Battery Model

4.2.1.1 Decoupled

A decoupled method that treats lithium-ion batteries as heat source in a standalone 3D symmetrical model can be used to evaluate the proposed BTM system performance (Fig. 4.3). The heat generation is modelled as a uniform heat generation inside the volume of each cell (Fig. 3.8), and the rate was obtained using Eq. 3.18 – 3.19. Decoupled model eliminates the dependency of temperature in battery heat generation serving as an effective tool in evaluating heat pipe performance as well as system cooling/preheating behaviour.

Figure 4.3: Decoupled model.

Table 4.3 presents the charge and discharge condition based on a rated capacity of 16.5 Ah lithium-ion cell demonstrated in Table 4.1. Note that the heat value per unit cell under corresponding charge and discharge conditions are calculated and demonstrated according to a representative value, i.e. 50%

DOD (depth of discharge). The electrical resistance depends on the current network that links collector and electrode active materials by a nonconductive metal (i.e. the separator) and the electrolyte among others [97]. 50% DOD represents the baseline value of the internal resistance ratio of 1. Electric conductivity increases with elevated battery temperature. But for simplicity,

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the heat generation rates calculated for decoupled model are based on an average value.

Table 4.3: Lithium-ion battery cell charge/discharge condition Current Equivalent charge

current (A)

Heat value/cell (W)

Charge Condition

Pre-charge 0.05C 0.825 -0.07

0.1C 1.65 -0.14

Standard/normal charge range

0.2C 3.3 -0.23

0.5C 8.25 -0.26

1C 16.5 +0.57

Max charge 2C 33 +5.50

Discharge Condition

Standard/normal discharge range

0.2C 3.3 +0.41

0.4C 6.6 +1.35

1C 16.5 +3.78

2C 33 +11.92

3C 49.5 +24.42

Max discharge (continuous)

4C 66 +41.27

Max discharge (peak < 10s)

6C 99 +88.04

Figure 4.4: Heat generation rate per unit cell under charge/discharge current.

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Negative sign indicates endothermic value, and positive, exothermic. Fig.

4.4 demonstrates how heat absorption and dissipation perform in the range from 0.05C (the minimum pre-charge current) to 2C (the specified maximum charge current). The heat value during endothermic process starts to increase from 0.05C to a peak value, and decrease to zero under approximately 0.75C where reaction heat is equal to the sum of polarisation heat and Joule heat. It will then release heat at 5.5 W/cell rate. The heat dissipation per unit cell under discharge rate from 0.2C to 6C is also indicated in Fig. 4.4. The heat generation during discharge is more substantial compared to that under charging. This is because the reaction heat becomes the most dominant factor and is positive during the entire process.

4.2.1.2 Coupled

A full 1D electrochemical model for lithium-ion batteries is also developed to calculate the average heat source in relation to the temperature profile of the battery cell. A 3D symmetrical model is used to model the conjugated heat transfer including laminar flow and heat transfer in solids (Fig.

4.5). Since the heat conductivity of the components of a lithium-ion battery is high compared to the heat generated, it is assumed that the battery will have a uniform temperature profile (Bi = 0.0047 ~ 0.18 < 1) and the battery chemistry will not be heavily affected by small temperature changes. The above two models will be coupled by the generated heat source and the average temperature based on lumped heat transfer.

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Figure 4.5: Coupling between the cell and flow and heat transfer model using the average values for the temperature and battery heat generation.

Figure 4.6: 1D isothermal lithium-ion battery model created in COMSOL Multiphysics 4.3b.

The cell model consists of 5 domains as illustrated below (Fig. 4.6):

 Negative current collector (copper, 7 m)

 Negative porous electrode (LixC6, 55 m)

 Separator (electrolyte 1:1 EC/DMC in LiPF⁶, 30 m)

 Positive porous electrode (Li^xMn2O4, 55 m)

 Positive current collector (Al, 10 m)

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The model involves 5 following processes [218]:

 Electronic current conduction in the electrodes

 Ionic charge transport in the electrodes and electrolyte/separator

 Material transport in the electrolyte, which allows to account for the effects of concentration on ionic conductivity and concentration overpotential (obtained from experiment)

 Material transport within the spherical particles that from the electrodes

 Butler-Volmer electrode kinetics using discharge curves (measured from experiment) to obtain the equilibrium potential

The boundary condition of 1D lithium-ion battery model will be summarised in Table 4.4. The electric potential in the electron conducting phase can be calculated using Ohm’s law. For the porous electrodes effective conductivities seff, it can be formulated using Eq. 4.1 where is the Bruggeman coefficient ( = 1.5 to indicate a packed bed of spherical particles).

(4.1) The ionic charge balances and material balances are modelled using Eq.

4.2 – 4.5 for 1:1 EC:DEC/LiPF6 electrolyte. Fickian diffusion equation (Eq.

