An investigation of irreversible heat generation in lithium ion batteries based on a thermo-electrochemical coupling method

Accepted Manuscript An investigation of irreversible heat generation in lithium ion batteries based on a thermo-electrochemical coupling method Shuanglong Du, Yanqing Lai, Liang Ai, Lihua Ai, Yun Cheng, Yiwei Tang, Ming Jia PII: DOI: Reference:

S1359-4311(17)31946-4 http://dx.doi.org/10.1016/j.applthermaleng.2017.04.077 ATE 10226

To appear in:

Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

24 March 2017 15 April 2017 18 April 2017

Please cite this article as: S. Du, Y. Lai, L. Ai, L. Ai, Y. Cheng, Y. Tang, M. Jia, An investigation of irreversible heat generation in lithium ion batteries based on a thermo-electrochemical coupling method, Applied Thermal Engineering (2017), doi: http://dx.doi.org/10.1016/j.applthermaleng.2017.04.077

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An investigation of irreversible heat generation in lithium ion batteries based on a thermo-electrochemical coupling method Shuanglong Dua, Yanqing Laia, Liang Aib, Lihua Aib, Yun Chenga, Yiwei Tanga, Ming Jiaa,b,* a

School of Metallurgy and Environment, Central South University, Changsha 410083, PR China

b

Hunan aihua group co., LTD, Yiyang 413002, PR China

Absract：Irreversible heat generation plays a dominant role in li-ion batteries, it is thus highly important to study its evolution in order to adapt the development of electronic devices. Internal irreversible heat generation mainly consists of two parts: one arises from the polarization and the other one from ohmic heat generation. A thermo-electrochemical coupling model was established here to study the production and evolution of the irreversible heat within li-ion batteries considering dynamic parameters and the electric double layer. Results show that the irreversible heat production rapidly increases with the discharge rate and the polarization heat production is the dominating factor. Ohmic heat production mainly resulted in the heating of the electrolyte, and the heating produced at the negative active materials result to be negligible respect to the one produced at the positive active materials. According to calculations, the ratio between the ohmic heat production and the total irreversible heat production increases from 24.2% at 3C to 32.8% at 8C, thus, the ratio related to the polarization heating decreases from 75.6% to 67.2%. In addition, effects of the particle size at the positive and negative electrodes at the rate of 3C were studied. Results show that the negative electrode particle size has thus a more significant impact on the irreversible heat production and the polarization heat production of the battery. Keywords: lithium ion battery; coupling model; irreversible heat; polarization; particle size 1. Introduction With the rapid development of Li-ion batteries for electronic devices, the problem of high-rate discharge becomes more and more serious; within them, safety is the most important issue. The

problem of heat production in the operating batteries is the main aspect affecting the device safety. In recent years, numerical simulation technology has played an important role in the simulation of energy storage devices[1]. Numerous studies have shown that Li-ion battery internal heat production is mainly composed of two parts, reversible heat production and irreversible heat production. While reversible heat production dominates at low-rate, irreversible heat production dominates at high-rate [2,3,4]. As the demand for high-rate power discharge is growing, the thermal safety issue at high-rate is particularly relevant. Therefore, it is of great significance to carry out the thermal analysis on the batteries operated at high-rate[5,6]. Many works have been devoted to the battery heat production till now. Song et al.[7] combined calculation based on an electrochemical model with a calorimeter and studied Li-ion batteries made with LiFePO4 as the positive electrode. Results showed that the battery heat production was directly related to the discharge rate and the working temperature; irreversible heat generation was shown to occupy the dominant role at 0.5 C and 1 C. Lai et al.[8] made a detailed analysis on heat generation behavior in Li-ion batteries with LiFePO4 as positive electrode through a simulation platform, indicating that reversible heat generation curve fluctuations influence the total heat generation fluctuation and that the irreversible heat generation at the negative electrode occupied the dominant position respect to the total reversible heat generation. Moreover, the design parameters also have great influence on the internal heat generation. Zhao et al.[9,10] established the model of electrochemical thermal coupling, focused on the evolution law of reversible heat generation and analyzed the effect of electrode thickness and size of the active materials on reversible heat generation. The study found that electrode thickness and particle size of the active material have important influence on the internal heat generation and reasonable electrode design should be used to optimize the thermal behavior. Marcicki et al.[11] quantified reversible heat generation factors based on experiment results and then established the real-time evolution of reversible heat generation in Li-ion battery under different working conditions. Reversible heat generation in batteries has been much more investigated in literature respect to irreversible heat, for which generation and evolution laws have been poorly reported. In this paper, we focused on the soft package lithium iron phosphate battery; the

