Composite Thermal Control Systems with Phase Change Material in Metal Foam for Lithium Batteries Cooling

The physical domain consists of a zone where, based on various simulations, air has been introduced, copper foam at 40 PPI with ε=0.934, the RT42 PCM melting at about 42°C [11] and the copper foam filled by the PCM.

The operating temperature of the lithium batteries is expected to remain around 318 K [12].

Figure 1 shows the 2D domain used in this study where the dimensions, obtained from a commercial electric battery, are L=310mm, H=40 mm.

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Figure 1. Physical Domain

The properties of the materials and the copper foam have been reported in Table 1 and Table 2.

The enthalpy-porosity method [14] is used to describe the melting phenomenon of PCM, where the solid-liquid interface is not drawn explicitly but the method defines a mixed solid-liquid region that represents the "pseudo" porous zone that is the liquid fraction. The liquid fraction β has a value of 0 when the zone is totally solid, 1 when it is totally liquid, and between 0 and 1 for the mixed region.

Table 1. Properties of materials

Table 2. Properties of copper foams [13]

$\left\{\begin{array}{l}\beta=0 \quad \text { for } T<T_{\text {solidus }} \\ \beta=\frac{T-T_{\text {solidus }}}{T_{\text {liquidus }}-T_{\text {solidus }}} \quad \text { for } T_{\text {solidus }}<T<T_{\text {liquidus }} \\ \beta=1 \quad \text { for } \quad T>T_{\text {liquidus }}\end{array}\right.$      (1)

In (1) T is the local temperature of the cell, T_liquidus is the upper temperature at which the domain is totally liquid, and T_solidus is the temperature below which it is totally solid. The solid part of the mixed zone is modeled by adding a source term in the momentum equation. The metal foam is modeled with the Darcy-Forchheimer extended Brinkman model because it behaves as a porous medium; The gravitational acceleration is along the y-axis and the Boussinesq approximation is considered to account for the buoyancy force due to natural convection. The thermal interaction between the metal foam and the flows is modeled with the LTE assumption, where a unique local temperature function is defined between the porous medium and the fluid flow. The reference equations are as follows:

-PCM

$\frac{\partial u_{p c m}}{\partial x}+\frac{\partial v_{p c m}}{\partial y}=0$     (2)

$\frac{\rho_{p c m}}{\varepsilon}\left(\frac{\partial u_{p c m}}{\partial t}+\frac{u_{p c m}}{\varepsilon} \frac{\partial u_{p c m}}{\partial x}+\frac{v_{p c m}}{\varepsilon} \frac{\partial u_{p c m}}{\partial y}\right)$$=-\frac{\partial p_{p c m}}{\partial x}+\frac{\mu_{p c m}}{\varepsilon}\left(\frac{\partial^{2} u_{p c m}}{\partial x^{2}}+\frac{\partial^{2} u_{p c m}}{\partial y^{2}}\right)+S_{x}$      (3)

$\frac{\rho_{p c m}}{\varepsilon}\left(\frac{\partial v_{p c m}}{\partial t}+\frac{u_{p c m}}{\varepsilon} \frac{\partial v_{p c m}}{\partial x}+\frac{v_{p c m}}{\varepsilon} \frac{\partial v_{p c m}}{\partial y}\right)$$=-\frac{\partial p_{p c m}}{\partial y}+\frac{\mu_{p c m}}{\varepsilon}\left(\frac{\partial^{2} v_{p c m}}{\partial x^{2}}+\frac{\partial^{2} v_{p c m}}{\partial y^{2}}\right)+S_{y}$      (4)

$\left[(1-\varepsilon)(\rho c)_{m f}+\varepsilon(\rho c)_{p c m}\right]$$\left(\frac{\partial T_{p c m}}{\partial t}+u_{p c m} \frac{\partial T_{p c m}}{\partial x}+v_{p c m} \frac{\partial T_{p c m}}{\partial y}\right)=$$=k_{e f f \mid p c m}\left(\frac{\partial^{2} T_{p c m}}{\partial x^{2}}+\frac{\partial^{2} T_{p c m}}{\partial y^{2}}\right)-\varepsilon \rho_{p c m} H_{L} \frac{\partial \beta}{\partial t}$      (5)

-Air

$\frac{\partial u_{\text {air }}}{\partial x}+\frac{\partial v_{\text {air }}}{\partial y}=0$      (6)

$\frac{\rho_{a i r}}{\varepsilon}\left(\frac{\partial u_{a i r}}{\partial t}+\frac{u_{a i r}}{\varepsilon} \frac{\partial u_{a i r}}{\partial x}+\frac{v_{a i r}}{\varepsilon} \frac{\partial u_{a i r}}{\partial y}\right)=$$-\frac{\partial p_{a i r}}{\partial x}+\frac{\mu_{a i r}}{\varepsilon}\left(\frac{\partial^{2} u_{a i r}}{\partial x^{2}}+\frac{\partial^{2} u_{a i r}}{\partial y^{2}}\right)$     (7)

