance variable, which includes both C_W and a non-temperature dependent double-layer capacitance (C_dl) as follows

C T = C W + C d l (14)

C_dl is found by

C d l = F c ρ l 2 M ( 1 − ε ) l l s (15)

where ρ is the density of the electrode material, M is the mass of the active material, ε is the porosity of the electrode material, l is the thickness of the electrode, and l_S is the thickness of the separator material between the electrodes.

These results showed moderate agreement between T_mod and T_exp, except at high power, as shown in Figure 4. However, all tests were performed in a room with an ambient temperature of 300 K. Thus the low power experimental temperature values and all model temperature computations, which hovered near 150 K, were suspect and called for a reassessment of the model’s computations.

In order to understand how T_a impacts V, a fit equation was computed from T_exp in order to determine T as a function of P as follows

T = − 0.0006 P 3 + 0.0617 P 2 + 3.1543 P + 131.0 (16)

Use of this equation improved V calculations as shown in Figure 5. Of note, improvements in the accuracy of T_a improve the model’s computation of V. However, improving T_a does not eliminate all error, especially at low power.

This data indicated that error in T computations only partially addressed the error in V. Equation (2), used to compute T_a, computes T_a as a function of η, i_d, and i_o, but is unaffected by R_W. By contrast, T_exp relies upon Equation (3), which considers R_W. Subsequent experiments reverted to computing T_a using Equation (2) and studied R_W’s effects upon the circuit because R_W had previously received little attention. Upon examination it was hypothesized that because R_W is divided into four smaller RC circuits, each of which has its own impact on temperature (Figure 2), T_exp may have been misinterpreted as a temperature change

across each of the RC circuits, implying

T m o d = 4 × T e x p (17)

In order for this to be accurate, R_W had to be adjusted to change with charge current. If changes in the model’s R_W input led to matches in experimental and modeled E and V, and if Equation (17) is correct, then the R_W value may be computed as the only missing variable. Under this assumption R_W showed a predictable relationship with respect to charge power, but this relationship broke down around 30 W, spiking at 60 W, as shown in Figure 6.

This relationship can be approximated as

R W = − 8 × 10 − 6 P 3 + 0.0013 P 2 − 0.0103 P + 1.7604 (18)

The reason for this spike in R_W at 60 W is probably the same as the culprit for C_T’s increases at very high power. This phenomenon of increasing C_T is probably due to electrolyte breaking down at high temperature and increasing the number of ions that could carry charge inside the LIC [10] [28]. An increase in C_T can be mathematically accommodated by an increase in R_W in accordance with Equations (3) and (12). Earlier studies indicate R_W increases with temperature, but these studies employed low charge currents (0.2 mA/cm²) and studied relatively low ambient temperatures (−20˚C - +70˚C) [31], so they may have not incorporated many of the capacitive aspects of a LIC. By contrast, at 30 W the charge current is approximately 360 mA/cm² and the T variable has increased by 30% over its values below 10 W. Experimentalists do not yet understand these high power dynamics. Below 30 W charge power, R_W can be approximated as

R W = − 4 × 10 − 5 P 3 − 0.0012 P 2 − 0.0222 P + 1.7195 (19)

These R_W values identified in Figure 6 and explained in Equation (18) and Equation (19) all rendered less than 1% error in voltage. They also yielded good agreement between experimentally-computed temperature (T_exp) and temperature computed from the model’s C_W(T_check). However, as shown in Figure 7, T_exp and T_check did not agree with T in accordance with Equation (17).

In order to verify T_a’s accuracy, temperature increase was computed across each of the four R_WC_W elements in series (R₁C₁, R₂C₂, R₃C₃, R₄C₄) using Equation (17) and R_ctC_dl using a modified Equation (2), as follows

T a = R C T n F i o R u (20)

This temperature increase was designated T_check2. T_check2 and T_check must obey Equation (17) in order to be valid. R_ctC_dl’s contribution to T_a is negligible at all charge powers. For example, at 60 W, where the temperature was computed to be highest,

T R C T ∥ C d l = 0.09 × 1 × 96485 × 7.6 × 10 − 8 8.314 = 7.9 × 10 − 5 (21)

Consequently, the following approximation is valid for T_check2

T c h e c k 2 = c i R 1 2 F 2 A s l 2 R C 1 + c i R 2 2 F 2 A s l 2 R C 2 + c i R 3 2 F 2 A s l 2 R C 3 + c i R 4 2 F 2 A s l 2 R C 4 (22)

T_mod, T_check, T_exp, and T_check2 are compared in Figure 8.

T_check2 is four times T_check at all charge powers, satisfying Equation (17). Also, T_check2 agrees well with T at all powers until P is 30 W, after which error increases, maximizing at 60 W, where R_W spikes. At 60 W and 120 W T_check2 continues to be four times as large as T_exp and T_check. But T decreases to within 50% of T_exp and T_check. Consequently, Equation (17) is only reliably satisfied below 30 W, where T is reliable. Further research into electrolyte breakdown at high temperatures and/or current is needed to provide more insight into this phenomenon and to enable effective modeling.

Using the methods described above, computing T from Equation (2), and computing R_W from Equations (18) or (19), error in V’s final value was less than 1%, and error in E remains negligible. Although the model gave accurate results regarding energy storage and voltage, it still had two serious deficiencies:

· There was no in-situ data to validate the computation of T. All T values had been theoretically computed from the Butler-Volmer equation and various derivations, Equations (1)-(3), (13), and (20). Some direct, experimental temperature data was needed in order to assess this.

