# A Comprehensive Study on the Effect of Inhomogeneous Heat Dissipation on Battery Electrochemical Performance

^{*}

## Abstract

**:**

## 1. Introduction

^{2}) to 200 W/(K·m

^{2}) are imposed on the battery pack to create various temperature distributions. The distribution of the current; the SOC; and the electrochemical parameters such as electronic concentration, potential difference, local volumetric current density, and so on are computed to quantitatively explain the effect of the temperature distribution among cells on the electrical and electrochemical performance of the battery and find a suitable thermal parameter to improve the electrical and electrochemical performance. Thirdly, the ambient temperatures vary from 10 °C to 30 °C are applied. They are combined with cases of uneven cooling to achieve the suitable range of the temperature gradients in the pack at different ambient temperatures for good consistencies of SOC electrochemical parameters among cells.

## 2. Electro-Electrochemical-Thermal Model of Battery Pack

#### 2.1. Cell Model

#### 2.1.1. Electrochemical Model

_{P}

_{2D}coordinate is used for the P2D model at the microscale, and its direction shown in Figure 2 is from the negative electrode to the positive electrode. The additional pseudo dimension r for the P2D model is along the radial distance of the electrode particle r.

_{P}

_{2D}coordinate is used to replace the x

_{P}

_{2D}coordinate in Table 1. Because z

_{P}

_{2D}is defined in the electrode and the diaphragm areas, respectively, axis z

_{P}

_{2D}is divided into z

_{P}

_{2D,n}for the negative electrode, z

_{P}

_{2D,s}for the separator, and z

_{P}

_{2D,p}for the positive electrode. Figure 2 shows the regions of z

_{P}

_{2D,n}, z

_{P}

_{2D,s}and z

_{P}

_{2D,p}, and Table 2 shows the P2D model in the z

_{P}

_{2D}coordinate.

- (a)
- Electrolyte concentration approximation

_{e}is the electrolyte concentration and the subscript n represents the negative electrode. The subscript s represents the separator, and the subscript p represents the positive electrode. In order to satisfy the boundary condition of the Li-ion diffusion equation given the third equation of Equation (6), the coefficients a

_{1,1}and a

_{3,1}are 0, and Equation (9) is simplified as

_{0}, a

_{1}, a

_{2}, a

_{3}, a

_{4}, a

_{5}, a

_{6}] in Equation (10) are time-varying parameters, which need to be calculated in each time step. Substituting the first four equations in Equation (8) with Equation (10) gives

_{e}is the liquid diffusion coefficient, D

_{e}

^{eff}is the effective liquid diffusion coefficient, and their relationship is D

_{e}= D

_{e}

^{eff}ε

_{e}

^{−1.5}. ε

_{e}is the volume fraction of Li-ion in the electrolyte, and ε

_{e}c

_{e}is the concentration of Li-ion. The mole number of Li-ion at the unit area in the regions of the negative electrode Q

_{e,n}(t), the separator Q

_{s}(t), and the positive electrode Q

_{e,p}(t) are obtained by integrating its concentration along the z-axis, and they are

_{f}along the z axis, and the expressions are

_{e,n}(t), Q

_{e,s}(t), and Q

_{e,p}(t), which are

^{+}is the transfer number of Li-ion.

_{e,n}(0), Q

_{e,s}(0), and Q

_{e,p}(0) can be achieved by solving

_{e,}

_{0}is the initial electrolyte concentration. In order to satisfy Equation (23), the initial polynomial coefficients [a

_{0}, a

_{1}, a

_{2}, a

_{3}, a

_{4}, a

_{5}, a

_{6}] in Equations (15)–(17) should be meet the requirement below.

_{e,n}(t), Q

_{e,s}(t), and Q

_{e,p}(t) in each time step. Then, these total mole numbers of Li-ion are used for Equations (11)–(17) to calculate [a

_{0}, a

_{1}, a

_{2}, a

_{3}, a

_{4}, a

_{5}, a

_{6}]. After the coefficient matrix is achieved, we can employ Equation (10) to approximate the distribution of the electrolyte concentration in the electrodes and separator.

- (b)
- Reaction flux approximation

_{f}is used to describe its distribution between the negative and positive electrodes. Because j

_{f}is distributed like a parabola, a quadratic polynomial is applied to approximating the j

_{f}. It is

_{s}is the potential of the solid phase. U is the electrode equilibrium potential, and R

_{SEI}represents the resistance of SEI film. φ

_{e}represents the potential of the electrolyte. a

_{s}means the specific interfacial surface area, and the subscript k is n for the negative electrode or p for the positive electrode. η can also be calculated by solving the Bulter–Volmer given in Equation (5), and it is

_{0}represents the exchange current density. Substituting the η in Equation (26) with Equation (27) gives

_{P}

_{2D,k}) is

_{P}

_{2D,n}and Z

_{P}

_{2D,p}, respectively, gives the expressions of the φ

_{s}and φ

_{e}. They are

^{eff}means solid-phase effective conductivity, t

^{+}is the transfer number of lithium-ion, and κ

^{eff}is the effective conductivity of the electrolyte.

