Understanding the thermal behavior of li-ion battery cells is fundamental to the design of effective thermal management systems for electric vehicles. This behavior encompasses heat generation characteristics, temperature field distribution, and the rate of temperature rise, collectively describing how heat is produced and spreads within a li-ion battery during operation. Accurate thermal models are essential for this purpose, which can be broadly categorized into two groups: macroscopic models and micro-mechanistic models.
Macroscopic models often rely on mathematical formulations to calculate the overall heat generation rate, relating it to the observable temperature change of the li-ion battery. Examples include models based on Fractal Theory to guide cell arrangement in air-cooled systems, or those derived from Fick’s and Fourier’s laws to establish theoretical heat generation rate models. Transient and steady-state thermal models based on heat conduction differential equations have also been developed to study the influence of thermal properties and convection coefficients on temperature distribution within a cylindrical li-ion battery. Other approaches, like lumped-mass thermal analysis models, simplify the battery to a single heat-generating point to predict its overall temperature evolution.
In contrast, micro-mechanistic thermal analysis models delve into the fundamental electrochemical processes inside a li-ion battery. Building upon foundational electrochemical models, the Bernardi model is a classic framework that breaks down the total heat generation into its components: Joule heating (irreversible ohmic heat), reversible entropic heat, heat of mixing, and heat from phase changes. Measuring these components individually, such as using step-current methods for Joule heat or equilibrium potential methods for entropic heat, can be a time-consuming process.
On the experimental side, specialized and often costly instruments like Accelerating Rate Calorimeters (ARC), Extended-volume ARC (EV-ARC), or Heat Flux Meters (HFM) are commonly employed to characterize the thermal behavior of li-ion battery cells. While these methods provide valuable data, their cost and complexity can be a barrier. This work proposes and validates a novel macroscopic testing methodology—the Calorimetric Calibration Method—for accurately characterizing the heat generation behavior of high-energy-density li-ion battery cells. This method offers a balance of accuracy, simplicity, and lower cost compared to some conventional techniques.
The Calorimetric Calibration Mathematical Model
The proposed Calorimetric Calibration Model is an enhanced version of a traditional ideal adiabatic model, incorporating a precise accounting for heat loss. It posits that the total heat generated $q_{cc}$ by a li-ion battery during operation consists of two parts: the heat $q_{cc-1}$ used to increase the battery’s own temperature, and the heat $q_{cc-2}$ dissipated from its surface to the surroundings.
The heat contributing to self-warming is governed by the conservation of energy:
$$ q_{cc-1} = c m \frac{dT_b}{dt} $$
where $c$ is the specific heat capacity, $m$ is the mass of the li-ion battery, $T_b$ is its temperature, and $t$ is time.
The heat dissipated to the environment follows Newton’s law of cooling:
$$ q_{cc-2} = \alpha A (T_b – T_i) $$
where $\alpha$ is the convective heat transfer coefficient of the battery surface, $A$ is its surface area, and $T_i$ is the ambient temperature.
Therefore, the total heat generation rate of the li-ion battery is:
$$ q_{cc} = c m \frac{dT_b}{dt} + \alpha A (T_b – T_i) $$
The key to this model is determining the convective heat transfer coefficient $\alpha$ without specialized equipment. This is achieved through a separate heat loss calibration experiment. By allowing a pre-heated li-ion battery to cool freely in a controlled environment and monitoring its temperature drop, we can relate the rate of temperature decrease to the instantaneous temperature difference between the battery and the ambient. The coefficient $\alpha$ is derived from the slope of the linear relationship between the cooling rate $\frac{dT_{drop}}{dt}$ and the temperature difference $(T_b – T_i)$, as established from the following energy balance during cooling:
$$ c m \frac{dT_{drop}}{dt} = -\alpha A (T_b – T_i) $$
Rearranged for analysis: $$ \frac{dT_{drop}}{dt} = – \frac{\alpha A}{c m} (T_b – T_i) $$
Experimental Methodology and Setup
The test subject was a high-energy-density 21700 cylindrical li-ion battery with a nickel-cobalt-manganese oxide (NCM) cathode and a graphite anode. Key specifications are summarized in Table 1.