· In order for the model to work properly, R_W had to be on the order of 1.5 Ω, as shown in Figure 6. But experimental EIS data from the previous study indicated R_W should be on the order of 0.015 Ω.

As in the previous study [10], a 200F LIC made by General Capacitor (product number LCA200G1, shown in Figure 9, was used to provide experimental data as a baseline to compare against the model. The reasons for this choice are twofold.

· At the time General Capacitor’s 200F product was widely marketed and was commercially available.

· The previous version of the model computed a theoretical temperature increase for LCA200G1, which needed to be validated or disproven.

LCA200G1 design specifications are shown in Table 1. All experimental data in this study came from a LCA200G1 made using the method described by Cao et al. [27]. Because LCA200G1 was a commercial, flagship product, much data had already been collected about its performance, including cycle life data, which can take months, if not years to collect. Additionally, many components and operational parameters, including separator material, electrolyte, operating voltage range, and current collector tab position and welding methods were well-known and likely to be repeated by General Capacitor. This eliminated much of the variability often encountered in laboratory-made LICs.

As discussed in section 1.3, the experimental validation of the theoretical T value was required in order to validate model’s accuracy. Because the test LICs were handled shortly after testing, it was doubtful that the actual ΔT was any higher than a few Kelvin. The initial hypothesis was that the actual ΔT was only slightly lower than the experimental value because of thermal insulating effects inside the LIC. Results indicated the temperature question is more complex than this, as will be explained. Results are shown in Figure 10. At high power charges no temperature increase was observed, as shown by the 10 W charge in Figure 9 and as anticipated by Equation (6). The ΔT values observed at all of these charge powers are compared in Figure 11.

In Figure 9 ΔT peaks then decrease again. This is probably a byproduct of current changes due to Ohm’s law. Because of voltage changes during a charge, a constant power charge will begin with a high current when the voltage is low and end with a low current when the voltage is high. Therefore, near the end of a charge, there is less current to affect the temperature of the LIC. In order to more directly assess the effects of current upon temperature, constant current measurements were made. At currents below 1.0 A, these results showed a linear temperature increase with time. At a higher current, 5.0 A, no temperature increase was measured. 5.0 A was the limit of test equipment. These results are shown in Figure 12. They indicate the Butler-Volmer equation’s temperature increase appears to accurately link current and temperature at relatively low current, but this relationship breaks down at higher currents, where more capacitive behavior dominates. There appears to be a hysteresis as charge current does not immediately affect temperature. Above a certain threshold the charge cycle finishes before current’s effects on cell temperature can be felt. The reason for this is not understood.

If the 5.0 A charging data is ignored, the slope of the charge temperature increases linearly with charge current and can be predicted by an equation

Δ T S l o p e = 0.0043 i (23)

which is plotted in Figure 13. In terms of current density, this is

Δ T S l o p e = 1 × 10 − 5 i d (24)

as demonstrated in Figure 14.

The results shown in Figures 9-14 indicated what the value of ΔT used in the model should be. These experimental ΔT values were used to calibrate the model. First, the initial ambient temperature was set at 298 K, the nominal ambient temperature of the environmental test chamber where the study was conducted. As the maximum ΔT was 2.24 K, the change in the T variable is less than 1% when operating at room temperature. Thus T may be treated as a constant in many circumstances.

As discussed earlier, another problem with the previous model is that R_W was considered static. In fact, it is not. The values found in Figure 6 and Equations (16) and (17) were two orders of magnitude higher than the experimental value of 0.011 Ω computed from potentiostatic EIS data, using a maximum current of 0.1 A. In order to improve the model, another approach was adopted, found by Greenleaf et al. in an earlier study [32]. This approach computes R_W as follows

R W = R u T δ n 2 F 2 C s 2 A 2 D s (25)

where D_s is the species’ diffusion coefficient, δ is the Nernst diffusion coefficient, found from the effective particle radius (r) of the negative electrode material as follows

δ = r 5 (26)

r for the hard carbon material in this anode is approximately 50 μm, making δ equal to 10 μm.

R_W values calculated using Equation (20) and experimental ΔT data agreed with experimental values found from the previous study’s EIS data that were used to build the previous study’s model. This data anticipates R_W values on an order of magnitude similar to the EIS data. Results are compared in Figure 15.

This R_W computation method was combined with the revised method of calculating ΔT. Once R_W was computed by Equation (20) and used accurate ΔT values, error in V diminished but was not eliminated. When the lowest computed R_W value was used, 0.013 Ω, error in the final V value was less than 5%. R_W and energy computations matched experimental values. Energy computations never appreciably changed. The voltage drop across R_S is very small, 5.4 mV, so it does not appreciably affect ΔV. These results also indicate that R_W should be treated as a constant and does not change with current. Voltage error is compared in Figure 16.

One limitation of this method, however, is that the model initially increases V very quickly with high error, although V eventually levels off, as shown in Figure 17. Upon examination of V_S, V_dl//CT, V_W, it is found V_dl//CT is the primary source of this error, as it increases by 1 V during the first several seconds of a charge but is then constant. More research is needed to determine why V_dl//CT behaves in this manner and to correct it. Because the problem has been isolated to V_dl//CT, it is believed the model may struggle with the non-Faradaic, capacitor behavior in an LIC more so than the Faradaic, battery behavior.

The only other unknown left in Equation (2) was η. η values that agreed with experimental results shown in Figures 9-11 were found and are shown in Figure 18. The method of computing R_W had minimal impact on η. Because T is relatively constant these results caused η to decrease by two orders of magnitude when compared to the previous model.