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An earlier study manipulated the Butler-Volmer equation to effectively model a lithium-ion capacitor’s (LIC) energy storage as a function of its constituent components and charge current. However, this model had several shortcomings: computed temperature values were too low, voltage was inaccurate, and the model required Warburg impedance values that were two orders of magnitude higher than experimental results. This study began by analyzing the model’s temperature and voltage computations in order to justify output values. Ultimately, these justifications failed. Therefore, in situ temperature rise was measured during charge cycles. Experimental results indicated that temperature increases minimally during a charge cycle (<1%). At high current densities (≥150 A ·kg<sup>-1</sup>) temperature increase is negligible. After it was found that LIC temperature change is minimal during a charge cycle, the model accurately computed LIC voltage during the charge cycle and computed Warburg impedance that agreed with values derived from earlier experimental studies, even falling within the measurements’ precision error.

Renewable energy and electrification of the energy sector have surged in popularity as the environmental, economic, and sustainability concerns associated with fossil fuels have called for alternative energy sources. Because renewable energy sources are intermittent and relatively unpredictable, their wholesale integration requires substantial energy storage infrastructure. Additionally, an electric or hybrid electric vehicle requires energy storage. These factors have caused energy storage technology to also see significant attention.

Lithium-ion batteries (LIB) are the most common type of electrochemical energy storage device under development and in use. LIBs are characterized by high specific energy (150 - 200 W∙h∙kg^{−1}) but low specific power (often below 1 kW∙kg^{−1}) and relatively low cycle lives (2000 cycles for lithium-sulfur, 5000 cycles for lithium titanium dioxide, often lower for other chemistries) [^{−1}), long cycle life (e.g. 300,000 cycles in laboratory experiments at General Capacitor) [^{−1}) [

Often customers seeking energy storage solutions do not base their products on available energy storage technology but have a product under development for which they seek a niche solution. Usually, when a customer considers LIC technology for an application a cell must be prototyped for evaluation, including designing a new form factor and estimating its energy storage, power, and other performance characteristics. There is often trial and error in this process. Building and testing prototypes can be time-consuming, expensive, and therefore a substantial barrier to market entry. Therefore, the LIC industry needs a predictive model that estimates a LIC’s performance based upon its design parameters and operating conditions.

Recent research has developed models that can describe both LIB and EDLC voltage and energy storage dynamics as a function of charge or discharge (dis/charge) current and time [

Few inroads have been made to dynamically predict LIC energy storage or voltage a function of the electrochemical interactions between the materials that comprise the constituent components of the device. Voltage has been demonstrated as a tool for estimating state of charge in LIBs, while studying their heat generation during dis/charge [

Last year Moye et al. published the first model to dynamically predict energy storage as a function of the constituent components that comprise a LIC [

The first LIC models simply described experimental results. A common means of doing this is building a Randles ECM to describe a premade, experimental LIC [_{S}), a double layer capacitance (C_{dl}), a charge transfer resistance (R_{ct}), and a capacitive element with its own resistance, referred to as the Warburg element (R_{W}) [_{W} can be represented in the time domain as shown in

The earliest LIC modeling studies considered LIC performance as part of a larger system. In 2013 Omar et al. [

The first predictive models involved studies of ambient temperature accelerating cycle life degradation by Uno et al. [

Earlier this year Moye et al. published a model intended to predict LIC performance as a function of constituent components [

This study is preceded by another that developed a model to predict LIC performance as a function of constituent components [

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 [

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 [

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

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 [

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. [

Voltage in the model should be computed as a sum of all voltage drops across major RC elements in the Randles ECM depicted in _{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

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 _{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 (_{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

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 [_{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/cm^{2}) and studied relatively low ambient temperatures (−20˚C - +70˚C) [^{2} 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 _{exp}) and temperature computed from the model’s C_{W}(T_{check}). However, as shown in _{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_{W}C_{W} elements in series (R_{1}C_{1}, R_{2}C_{2}, R_{3}C_{3}, R_{4}C_{4}) using Equation (17) and R_{ct}C_{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_{check}_{2}. T_{check}_{2} and T_{check} must obey Equation (17) in order to be valid. R_{ct}C_{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_{check}_{2 }

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_{check}_{2} are compared in

T_{check}_{2} is four times T_{check} at all charge powers, satisfying Equation (17). Also, T_{check}_{2} 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_{check}_{2} 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 _{W} should be on the order of 0.015 Ω.

