expanded graphite composite

International Journal of Heat and Mass Transfer 149 (2020) 119199

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/hmt

Thermal management of electronic devices using pin-fin based cascade microencapsulated PCM/expanded graphite composite Qinlong Ren a,∗, Penghua Guo b, Jianjun Zhu a a b

Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China Department of Fluid Machinery and Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China

a r t i c l e

i n f o

Article history: Received 9 July 2019 Revised 22 November 2019 Accepted 9 December 2019 Available online 13 January 2020 Keywords: Electronic thermal management Microencapsulated PCM Pin-fins Expanded graphite GPU-based lattice Boltzmann modeling

a b s t r a c t Microencapsulated phase change material (MEPCM) could be used for effective thermal management of electronic devices due to its large latent heat, phase change at nearly constant temperature, low volume expansion, and anti-leakage characteristics. Unfortunately, the low thermal conductivity of MEPCM hinders the heat dissipation rate from electronic devices especially at high heat flux conditions. Although the heat transfer capability of MEPCM could be accelerated by adding expanded graphite (EG) or inserting high thermal conductivity pin-fins, the latent heat energy storage capacity of MEPCM composite becomes less which reduces the corresponding operating time of limiting the electronic device temperature rise through solid–liquid phase change. In purpose of clarifying and optimizing this tradeoff effect on electronic device thermal management using MEPCM–EG composite with pin-fins, a numerical study is carried out through 3D lattice Boltzmann method with respect to different pin-fin configurations, EG content, PCM melting temperature, and heat flux conditions. The results indicate that the pin-fin array with medium fin number and fin thickness is beneficial for balancing the increased heat transfer capability and the decreased latent heat of MEPCM so that its optimum thermal performance is achieved. At the early working stage of electronic device, the pin-fin based MEPCM is demonstrated to be more effective than MEPCM–EG composite for controlling the electronic device temperature rise because of the direct contact between pin-fins and electronic heat sink base. However, as the electronic device working time evolves, the MEPCM–EG composite with network heat transfer channel is found to be more efficient for dissipating heat out of electronic device due to the relatively high average thermal conductivity throughout the whole heat sink system. Furthermore, the thermal performance of electronic device is found to be improved by inserting MEPCM–EG composite with cascade melting temperature decreasing from the heat sink base to heat sink cover. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction With the rapid progress of manufacture technologies during the past decade, the electronic devices exhibit a developing trend of multi-functionality and miniaturization that simultaneously requires larger electronic power and smaller chip component sizes. Hence, there exists an increasing challenge for preventing the electronic devices from overheating under which situation their working efficiency and lifetime could be seriously deteriorated. Based on this, an effective thermal management approach is essential for guaranteeing the normal working efficiency and safety of different type electronic devices. In general, the thermal management techniques of electronic devices could be classified into two categories

∗

Corresponding author. E-mail address: [email protected] (Q. Ren).

https://doi.org/10.1016/j.ijheatmasstransfer.2019.119199 0017-9310/© 2019 Elsevier Ltd. All rights reserved.

including active cooling and passive cooling. However, the active cooling becomes highly limited for miniaturized electronic devices because it usually requires sufficient space and additional power, and it also has the issues of high noise levels and difficult maintenance. Due to this reason, the reliable passive cooling approaches become indispensable and attractive for ameliorating the thermal performance and life cycle of small electronic devices. Phase change materials (PCMs) have gained tremendously increasing attention for managing the thermal issues of electronic components and portable batteries in recent years due to its large latent heat and nearly constant solid–liquid phase change temperature [1]. Unfortunately, the low thermal conductivity of PCMs hinders the heat dissipation rate from electronic devices which brings about a significant problem for designing highly efficient PCMbased electronic device cooling systems. To overcome this undesired characteristic, the thermal conductivity of PCM-based cooling

2

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199

Nomenclature cp Fo fl H hc k n q Ste T t x y z

specific heat Fourier number liquid fraction of MEPCM enthalpy latent heat of MEPCM–EG composite thermal conductivity normal direction of surfaces input heat flux Stefan number temperature time horizontal Cartesian coordinate Cartesian coordinate perpendicular to x and z vertical Cartesian coordinate