$\frac{\rho_{\text {air }}}{\varepsilon}\left(\frac{\partial v_{\text {air }}}{\partial t}+\frac{u_{\text {air }}}{\varepsilon} \frac{\partial v_{\text {air }}}{\partial x}+\frac{v_{\text {air }}}{\varepsilon} \frac{\partial v_{\text {air }}}{\partial y}\right)=$$-\frac{\partial p_{\text {air }}}{\partial y}+\frac{\mu_{\text {air }}}{\varepsilon}\left(\frac{\partial^{2} v_{\text {air }}}{\partial x^{2}}+\frac{\partial^{2} v_{\text {air }}}{\partial y^{2}}\right)+S_{\text {air }}$      (8)

$\left[(1-\varepsilon)(\rho c)_{m f}+\varepsilon(\rho c)_{a i r}\right]$$\left(\frac{\partial T_{a i r}}{\partial t}+u_{a i r} \frac{\partial T_{a i r}}{\partial x}+v_{a i r} \frac{\partial T_{a i r}}{\partial y}\right)=$$=k_{\text {eff } \mid a i r}\left(\frac{\partial^{2} T_{\text {air }}}{\partial x^{2}}+\frac{\partial^{2} T_{\text {air }}}{\partial y^{2}}\right)$     (9)

The heat generation rate simulates a typical battery dissipated power of 5.3 KW.

In the equations, ρ and c are the density and specific heat, respectively; u and v are the velocity in the x and y directions, respectively; ε is the porosity of the metal foam; p is the relative pressure; μ is the dynamic viscosity of the PCM or air; S is the source term; V is the velocity vector of the PCM in the liquid or air phase; and et is the time. The subscripts Lit, pcm, and air referred to the battery, phase change material, and air, respectively. Regarding the local temperature, the LTE approach is used and thus the temperature foam in the PCM domain is equal to the PCM temperature and the same assumption is set in the air domain. 

The source terms in the PCM momentum equations are:

$S_{x}=\left(\begin{array}{l}\frac{(1-\beta)^{2}}{\left(\beta^{3}+0.001\right)^{3}} A_{\text {mush }} \\ +\frac{\mu_{p c m}}{K}+\frac{C_{F}}{\sqrt{K}} \rho_{p c m}\left|\vec{V}_{p c m}\right|\end{array}\right) v_{p c m}$      (10)

$S_{y}=\left(\frac{(1-\beta)^{2}}{\left(\beta^{3}+0.001\right)^{3}} A_{m u s h}+\frac{\mu_{p c m}}{K}+\frac{C_{F}}{\sqrt{K}} \rho_{p c m}\left|\vec{V}_{p c m}\right|\right)$

$v_{p c m}+\rho_{p c m} g \gamma_{p c m}\left(T_{p c m}-T_{0}\right)$      (11)

The first term is the Kozeny - Carman term that models the presence of the solid part of the PCM during melting. A_mush is the mushy zone constant that takes into account the velocity damping during solidification. Its value is fixed at 105 kg/ (m³s). The second term is the Darcy term where K is the permeability of the porous medium and the third term is the Forchheimer term, where C_F is the inertial strength factor.

The details of equation 12 can be found in [15].

Permeability and resistance coefficients are calculated from:

$\left\{\begin{array}{l}K=0.00073(1-\varepsilon)^{-0.224}\left(\frac{d_{f}}{d_{p}}\right)^{-1.11} d_{p}^{2} \\ C_{F}=0.00212(1-\varepsilon)^{-0.132}\left(\frac{d_{f}}{d_{p}}\right)^{-1.63} \\ \frac{d_{f}}{d_{p}}=1.18 \sqrt{\frac{1-\varepsilon}{3 \pi}}\left(\frac{1}{1-e^{1(1-\varepsilon) / 0.04}}\right) \\ d_{p}=\frac{0.0224}{\omega}\end{array}\right.$       (12)

The last term is the Boussinesq approximation where g is the acceleration modulus of gravity, γ_pcm is the thermal expansion coefficient of the PCM, and T0 is the operating temperature, set to 310K. k_eff is the effective thermal conductivity calculated by [16]:

