Analysis of the thermal effect of a lithium iron phosphate battery cell and module

Abstract

Based on the theory of porous electrodes and the properties of lithium iron batteries, an electrochemical‐thermal coupling model of a single cell was established. The model was mainly used to study the temperature rise and temperature distribution characteristics in different regions of lithium iron batteries under different working conditions. In addition, a heat dissipation comparison analysis was carried out for different types of liquid cold runners, the optimal runner scheme was selected, and a structural optimization analysis was carried out. The simulation results show that the lithium iron battery discharges under the same ambient temperature and different C rates, and the battery temperature continuously increases with C. The temperature rise is mainly affected by Joule heat, and when the lithium iron battery is discharged at the same C but different ambient temperatures, the temperature rise of the lithium iron battery shows a decreasing trend with the increase in ambient temperature in a certain temperature range. The serial flow channel solution induces the best thermal behavior. Using the response surface optimization analysis, the three independent variable factors at which the temperature equilibrium R reaches the largest value under the liquid cooling module model are the rectangular flow channel length‐to‐width ratio, flow rate, and thermal conductivity.

INTRODUCTION

The core component of electric vehicles is the power battery pack. The quality of the power battery pack directly affects the performance of the vehicle. Lithium iron batteries have many advantages, such as energy density, no memory effect, low self‐discharge rate, and long life spans. Therefore, lithium iron batteries have become an ideal power source for electric vehicles.¹ However, the thermal safety problems of lithium iron battery cannot be ignored. If the heat generated by the battery cannot be dissipated in time, it will cause the battery temperature to rise, or even thermal runaway. When the temperature rises up to 50°C, the electrochemical performance and cycle life of lithium iron battery will be seriously decreased.² Therefore, ensuring a power density and high energy density simultaneously has become increasingly challenging.³

Some scholars have used the finite element method to model the multiphysics of lithium iron battery. However, this modeling is too complicated. Furthermore, most of these simulations focus on the temperature rise of the battery surface and the internal temperature field distribution and do not take into account the physical structure characteristics of the battery. For different discharge conditions, comparative analysis of the thermal characteristics of the same lithium iron battery has rarely been performed. In addition, few comparative analyses of the influence of different liquid‐cooled pipeline designs on the temperature field of the battery module have been presented. This issue should be seriously considered. Therefore, the focus of this paper is to study the heat generation characteristics of lithium iron phosphate battery and to analyze the heat generation characteristics and temperature rise under different working conditions. In this paper, a one‐dimensional electrochemical model is used to calculate the electrochemical heat of the porous electrode region, and by combining the Joule heat generated by the current collector plate and the electrolyte, a heat generation model is established. In addition, a three‐dimensional heat dissipation model is established for a lithium iron phosphate battery, and the heat generation model is coupled with the three‐dimensional model to analyze the internal temperature field and temperature rise characteristics of a lithium iron battery. Additionally, the influences of four different types of liquid‐cooled pipelines on the temperature field are compared, and the optimal cooling pipeline scheme is obtained. The study can provide reference for thermal management for lithium iron phosphate battery.

NUMERICAL MODEL FOR ELECTROCHEMICAL MODEL

The lithium iron battery internally relies on an electrochemical reaction to release or store electrical energy. However, the electrochemical system is complicated. Due to the diffusion and migration of lithium ions inside the battery, the internal state of the battery cannot be calculated as described in the chemical design.⁴

This paper is based on the P2D model proposed by Doyle et al.⁵ The electrochemical one‐dimensional model is constructed with COMSOL Multiphysics software. The schematic diagram is shown in Figure 1. L_n, L_sp, and L_p represent the thickness of the negative active material, the thickness of the separator, and the thickness of the positive active material, respectively. In the model, the positive and negative active material is represented as uniformly distributed small spheres, and the areas of the positive and negative active material and separator are full of electrolytes. Considering the actual electrochemical reaction process, this numerical model takes on the following assumptions⁵:

1. The intraelectrolyte transport process conforms to the theory of concentrated solution.
2. There is no generation of side reaction heat in the lithium iron battery.
3. The positive and negative active material is composed of particles of uniform size.
4. The change in the volume of the electrode during the reaction is negligible, and the electrode has a constant porosity.
5. The electric double‐layer capacitor effect and the gas phase formation in the reaction are ignored.
6. The transport of ionic species in the electrolyte solution is carried out by diffusion and electromigration only, regardless of the convection phenomenon of the ionic species.