Greek symbols ρ density  volume fraction of expanded graphite ɛk ratio of thermal conductivity ε (ρ c p ) ratio of multiplication of density and specific heat Subscripts c PCM–EG composite cs PCM–EG composite at solid state cl PCM–EG composite at liquid state CaCO3 calcium carbonate EG expanded graphite f heat sink shell and pin-fins by copper MEPCM microencapsulated phase change material PCM paraffin ref reference value

or energy storage systems could be improved by adding thermal conductivity enhancers (TCE) including extended internal fins [2– 9], carbon and metal foams [10–14], heat pipes [15–18], expanded graphite [19–21], carbon nanotube [22-23], and other nanofillers [24-25]. With the advantages of simple manufacture and low cost, extended internal fins are commonly used as TCE in purpose of overcoming the low thermal conductivity drawback of PCMs in electronic device thermal management. Although the high thermal conductivity internal fins could enhance the heat transfer rate and depth of PCMs, it also decreases the average latent heat of PCM-based cooling system so that the operating time of controlling the temperature rise in electronic devices depending on solid– liquid phase change becomes shorter. Due to this reason, the configurations of internal fins as TCE in PCM-based electronic cooling system need to be carefully designed and optimized. Kalbasi et al. investigated the optimum thermal performance of PCM-based heat sink with vertical fins by choosing the objective function as the longest safe operating time of electronic devices before reaching the critical temperature [2]. The results indicated that the optimum fin number is comprehensively influenced by several parameters such as heat sink height and width, fin spacing and fin thickness, and heat flux conditions. Ping et al. numerically studied the PCM-fin structure performance on cooling the Li-ion battery module in terms of PCM species, PCM thickness, fin thickness, and fin spacing using a 1D-electrochemical and 3D-thermal coupled model [3], and they found that utilizing the PCM-fin structure could achieve minimum temperature rise and most uniform temperature distribution in Li-ion battery in comparison with the air cooling or pure PCM cooling because of the PCM latent heat and the enhanced heat transfer area by using fins. However, the aforementioned plate or crossed fins usually occupy a lot of space in

the PCM-based heat sink. As a consequence, the latent heat of heat sink dramatically decreases which reduces the corresponding operating time for electronic device before its temperature rises up to the critical value. In order to tackle this drawback, more compact pin-fins are widely used instead of plate fins to keep the balance between the increased heat transfer capability and the decreased latent heat energy storage capacity of electronic heat sink. Pakrouh et al. optimized the PCM-based pin-fin heat sink for maximum operating time using Taguchi method [4], and they found that 100 pin-fin heat sink is appropriate for all critical temperatures while the fin-thickness should be varied from 2 mm to 4 mm according to different critical temperatures. It was also demonstrated that number and height of pin-fins are the key factors for determining the thermal performance of heat sink, nevertheless the heat sink base thickness has the minimum effect on controlling its temperature variations. Ashraf et al. experimentally investigated the influence of pin-fin geometries on PCM-based passive electronic cooling [5], and it was found that the pin-fins with circular cross-section exhibit better thermal behavior than those with rectangle crosssection while the inline pin-fins are highly recommended rather than staggered pin-fins. Ali et al. further analyzed the triangular, rectangular, and circular pin-fin heat sinks with respect to various PCMs [6], and their results showed that triangular pin-fins are most effective for managing the electronic device temperature due to its less surface area ratio with more number of pin-fins at a specified PCM volume fraction. Besides, to cope with ultra-high thermal shock, Yang et al. applied the low melting point metal as the PCM to control the temperature of heat sink by comparing the performances of plate fin, crossed fin, and pin-fin [7]. It was demonstrated that the low melting point metals own high cooling capability over conventional PCMs because of its outstanding thermal conductivity as well as its high volumetric latent heat. They also presented that the thermal performance of heat sink improves with the increasing number of fins until it reaches a limit value. From the aforementioned recent researches, there is no doubt that the pin-fin enhanced PCM could be effective for thermal management of electronic devices after careful design of pin-fin configurations and PCM thermophysical properties. Even though the heat transfer area between pin-fins and PCM is increased by extending the pin-fin length, it also increases the conductive thermal resistance inside the pin-fins. Furthermore, the pin-fins are usually directly contacted with the base of heat sink which means that the thermal conductivity of lower part heat sink is higher than that of its upper part. Therefore, the heat transfer rate is limited in the region of PCM without pin-fins. As a comparison, the thermal conductivity of PCMs could be uniformly enhanced by adding expanded graphite (EG) to form a carbon network structure for achieving continuity of heat conduction. Wu et al. used paraffin and expanded graphite as PCM and TCE matrix respectively to prepare the PCM–EG composite [19], and they presented that the thermal conductivity of PCM composite is dramatically enhanced by adding 20% EG with slightly decreased latent heat compared with pure PCM making the PCM–EG composite highly suitable for electronic thermal management. Huang et al. reported electronic thermal management by using Wood’s alloyEG composite [20], and the results indicated that decreasing the Wood’s alloy content and the compacting density could lead to an improvement on thermal conductivity of PCM composite, nevertheless its corresponding form-stable performance is changed. Hence, a reasonable mass fraction of Wood’s alloy and EG is essential for its potential application in electronic thermal management. Xu et al. introduced a paraffin (PA)-EG-graphene composite material in order to improve the thermal performance of electronic devices [21], and their results demonstrated that the thermal conductivity of PA-EG composite could be further ameliorated using graphene. Due to the distinct heat transfer enhancement charac-