electrochemical thermal coupling model was established and the thermal behavior at high discharge rates was simulated. Furthermore, the evolution of irreversible heat generation and the thermal evolution of different electrode materials particles morphology under irreversible heat generation under were simulated. This research is expected to provide an effective guidance for the thermal analysis of Li-ion battery and the design of the thermal management system for the battery packs. 2. The thermo-electrochemical coupling model At present, Li-ion battery models are mainly established on the basis of the Newman model[12,13], that is, the Li-ion battery is projected in the x-axis, while positive and negative electrodes are assumed to be composed of homogeneous particles. During the working process, the thermal behavior of the battery has a serious influence on its dynamic parameters, which can affect the battery charging and discharging behaviors. As a result, the influence of the temperature within the battery must be considered in the simulation process. Based on the Newman model, the dynamic behavior of battery during operation was objectively described by a coupled 3D thermal model. The coupling mechanism of these two models is as follows. The heat generation rate calculated by the electrochemical model is used as the input and introduced into the 3D heat transfer model. The temperature calculated by the 3D heat transfer model is then used as a feedback, which will affect the numerical value of the electrochemical model. This coupled model is denoted as the electrochemical thermal coupling model. Figure 1 shows the schematic diagram of mechanism of the soft package Li-ion battery.

Figure 1 schematic diagram of lithium ion battery

2.1 The quasi-two-dimensional electrochemical model Since it is also required to consider the diffusion process of the electrode particles in the radial direction when the battery is projected to one direction, the model can be called as a quasi-two-dimensional electrochemical model. The electrochemical model is established based on the conservation of materials and charge and used to describe the transmission process of the battery by a series of partial differential equations. One of the most important equation is the Butler-Volmer equation which describes the interface between positive and negative electrode particles and the electrolyte, as well as the electric double layer effect formed at particles surface. Since many previous works described these equations in detail, this will not be repeated in this paper. All equations related to electrochemical processes and boundary conditions are listed in Table 1.

Table 1 Govering equations and boundries Physics

Govering equation and boundries

τ2R i

mass balance in

t



D1,i    R i  τ

mass balance in

ε2



 2 c1,i  τ   τ  

c1, i

τ = r/Ri

solid phase

electrolyte phase

c1,i

τ

 0 τ0



Sa, i jloc, i dc 2 1  t       Deff 2 c 2  dt F

γ

Deff  D2ε 2 2 2

  - σ c φ c   -J i

  φ φ     - k1eff φ1  -Sa, i  jloc, i  Cdl  1 - 2   t    t 





electron transport in

3ε1, i

Sa, i 

solid phase

rp,1

; k 1eff  k 1 ε γ11

φc x  0  0 ; - σcφc xL - k1eff φ1

ionic transport in solid phase

Electrochemical kinetics

x  L ncc  L n

 0;

ncc

 -Iapp

 L n  L s  L p  L pcc

- k1eff φ1

x  L ncc  L n  L s

 0

   2 RT  ln f   1  t    c 2    Sa,i jloc,i   k 2eff - φ2  1   F  ln c 2  c2     

φ 2 x

 x  L ncc

φ 2 x

 0 x  L ncc  L n  L s  L p

  α η F   α η F   j loc, i  j0, i exp  a, i i   exp  c, i i     RT   RT  

ηi  φ1,i  φ 2,i  U i i j0, i  Fk i c 2 a, i c1, max, i  c1, surf, i  a, i c1,c,surf, i