_{2,n}, c

_{1,n}and c

_{0,n}in Equation (25) are reached. The subscript k in Equations (26)–(31) becomes n. Substituting [∂φ

_{s}(Z

_{P}

_{2D,n})/∂Z

_{P}

_{2D,n}] and [∂φ

_{e}(Z

_{P}

_{2D,n})/∂Z

_{P}

_{2D,n}] in Equation (28) with Equations (30) and (31) gives

_{P}

_{2D,n}) is

_{P}

_{2D,n}is 0, Equation (32) is solved by replacing j

_{f}(Z

_{P}

_{2D,n}) with Equation (25), and the result is

_{P}

_{2D,n}is L

_{n}, the difference between the first second term on right side of Equation (32) is 0 according to Equation (28) and the fifth equation in Equation (6), and the integral value in the four term is (I/A) according to Equation (18). With the calculation above, Equation (32) under the condition of Z

_{P}

_{2D,n}= L

_{n}becomes

_{2,n}and c

_{1,n}in Equations (34) and (35), it is necessary to calculate the electrode equilibrium potential U(Z

_{P}

_{2D,n}) at Z

_{P}

_{2D,n}= 0 and Z

_{P}

_{2D,n}= L

_{n}. According to Ref. [24], U(Z

_{P}

_{2D,n}) in the negative electrode is decided by the stoichiometry of the electrode concentration θ

_{n}, and it is

_{n}(Z

_{P}

_{2D,n}) is

_{2,n}, c

_{1,n,}and c

_{0,n}in Equation (25) is achieved by integrating Equation (18), and it is

_{2,n}, c

_{1,n,}and c

_{0,n}in Equation (25) and the local volumetric current density in the negative electrode. Because the cell is in a stable state in the beginning of discharging or charging, the j

_{f}is evenly distributed and the initial j

_{f}for Equation (25) is

_{f}in the positive electrode is almost same as that in the negative electrode. The only difference is that the current I in the positive electrode should be multiplied by −1 because the direction of the Z

_{P}

_{2D,p}-axis is opposite to that of the Z

_{P}

_{2D,n}-axis.

- (c)
- Solid-phase surface concentration approximation

_{s,e}(t) is not only related to its volume-averaged concentration ${\overline{c}}_{s}\left(t\right)$ but also to its volume-averaged concentration flux $\overline{q}\left(t\right)$, and it can be calculated through [25]

_{s}is the solid-phase diffusivity of lithium-ion, R

_{s}is the particle radius, and the subscript i is n for the negative electrode or p for the positive electrode.

_{s,n}(0) and $\overline{c}$

_{s,p}(0) are the volume-averaged concentration of lithium-ion in the positive and negative electrodes; SOC

_{ini}is the initial SOC, c

^{n}

_{s,max}and c

^{p}

_{s,max}are the maximum solid-phase concentration of lithium-ion in the positive and negative electrodes; θ is the stoichiometry of electrode concentration; superscripts n and p represent the positive and negative electrode, and subscript 0 and 100 are 0% and 100% of the cell SOC, respectively.

- (d)
- Other parameters for PP2D model

_{e}is the electrolyte potential; (0) is the current collector; and the subscript p and n represent positive and negative electrode separately. Terminal voltage V of cell is reached by solving

_{e}is the solid-phase potential. Combining Equations (26), (37), (48), and (49) gives the expression of the cell terminal voltage. It is

_{p}(0)” and “

_{n}(0)” refer to the variable values at Z

_{P}

_{2D,p}= 0 and Z

_{P}

_{2D,n}= 0, respectively. A 50Ah NCM/graphite prismatic cell is used in this paper, and its electrochemical parameters are given in Table 3 and Table 4.

#### 2.1.2. Thermal Model

_{p}are cell’s density and specific heat capacity, and T represents temperature. t means time, and q

_{v}is the volume-averaged heat generation rate. k

_{x}, k

_{y}, and k

_{z}represent the thermal conductivities along the direction of x, y, and z axes separately. The heat generation rate inside the cell is composed of the reaction heat generation q

_{rea}, ohmic heat generation q

_{ohm,}and active heat generation q

_{act,}which are calculated by

_{v}is

_{b}is the cell volume.

#### 2.1.3. Specific Heat Capacity and Thermal Conductivities

_{p}was tested, the cell was covered with aluminum foil to enhance the heat transfer and ensure that its temperature was homogeneously distributed. The PTC heating plate was fixed on the surface of the aluminum to heat the cell. The whole device was put in the accelerating rate calorimeter (ARC), which was used to make the ambient temperature the same as the cell temperature and avoid heat transfer among cell and air. Figure 3b presents the measured cell temperature increase. The slope of the temperature line is used to calculate the C

_{p,}which is expressed by

_{heat}is the heating power of the PTC, m

_{b}is the cell mass, and ΔT/Δt is the slope of the temperature line in Figure 3b. The tested C

_{p}of the 50 Ah prismatic cell is 989 J/(kg∙K).

_{x}is 1.26 W/(m·K), and the thermal conductivities along the cell width and height k

_{y}and k

_{z}are 23.36 W/(m·K).

#### 2.2. Pack Model

#### 2.2.1. Current Distribution Model

_{j}is total terminal voltage of the PCBS_j, V

_{4,i}(i = 1, 2 and 3) is the cell’s terminal voltage in the PCBS_4, I

_{4,i}(i = 1, 2 and 3) is the branch current through the cell i in the PCBS_j, and r is the welding resistance. The terminal voltage of cell i in the PCBS_j V

_{j,i}is

_{OCV}is the open-circuit voltage (OCV), R is the resistance, and the subscript (j,i) is the ith cell in PCBS_j. The current distribution in the PCBS_j is achieved by solving Equations (57) and (58). It is

_{OCV}can be calculated by

_{bp}and cell resistance R

_{j,i}. Firstly, the R

_{bp}and R

_{j,i}are measured by the test of HPPC (hybrid pulse power characterization) at 1 C discharging rate, SOC of 100%, and 20 °C ambient temperature. Then, Equation (61) is applied to calculate the r of the 3P4S pack, and r is 0.717 mΩ.