As in the previous study [

· 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

Parameter | Value |
---|---|

Lithium Source | Foil Strips |

Positive Electrode Active Material | Activated Carbon^{ } |

Positive Electrode Active Layer Thickness | 100 µm |

Double-Sided Positive Electrodes^{ } | 7 |

Negative Electrode Active Material | Hard Carbon |

Negative Electrode Active Layer Thickness | 90 - 95 µm |

Double-Sided Negative Electrodes | 6 |

Single-Sided Negative Electrodes | 2 |

Negative Electrode Lithium Loading | 8.81% |

Negative Electrode Porosity | 44.16% |

Positive Electrode : Negative Electrode Mass Ratio | 0.678 |

Lithium Source | Foil Strips |

Positive Electrode Active Material | Activated Carbon |

Positive Electrode Active Layer Thickness | 100 µm |

Double-Sided Positive Electrodes | 7 |

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

In

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

Δ T S l o p e = 1 × 10 − 5 i d (24)

as demonstrated in

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 _{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

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

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 _{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 _{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.

The previous study’s model was improved with experimental data and additional theoretical relationships. The first step was to compute temperature values from the model’s predicted energy storage and from experimentally measured energy storage. This method used energy storage’s implications on Warburg capacitance and Warburg capacitance’s implications on cell temperature. These temperature values were analyzed, but they gave values that were lower than expected. Experimental measurements were taken and used to validate the temperature variables. The experimental results indicated that although the temperature does increase as an LIC is charged, this increase is small, <1%. However, the model could only explain these results at relatively low current, where battery behavior dominates. Where capacitive behavior is more pronounced, the model breaks down and indicates negligible temperature increases. In order for the model to accurately reflect the experimental temperature values, the overpotential variable needed to be significantly reduced from the previous study. In fact, the temperature increases are so small that the overpotential variable has little impact on the overall model. The model was adjusted to compute Warburg impedance, which then agreed with the EIS values published in the previous study. It was discovered that Warburg impedance behaves as a constant variable. The result is an improved Butler-Volmer based model that computes energy storage as a function of charge power, also accurately computing voltage and temperature.

There are several shortcomings to model as it now stands. Additional experimental data would be useful to improve the calibration of the model’s temperature variables. However, it is unlikely that additional experimental data would cause a significant improvement in voltage and energy storage modeling below 10 W or 3.3 A. Above 10 W the model’s voltage value initially overshoots experimental data, resulting in voltage error that is initially very high. But the model’s voltage becomes accurate near the end of the charge cycle. This initial overshoot appears to be caused by the model’s treatment of the non-Faradaic, capacitor behavior in the LIC and merits further investigation.

Specific findings from this study were:

· Improved understanding of temperature changes during LIC charging. Specifically, LIC temperature increases little during a charge cycle. This increase is linear vs. time and charge current. However, above a certain threshold current, temperature rise is negligible. In all cases temperature rise is >1%. Future studies may approximate temperature as a constant.

· Mathematical verification that R_{W} values in LIC models are on the order of 0.01 - 0.02 Ω.

· An improved model that computes LIC energy storage and voltage as a function of the LIC’s constituent components, ambient temperature, and charge current.

This research was performed using the resources of General Capacitor LLC and Moye Consultants LLC.

The authors declare no conflicts of interest regarding the publication of this paper.

Moye, D.G., Moss, P.L., Kannan, D.R.R., Chen, X.J., Bolufawi, O., Cao, W.J. and Foo, S.Y. (2020) Improvements to Temperature, Warburg Impedance, and Voltage Computations for a Design-Based Predictive Model for Lithium-Ion Capacitors. Materials Sciences and Applications, 11, 347-369. https://doi.org/10.4236/msa.2020.116024