α

α

α

In the discharge process of the battery, the temperature and the concentration of Li ions in different internal regions will change, causing the change of some important physical parameters such as the diffusion coefficient and the electrical conductivity. As these parameters are highly non-linear, the dynamic change of these parameters must be taken into account to get accurate simulation results. In this study, the effective diffusion coefficient of Li ions in the

solid and the liquid phase were modified by using the Bruggman coefficient, while temperature effects on the diffusion coefficients were described by the Arrhenius equation. Resulting dynamic parameters are summarized in Table 2. Table 2 Dynamic physical properties parameters Parameters Diffusion coefficient of the solid phase

Diffusion coefficient of the liquid phase

Expression D1,p 

1.18 1018

1  y 

1.6

 E1, D ,p exp   R 

1  1    T 298.15  

 E1, D ,n  1 1  D1,n  3.9  10 14 exp      R T 298 .15    

 E k1, n  3  10 11exp   1, k, n R 

[14]

1 1      T 298.15  

 E 1 1  k1,p  1.4  10 12 exp - 3 y exp   1,k ,p    R  T 298.15   



[14]



thermodynamic factor relating to electrolyte activity

[15]

The ionic conductivity

  10.5  0.074T  6.69  105 T 2  6.68  104 C  1.78  105 CT   k  1  10- 4   8 2 7 2 10 2   2.8  10 CT  4.49  10 C  8.86  10 C T 

ν  0.601  0.24 10 3 C  0.982 1 - 0.0052T - 294.0 10 -9 C 3

[16]

The Li+ transference number

 833  C   653   C   49.6     3.09  10 3 exp   *    0.517exp    t   2.67  10 4 exp  T 1000 T 1000 T         

[15,17]



U ref, cathode  3.4323 - 0.4828 exp - 80.2493 1 - y 



1.3198

 3.2482  10  6 exp 20.2646 1  y 

Open circuit potential

3.7995



  3.2474  10

6



exp 20.2645 1  y 



3.8003

U ref, anode  0.6379  0.5416 exp - 305.5309 x   0.044tanh  x - 0.1958   x - 1.0571   x  0.0117   x - 0.5692  [18]  - 0.1088  - 0.1978 tanh 0.0854  - 0.6875 tanh 0.0529  - 0.0175 tanh 0.0875         

The value of E1, D, p and E1, D, n are both 35 kJ mol-1[19,20], which represents the activation energy for diffusion of lithium respectively in the positive and negative electrodes. Likewise, the E1, k, p and E1, k, n are 20 kJ mol-1, and 30kJ mol-1[19,21] respectively, which depicts the activation energy for the rate constant of the positive and negative electrodes. 2.2 The mechanism of 3D thermal model

In the simulation of Li-ion batteries, another important conservation relation is the energy conservation. Continuous temperature change occurs during the working process, which affects internal complex electrochemical processes. The following energy conservation relational expression was mainly used for the Li-ion battery simulation:

ρiC p,i

T     kiT   q re  qirr t

(1)

Internal heat generation is mainly divided in two parts, the reversible heat qre and the irreversible heat qirr. Reversible heat is mainly related to the entropy change due to the de-intercalation or intercalation processes; its numerical value is related to the entropy change due to chemical reactions. The effect of the internal battery reversible heat generation on the overall thermal behavior of the battery has been intensively studied [8,10]. Reversible heat generation is expressed as:

q re  S a,i jloc,i T

S nF

(2)

U

S  nF( Ti )

(3)