1 FIGURE. Electrochemical model structure diagram

Lithium‐ion diffusion in the solid phase of positive and negative active material particles

Solid phase lithium‐ion diffusion is a mass transfer process that occurs inside solid particles, as expressed by Fick's second law⁶: [Image Omitted. See PDF]where C_1,i is the solid phase lithium‐ion concentration, D_1,i is the solid phase diffusion coefficient; and r_i is the distance of the particles along the radial direction. The boundary conditions are as follows: (a) the initial concentration of lithium ions in the spherical particles is uniform; (b) the lithium‐ion concentration at the center of the active material particles does not change; (c) the gradient of the lithium‐ion concentration on the surface of the active material particles and the solid phase diffusion rate determine the density of the lithium‐ion current at the particle surface.

Electrochemical reactions on the interface of solid phase and liquid phase

As the lithium iron battery functions, an electrochemical reaction occurs on the spherical surface of the electrode. According to the operating current of the battery, the density of the reactive lithium‐ion on the surface of each particle can be calculated. The Butler–Volmer kinetic equation can be obtained: [Image Omitted. See PDF]where $a_v$ is the particle specific surface area, F is the Faraday constant, $\eta$ represents the overpotential, R is a constant equal to 8.314 J/mol K, i₀ is the exchange current density, and J_c represents the lithium‐ion current generated by the electrochemical reaction of the particle surface.

Lithium‐ion diffusion and migration in electrolytes

The electrolyte solution fills the entire active electrode region, including the separator and the porous electrode. The diffusion of particles in the porous electrode is not the same as that in the separator region, and the pore structure in the porous electrode is difficult to diffuse through. Therefore, the electrolyte salt transfer coefficient in the porous electrode and the conductivity must be corrected. The diffusion is described by Nernst–Planck, and the modeling does not consider the convective mass transfer process. The simplified equation is as follows: [Image Omitted. See PDF]where ${\epsilon }_e$ is the electrolyte volume fraction, $c_l$ is the liquid phase lithium‐ion concentration, $t_{+}^0$ is the migration number in the electrolyte, $D_l^{\mathit{eff}}$ is the corrected liquid phase diffusion coefficient, and ${\sigma }_l^{\mathit{eff}}$ is the corrected conductivity.

Charge conservation and heat generation rate in actual calculation

The lithium iron battery model must also satisfy the conservation of charge. The solid phase and liquid phase charge conservation expressions are as follows: [Image Omitted. See PDF]

The boundary control conditions of the equation must be satisfied: the liquid phase current density of the positive and negative current collector boundaries 1 and 4 is $i_e^{‐}\left(1,t\right)=i_e^{+}\left(4,t\right)=0$. When x is in the separator region between boundaries 2 and 3, the liquid phase current density is equal to the current density $i\_app$ at the time of operation, that is, $x\in \left(L_n,L_n+L_{\mathit{sp}}\right)$, $i_e=i\_app$. The solid phase current density at positive collector boundary 4 and negative current collector boundary 1 is equal to the current density $i\_app$ at the time of battery operation, that is, $i_s^{+}\left(4,t\right)=i\_app$ and $i_s^{‐}\left(1,t\right)=i\_app$.

The lithium iron battery model needs to satisfy the energy conservation equation. The actual charge and discharge heat generation rate $Q_{\mathit{ac}}$ mainly consists of four parts: [Image Omitted. See PDF]where Q_r represents electrochemical reaction heat. Q_s represents electrochemical side reaction heat. Q_p represents electrochemical polarization heat. Q_j represents the joule heat generated by ohmic resistance.