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199

teristics of pin-fins and expanded graphite for PCM-based electronic cooling system, it is deserved to compare the thermal performances of pin-fin based PCM electronic cooling and PCM–EG composite based electronic cooling in order to satisfy the thermal requirement of electronic devices under different working conditions. Although the PCM based cooling with heat transfer enhancement technologies is successfully demonstrated as an effective approach for passive electronic thermal management, the volume expansion and shrinkage of PCMs during solid–liquid phase change cause the PCM leakage from heat sink which could seriously damage the component of electronic devices versus chemical reaction. Hence, the anti-leakage property of PCM-based heat sink needs to be highly consolidated before its wide applications in industrial fields. Microencapsulated phase change material (MEPCM) with high resistance to mechanical and thermal stresses is an excellent alternative to traditional PCMs in purpose of tackling the volume variation and leakage issues because of the formed micro shells around the desired PCMs [26-27]. With the existence of high thermal conductivity inorganic shells, the effective thermal conductivity of MEPCM is usually higher than that of pure PCM. Unfortunately, it is still inadequate to satisfy the requirement of heat dissipation rate from modern electronic devices so that the heat transfer enhancement technologies are also necessary for MEPCM-based heat sinks [28–30]. During recent years, the MEPCM with expanded graphite has been successfully prepared for thermal energy storage applications [31]. However, according to the author’s knowledge, thermal management of electronic devices using ME PCM–EG composite is seldom investigated, and its comparison with pin-fin based MEPCM cooling technique is also not reported even though their heat transfer enhancement mechanism is highly different. Besides, the previous PCM based thermal management of electronic devices commonly uses the PCM with a fixed melting temperature. The cascade PCM which is widely studied for thermal energy storage applications also deserves further investigation to figure out whether PCMs with cascade solid–liquid phase change temperature is beneficial for electronic thermal management. Lattice Boltzmann method (LBM) has been developed as an effective approach for simulating complex heat transfer and fluid dynamic problems during recent years [32–34]. For solid–liquid phase change phenomenon, the enthalpy-based LBM is developed and widely used due to its high computational efficiency and excellent stability [35–38]. In addition, the highly parallel nature of LBM makes it suitable for graphic process units (GPU) accelerated modeling which significantly reduces the computational time especially for three dimensional (3D) simulations [39–42]. Based on the above several advantages, the 3D enthalpy-based LBM with GPU acceleration is used in the current work to simulate the solid–liquid phase change of PCM and the conjugate heat transfer between heat sink shell, pin-fins, and PCM–EG composite. In this paper, the cascade ME PCM–EG composite based electronic thermal management with pin-fins is investigated with respect to pin-fin configurations, expanded graphite content, heat flux conditions, and PCM melting temperature in order to achieve its optimum thermal performance. The remainder of the paper is organized as follows. In Section 2, the physical model and mathematical formulation for solid–liquid phase change of PCM–EG composite and conjugate heat transfer between PCM–EG composite, heat sink shell, and pin-fins are presented. In Section 3, the PCM–EG composite based electronic thermal management with pin-fins is discussed and optimized in detail in terms of different parametric conditions. A conclusion is drawn in Section 4. Besides, the details about 3D enthalpy-based LBM for solid–liquid phase change and conjugate heat transfer as well as the CUDA code validation are presented in the Appendix.