| Parameter | Value |
|---|---|
| Cathode Material | Li(Ni0.8Co0.1Mn0.1)O2 |
| Anode Material | Artificial Graphite |
| Mass ($m$) | 66.7 g |
| Specific Heat Capacity ($c$) | 1097.8 J/(kg·°C) |
| Nominal Capacity | 4.6 Ah |
| Nominal Voltage | 3.6 V |
To measure the core temperature of the li-ion battery, a minor modification was necessary. The cell was carefully disassembled after being fully discharged. A T-type thermocouple (0.25 mm wire diameter) was inserted into the central hollow core of the 21700 li-ion battery to half its height. The opening was sealed with vacuum putty and secured with polyimide tape. Three additional thermocouples were attached to the outer surface at axial and circumferential intervals to monitor surface temperature distribution and calculate an average surface temperature.
The instrumented li-ion battery was then mounted in a fixture and insulated on all sides using a 20-mm thick aerogel blanket to minimize uncontrolled heat loss. This entire assembly was placed inside a polystyrene foam insulation box to create a stable, low-convection test environment. The box was housed within a temperature-controlled chamber to set and maintain the initial and ambient temperature $T_i$. A battery cycler was used to perform charge/discharge protocols, while a data acquisition system recorded voltage and temperature data from all thermocouples.
Experimental Procedure
1. Heat Loss Calibration: The insulated li-ion battery assembly and chamber were stabilized at 0°C. The battery was then subjected to charge/discharge cycles at 2C until its temperature rose significantly. The power was disconnected, and the battery was allowed to cool freely while temperatures were logged. The free cooling curve, plotting battery temperature against time, was used to determine the relationship between cooling rate and temperature difference. The slope of this linear relationship, as per Equation derived from the cooling energy balance, yields the value of $\frac{\alpha A}{cm}$, from which the effective heat loss characteristics for the subsequent tests are defined.
2. Discharge Heat Generation Tests: Starting from a fully charged state (SOC=100%) and a stabilized chamber temperature $T_i$, the li-ion battery was discharged at a specified constant current (C-rate) until its cutoff voltage was reached. Temperature data from all thermocouples were recorded throughout. This was repeated for various ambient temperatures ($T_i$ = -20, -10, 0, 10, 20, 30°C) at a 1C discharge rate, and for various discharge rates (1C, 1.5C, 2C, 2.5C) at a fixed ambient temperature of 30°C.
Results and Discussion on Li-ion Battery Thermal Behavior
Heat Generation and Temperature Rise Characteristics
The internal temperature of the li-ion battery was consistently higher than its surface temperature during discharge, confirming a thermal gradient from the core outward due to internal heat generation and external heat loss. The rate of temperature rise of the core $\frac{dT_{in}}{dt}$ was not constant. Initially, it decreased slightly as the battery warmed and its internal resistance dropped. Towards the end of discharge (at low State-of-Charge, SOC), it increased sharply due to a significant rise in polarization resistance, a common phenomenon in li-ion battery operation.
The instantaneous heat generation rate of the li-ion battery core, calculated using the Calorimetric Calibration Model, showed a clear dependence on SOC. At an ambient temperature of 0°C and a 1C discharge rate, the heat generation rate decreased in the mid-SOC range (approximately 0.6 to 1.0) as the cell warmed, then increased markedly in the final stage (SOC < 0.6), exhibiting the characteristic “uptick” at end-of-discharge.
The maximum temperature rise of the li-ion battery core and the delivered discharge capacity under different conditions are critical performance metrics. As shown in Table 2, for a 1C discharge, lower ambient temperatures lead to a higher core temperature rise but a lower delivered capacity. Similarly, at a fixed high temperature (30°C), higher discharge rates (C-rates) cause a greater temperature rise and reduce the usable capacity. This data underscores the negative impact of low temperatures and high currents on both the thermal performance and efficiency of a li-ion battery.