#### 2.2.2. Thermal Model of Pack

_{w}is heat production of welding point. It is

_{W}represents the welding point’s volume. The third kind of heat transfer condition [35] is employed for the pack thermal model, and it is

_{w}means the temperature of the pack surface. T

_{c}is the coolant temperature close to the pack. h represents the heat-transfer coefficient.

#### 2.3. Solution of Electrochemical-Thermal Model for the Pack

## 3. Experimental Validation

#### 3.1. Experimental Devices

#### 3.2. Verification Results

## 4. Influence of Inhomogeneous Cooling on Pack

#### 4.1. Uneven Cooling Cases

^{2}) to 220 W/(K·m

^{2}) and coolant temperature (T

_{c}) from 10 °C to 30 °C are imposed on both sides of the pack. The other surfaces of the pack are assumed to be adiabatic. When h is ≤32 W/(K·m

^{2}), air cooling is applied. When h is ≥100 W/(K·m

^{2}), water cooling is adopted. Moreover, in the simulation, the coolant temperature is considered the same as the pack initial temperature. The discharging rate is 1 C, and the SOC range is from 1 to 0.

#### 4.2. Temperature Distribution of the Pack

_{max_diff}) between cells and the average temperature of the package (T

_{ave_pack}) at the end of discharge. According to the information in the table, with the increasing of heat-transfer coefficient, temperature difference inside the pack increases and the temperature of the pack decreases. For example, when T

_{c}= 10 °C, ΔT

_{max_diff}increases to 10.64 °C, and T

_{ave_pack}drops to 31.06 °C with the increase of h because the temperature gradient of the pack is required to be wide to quickly transfer heat. Therefore, better cooling will cause lower T

_{ave_pack}, but the temperature between batteries changes greatly, which greatly affects the consistency of SOC between batteries.

#### 4.3. Current Distribution of the Pack

_{max_diff}) between batteries in the battery pack. With the heat transfer progressively strengthening, current difference increases. When T

_{c}= 10 °C, ΔI

_{max_diff}grows by 1506.2% with the increase of h. The results show that the enhanced cooling of can lower the temperature, but it will cause poor temperature homogeneity and a large current gap in the parallel branch in the battery pack.

#### 4.4. SOC Distribution of the Pack

_{max_diff}) in PCBS_1 is 4.89%, and that of PCBS_2 is only 0.40%. In each PCBS, the SOC decreases with the expansion of the distance from the battery to the coolant, which is because of the bigger current caused by the higher temperature.

_{max_diff}). During the discharging process, it can be seen from Table 9 that due to the huge current change caused by cooling, the coolant with bigger h has a larger SOC non-uniformity on the module. When T

_{c}= 10 °C, ΔSOC

_{max_diff}increases by 1530% with the increase of h.

_{c}, the maximum temperature difference ΔT

_{max_diff,}and the maximum SOC difference ΔSOC

_{max_diff}under the inhomogeneous cooling case 13. For the pack designed as the maximum 1C discharge rate, the cell equalization goal of the control objective is to let ΔSOC

_{max_diff}less than 2%. Consequently, take ΔSOC

_{max_diff}= 2% as the standard to determine the temperature difference of cells in the pack, hoping objective ΔT

_{max_diff}can always make ΔSOC

_{max_diff}remain less than 2%. In Figure 16, as T

_{c}increases from 10 °C to 30 °C, the objective ΔT

_{max_diff}boundary increases from 5.21 °C to 7.94 °C. The result illustrates that the objective ΔT

_{max_diff}for the cell equalization is not a stable value but increases with the coolant temperature.

#### 4.5. Electrochemical Parameters of the Pack

_{OCV}= U

_{p}− U

_{n}, the electrolyte potential difference Δϕ

_{e}= ϕ

_{e,n}(0) − ϕ

_{e,p}(0), and the activation overpotential difference Δη = η

_{n}(0) − η

_{p}(0). In this paper, the R

_{SEI}of formula (54) is 0, so the terminal voltage V is only affected by the open-circuit voltage U

_{OCV}, the electrolyte potential difference Δϕ

_{e,}and the activation overpotential difference Δη. Formula (54) is simplified to V = U

_{OCV}− Δϕ

_{e}− Δη.

_{1}and cell

_{3}. Figure 16 shows the difference of open-circuit voltage (U

_{OCV}

_{,}

_{1}− U

_{OCV}

_{,}

_{3}), the difference of the activation overpotential difference (Δη

_{1}− Δη

_{3}), and the difference of the electrolyte potential difference (Δϕ

_{e,}

_{1}− Δϕ

_{e,}

_{3}) between cell 1 and cell 3 under the inhomogeneous cooling case 13.

_{OCV}

_{,}

_{1}− U

_{OCV}

_{,}

_{3}) and the difference of the activation overpotential difference (Δη

_{1}− Δη

_{3}), the difference of the electrolyte potential difference (Δϕ

_{e,}

_{1}− Δϕ

_{e,}

_{3}) is one order of magnitude smaller, so it can be ignored. Therefore, the change of terminal voltage is mainly determined by the open-circuit voltage and the activation overpotential difference.

_{OCV}is determined by SOC. The activation overpotential is calculated by Formula (26), so it is mainly affected by the electrochemical parameters j

_{f}and i

_{0}. The activation overpotential η is positively correlated with j

_{f}and negatively correlated with i

_{0}. Similarly, due to the negative sign in front of η

_{p}(0), which is just offset by the negative signs of j

_{f,p}, the activation overpotential difference Δη is positively correlated with the absolute value of j

_{f}and negatively correlated with i

_{0}.

_{f}of PCBS_1 under the inhomogeneous cooling case 13. It can be seen from Formula (42) that j

_{f}is mainly affected by the current. This is also consistent with the results obtained. As shown in the Figure 16, the changing trend of the absolute value of j

_{f}is consistent with the current but is little affected by temperature.