THREE‐DIMENSIONAL HEAT DISSIPATION MODEL AND PARAMETER DETERMINATION

Establishment of the three‐dimensional heat dissipation model

The research object is a 26650 lithium iron phosphate battery, which capacity of 4500 mA h and a maximum discharge current of 9.6 A. The model is simplified as shown in Figure 2. The 26650 lithium iron phosphate battery is mainly composed of a positive electrode, safety valve, battery casing, core air region, active material area, and negative electrode. The model has an extremely uniform composition, wherein the main heat source is the active material; the areas of active material transfer heat from other parts through heat conduction. Finally, heat is exchanged with the external environment through the battery casing to dissipate heat.

2 FIGURE. The internal structure of the 26650 lithium iron phosphate battery

The heat generation and heat dissipation process of the battery module is an unsteady conduction process. The objective of the battery thermal model is to maintain the entire energy balance of the battery module. To facilitate modeling, the following assumptions are made for the battery model:

1. The material inside the battery is uniform.
2. The specific heat capacity of the material is uniform, and the thermal conductivity of the material is uniform in any direction.

The model of a 26650 cylindrical lithium iron phosphate battery and is an ax symmetric model. The temperature field inside the battery is calculated by solving the following equation, which can be established in the polar coordinate system. The actual solution must be determined by solving two problems. [Image Omitted. See PDF]where p indicates the battery density, C_p indicates the specific heat capacity of the battery,$\lambda$ indicates the internal thermal conductivity of the battery, $T$ indicates the temperature, $t$ indicates the time, and $Q_{\mathit{mod}}$ indicates the heat generation rate in modeling.

Heat generation rate in modeling

As shown in Figure 3, the heat generation rate is calculated using a one‐dimensional electrochemical model. The simplified heat generation rate $Q_{\mathit{mod}}$ can be expressed as: [Image Omitted. See PDF]where $Q_{\mathit{mod}}$ represents the algebraic sum of the heat produced by the two electrodes during the chemical reaction, $\text{kJ}\text{mol}$. $F$ is the Faraday constant, 96 484.5 $\mathrm{C}/\text{mol}$. $I$ represents the charge and discharge current, A. $R_{\mathit{act}}$ is the polarized internal resistance, Ω. $R_{\mathit{ohm}}$ is the ohmic internal resistance, Ω.

3 FIGURE. Primary heat generation method of the electrochemical‐thermal coupling model

Calculation of thermal physical parameters

Building a three‐dimensional model of a battery requires an accurate calculation of the specific heat capacity and thermal conductivity of the internal active area. Table 1 lists the specific physical parameters of the various materials inside a battery.^4,7–9

1 TableInternal physical parameters of a battery

Battery composition	Material	Density/$\text{kg}/{\mathrm{m}}^3$	Specific heat capacity/$\mathrm{J}/\text{kg k}$	Thermal conductivity/$\mathrm{W}/\text{mk}$
Safety valve	Aluminum alloy	2700	900	237
Battery case	Stainless steel	7900	500	16.3
Core air zone	Air	1.225	1006.43	0.0242
Upper and lower gasket	ABS	1200	1185.3	0.21

Fixed parameters

The fixed geometric parameters and electrochemical parameters of the one‐dimensional electrochemical model are shown in Tables 2‐4.^4,7–9

2 TableModel component thicknesses

Positive electrode/μm	Exchange membrane/μm	Positive current collector/μm	Negative electrode/μm	Negative current collector/μm
55	30	20	45	16

3 TableBattery electrode plate characteristic parameters

Parameter	Units	Positive electrode	Negative electrode
Conductivity	S/m	91	100
Porosity	1	0.42857	0.38
Active particle radius	m	$5.2\times {10}^{‐8}$	$1.1\times {10}^{‐5}$
Maximum lithium‐ion concentration	$\text{mol}/{\mathrm{m}}^3$	22 800	31 507
Inactive additive volume fraction	1	0.3	0.12