The experimental value of entropy change is usually derived, at specific charging states, by using the reference electrode as the counter electrode and measuring the open circuit voltage at different temperatures. This process is very complex. In this paper, a lithium iron phosphate battery was considered; changes in the entropy at positive and negative electrodes were obtained by using the data available from the literature [22,23], the expressions are reported in Eq. 4 and 5 and the corresponding curves are shown in Figure 2. dU n exp  32.9633287x  8.316711484   344.1347148  dT 1  749.0756003 exp  34.79099646x  8.887143624 

(4)

 0.8520278805x  0.362299229x  0.2698001697 2

dU p

 0.35376y 8  1.3902y 7  2.2585y 6  1.9635y 5  0.98716y 4

dT  0.28857y 3  0.046272y 2  0.0032158y  1.9186  10  5

(5)

x and y in the above equation refer to the particles concentration at the electrode surface and the maximum concentration of the electrolyte in the charged state during the battery discharge cycle. The specific expression is shown as follows, the corresponding fitting curves are shown in

SOC（xory）

C sur C max (6)

Figure 2 Entropy fitting curves (dU/dT) for LixC6 (panel a, Eq 4) and for LiyFePO4 (panel b, Eq 5). The internal irreversible heat generation qirr mainly includes the polarization heat generation qp due to electrochemical reactions and the ohmic heat generation qohm generated by the internal ohmic resistance of the battery:

qirr  q p  qohm

(7)

The specific expression of the polarization heat generation qp is as follows:

q p  S a,i jloc,i φ1,i  φ2,i  U i 

(8)

In the above formula, Ui is used to describe the equilibrium potential of positive and negative electrodes as a function of the charge state of the battery. Considering the effect of the temperature on the battery discharge process, a Taylor expansion was carried out at the reference temperature:

U i  U ref,i  T  Tref 

dU i dT

(9)

Ohmic heating is mainly generated by three factors: (1) ohmic heat created from the solid-state active material resistance qi,s, (2) heat generated by ions transport in the electrolyte qe and (3) current-collector heat generation qi,c. In the following text, the internal polarization heat generation and the ohmic heat generation will be analyzed in depth. Numerical expressions of the three kinds of heat generations are shown below:

qi,s  σieff φ1,i  φ1,i qe  k φ2,i  φ2,i eff 2

2RTk 2eff t  11  lnf   lnc2,i   φ2,i  F lnc2,i  

(10)

(11)

qi,c  σi,cφi,c  φi,c

(12)

Newton’s cooling law was used to describe convective heat transfer phenomena in this study and the Stephan Boltzmann equation was used to describe the radiation heat transfer process. Based on interfacial heat conservation, the thermal boundary conditions were obtained and shown in Eq.13:

- kT  h(Tamb  T )   (T 4amb  T 4 )

(13)

In the above formula, k represents the thermal conductivity coefficient of the battery and h represents the heat exchange coefficient between the battery surface and the surrounding environment. It has been shown that the typical value of h under natural convection conditions is 7.17 W•m-2•K-1 [24]. ε represents the battery surface emissivity (ε=0.8[24]) and σ represents the Boltzmann constant with the set value of 1.38×10-23 J/K. Finally, Tamb represents the ambient temperature and T represents the surface temperature of the battery. Due to the layered structure of the battery, the thermal conductivity varies with the direction. Along the out-of-plane direction, the battery can be considered as a series connection between different components and along the in-plane direction it can be considered as a parallel connection of each component, which is similar to the series or parallel connections of resistors. The thermal conductivity in the three direction of x,y and z are reported in Eq.14 and Eq.15:

k x  kz 

k p Lx p  k n Lx n  ks Lx s

ky 

Ly

Lx p kp



Ly Lx n kn

(14)



Lx s ks

(15)

The average specific heat capacity of the Li-ion battery can be calculated by Eq. 16:

ρccell 

 ρ cV V

i i i i

The parameters involved in all formulas above are summarized in Table 3. Table 3. Thermal properties of the battery components

(16)

Heat conducti Materials

Density -3

(Kg·m )