_{f}

_{,max_diff}) in the discharge process. As seen from Table 10 and Table 11, because of the huge current change caused by cooling, the coolant with a large heat-transfer coefficient has large Δj

_{f}

_{,max_diff}. When T

_{c}= 10 °C, Δj

_{f,n,max_diff}and Δj

_{f,p,max_diff}increase by 1438% and 1414%, respectively, with the increase of h.

_{0}of PCBS_1 under the inhomogeneous cooling case 13. It can be seen that i

_{0}increases with temperature increase, which is primarily due to the effect that reaction rate constant k increases with temperature.

_{0}

_{,max_diff}) in the discharge process. It can be seen from Table 12 and Table 13 that because of the huge current change caused by cooling, the coolant with a large heat-transfer coefficient has large Δi

_{0}

_{,max_diff}on the module. When T

_{c}= 10 °C, Δi

_{0,n,max_diff}and Δi

_{0,p,max_diff}increase by 1100% and 875%, respectively, with the increase of h.

_{0}of the battery with lower temperature is lower, resulting in the activation overpotential difference Δη higher, which affects the gradual decrease of its discharge current. In the middle discharge stage of discharging, open-circuit voltage U

_{OCV}begins to play a leading role due to the gradually increasing SOC gap. Due to the higher SOC, the discharge current of the battery with lower temperature began to increase and gradually became a larger current in the later stage of discharge, and the SOC gap between batteries began to narrow. In the later stage of discharging, the battery with lower temperature has higher current, so its i

_{0}is lower and its j

_{f}is higher, resulting in a larger activation overpotential difference Δη. Although the SOC gap between batteries is narrowing, the U

_{OCV}accelerates to decrease with the decrease of SOC at the later stage of discharging. As a result, the open-circuit voltage gap between batteries gradually becomes larger, and the U

_{OCV}still plays a leading role. The SOC of batteries with lower temperature is higher, and the discharge current continues to increase due to higher U

_{OCV}. Therefore, in general, the discharge current of the battery with lower temperature first decreases in the early stage of discharge, increases in the middle stage of discharge, gradually exceeds the current of the battery with higher temperature, and becomes a larger current in the late stage of discharge.

## 5. Conclusions

_{0}of the battery with lower temperature is lower, resulting in the activation overpotential difference Δη higher, which affects the gradual reduction of its discharge current. The open-circuit voltage U

_{OCV}plays a leading role in the middle and late stages of discharge. The open-circuit voltage U

_{OCV}of batteries with higher SOC is higher, so the discharge current of batteries with lower temperature is further increased. During the whole discharge process, the SOC gap in the battery pack first increases and then decreases.

_{c}= 10 °C, ΔT

_{max_diff}increases by 1120% and ΔI

_{max_diff}increases by 1506.2% with the increase of h.

_{c}= 10 °C, ΔSOC

_{max_diff}increases by 1530% with the increase of h. Therefore, the control of battery pack temperature and temperature difference is contradictory and should be balanced. In addition, when the initial temperature, coolant temperature, and ambient temperature are set to the same value, the objective maximum temperature difference ΔT

_{max_diff}most suitable for controlling the uniformity of SOC increases with the increase of T

_{c}. When T

_{c}increases from 10 °C to 30 °C, the objective ΔT

_{max_diff}boundary to maintain ΔSOC

_{max_diff}within 2% increases from 5.21 °C to 7.94 °C.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | Electrode plate area, m^{2} |

α_{s} | Specific interfacial surface area, m^{−1} |

c_{e} | Electrolyte concentration, mol m^{−3} |

c_{e,0} | Average electrolyte concentration, mol m^{−3} |

c_{s} | Solid-phase concentration, mol m^{−3} |

c_{s,e} | Solid-phase surface concentration, mol m^{−3} |

$\overline{c}$ | Volume-averaged solid-phase concentration, mol m^{−3} |

D_{s} | Solid-phase diffusivity, m^{2}s^{−1} |

D_{e} | Electrolyte phase diffusivity, m^{2}s^{−1} |

F | Faraday’s constant, 96,487 C mol^{−1} |

I | Current, A |

i_{0} | Exchange current density, A m^{−2} |

j | Reaction flux, mol m^{−2} s^{−1} |

j_{f} | Local volumetric current density, A m^{−3} |

k | Reaction rate, mol^{−0.5}m^{2.5}s^{−1} |

L | Thickness, m |

Q | Total amount of lithium-ion in each region, mol m^{−2} |

$\overline{q}$ | Volume-averaged concentration flux, mol m^{−4} |

R | Battery resistance, Ω |

R_{s} | Particle radius, m |

T | Temperature, K |

r | Welding resistance, Ω |

t | Time, s |

t^{+} | Lithium-ion transfer number |

U | Electrode equilibrium potential, V |

U_{OCV} | Open-circuit voltage, V |

V | Terminal voltage of battery, V |

x | lD coordinate across the cell, m |

z | lD coordinate across electrode/separator, m |

ε_{s} | Active material volume fraction |

ε_{e} | Electrolyte volume fraction |

θ | Stoichiometry of electrode concentration |

θ_{0} | Stoichiometry at 0% SOC |

θ_{100} | Stoichiometry at 100% SOC |

α_{a}, α_{c} | Charge transfer coefficient |

σ | Solid-phase conductivity, S m^{−1} |

κ | Electrolyte phase conductivity, S m^{−1} |

φ_{s} | Solid-phase potential, V |

φ_{e} | Electrolyte phase potential, V |

η | Activation overpotential, V |

dlnf_{±}/dlnc_{e} | Activity dependence |

Δt | Sampling step, s |

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**Figure 2.**Charge transfer in the battery cell [23].