4 TableElectrochemical parameters of the diaphragm

Parameter	Units	Value
Porosity	1	0.54
Lithium‐ion mobility	1	0.4
Initial lithium‐ion concentration	$\text{mol}/{\mathrm{m}}^3$	1000
Faraday constant	$\mathrm{C}/\text{mol}$	96 487

Variable parameters

The diffusion coefficient of the positive and negative electrode regions changes with temperature, and the lithium‐ion electrolyte migration number $t_{+}^0$, electrolyte conductivity, positive and negative reaction rate, and lithium‐ion transfer coefficient in the electrolyte solution are variable parameters. The model variable parameter settings are as follows^10–12:

1. Variation in positive and negative diffusion coefficients with temperature and SOC

[Image Omitted. See PDF]

• Electrolyte solution conductivity

[Image Omitted. See PDF]

• Electrolyte migration number

[Image Omitted. See PDF]

• Lithium‐ion transfer coefficient in the electrolyte solution

[Image Omitted. See PDF]

• Positive and negative reaction rates

[Image Omitted. See PDF]

• Schematic diagram of the equilibrium potential and temperature coefficient with the SOC of a lithium iron phosphate battery, as shown in Figure 4.^13–15

4 FIGURE. Schematic diagram of positive and negative balance potential and temperature coefficients. (A) Positive and negative equilibrium potentials. (B) Positive and negative temperature conductivity coefficients

The electrolyte migration number $t_{+}^0$ uses the values in the COMSOL Multiphysics material library, the value is 0.37 at 200 seconds and 0.11 at 2000 seconds. The relationship between $t_{+}^0$ and time is a curve that decreases, then remains unchanged and then decreases. In the above formula, T is temperature, $c_l$ is the electrolyte salt concentration, and SOC is the state of charge. The equivalent replacement method of the SOC in the simulation process is explained in Equation (13). In this paper, the SOC is defined as the ratio between the surface concentration of the positive and negative electrode active material particles and the theoretical maximum concentration. The equation is as follows: [Image Omitted. See PDF]

ELECTROCHEMICAL‐THERMAL COUPLING MODEL

The performance of a single cell has a great impact on the overall performance of the battery module. The electrochemical‐thermal coupling model can simulate the thermal characteristics of a single cell and reduce the development time of the product. In this section, the voltage and temperature rise characteristics of lithium iron battery are simulated at different discharge rates, the temperature rise of various areas inside a single cell under different discharge rates are studied, and the heat production of lithium iron battery under different working conditions is calculated while simulating an ambient temperature to reflect the effects of the increase in discharge temperature and the voltage platform of the lithium iron battery.

Meshing

This paper uses the mesh analysis module of COMSOL Multiphysics to perform the related mesh analysis. As shown in Figure 5, after importing the lithium iron battery geometry model, the model area module is used to divide the geometry model so that it geometrically correlates to the grid in the three‐dimensional model. The meshing is performed by the sweeping method.

5 FIGURE. Lithium iron battery grid model. (A) Battery middle section. (B) Back of the battery

Thermal simulation analysis comparison

When the battery is discharged at a constant current but varying rate, the inside of the battery generates heat due to the migration and reaction of lithium ions. As shown in Figure 6, this is characterized by heat generation in the cell region and heat transfer to the battery case, battery tip, bottom end region, and inner air region.

6 FIGURE. Lithium iron battery 1 C discharge stage temperature contours. (A) Time = 1200 s, (B) Time = 3600 s, (C) Time = 7200 s

The heat of the lithium iron battery is mainly concentrated in the cell area. When the heat generation rate of the battery is greater than the heat dissipation rate, the heat inside the active material is more likely to accumulate, and part of the cell area will undergo the maximum temperature rise when the discharge is terminated; the minimum temperature of the battery occurs between the negative electrode and the safety valve. The heat transferred by the active material will first exchange heat with the outside through the battery case, causing the central area between the negative and the safety valve to undergo very little heat transfer, resulting in a minimum temperature rise in that area.