Specific heat -1

-1

(J·Kg ·K )

vity coefficie

Electrical conductivity (S·m-1)

nt (J·m-1·K1

)

Separator Positive electrode Negative electrode Aluminum foil

492.00

1978.16

0.334

-

1500.00

1260.20

1.48

0.5

2660.00

1437.40

1.04

100

2702.00

903.00

238.00

-0.0325T3+37.07T2-15000T+2.408× 106 (s/cm) 3

Copper foil

-0.04889T +54.65T2-218.00T+3.52×

8933.00

385.00

398.00

1636.000

1376.947

0.427

-

Positive tab

2702.000

903.000

238.000

3.774e7

Negative tab

8933.000

385.000

398.000

5.998e7

Electrolyte

1290.000

133.900

0.450

Seen Table 2

106 (s/cm)

Aluminum laminate film

3. Results and discussion In general, the contribution from reversible and irreversible heat generation to the total internal heat production will change with the charging and discharging rate. Generally, reversible heat dominates in the low discharge rate, while irreversible heat dominates at high discharge rate. The evolution laws of reversible heat generation have been studied in a large amount in the literature, while few works were dedicated to the irreversible heat generation. However, with the expansion of the applications of electronic devices, the problem of heat management in the discharge process at high-rate becomes more and more significant and it is very important to investigate the laws describing irreversible heat generation. Among them, the polarization heat generation is the main contribution to the irreversible heat generation induced by electrochemical reactions. While the ohmic heating is generated during the transportation process of electrons passing through the active material and ions in the electrolyte, whose value is related to the current and to the active material and the electrolyte properties. This paper will also make a detailed discussion on these two contributions to the ohmic heat generation.

3.1 Model validation Before discussing the simulation of Li-ion batteries, the validity of the established model should be verified. Experiments were carried out at different discharge rates using an electrochemical workstation (model ht-5c200d200-4). Experimental electrochemical curves were compared with the actual discharge curves, as shown in Figure 3. In the discharge process that the experimental curve resulted in a higher fitting degree with the simulation curve especially at discharge rates of 1C and 3C. However, there is still a small degree of discrepancy occurring at the early and late periods of the discharge process, appearing most evident for 5C. An explanation may come from two aspects: (1) most of the parameters come from literatures [25] which differ from the aspects of this experimental device and (2) with higher C-rate discharge processes, the internal equilibrium assumption may not exist specifically when the processes are in the latter or very early periods [2]. In order to further validate the thermal behaviors at different discharge rates, the simulation results are compared with the infrared imagery at different discharge rates (1C, 3C, 5C) at 25℃ (shown as Figure 4), because this battery is designed for high power density, its typical discharge rates are less than 5C. Figure 3 and 4 show that simulation results are in good agreement with experimental results; this illustrates that the proposed coupled model can effectively simulate the thermal response behavior of the battery.

Figure 3 Validation between simulation and experimental data

Figure 4 The contrast of contours between infrared imagery and simulated temperature distribution in natural cooling condition (a) 1 C discharge; (b) 3 C discharge; (c) 5 C discharge.

3.2 Internal irreversible heat generation analysis at 3 C rate The analysis of the irreversible heat generation in the discharge process of the battery will be carried out in the following text. Figure 5 shows the total irreversible heat generation rate of positive and negative electrodes in the battery at the rate of 3 C and for the five time intervals of 10, 300, 600, 900 and 1185 s. Calculations of the heat generation rate follow Eqs 6-12. It can be seen from Figure 5 that the irreversible heat generation at positive and negative electrodes were

different and exhibited certain regularity. When advancing in the discharge process, the irreversible heat generation of the negative electrode rapidly increases; this is especially obvious in the range between 900 and 1185 s. This is mainly due to the too large lithium consumption at the negative electrode at the end of the discharge process, resulting in a “low lithium state” and in the increasing polarization within the negative electrode. This phenomenon can be confirmed by the heat generation curve of the polarization process shown in Figure 6. It also can be seen from Figure 5 that the irreversible heat generation evolution at the positive electrode was relatively small; the irreversible heat generation variation at the negative electrode was larger, and the heat generation rate near the separator was higher than that at the collector.