**Figure 3.**Test device for cell C

_{p}and test results: (

**a**) experiment devices, (

**b**) cell temperature increase.

**Figure 9.**The verification of single battery under different discharge rate and different ambient temperature. (

**a**) Temperature increase; (

**b**) terminal voltage.

**Figure 10.**The verification of battery pack at different ambient temperature with 1C discharge rate: (

**a**) temperature increase, (

**b**) terminal voltage.

**Figure 15.**Relationship among T

_{c}, ΔT

_{max_diff}and ΔSOC

_{max_diff}under the inhomogeneous cooling case 13.

**Figure 16.**The difference of open-circuit voltage, the activation overpotential difference, and the electrolyte potential difference between cell 1 and cell 3 under the inhomogeneous cooling case 13.

Negative/Positive Electrode | |||
---|---|---|---|

Governing equations | Boundary conditions | ||

$\left\{\begin{array}{c}\frac{\partial {c}_{s}}{\partial t}=\frac{{D}_{s}}{{r}^{2}}\frac{\partial}{\partial r}\left({r}^{2}\frac{\partial {c}_{s}}{\partial r}\right)\\ {\epsilon}_{e}\frac{\partial {c}_{e}}{\partial t}={D}_{e}^{eff}\frac{{\partial}^{2}{c}_{e}}{\partial {x}_{P2D}^{2}}+\left(1-{t}^{+}\right)\frac{{j}_{f}}{F}\\ {\sigma}^{eff}\frac{{\partial}^{2}{\phi}_{s}}{\partial {x}_{P2D}^{2}}={j}_{f}\\ {\kappa}^{eff}\frac{{\partial}^{2}{\phi}_{e}}{\partial {x}_{P2D}^{2}}+\frac{2RT{\kappa}^{eff}\left({t}^{+}-1\right)}{F}\left(1+\frac{dIn{f}_{\pm}}{dIn{C}_{e}}\right)\frac{{\partial}^{2}In{c}_{e}}{\partial {x}_{P2D}^{2}}+{j}_{f}=0\\ {j}_{f}={a}_{s}{i}_{0}\left[\mathrm{exp}\left(\frac{{\alpha}_{a}F}{RT}\eta \right)-\mathrm{exp}\left(-\frac{{\alpha}_{c}F}{RT}\eta \right)\right]\\ {i}_{0}=Fk{c}_{e}^{{\alpha}_{a}}{\left({c}_{s,\mathrm{max}}-{c}_{s,e}\right)}^{{\alpha}_{a}}{c}_{s,e}^{{\alpha}_{c}}\end{array}\right.$ | (1) | $\left\{\begin{array}{c}{D}_{s}\frac{\partial {C}_{s}}{\partial r}{|}_{r=0}=0\\ {D}_{s}\frac{\partial {C}_{s}}{\partial r}{|}_{r={R}_{s}}=-\frac{{j}_{f}}{{a}_{s}F}\\ \frac{\partial {c}_{e}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}=0}=\frac{\partial {c}_{e}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}=L}=0\\ -{\sigma}^{eff}\frac{\partial {\phi}_{s}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}=0}={\sigma}^{eff}\frac{\partial {\phi}_{s}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}=L}=\frac{I}{A}\\ \frac{\partial {\phi}_{s}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}={L}_{n}}=\frac{\partial {\phi}_{s}}{\partial x}{|}_{{x}_{P2D}={L}_{n}+{L}_{s}}=0\\ \frac{\partial {\phi}_{e}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}=0}=\frac{\partial {\phi}_{e}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}=L}=0\end{array}\right.$ | (2) |

Separator | |||

Governing equations | Boundary conditions | ||

$\left\{\begin{array}{c}{\epsilon}_{e}\frac{\partial {c}_{e}}{\partial t}={D}_{e}^{eff}\frac{{\partial}^{2}{c}_{e}}{\partial {x}_{P2D}^{2}}\\ {\kappa}^{eff}\frac{{\partial}^{2}{\phi}_{e}}{\partial {x}_{P2D}^{2}}+\frac{2RT{\kappa}^{eff}\left({t}^{+}-1\right)}{F}\left(1+\frac{dIn{f}_{\pm}}{dIn{c}_{e}}\right)\frac{{\partial}^{2}In{c}_{e}}{\partial {x}_{P2D}^{2}}=\frac{I}{A}\end{array}\right.$ | (3) | $\left\{\begin{array}{c}{c}_{e}{|}_{{x}_{P2D}={L}_{n}{}^{-}}={c}_{e}{|}_{{x}_{P2D}={L}_{n}{}^{+}}\\ {D}_{e,n}^{eff}\frac{\partial {C}_{e}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}={L}_{n}{}^{-}}={D}_{e,s}^{eff}\frac{\partial {C}_{e}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}={L}_{n}{}^{+}}\\ {c}_{e}{|}_{{x}_{P2D}={L}_{n}+{L}_{s}{}^{-}}={c}_{e}{|}_{{x}_{P2D}={L}_{n}+{L}_{s}{}^{+}}\\ {D}_{e,s}^{eff}\frac{\partial {C}_{e}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}={L}_{n}+{L}_{s}{}^{-}}={D}_{e,s}^{eff}\frac{\partial {C}_{e}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}={L}_{n}+{L}_{s}{}^{+}}\\ {\phi}_{e}{|}_{{x}_{P2D}={L}_{n}{}^{-}}={\phi}_{e}{|}_{{x}_{P2D}={L}_{n}{}^{+}}\\ {\kappa}_{e,n}^{eff}\frac{\partial {\phi}_{e}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}={L}_{n}{}^{-}}={\kappa}_{e,s}^{eff}\frac{\partial {\phi}_{e}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}={L}_{n}{}^{+}}\\ {\phi}_{e}{|}_{{x}_{P2D}={L}_{n}+{L}_{s}{}^{-}}={\phi}_{e}{|}_{{x}_{P2D}={L}_{n}+{L}_{s}{}^{+}}\\ {\kappa}_{e,s}^{eff}\frac{\partial {\phi}_{e}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}={L}_{n}+{L}_{s}{}^{-}}={\kappa}_{e,p}^{eff}\frac{\partial {\phi}_{e}}{\partial {x}_{P2D}}{|}_{{x}_{P2D}={L}_{n}+{L}_{s}{}^{+}}\end{array}\right.$ | (4) |