Effect of discharge rate and ambient temperature

At the same ambient temperature, the lithium iron battery is discharged to the cutoff voltage at 1 C and 3 C, and the average increase in the temperature of the lithium iron battery cell area reaches 4.5 K and 15 K, respectively. When the lithium iron battery is discharged to the cutoff voltage at 0.5 C, the average temperature rise of the cell area is the lowest, only 1.7 K. This is because in the lithium iron battery simulation, the discharge is performed with different currents, and a large current will generate higher Joule heat, causing the temperature of the lithium iron battery to rise sharply. When a cell is discharged with a large current, the Joule heat is increased due to the large solid phase lithium‐ion migration, which dominates the total heat generation.

When the battery is subjected to constant current discharge at 1 C, 3 C, and 5 C, the temperature rise of each region continues to increase throughout the discharge period. Figure 7 shows that when the lithium iron battery is subjected to constant current discharge at 0.5 C, the reaction heat of lithium iron battery discharge at low rate current is obviously greater than Joule heat. In the discharge stage of 3500 seconds to 4000 seconds, it can be clearly found that the reaction heat of lithium iron battery decreases, resulting in a decrease in total heat. The discharge time is 3500 seconds to 4000 seconds. Meanwhile, the temperature in each region of the lithium iron battery decreases. The direct cause of the temperature decrease is that the heat dissipation rate is greater than the heat generation rate. Figure 8A shows that the temperature rise of the lithium iron battery is very different under different ambient temperatures when the lithium iron battery is discharged at a constant current of 1 C. It can be observed that when the ambient temperature is 15°C, the lithium‐ion generates the largest amount of heat, and it is discharging at the end of the period, the temperature rise of the cell area is as high as 5.7 K; when the ambient temperature is 45 ℃, the heat generation of the lithium iron battery is the smallest. At the end of the discharge, the temperature rise of the cell area is only 2.6 K; at the ambient temperature when it is 25 ℃, the temperature rise of the battery cell area reaches 4.4 K, which is an intermediate value. Figure 8B shows that when the lithium iron battery is discharged to the cutoff voltage at a rate of 5 C, the lithium iron battery is discharged for a total of 720 seconds. During the discharge termination period, the average temperature rise of the lithium iron battery cell area reaches the highest, reaching 24 K, which has exceeded the optimal operating temperature range of the lithium iron battery; lithium iron battery is discharged to the cutoff voltage at 1 C and 3 C, and the average temperature rise of the lithium iron battery cell area reaches 4.5 K and 15 K, respectively; and when the lithium iron battery is discharged to the cutoff voltage at a rate of 0.5 C, the average temperature rise of the cell area is only 1.7 K. This is because when the lithium iron battery is simulated, the current of different magnitudes is used for discharging, and the large‐rate current will generate higher Joule heat, which will cause the temperature of the lithium iron battery to rise sharply. When a lithium iron battery is discharged with a high‐rate current, the main heat generation method is Joule heat generated by the migration of solid and liquid lithium ions. When the total heat is less than the exchange between the battery and the external environment heat dissipation, the temperature of each area of the battery will drop.

7 FIGURE. 0.5 C discharge heat power change over time

8 FIGURE. Effect of discharge rate and ambient temperature on battery heating

Heating power calculation

Since the capacity and voltage of the monomer 26650 lithium iron phosphate battery are low, to use the battery as a power battery to supply electric vehicles, a large number of single battery strings must be connected in parallel to meet the power demand of an electric vehicle. During the operation of an electric vehicle, the maximum rate of discharge of a typical single battery is 3 C.

To improve the research based on CFD battery air cooling and liquid cooling heat dissipation modules, it is necessary to solve the average heating power of single cells at different discharge rates. As shown in Figure 9, the average heating power at different discharge rates (0.5 C, 1 C, 3 C, 5 C) of the lithium iron battery can be determined by utilizing the heat generation characteristics of the lithium iron battery model.

