Journal of Materials Science

, Volume 53, Issue 15, pp 10987–11001 | Cite as

A coupled electrochemical–thermal–mechanical model for spiral-wound Li-ion batteries

  • Xiting Duan
  • Wenjuan JiangEmail author
  • Youlan Zou
  • Weixin Lei
  • Zengsheng MaEmail author


In order to clarify the interaction of electrochemistry, thermal and diffusion-induced stress, in this work, we present a coupled electrochemical–thermal–mechanical model for spiral-wound Li batteries by coupling the mass, charge, energy and mechanics conservations as well as the electrochemical kinetics. A series of temperatures and Li concentration parameters on the reaction rate and Li+ transport are employed in this model. The results show that this model is validated for both the electrochemical performances and thermal behaviors at a constant discharge current by finite element simulation. Furthermore, the heat generation of three thermal sources and stress analysis are also discussed. This work is helpful to the battery structural design and battery thermal management.

List of symbols

\( c_{2} \)

Li-ion concentration in the electrolyte (mol m−3)

\( c_{1} \)

Concentration of Li in the active material (mol m−3)

\( c_{1}^{0} \)

Initial concentration of Li in the active material (mol m−3)

\( c_{1}^{\text{surf}} \)

Li-ion concentration on the surface of the active particle

\( c_{1}^{ \hbox{max} } \)

Maximum concentration of Li in the active material (mol m−3)

\( c_{2}^{0} \)

Initial electrolyte concentration (mol m−3)

\( \Delta c \)

Li concentration change (mol m−3)


Normalized concentration


Heat capacity [J (kg K)−1]


Diffusion coefficient of lithium in the active material (m2 s−1)


Diffusion coefficient of electrolyte (m2 s−1)

\( E_{\text{cell}} \)

Working voltage of the battery (V)


Diffusion activation energy (J mol−1)


Reaction activation energy (J mol−1)


Young’s modulus (GPa)


Faraday’s constant (96487/C mol−1)


Convective heat transfer coefficient [W (m2 K)−1]


Applied current density (A m−2)

\( i_{1} \)

Solid-phase current density (A m−2)

\( i_{2} \)

Current density in the electrolyte (A m−2)


Exchange current density (A m−2)


Charge transfer current density at the interface (A m−2)

\( J_{2} \)

Molar flux of Li ions (mol m−2 s−1)


Thermal conductivity [W (m2 K)−1]

\( k_{0} \)

Reaction rate of active material

\( Q \)

Heat generation rate per unit volume (J m−3)

\( Q_{\text{act}} \)

Active heat generation (J m−3)

\( Q_{\text{ohm}} \)

Ohmic heat generation (J m−3)

\( R_{0} \)

Radius of electrode particles (m)


Radial coordinate inside a spherical particle (m)


Gas constant [8.314/J (K mol)−1]

\( \Delta S \)

Entropy change (J mol−1 K−1)


Specific surface area of the electrode (m−1)


State of charge


Time (s)


Transport number of Li+


Temperature (K)


Ambient temperature (K)

\( \Delta T \)

Temperature variation (K)

\( U_{\text{eq}} \)

Open circuit potential of the electrode (V)

\( U_{{{\text{eq}},{\text{ref}}}} \)

Open circuit potential under the reference temperature (V)


Displacement (m)

Greek letters


Poisson’s ratio

\( \varepsilon_{\text{p}} \)

Volume fraction of polymer phase

\( \varepsilon_{\text{l}} \)

Volume fraction of electrolyte

\( \varepsilon_{\text{f}} \)

Volume fraction of conductive filler additive


The local surface overpotential

\( \alpha_{\text{a}} \)

Anodic transfer coefficient

\( \alpha_{\text{c}} \)

Cathodic transfer coefficient

\( \phi_{2 } \)

Liquid-phase potential (V)

\( \phi_{1 } \)

Solid-phase potential (V)

\( \phi_{2}^{0} \)

Initial liquid-phase potential (V)

\( \phi_{1}^{0} \)

Initial solid-phase potentia (V)

\( \sigma_{1} \)

Solid-phase conductivity of electrodes (S m−1)

\( \sigma_{2} \)

Effective ionic conductivity of electrolyte (S m−1)

\( \sigma_{\text{c}} \)

Effective electrical conductivity of current collectors (S m−1)

\( \rho \)

Density (kg m−3)

\( \sigma_{\text{h}} \)

Hydrostatic stress (GPa)

\( \varOmega \)

Partial molar volume (m3 mol−1)


Strain components


Stress components (GPa)

\( \delta_{ij} \)

Dirac delta function

\( \sigma_{\text{r}} \)

Radial stress (GPa)

\( \sigma_{\theta } \)

Hoop stress (GPa)

\( \alpha \)

Coefficient of thermal expansion (K−1)

\( \xi \)

Emissivity of the outer can material

\( \beta \)

Stefan–Boltzmann constant (5.670400 × 10−8 W m−2 K−4)

Subscripts and superscripts


Initial or equilibrated state


Solid phase


Liquid phase


Ambient temperature


Negative electrode


Positive electrode








Eigen strain due to thermal expansion


Eigen strain due to intercalation





This study was funded by the National Natural Science Foundation of China (Grant Nos. 11702234 and 11602213), Natural Science Foundation of Hunan Province (Grant No. 2017JJ3301) and Key Fund Project of Hunan Provincial Department of Education (Grant No. 17A206).


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Authors and Affiliations

  1. 1.National-Provincial Laboratory of Special Function Thin Film Materials, School of Materials Science and EngineeringXiangtan UniversityXiangtanChina

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