Negative/Positive Electrode (k = p or n for Positive and Negative Electrode Separately) | |||
---|---|---|---|

Governing equations | Boundary conditions | ||

$\left\{\begin{array}{c}\frac{\partial {c}_{s}}{\partial t}=\frac{{D}_{s}}{{r}^{2}}\frac{\partial}{\partial r}\left({r}^{2}\frac{\partial {c}_{s}}{\partial r}\right)\\ {\epsilon}_{e}\frac{\partial {c}_{e}}{\partial t}={D}_{e}^{eff}\frac{{\partial}^{2}{c}_{e}}{\partial {z}_{P2D,k}^{2}}+\left(1-{t}^{+}\right)\frac{{j}_{f}}{F}\\ {\sigma}^{eff}\frac{{\partial}^{2}{\phi}_{s}}{\partial {z}_{P2D,k}^{2}}={j}_{f}\\ {\kappa}^{eff}\frac{{\partial}^{2}{\phi}_{e}}{\partial {z}_{P2D,k}^{2}}+\frac{2RT{\kappa}^{eff}\left({t}^{+}-1\right)}{F}\left(1+\frac{dIn{f}_{\pm}}{dIn{C}_{e}}\right)\frac{{\partial}^{2}In{c}_{e}}{\partial {z}_{P2D,k}^{2}}+{j}_{f}=0\\ {j}_{f}={a}_{s}{i}_{0}\left[\mathrm{exp}\left(\frac{{\alpha}_{a}F}{RT}\eta \right)-\mathrm{exp}\left(-\frac{{\alpha}_{c}F}{RT}\eta \right)\right]\\ {i}_{0}=Fk{c}_{e}^{{\alpha}_{a}}{\left({c}_{s,\mathrm{max}}-{c}_{s,e}\right)}^{{\alpha}_{a}}{c}_{s,e}^{{\alpha}_{c}}\end{array}\right.$ | (5) | $\left\{\begin{array}{c}{D}_{s}\frac{\partial {C}_{s}}{\partial r}{|}_{r=0}=0\\ {D}_{s}\frac{\partial {C}_{s}}{\partial r}{|}_{r={R}_{s}}=-\frac{{j}_{f}}{{a}_{s}F}\\ \frac{\partial {c}_{e}}{\partial {z}_{P2D,n}}{|}_{{z}_{P2D,n}=0}=\frac{\partial {c}_{e}}{\partial {z}_{P2D,p}}{|}_{{z}_{P2D,p}=0}=0\\ -{\sigma}^{eff}\frac{\partial {\phi}_{s}}{\partial {z}_{P2D,n}}{|}_{{z}_{P2D,n}=0}={\sigma}^{eff}\frac{\partial {\phi}_{s}}{\partial {z}_{P2D,p}}{|}_{{z}_{P2D,p}=0}=\frac{I}{A}\\ \frac{\partial {\phi}_{s}}{\partial {z}_{P2D,n}}{|}_{{z}_{P2D,n}={L}_{n}}=\frac{\partial {\phi}_{s}}{\partial {z}_{P2D,p}}{|}_{{z}_{P2D,p}={L}_{p}}=0\\ \frac{\partial {\phi}_{e}}{\partial {z}_{P2D,n}}{|}_{{z}_{P2D,n}=0}=\frac{\partial {\phi}_{e}}{\partial {z}_{P2D,p}}{|}_{{z}_{P2D,p}=0}=0\end{array}\right.$ | (6) |