Figure 5 The total irreversible heat generation of the positive electrode and negative electrode at the discharge rate of 3C

Figure 6 Total polarization heat generation of the positive electrode and negative electrode at the

discharge rate of 3C

Figure 7 Total ohmic heat generation at the positive electrode and negative electrode at the discharge rate of 3C.

Figure 7 shows the variation of the total internal ohmic heat generation in the battery. It can be seen that the ohmic heat generation from the collector to the diaphragm shows a rapid increase and that in the whole discharge process, the ohmic heat generation at the negative electrode is relatively stable, while at the positive electrode it gradually increases during the discharge. However, the difference between the heat generation rate at the positive and the negative electrodes at the end of the discharge and near the separator is only of 55 kW/m3.

Figure 8 The ohmic heat generation of active materials of the positive electrode and negative electrode at the discharge rate of 3C

Ohmic heat is mainly composed of two parts, one being the heat generated by the resistance of electrodes active materials, as shown in Figure 8. Due to the low value of the heat generation rate, a logarithmic scale was used on the time axis. As shown in Figure 8, the ohmic heat

generated at the negative electrode is much lower than that at the positive electrode. This is mainly due to the self-characteristics of the electrode materials, namely the thermal resistance rate of graphite (negative electrode material) is far lower than that of the positive electrode material. According to the calculation, the average ohmic heat generation rate at the negative electrode is 13.1, 13.3 and 13.7 W/m3 at the time of 600, 900 and 1185 s, respectively, while it is 3.71, 3.84 and 4.06 kW/m3, respectively, at positive electrode. In terms of magnitude, the ohmic heat generation rate at the negative electrode can thus be ignored. In addition, the ohmic heat generation rate near the collector is slightly higher than that near the separator.

Figure 9 Ohmic heat generation of the electrolyte at the positive and negative electrodes at the discharge rate of 3C

The second important factor contributing to the ohmic heat generation is the ohmic heat generation due to the electrolyte. Figure 9 shows the ohmic heat generation at the interface between the positive and negative electrodes and the electrolyte, the trend is basically similar to the one of ohmic heat. Results from the calculation of the battery internal irreversible heat generation show that at the end of the 3 C discharge process (data extracted from Figure 10), the ohmic heat generation in the electrolyte accounted for 70.1% of the total ohmic heat generation, while it accounted for 68.9% at the 8 C rate. Compared with the ohmic heat generation due to active materials, ohmic heat generation at the electrolyte results in the dominant position (see data in Figure 11). 3.3 Effect of the discharge rate on the irreversible heat generation of the battery The discharge current has a significant influence on the irreversible heat generation of the battery. Generally, with the increasing of the discharge rate, the internal battery heat generation

rapidly increases, and the irreversible heat generation becomes more sensitive to the discharge rate. The heat generation within battery at 3 and 8 C rates was studied as follows. Figure 10a shows the average variation of the internal heat generation rate during the discharge at 3C. It can be seen that the total irreversible heat generation displayed a relatively stable state in the up to1000 s, and then the heat generation rate increased rapidly from 1000 to 1185 s. This is because at the end of the discharge process, the polarization within the internal battery increases and thus the polarization heat generation increased accordingly. In order to describe proportions between various parts of the internal heat generation more clearly, average heat generation rates of different types of heat in the whole discharge process were calculated. The total irreversible heat generation rate inside the battery was 36015.0 W/m3; among this, the polarization heat generation rate was 27231.9 W/m3, the heat generation rate at positive and negative electrode active materials was 2644.5W/m3 and the heat generation rate due to the electrolyte was 6138.7 W/m3.The calculated proportions are shown in Figure 10b; the polarization heat generation result to account for 75.6% of the total irreversible heat generation. Electrolytic heat results the dominant factor within the ohmic heating, with a proportion of about 17.1%.