9 FIGURE. Heat flux at different discharge rates

The 26650 lithium iron phosphate battery discharges to the cutoff voltage at different discharge rates. The heat flux density varies in different periods, and the heat flux density curve does not show a linear correlation. It is difficult to directly fit the function of heat flux density with time. Therefore, this paper will use COMSOL Multiphysics for this task. A time averaged function is used to calculate the average heating power during the entire discharge time of the lithium iron battery, and the final calculation result is shown in Table 5.¹⁶

5 TableAverage heating power under different discharge rates

Discharge rate (C)	Average power/$\mathrm{W}/{\mathrm{m}}^3$
0.5	1975
1	8520
3	72 100
5	156 320

NUMERICAL SIMULATION AND OPTIMIZATION OF HEAT DISSIPATION FOR LIQUID‐COOLED BATTERY MODULES

The liquid‐cooled battery module is modeled with Creo3.0. Similar to the air‐cooled heat‐dissipating module, the liquid‐cooled battery module is simplified as a 64‐cell 26650 single‐cell battery. As shown in Figure 10, the length of battery module is 280 mm, the width is 245 mm and the height is 70 mm, the cold plate is arranged around the batteries. The gap between the batteries is 5 mm, and the distance between the cells of each row is the same, 10 mm. The single cells are in contact with the cold plate through the thermal conductive silica gel, wherein the cold plate has a liquid‐cooled pipeline. The commonly used liquid‐cooled flow passages are circular flow passages and rectangular flow passages. The model adopts rectangular single‐flow cooling passages, and the height and width of the rectangular flow passages are 50 mm and 3 mm, respectively.

10 FIGURE. Simplified schematic diagram of the liquid‐cooled battery module

Comparative analysis of various liquid‐cooled pipelines

The cooling method of the liquid cooling module is divided into cold plate cooling, microtube liquid cooling, and cold tube cooling. Combined with the effect of each liquid cooling module, this section focuses on cold tube cooling. In the liquid cooling system, the module is placed in a closed insulation box, and the generated heat is removed by the cooling liquid in the liquid cooling pipeline. In combination with previous studies, three battery modules are designed in this paper. The structure and dimensions of the cooling pipeline runner scheme are shown below.

Figure 11 shows three different flow channel design schemes for the liquid cooling module. Parallel flow channel schemes I and II arrange the flow channels in parallel. Each inlet of the square flow channel is cooled by the same flow channel, and only the coolant outlet is different; the inlets of parallel flow channel schemes I and II are both at the bottom left; the outlet of parallel flow channel scheme I is at the bottom right; the outlet of parallel flow channel scheme II is at the top right; the cold plate in the serpentine runner channel scheme curves so that it is in direct contact with the battery, without requiring thermal silica gel between the cold plate and the battery. The battery modules of different liquid‐cooled pipes are discharged at a 3 C rate. The inlet flow rate of the coolant is 0.05 m/s, the temperature is 25°C, the liquid‐cooled flow channel is unified into a single‐flow channel, and the rectangular flow channel section is 150 mm². The length‐to‐width ratio is 50:3. The battery module temperature field is shown in Figure 12.

11 FIGURE. Three different flow plan diagrams

12 FIGURE. Temperature field comparisons of different flow channel battery modules

The temperature field comparison chart shows that different flow channel schemes result in significantly different heat dissipation effects of the battery module. As shown in Table 6, the highest temperature of the liquid‐cooled battery modules follows the order of parallel flow channel scheme I > parallel flow channel scheme II > serpentine runner channel scheme > serial runner channel scheme. The serial flow channel scheme module has the lowest temperature rise.

6 TableDifferent cooling pipe structure cooling temperatures (K)

Runner scheme	Serpentine flow channel	Parallel flow channel scheme I	Parallel flow channel scheme II	Serial flow channel solution
Average	301.9	304.3	303.5	301.5
Maximum	305.3	309.3	307.6	303.5
Minimum	298.5	399.3	299.5	299.5
Difference	6.8	10	8.1	4

For the liquid‐cooled battery, the module battery temperature difference follows the order of parallel flow channel scheme I > parallel flow channel scheme II > serpentine runner channel scheme > serial runner channel scheme. The balance among the batteries in the apparent serial runner channel scheme is better than those of the other schemes. When the maximum temperature value satisfies the working range of the lithium iron phosphate battery, a scheme with better battery temperature balance is selected, and the serial flow channel scheme is selected from the comparison.