Separator | |||

Governing equations | Boundary conditions | ||

$\left\{\begin{array}{c}{\epsilon}_{e}\frac{\partial {c}_{e}}{\partial t}={D}_{e}^{eff}\frac{{\partial}^{2}{c}_{e}}{\partial {z}_{P2D,s}^{2}}\\ {\kappa}^{eff}\frac{{\partial}^{2}{\phi}_{e}}{\partial {z}_{P2D,s}^{2}}+\frac{2RT{\kappa}^{eff}\left({t}^{+}-1\right)}{F}\left(1+\frac{dIn{f}_{\pm}}{dIn{c}_{e}}\right)\frac{{\partial}^{2}In{c}_{e}}{\partial {z}_{P2D,s}^{2}}=\frac{I}{A}\end{array}\right.$ | (7) | $\left\{\begin{array}{c}{c}_{e}{|}_{{z}_{P2D,n}={L}_{n}}={c}_{e}{|}_{{z}_{P2D,s}=0}\\ {D}_{e,n}^{eff}\frac{\partial {C}_{e}}{\partial {z}_{P2D,n}}{|}_{{z}_{P2D,n}={L}_{n}}={D}_{e,s}^{eff}\frac{\partial {C}_{e}}{\partial {z}_{P2D,s}}{|}_{{z}_{P2D,s}=0}\\ {c}_{e}{|}_{{z}_{P2D,s}={L}_{s}}={c}_{e}{|}_{{z}_{P2D,p}={L}_{p}}\\ {D}_{e,s}^{eff}\frac{\partial {C}_{e}}{\partial {z}_{P2D,s}}{|}_{{z}_{P2D,s}={L}_{s}}=-{D}_{e,p}^{eff}\frac{\partial {C}_{e}}{\partial {z}_{P2D,p}}{|}_{{z}_{P2D,p}={L}_{p}}\\ {\phi}_{e}{|}_{{z}_{P2D,n}={L}_{n}}={\phi}_{e}{|}_{{z}_{P2D,s}=0}\\ {\kappa}_{e,n}^{eff}\frac{\partial {\phi}_{e}}{\partial {z}_{P2D,n}}{|}_{{z}_{P2D,n}={L}_{n}}={\kappa}_{e,s}^{eff}\frac{\partial {\phi}_{e}}{\partial {z}_{P2D,s}}{|}_{{z}_{P2D,s}=0}\\ {\phi}_{e}{|}_{{z}_{P2D,s}={L}_{s}}={\phi}_{e}{|}_{{z}_{P2D,p}={L}_{P2D,p}}\\ {\kappa}_{e,s}^{eff}\frac{\partial {\phi}_{e}}{\partial {z}_{P2D,s}}{|}_{{z}_{P2D,s}={L}_{s}}={\kappa}_{e,p}^{eff}\frac{\partial {\phi}_{e}}{\partial {z}_{P2D,p}}{|}_{{z}_{{z}_{P2D,p}}={L}_{p}}\end{array}\right.$ | (8) |

Parameter | Negative Electrode | Separator | Positive Electrode | Reference |
---|---|---|---|---|

L (m) | 73 × 10^{−6} | 13 × 10^{−6} | 61 × 10^{−6} | [26] |

A (m^{2}) | / | 2.14 | / | Measured |

R_{s} (m) | 9.93 × 10^{−6} | / | 6.32 × 10^{−6} | [27] |

ε_{s} | 0.65 | / | 0.547 | [28,29] |

ε_{e} | 0.315 | 0.5307 | 0.332 | [28,29,30] |

c_{s,max} (mol m^{−3}) | 31,389 | / | 48,396 | [24] |

θ_{0} | 0.01 | / | 0.955 | [24] |

θ_{100} | 0.785 | / | 0.415 | [24] |

c_{e,}_{0} (mol m^{−3}) | / | 1200 | / | [30] |

α_{a}, α_{c} | 0.5, 0.5 | / | 0.5, 0.5 | [24] |

t^{+} | / | 0.363 | / | [30] |

σ (Sm^{−1}) | 100 | / | 100 | [23] |

R_{SEI} (Ωm) | 0 | / | 0 | [23] |

dlnf_{±}/dln c_{e} | / | 0 | / | [30] |

Parameter | Equation | Reference |
---|---|---|

Diffusion coefficient of Li-ion in the electrode | ${D}_{s,p}={10}^{-14}\mathrm{exp}\left[-\frac{30000}{8.314}(\frac{1}{T}-\frac{1}{298.15})\right]$ ${D}_{s,n}=1.4523\times {10}^{-13}\mathrm{exp}\left[-\frac{30000}{8.314}(\frac{1}{T}-\frac{1}{298.15})\right]$ | [31] |

Diffusion coefficient of Li-ions in the electrolyte | ${D}_{e}={10}^{\left(-8.43-\frac{54}{T-229-0.005{c}_{e}}-0.22\times 0.001{c}_{e}\right)}$ | [31] |

Reaction rate of the electrode | ${k}_{n}=2\times {10}^{-11}\mathrm{exp}\left[-\frac{30000}{8.314}(\frac{1}{T}-\frac{1}{298.15})\right]$ ${k}_{p}=2\times {10}^{-11}\mathrm{exp}\left[-\frac{30000}{8.314}(\frac{1}{T}-\frac{1}{298.15})\right]$ | [30] |

Ionic conductivity | $\begin{array}{c}\kappa =1.254{c}_{e}\times {10}^{-4}(-8.248+0.05324T-2.987\times {10}^{-5}{T}^{2}+\\ 0.2623\times {10}^{-3}{c}_{e}-0.009306\times {10}^{-3}{c}_{e}T\\ +0.000008069\times {10}^{-3}{c}_{e}{T}^{2}+0.22\times {10}^{-6}{c}_{e}^{2}-0.0001765\times {10}^{-6}{c}_{e}^{2}T)\end{array}$ | [31] |

State of charge (SOC) | ${\theta}_{i}=\frac{{c}_{e,i}}{{c}_{e,i,\mathrm{max}}},\left(i=p,n\right)$ | [31] |

Open circuit potential | ${U}_{p}=4.3655+5.3596{\theta}_{p}-23.8949{\theta}_{p}^{2}+30.4942{\theta}_{p}^{3}-12.7557{\theta}_{p}^{4}$ ${U}_{n}=0.6554-5.8181{\theta}_{n}+22.5962{\theta}_{n}^{2}-36.1670{\theta}_{n}^{3}+20.0406{\theta}_{n}^{4}$ | [24] |

Temperature derivative of open circuit potential | $\frac{d{U}_{p}}{dT}=-7.225\times {10}^{-5}$ $\frac{d{U}_{n}}{dT}=0.00305-0.002762{\theta}_{n}+0.005726{\theta}_{n}^{2}-0.004453{\theta}_{n}^{3}$ | [24] |

Case | T_{c} (°C) | h (W/K·m^{2}) | Case | T_{c} (°C) | h (W/K·m^{2}) |
---|---|---|---|---|---|