4.5) describes the transport in the spherical particles and is expressed for the material balance of lithium in the particles in spherical coordinates. Butler-Volmer electrode kinetics (Eq. 4.14 – 4.16) can be used to obtain the local charge transfer current density in the electrodes.

s





_s^eff  _s

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condition Electrolyte conductivity l = (depends on salt concentration c, see Fig. 4.7

Electrolyte salt diffusivity Dl = 3e-10 (m²/s) Transport number t+ = 0.363

Activity dependence = 0 Domain 1-3

Negative and Positive Current Collector (Domain 1, 5) Equation

(4.6) (4.7) Negative and Positive Porous Electrode (Domain 2, 4)

Equation

85 Properties Anodic transfer coefficient a = 0.5

Cathodic transfer coefficient c = 0.5 Anodic rate constant ka = 2e-11 m/s Cathodic rate constant kc = 2e-11 m/s

Electrolyte reference concentration cl,ref = 1 mol/m³ Domain 4, 5

Initial condition

Electrolyte potential = -0.1 V

Electrolyte salt concentration cl = cl_0 = 2,000 mol/m³ Electric potential = 3.6 V

The electrolyte conductivity and the equilibrium potential of the electrodes plotted in Fig. 4.7 and Fig. 4.8 are from experimental measured data stored in COMSOL. The equilibrium potential for the negative and positive electrodes can be expressed as a function of the measured state of charge (SOC) (Eq.

4.17).

(4.17)

The initial values of SOC for the negative and positive electrodes are 0.17 and 0.56 respectively. This translates to an open circuit cell voltage of 4.27 V, which indicates a fully charged battery. Fig. 4.9 demonstrates the discharge

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different discharge rates and the end-of-discharge can be defined as the time when the cell voltage drops below 3 V. High discharge rates such as 4C and 6C make the battery capacity deliver less than half of the theoretical capacity of 16.5 Ah/m² obtained from 1C (Fig. 4.9).

Figure 4.7: 1:1 EC:DEC/LiPF6 electrolyte conductivity obtained from experimentally measured data (COMSOL stored data) using interpolation function according to concentration.

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Figure 4.8: The voltage of the electrode materials measured from experiment.

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Figure 4.9: Discharge curves based on 0.2C, 1C, 2C, 3C, 4C & 6C rate.

The temperature is assumed to be the mean temperature of the battery using an integral function of component coupling (set as aveop(mod.T) in COMSOL). The applied current (i_app) is normally user-defined. The default expression will be expressed as a square wave function (wv) (Fig. 4.10) with an alternating charge/discharge current at 4C rate (continuous max) under a cycle time of 600 s. i_app can be written as:

(4.18) ])

1 / [ ( _

_app i load wvt

i  

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Figure 4.10: Wave function (angular frequency = 2×pi/600).

Figure 4.11: Cell potential and battery load at 4C rate under 600 s cycle time.

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(a) t = 300 s, 600 s

(b) t = 299.95 s, 599.95 s

Figure 4.12: Total power dissipation density (W/m³) simulated from 1D lithium-ion battery under a 4C charge-discharge cycle of 600 s: (a) at 300 s and 600 s; (b) at 299.95 s and 599.95 s.

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Fig. 4.11 demonstrates the cell potential and battery load at 4C rate under a cycle time of 600 s based on the 1D electrochemical model. Fig. 4.12 shows the total heat dissipation resulted from negative electrode (Domain 2), positive electrode (Domain 4), separator (Domain 3) and the battery (Domain 1–5) at the end of each charge/discharge (i.e. at 300 s and 600 s) and 0.05 s before each charge/discharge (i.e. 299.95 s and 599.95 s) along corresponding 1D dimension arc length.

The heat generated from both current collectors (negative and positive) can be neglected. The heat comes mostly from the negative and positive electrode and is exothermic at all times (+1,000 W/m³ for charge at 300 s and +4,000 – 8,000 W/m³ under discharge at 600 s) for negative electrode. For positive electrode, the heat value is endothermic during charge (-2,000 W/m³ at 300 s) and exothermic under discharge (+12,000 – 16,000 W/m³ at 600 s).

From Fig. 4.12 (a) for all domains, the heat added up during charge condition is negative, which means the battery absorbs heat during charge (-1,000 W/m³ at 300 s, i.e. -0.23 W/cell). For discharge condition, the heat dissipated from all domains becomes exothermic and approximately equals to +16,000 – 24,000 W/m³ at 600 s (equivalent to 3.71 – 5.56 W/cell).

Fig. 4.12 (b) shows the heat generated 0.05 s prior to each charge/discharge. The value changes significantly compared to that under 300s and 600s. Even the heat generated by the separator rises up from none to +10,000 – 10,260 W/m³ (equivalent to 2.32 – 2.38 W/cell). Both negative and positive electrodes dissipate heat, ranging from +1.4×10⁵ – 2.4×10⁵ W/m³

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(equivalent to 32.44 – 55.62 W/cell) and +1.1×10⁵ – 3×10⁵ W/m³ (equivalent to 25.49 – 69.52 W/cell) respectively.

The variations in heat generation during the load cycle is huge, but may not affect the overall temperature change as a considerable amount of time period is needed before reaching to thermal balance. The values obtained serve as references in comparison with the decoupled model, which accounts for an average heat value from -0.23 – 5.50 W/cell (-992 – 23,733 W/m³) during 0.2 – 2C charge, and 3.78 – 41.27 W/cell (+16,182 – 178,084 W/m³) under 1 – 4C discharge.