Response surface optimization analysis

In the design of the liquid‐cooled battery module, the influence of various parameters on the temperature field of the battery module must be considered. The thermal conductivity of silica gel with different thermal conductivities, the length and width of the cold‐end inlet and the coolant flow rate are compared. To improve the temperature uniformity of the battery in the liquid cooling module, the optimization target equation is the uniformity of the cell temperature. At the same time, it is necessary to constrain the various influencing parameters, and reasonable constraints can reduce the calculation time and steps.^17,18[Image Omitted. See PDF]where a and b are the length and width of the rectangular flow channel (mm), v is the flow velocity of the coolant (m/s), and $C_{p1}$ is the heat capacity of the thermal silica gel. The optimization objective equation is:¹⁹[Image Omitted. See PDF]

where $R_e$ is the module battery temperature equilibrium, $T_u$ is the highest regional temperature average of a single cell in the module, $T_{\mathit{max}}$ is the highest temperature rise of the battery, and $T_{\mathit{min}}$ is the lowest temperature rise of the battery. Iterative is carried out according to the method above, but due to the difficulty of automatic iterative calculation, this paper considers the response surface analysis method for the above situation. Response surface analysis considers the data obtained by a series of experiments and simulations. The least squares method is used to build models to ultimately determine the response relationship between the multivariate and dependent variables, determine the interaction between the multiple variables, and obtain the optimal results.²⁰

Design‐Expert can help users improve their experimental products or processes by allowing users to view the response surfaces at all angles and filter out important factors to determine the best settings and recipes. Based on the influence of a single factor on the simulation results, the effects of three factors on the temperature field of the liquid‐cooled battery module were investigated: the rectangular flow channel length and width, the flow velocity, and the thermal conductivity of the thermal silica. ²¹ The experimental data were recorded by Fluent software. ²² The BBD test method is used to establish the Box–Behnken model to perform accurate statistical analysis of experimental data and provide image analysis with continuity features to intuitively understand the correlation between the studied factors and responses. When using the BBD test method in Design‐Expert, the length‐to‐width ratio of the rectangular flow channel, the coolant flow rate, and the thermal conductivity of the thermal silica gel were considered three independent variables and were represented by the letters A, B, and C, respectively. The details are shown in Table 7.

7 TableFactors of the independent variables

Test number	Runner length‐to‐width ratio	Flow rate/m/s	Thermal conductivity/W/mK	Temperature equilibrium $R_e$/%
1	10	0.06	2.5	95.67
2	20	0.04	2.5	96.19
3	10	0.04	2	95.75
4	20	0.06	2	96.83
5	30	0.03	2.5	96.25
6	10	0.04	3	95.84
7	20	0.04	2.5	96.19
8	20	0.04	2.5	96.19
9	20	0.04	2.5	96.19
10	20	0.03	2	95.59
11	30	0.04	2	96.67
12	20	0.06	3	96.88
13	10	0.03	2.5	94.77
14	30	0.06	2.5	97.27
15	30	0.04	3	96.73
16	20	0.03	3	95.68
17	20	0.04	2.5	96.19

After the simulation data in Table 7 were imported to Design‐Expert, a regression analysis was performed on the temperature equilibrium $R_e$ using the least squares method with the BBD test method. The model is established as follows:²³[Image Omitted. See PDF]

In the above model, the variation in temperature equilibrium $R_e$ with the channel length‐to‐width ratio, flow velocity and thermal conductivity is given. In the software prediction analysis, the maximum value of Equation (16) is solved to obtain the temperature equilibrium $R_e$ of the model. The three independent variable factors with the highest values were the rectangular flow channel length‐to‐width ratio of 29.74, flow rate of 0.06, and thermal conductivity of 2.17. At this time, the predicted value of the liquid‐cooled battery module temperature equilibrium $R_e$ was 97.28%. Figure 13 shows that the values predicted by the software simulation are not considerably different from the actual values. Most of the predictions in the figure plot close to the fitted straight line; thus, the predicted value of the model is accurate.