| Condition | Max Core Temp. Rise (°C) | Delivered Capacity (Ah) |
|---|---|---|
| Varying Ambient Temperature at 1C Discharge | ||
| -20°C | 34.8 | ~3.9 (≈85% of nominal) |
| 0°C | ~24 | ~4.4 |
| 30°C | ~11 | ~4.5 |
| Varying Discharge Rate at 30°C Ambient | ||
| 1.0C | ~11 | ~4.5 |
| 2.5C | ~48 | ~4.2 |
The total heat generation rate (including contributions from the core and tabs) showed distinct trends. When discharging at a constant 1C rate, the heat generation rate of the li-ion battery increased as the ambient temperature decreased. This is primarily attributed to higher internal resistance at lower temperatures, leading to greater Joule heating. Conversely, during high-rate discharges at a constant 30°C ambient, the heat generation rate increased with the discharge current, as Joule heating is approximately proportional to the square of the current ($I^2R$). The end-of-discharge “uptick” was present in all tested scenarios for the li-ion battery.
By integrating the instantaneous heat generation rate over the entire discharge cycle, the average heat generation rate $\bar{q}$ for the li-ion battery was calculated for each test condition:
$$ \bar{q} = \frac{1}{SOC_{0\%} – SOC_{100\%}} \int_{SOC_{100\%}}^{SOC_{0\%}} q(SOC) \, d(SOC) $$
The relationship between the average heat generation rate of the li-ion battery and the operating conditions was found to be well-described by quadratic functions, as summarized in Table 3.
| Independent Variable | Quadratic Relationship for $\bar{q}$ | R² Value |
|---|---|---|
| Ambient Temperature $T_a$ (°C) at 1C | $\bar{q} = 3.4 \times 10^{-4}T_a^2 – 2.97 \times 10^{-1}T_a + 1.13$ | 0.9945 |
| Discharge Rate $C$ at 30°C | $\bar{q} = 0.91C^2 – 0.86C + 0.47$ | 0.9992 |
Method Validation: Comparison with Heat Flux Meter
To validate the accuracy of the proposed Calorimetric Calibration Method, its results were compared against measurements from a conventional Heat Flux Meter (HFM). The HFM directly measures the heat flux $q”$ from the li-ion battery surface, and the total heat generation is calculated as:
$$ q = c m \frac{dT}{dt} + q” A $$
The trends in heat generation rate versus SOC obtained from the HFM were very similar to those from the Calorimetric Calibration Method for the li-ion battery.
A direct comparison of the calculated average heat generation rates from both methods is shown in Table 4. The excellent agreement, with a maximum deviation of less than 5.4%, confirms the high accuracy and reliability of the proposed Calorimetric Calibration Method for characterizing the thermal behavior of a li-ion battery.
| Test Condition | Avg. Heat Rate (Calibration Method) | Avg. Heat Rate (HFM) | Deviation |
|---|---|---|---|
| 1C @ -20°C | 2.15 W | 2.05 W | +4.9% |
| 1C @ 30°C | 0.42 W | 0.41 W | +2.4% |
| 2.5C @ 30°C | 4.35 W | 4.58 W | -5.0% |
Conclusion
This work presented and validated a novel Calorimetric Calibration Method for experimentally characterizing the heat generation behavior of high-energy-density cylindrical li-ion battery cells. The study leads to the following key conclusions regarding the thermal behavior of the tested li-ion battery:
1. The heat generation rate and core temperature rise of the li-ion battery increase with decreasing operating temperature and increasing discharge current. Correspondingly, the discharge efficiency (usable capacity) decreases under these stressful conditions. The average heat generation rate exhibits a quadratic relationship with both ambient temperature and discharge current.
2. The proposed Calorimetric Calibration Method, which requires only a simple preliminary heat loss calibration experiment, provides results in excellent agreement (within 5.4%) with those from a standard Heat Flux Meter. This demonstrates its high accuracy for thermal analysis of li-ion battery systems.
3. Compared to conventional methods that rely on specialized instrumentation like heat flux meters, the Calorimetric Calibration Method offers significant advantages in terms of flexibility, lower cost, and ease of implementation, while maintaining high accuracy. This makes it a practical and accessible tool with strong potential for widespread application in the analysis and thermal management design of li-ion battery packs.