Figure 10 Average variation of the internal heat generation rate during the discharge at 3C

Figure 11 Average variation of the internal heat generation rate during the discharge at 3C Figure 11a shows the evolution of the composition of irreversible heat generation within the internal battery under 8 C discharge rate. Compared with Figure 10a, it can be seen that at the 8 C rate the heat generation of each factor is higher than the one at 3 C. Moreover, average heat generation curves in two charts show a similar trend. The average heat generation rate of the

battery during the whole discharge process was calculated; the total average irreversible heat generation at a rate of 8 C during the discharge process was 187633.8 W/m3. Among it, the average polarization heat generation rate was 126133.1 W/m3, the heat generation rate of positive and negative electrode active material was 19128.89 W/m3 and the heat generation rate at the electrolyte was 42371.82 W/m3. The calculated proportion table is shown in Figure 11b; the polarization heat generation results to account for 67.2% of the total irreversible heat generation and the electrolytic heat generation was the dominant factor within ohmic heat generation with the value of about 22.6%. The comparison between proportions shows that with the increase of the discharge rate, the proportion of the polarization heat generation decreased while it increased for the ohmic heat generation. The proportion of the electrolyte heat generation within the ohmic heat generation increased by 5.5%. 3.4 Effect of the particle size on the irreversible heat generation of the battery Except external factors such as the discharge rate and material parameters, the battery design also has an important influence on heat generation. The effect of the particle size at positive and negative electrode materials on the irreversible heat generation of the battery were studied as shown in following figures.

Figure 12 Average internal irreversible heat generation at the particle size of 0.5rp、rp、2rp

As can be seen from Figure 12, the total internal irreversible heat generation increased with the increasing of the positive electrode particle size. By calculating the average heat generation of the whole discharge process, it shows that when the particle diameter is increased to 2rp, the total internal heat generation rate of the battery increases by 11.1%, the polarization of heat

generation increases by 14.7%. In this condition, the polarization heat generation accounted for 78.1% of the total heat generation. When the particle size is reduced to 0.5 rp, the total internal heat generation rate of the battery decreases by 4.3%, the polarization of heat generation decreases by 6.1% and the polarization heat generation accounted for 74.2% of the total heat generation. Overall, the proportion of the polarization heat generation increased gradually with the increasing of the particles size at the positive electrode.

Figure 13 Average internal irreversible heat generation at the particle size of 0.5rn、rn、2rn

Figure 13 shows that with the increasing of the particle size at the negative electrode, the total irreversible heat generation inside the battery is increased. The calculation of the average heat generation rate of the whole discharge process showed that when the particle diameter increases to 2 rn, the total internal heat generation rate of the battery increases by 26.4%, the polarization of heat generation rate increases by 35.6% and the polarization heat occupied 80.8% of the total heat. When the particle size is reduced to 0.5 rn, the total internal heat generation rate of the battery decreases by 31.5%, the polarization of heat generation rate decreases by 31.9% and the polarization heat generation occupies 67.4% of the total heat generation. In the integrated point of view, both the proportion of total irreversible heat generation rate and the polarization heat generation increased rapidly when increasing the particle size at the negative electrode. Moreover, it can be seen from the figure above that when the particle size at negative electrode is rn, the total heat generation rate is close to the polarization heat generation rate obtained for a particle size of 0.5 rn. 4. Conclusions