$\left\{\begin{array}{l}K=0.00073(1-\varepsilon)^{-0.224}\left(\frac{d_{f}}{d_{p}}\right)^{-1.11} d_{p}^{2} \\ C_{F}=0.00212(1-\varepsilon)^{-0.132}\left(\frac{d_{f}}{d_{p}}\right)^{-1.63} \\ \frac{d_{f}}{d_{p}}=1.18 \sqrt{\frac{1-\varepsilon}{3 \pi}}\left(\frac{1}{1-e^{1(1-\varepsilon) / 0.04}}\right) \\ d_{p}=\frac{0.0224}{\omega}\end{array}\right.$      (13)

$\left\{\begin{array}{c}\boldsymbol{k}_{\text {eff } \mid \boldsymbol{p c m}}=(\mathbf{1}-\boldsymbol{\varepsilon}) \boldsymbol{k}_{m f}+\boldsymbol{\varepsilon} \boldsymbol{k}_{p c m} \\ \boldsymbol{k}_{\text {eff } \mid \text { air }}=(\mathbf{1}-\boldsymbol{\varepsilon}) \boldsymbol{k}_{m f}+\varepsilon \boldsymbol{k}_{\text {air }}\end{array}\right.$      (14)

In the following research work, simulations were performed to understand the thermal performance of various materials used for thermal control of a modern electric vehicle powered by a lithium-ion battery. Initially, the thermal performance obtained using only air was analyzed, then the analysis of only copper foam 40 PPI (Pores per Inch) with a porosity ε = 0.934, then the analysis of only PCM where a paraffinic phase change material from Rubitherm was used, and finally analysis was conducted on the copper foam filled with a paraffin PCM.

These materials were analyzed both individually and compared to each other to understand the differences. These analyses were conducted with a heat transfer coefficient of h=0, 20, 200 (W/m²K) and they have covered the liquid fractions for only PCM and copper foam filled PCM (Figure 2), the average temperatures of both layers Top Domain and Bottom Domain (Figure 3), average temperatures of the entire domain (Figure 4), the temperature profiles along the bottom layer of the domain (Bottom), and the temperature ranges of only air, only copper foam, and PCM-filled copper foam were obtained for h=20, 200 (W/m²K).

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Figure 2. Liquid fraction of domain for only PCM and 40 PPI copper foam filled by PCM h=0, 20, 200 [W/m²K]

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(a)

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(b)

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(c)

Figure 3. Average temperatures of Bottom and Top layers for only air, 40 PPI copper foam, only PCM, and 40 PPI copper foam filled by PCM (a) h=0 [W/m²K], (b) h=20 [W/m²K], h=100 [W/m²K]

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(a)

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(b)

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(c)

Figure 4. Average temperatures of domain for only air, 40 PPI copper foam, only PCM, and 40 PPI copper foam filled by PCM (a) h=0 [W/m²K], (b) h=20 [W/m²K], h=100 [W/m²K]

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Figure 5. Temperature profiles of bottom domain for only air, only 40 PPI copper foam, only PCM, and 40 PPI copper foam filled with PCM

In fact, while with h=0 (W/m²K) the average temperatures of the Top and Bottom layers are close, despite having the highest average temperatures, as the heat transfer coefficient h=20, 200 (W/m²K) increases, the gap between the average temperatures of the two zones increases greatly due to the triggering of convection. The most pronounced differences between the average temperatures of the Top and Bottom domains were found with the use of only PCM, in fact while the average temperature measured on the Bottom layer of the domain, which receives the heat flow, turns out to be at extremely high temperatures and therefore dangerous for the thermal safety parameters of lithium batteries, it completely liquefies, the areas above the Bottom layer up to the Top part of the domain turn out to be at extremely lower average temperatures, not triggering the transition between solid and liquid. The average temperatures obtained using only 40 PPI copper foam ε = 0.934 for the Top and Bottom layers, while less pronounced than previous cases, find extremely high average temperatures.  The best performance regarding average temperatures was obtained with the simulations with PCM-filled copper foams, which in addition to having lower average temperatures than the other cases analyzed, keep the two Top and Bottom layers at closer average temperatures. This happens because as the battery temperature approaches the critical temperature, the PCM begins to melt and thus the temperature tends to be maintained at levels more in tune with the battery safety parameters.As with the simulations conducted with only air, different trends were found depending on the heat transfer coefficient used.

To better understand the thermal effects occurring at the leading edge between the battery and the domain we did, analyses were performed on the temperature profiles for the bottom layer of the domain (Figure 5) for h= 20, 200 (W/m²K), using air only, 40 PPI copper foam only, PCM only, and paraffin-filled copper foam PCM. Again, the lowest temperatures found were achieved through the combined use of PCM and copper foam.

This research was partially funded by MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca), grant number PRIN-2017F7KZWS and by Università degli Studi della Campania “Luigi Vanvitelli” with the grant number D.R. n. 138 under NanoTES project - V:ALERE program 2020.