3.1.2. Specific Heat Capacity

The type 18650 battery is not homogeneous, so, the SHC *CM* refers to the equivalent SHC of a single battery. Assuming that the equivalent SHC of the battery is *CM*, according to the definition of SHC, it is described by Equation (4):

$$\frac{d\phi}{dt} = \mathcal{C}\_M M \frac{dT}{dt} \tag{4}$$

It is known that the value of the heat generated (i.e., temperature rise) by the battery is directly proportional to the equivalent SHC of the battery. In fact, the SHC is computed by the law of conservation of energy (COE). In an adiabatic environment, the value of the heat generation of a battery equals its stored heat. The accumulation of heat storage will lead to the rise in battery temperature. By measuring the temperature rise in a specific period time, the SHC of the battery can be calculated according to the above formula. In the actual experiment, in order to reduce the experimental error, the SHC of the battery under different discharge ratios (i.e., different value of heat generation) was measured and the SHC of the battery was calculated by linear regression. The experimental procedures have been described in detail in [23].

### 3.1.3. Heat Generation of Single-Cell and Thermal Conductivity

On this basis, Bernardi et al. (1984) provided a methodology to calculate the heat production rate of batteries, which is expressed by Equation (5).

$$P\_{\text{total}} = -IT\frac{dE}{dT} + I(E - \mathcal{U})\tag{5}$$

where *P* is the power of the heat generation, *I* is current in circuit, *T* is the real-time temperature of the battery, *E* refers to the open circuit voltage of battery, and *U* refers to the average terminal voltage of battery.

In Equation (5), the first item *IT dEdT* on the right side of equation is the formula to calculate the power of RH, which equals the *P*total in Equation (1). The second item refers to summary of JH and PH and it shows the voltage attribution. The voltage drops (*E* − *U*) in the open circuit voltage and terminal voltage are attributed to internal DC resistance and their relationship is represented by Equation (6).

$$I(E - \mathcal{U}) = I^2(R\_\Omega + R\_P) \tag{6}$$

*R* Ω + *RP* make up the internal DC resistance, which can be expressed by *R*. So, the heat production Equation (5) can be expressed by Equation (7),

$$P\_{\text{total}} = -IT\frac{dE}{dT} + I^2R\tag{7}$$

On the premise that the equivalent SHC is known (calculated in Section 3.1.2), the heat production in a certain period of time can be solved according to the definition of the SHC [23]. In this experiment, the heat dissipation e ffect of the battery within 15 min is discussed, so the heat generation of the battery should also be the heat generation within 15 min. The discharging current rate is set at 2C. Twenty single batteries were selected for heat yield measurement. The specific operational steps are as follows:


For each cell, the temperatures were recorded. During battery discharge, oxidation occurs in the negative electrode, and lithium is separated from the carbon rod and releases energy, which causes the temperature of the negative electrode to rise. At the same time, there is a reduction at the positive pole. Lithium ions precipitate at the positive electrode and absorb a certain amount of energy, so the temperature of the positive pole decreases. Therefore, during the whole discharge process, the temperature gradually decreases from the negative electrode to the positive electrode. Thus, it is assumed that the temperature of the positive pole is transferred from the negative electrode and the equivalent TC of the whole battery is calculated based on the battery temperature distribution. The formula to calculate TC is illustrated in Equation (8):

$$
\lambda = \frac{q\delta}{\Delta t} \tag{8}
$$

where λ is TC, *q* is conducted heat, and δ is the distance from the negative pole to the positive pole. Δ*t* refers to the temperature di fference (TD) between the two poles. In fact, the conductivity of the battery is anisotropic, which means the TC is di fferent in the surface and the thickness directions. However, the TC in the surface direction is much less than that in the thickness direction and it is ignored in this model [23,24].

In order to avoid contingency and remove outliers, 20 type 18650 batteries were tested under the same conditions. After calculating the average value of the 20 batteries, the thermodynamic parameters are listed in Table 1.


**Table 1.** The thermodynamic parameters of the battery.