1 | 10 | 5 | 9 | 30 | 100 |

2 | 20 | 5 | 10 | 10 | 175 |

3 | 30 | 5 | 11 | 20 | 175 |

4 | 10 | 32 | 12 | 30 | 175 |

5 | 20 | 32 | 13 | 10 | 220 |

6 | 30 | 32 | 14 | 20 | 220 |

7 | 10 | 100 | 15 | 30 | 220 |

8 | 20 | 100 |

T_{c} = 10 °C | T_{c} = 20 °C | T_{c} = 30 °C | |
---|---|---|---|

h = 5 | 0.95 | 0.80 | 0.72 |

h = 32 | 4.50 | 3.93 | 3.45 |

h = 100 | 8.32 | 7.29 | 6.40 |

h = 175 | 10.03 | 8.78 | 7.70 |

h = 220 | 10.64 | 9.30 | 8.15 |

T_{c} = 10 °C | T_{c} = 20 °C | T_{c} = 30 °C | |
---|---|---|---|

h = 5 | 36.66 | 42.75 | 49.36 |

h = 32 | 34.72 | 40.91 | 47.89 |

h = 100 | 32.49 | 38.96 | 46.19 |

h = 175 | 31.44 | 38.05 | 45.39 |

h = 220 | 31.06 | 37.72 | 45.11 |

T_{c} = 10 °C | T_{c} = 20 °C | T_{c} = 30 °C | |
---|---|---|---|

h = 5 | 1.12 | 0.75 | 0.51 |

h = 32 | 5.99 | 4.04 | 2.70 |

h = 100 | 12.45 | 8.38 | 5.57 |

h = 175 | 15.70 | 10.57 | 7.01 |

h = 220 | 16.87 | 11.36 | 7.53 |

T_{c} = 10 °C | T_{c} = 20 °C | T_{c} = 30 °C | |
---|---|---|---|

h = 5 | 0.30 | 0.21 | 0.14 |

h = 32 | 1.65 | 1.10 | 0.73 |

h = 100 | 3.52 | 2.31 | 1.52 |

h = 175 | 4.52 | 2.95 | 1.92 |

h = 220 | 4.89 | 3.19 | 2.07 |

**Table 10.**Maximum local volumetric current density difference at the negative electrode (Δj

_{f,n,}

_{max_diff}) in the discharge process (10

^{4}A/m

^{3}).

T_{c} = 10 °C | T_{c} = 20 °C | T_{c} = 30 °C | |
---|---|---|---|

h = 5 | 0.76 | 0.53 | 0.34 |

h = 32 | 4.10 | 2.75 | 1.81 |

h = 100 | 8.55 | 5.72 | 3.75 |

h = 175 | 10.81 | 7.23 | 4.73 |

h = 220 | 11.62 | 7.78 | 5.09 |

**Table 11.**Maximum local volumetric current density difference at the positive electrode (Δj

_{f,p,}

_{max_diff}) in the discharge process (10

^{4}A/m

^{3}).

T_{c} = 10 °C | T_{c} = 20 °C | T_{c} = 30 °C | |
---|---|---|---|

h = 5 | 0.85 | 0.57 | 0.39 |

h = 32 | 4.57 | 3.09 | 2.05 |

h = 100 | 9.50 | 6.41 | 4.24 |

h = 175 | 11.98 | 8.09 | 5.34 |

h = 220 | 12.87 | 8.70 | 5.74 |

**Table 12.**Maximum exchange current density difference at the negative electrode (Δi

_{0,n,}

_{max_diff}) in the discharge process (A/m

^{2}).

T_{c} = 10 °C | T_{c} = 20 °C | T_{c} = 30 °C | |
---|---|---|---|

h = 5 | 0.02 | 0.02 | 0.03 |

h = 32 | 0.11 | 0.11 | 0.12 |

h = 100 | 0.19 | 0.20 | 0.23 |

h = 175 | 0.22 | 0.24 | 0.27 |

h = 220 | 0.24 | 0.26 | 0.29 |

**Table 13.**Maximum exchange current density difference at the negative electrode (Δi

_{0,p,}

_{max_diff}) in the discharge process (A/m

^{2}).

T_{c} = 10 °C | T_{c} = 20 °C | T_{c} = 30 °C | |
---|---|---|---|

h = 5 | 0.04 | 0.04 | 0.05 |

h = 32 | 0.18 | 0.20 | 0.22 |

h = 100 | 0.32 | 0.35 | 0.39 |

h = 175 | 0.37 | 0.41 | 0.46 |

h = 220 | 0.39 | 0.43 | 0.48 |

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## Share and Cite

**MDPI and ACS Style**

Xie, Y.; Mu, X.; Deng, Z.; Zhang, K.; Chen, B.; Fan, Y.
A Comprehensive Study on the Effect of Inhomogeneous Heat Dissipation on Battery Electrochemical Performance. *Electronics* **2023**, *12*, 1266.
https://doi.org/10.3390/electronics12061266

**AMA Style**

Xie Y, Mu X, Deng Z, Zhang K, Chen B, Fan Y.
A Comprehensive Study on the Effect of Inhomogeneous Heat Dissipation on Battery Electrochemical Performance. *Electronics*. 2023; 12(6):1266.
https://doi.org/10.3390/electronics12061266

**Chicago/Turabian Style**

Xie, Yi, Xingyu Mu, Zhongwei Deng, Kaiqing Zhang, Bin Chen, and Yining Fan.
2023. "A Comprehensive Study on the Effect of Inhomogeneous Heat Dissipation on Battery Electrochemical Performance" *Electronics* 12, no. 6: 1266.
https://doi.org/10.3390/electronics12061266