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An Immersed Boundary Method for Conjugate Heat Transfer Involving Melting/Solidification

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Immersed boundary method (IBM) was developed that can solve conjugate heat transfer (CHT) including phase change to simultaneously analyze phase change material (PCM) and forced convection. The CHT was analyzed by determining the effective conductivity to satisfy the continuity of temperature and heat flux at the interface. The PCM was included in the energy equation using the fact that the liquid fraction is proportional to temperature. By applying the method developed in this study, flow, conjugate heat transfer, and melting/solidification can be considered in the unitary domain without the aid of the lattice Boltzmann method (LBM). For code verification, simulations were performed for the one-dimensional solidification of a binary mixture and battery package including the PCM. The IBM simulation results were in good agreement with the results of the analytical solution and finite element method (FEM). A hybrid cooling system with forced convection and PCM was proposed, and a simulation was performed with the code developed in this study. When forced convection is added, the temperature rise can be effectively suppressed but results in temperature non-uniformity.

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Data availability

Data are contained within this article.


C :

Heat capacity ratio (= (ρcp)f/(ρcp)s)

D :

Diameter of battery


Depth of discharge

f i :

Momentum forcing

g k :

Fraction of k phase

H sl :

Heat of fusion

I :

Charge/discharge current

k :

Thermal conductivity

K :

Thermal conductivity ratio (= ks/kf)

m :

Mass source/sink


Open circuit voltage

Q irr :

Irreversible heat generation rate

Q rev :

Reversible heat generation rate

R η :

Internal equivalent resistance


State of charge

ΔS :

Entropy change inside a battery

t cA :

Time for active cooling system (without PCM) to reach critical temperature

t cH :

Time for hybrid cooling system (with PCM) to reach critical temperature

T b,max :

Maximum temperature at battery pack

ΔT b,max :

Maximum temperature deviation across the battery pack

ε L :

Liquidus line

ε s :

Solidus line

η :

Similarity variable

θ :

Dimensionless temperature

ω :

Index function between the solid and the fluid


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This work was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry & Energy (MOTIE) of the Republic of Korea (No. 20212020800070).

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Ahn, J., Song, J.C. & Lee, J.S. An Immersed Boundary Method for Conjugate Heat Transfer Involving Melting/Solidification. Int. J. Aeronaut. Space Sci. 24, 1032–1041 (2023).

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