In this paper, a three-dimensional thermal model coupled with quasi-two-dimensional electrochemical model was established to model the internal generation and evolution of irreversible heat within a Li-ion battery at different discharge rates considering dynamic parameters changes and double layer effects. Results have shown that with the increasing of the charge/discharge rate, irreversible heat generation rate increases rapidly within the internal battery. Polarization heat generation results to occupy the dominant position in irreversible heat generation. The ohmic heat is found to be mainly generated within the electrolyte, and the heat generated from the negative active material is so small that it can be ignored. Through the comparison of proportions at different discharge rates, the proportion of polarization heat generation to the total irreversible heat generation decreased from 75.6% at a rate of 3 C to 67.2% at 8 C, while the proportion of ohmic heat generation to the total heat generation increased from 24.2% at 3C to 32.8% at 8C. In addition, the particle size of positive and negative electrode active materials also was fond to play a very important influence on the irreversible heat generation of battery. At the 3 C constant current discharge, the proportion occupied by the irreversible heat generation and the polarized heat generation is decreased by 4.3% and 6.1% when the particle radius is lowered to 0.5 rn, respectively. The proportion of the irreversible heat generation and the polarized heat generation result 11.1% and 14.7% when the particle radius is 2rn, respectively. Proportions of irreversible and polarization heat generation resulted to decrease by 31.5% and 31.9% at 0.5 rn, respectively. Furthermore, the values changed to 26.4% and 35.6% ar 2 rn, respectively. The negative electrode particle size resulted in a more significant influence on the irreversible heat generation and polarization heat generation in the battery. In summary, with the development of electronic devices, the internal heat generation of the batteries at high discharges rate should be solved urgently. In this paper, the evolution law of the irreversible heat generation in lithium iron phosphate battery and its constituent parts for different rates and particle sizes were analyzed in detail. The temperature distribution and thermal management caused by the heat generation of the battery will be described in another paper due to the length limitation.

Acknowledgment This

work

is

supported

by

Hunan

Provincial

Innovation

Foundation

For

Postgraduate(No.CX2015B043) and the National Natural Science Foundation of China (No. 51204211 and No. 51222403), which are greatly appreciated. Nomenclature symbols

description

Acell

area of the positive electrode (both sides) (m2)

c1,i

lithium in active material (mol m-3)

C1,max,i

maximum concentration (mol m-3)

C1,surf,i

Li+ concentration on the surface of active material particles (mol m-3)

Cp,i

heat capacity (J (kg K)-1)

D1,i

solid phase diffusivity (m2 s-1)

D10,i

reference solid phase diffusivity (m2 s-1)

Ea,D,i

diffusion active energy (kJ mol-1)

Ea,k,i I

reaction active energy (kJ mol-1) heat transfer coefficient (W m-2 K-1) current (A)

Iapp

cell current density related to Acell (A m-2)

j0,i

exchange current density (A m-2)

jloc,i

local current density (A m-2)

k0,i

reaction rate constant (m2.5 mol-0.5 s-1)

ki Li

thermal conductivity (W (m K)-1) thickness of component (m)

Qact

active heat generation (J m-3)

Qohm

ohmic heat generation (J m-3)

Qrea

reaction heat generation (J m-3) radius distance variable of particle (m) characteristic radius of electrode particles (m)

h

r Ri Sa,i SOC0,I t t+ T Tamb Ui v x

specific surface area (m-1) initial state of charge time (s) Li+ transference number absolute temperature (K) ambient temperature (K) open circuit voltage (V) thermodynamic factor relating to electrolyte activity distance from half negative foil along negative-positive direction (m)

Greek letters αa,i transfer coefficient for anodic current αc,i transfer coefficient for cathodic current ε1,i active material volume fraction

ε2,i φi γi

volume fraction electric potential (V) Bruggeman exponent

k

ionic or electronic conductivity (S m-1)

ρi

density (kg m-3)

σi

solid phase conductivity (S m-1)

Subscripts and superscripts 0 initial or equilibrated state 1 solid phase 2 liquid phase amb ambient temperature n negative electrode p positive electrode irr irreversible re reversible s separator

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Highlights: 1) A thermo-electrochemical model is established considering dynamic parameters. 2) The coupling model is validated through discharge curves and infrared imagery. 3) Every part of irreversible heat generations and evolutions are analyzed. 4) It provides an effective method when calculating the irreversible heat generation.