13 FIGURE. Corresponding graph of the model‐predicted and actual results

In the BBD test, the variance in the flow channel, flow velocity, and thermal conductivity are considered the independent variables for the regression analysis of the temperature equilibrium $R_e$, which is the dependent variable, as shown in Table 8. As seen from the test results, the $F$ value in this model is 55.76, and the $P$ value is less than 0.0001. The $P$ value and the $F$ value are the correlation coefficients of the regression model; when the $P$ value is less than 0.05, the model is considered to be significant.

8 TableVariance analysis of regression models

Factor	Sum of squared errors	Degrees of freedom	Mean squared error	F value	P value
Model	5.38	3	1.79	55.76	<.0001
A‐A	2.99	1	2.99	93.01	<.0001
B‐B	2.38	1	2.38	73.94	<.0001
C‐C	0.011	1	0.011	0.33	.5771
Residual	0.42	13	0.032
Total error	5.79	16

Note: R² = 0.9279, Adj R² = 0.9112, Adeq precision = 26.594.

The analysis of residuals is crucial in model prediction, as residuals contain important information related to the model design. If the model is established correctly, the residuals can be regarded as observations of errors. By analyzing the internal residuals, the rationality of the hypothesis of the model and the confidence of the model prediction can be investigated. The confidence analysis results of the regression model are shown in Table 9.

9 TableConfidence analysis of the regression model

Factor	Regression coefficients	Degrees of freedom	Standard error	95% confidence limit (lower)	95% confidence limit (upper)
Intercept	96.17	1	0.043	96.08	96.26
A	0.61	1	0.063	0.47	0.75
B	0.55	1	0.063	0.41	0.68
C	0.036	1	0.063	−0.1	0.17

Furthermore, in conjunction with the normal distribution map of the residuals in Figure 14A and the corresponding map of residual values and predicted values in Figure 14B, reliability analysis can be performed on the residuals of the model.

14 FIGURE. Schematic diagram of model reliability analysis

In the residual normal distribution map, the data points are generally distributed on both sides of the straight line, which shows that the residual data points in the model have a good linear fit. Therefore, the residuals can be considered to follow a normal distribution. Figure 14B shows the correspondence between the residuals and the predicted values of the equation. The data points in the graph are very scattered and irregularly distributed, which proves that other variables do not interfere with the residuals in the model. The performance is good, so the prediction accuracy of the model can be considered high.

After the liquid cooling module is optimized, the battery module is discharged at a rate of 5 C. The maximum temperature of the module occurs in the middle two rows, and the maximum value is only 307.2 K, and the maximum temperature difference between the battery modules is only 7.1 K. The temperature difference between the individual cells is significantly reduced, as shown in Table 10.

10 TableOptimization of temperature field evaluation parameter values under different discharge rates

Discharge rate	0.5 C	1 C	3 C	5 C
Maximum (K)	298.3	298.4	302.6	307.2
Minimum (K)	298.2	298.2	299.2	300.1
Difference (K)	0.1	0.2	3.4	7.1

CONCLUSION

1. The heat generation process of 26650 lithium iron phosphate battery was simulated, and the area with the highest and lowest temperature rise was analyzed. For different battery discharge rates, the factors affecting battery heat generation are analyzed. These factors mainly include reaction heat, Joule heat, and ambient temperature.
2. Through the research on the module temperature rise and battery temperature difference of the four flow channel schemes, it is found that the battery with the serial runner scheme is better balanced and can better meet the operating temperature requirements of lithium iron phosphate batteries.
3. The response surface optimization method is used to solve the temperature equilibrium to obtain the optimal value of the rectangular flow channel length‐to‐width ratio, flow rate, and thermal conductivity.

ACKNOWLEDGMENTS

This work is partially supported by the National Natural Science Foundation of China (Grant No. 51205051)

CONFLICT OF INTEREST

The authors declare no conflict of interest.

Funding information

National Natural Science Foundation of China (Grant No. 51205051).