3

2. Physical model and mathematical formulation The schematic diagram of PCM–EG composite based thermal management of electronic device heat sink assembly with pin-fins is shown in Fig. 1. The heat sink is composed of base, shell, and pin-fins which are all made by copper. The commercial paraffin with volume fraction of 61.63% is used as the core of MEPCM, and the calcium carbonate (CaCO3 ) with volume fraction 38.37% is utilized as the shell material of MEPCM. The expanded graphite is mixed with MEPCM to form the MEPCM–EG composite, and it is then filled into the heat sink assembly for thermal management depending on the latent heat of paraffin. The dimension of heat sink assembly is 112 mm × 112 mm × 27 mm with the thicknesses of base and shell to be 2 mm. A heater with size of 112 mm × 112 mm at constant input power varied from 4.5 kW/m2 to 7.5 kW/m2 is applied on the bottom surface of heat sink assembly. Due to the rubber insulation and the glass cover as shown in Fig. 1, all the other surfaces of heat sink assembly are assumed to be adiabatic. The thermophysical properties of copper, paraffin, calcium carbonate, and expanded graphite are presented in Table 1. Specifically, the thermal conductivities of PCM–EG composite and their corresponding latent heat with different EG content are referred to the data in reference [31]. Due to its symmetric configuration, a quarter of heat sink assembly is chosen as the domain for numerical simulation in order to save computational time. In Fig. 2, the heat sink assemblies with or without different pin-fins and EG contents are displayed. With the assumption of negligible convective heat transfer inside MEPCM and constant material thermophysical properties, the solid–liquid phase change heat transfer of MEPCM–EG composite is governed by the enthalpy-based energy equation as:

ρc

∂ Hc = kc ∇ 2 Tc ∂t

(1)

The density and specific heat of MEPCM are calculated by interpolating the thermophysical properties of paraffin (61.63%) and calcium carbonate (38.37%) as:

ρMEPCM = 0.6163 ∗ ρPCM ± 0.3837 ∗ ρCaCO3

(2)

(ρ c p )MEPCM = 0.6163 ∗ (ρ c p )PCM ± 0.3837 ∗ (ρ c p )CaCO3

(3)

Similarly, the density and specific heat of MEPCM–EG composite are expressed by:

ρc = (1 − )ρMEPCM ± ρEG

(4)

(ρ c p )c = (1 − )(ρ c p )MEPCM ± (ρ c p )EG

(5)

During the computational procedure for solid–liquid phase change, the liquid fraction fl is updated with respect to the enthalpy of PCM–EG composite at solid or liquid states:



0 Hc ≤ Hcs

fl =

Hc −Hcs Hcl −Hcs

Hcs < Hc < Hcl

(6)

1 Hc ≥ Hcl The temperature of PCM–EG composite is computed by the following equation:

Tc =

⎧ ⎨Tm −

Hcs −Hc cpc

Hc ≤ Hcs Tm Hcs < Hc < Hcl ⎩ T + H−Hcl H ≥ H m c cl cp

(7)

c

The heat transfer in heat sink shell and pin-fins is described by the heat conduction equation given by:

ρf

∂Hf = k f ∇ 2 Tf ∂t

(8)

4

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199 Table 1 The thermophysical properties of heat sink materials. Property

Copper

Paraffin

Calcium Carbonate

Expanded Graphite

MEPCM (0%EG)

MEPCM (5%EG)

MEPCM (10%EG)

ρ [kg/m3 ]

8954 383 400

880 2000 −

2800 856 −

520 710 −

− − 0.814 143.6

− − 2.047 137.4

− − 3.884 132.7

Cp [J/kg · K] k [W/m · K] hc [kJ/kg]

Fig. 1. Schematic diagram of electronic device heat sink assembly.

Fig. 2. A quarter of electronic device heat sink assembly. (a) Filled with pure MEPCM. (b) Filled with pure MEPCM with 100 pin fins of 1 mm2 cross section. (c) Filled with pure MEPCM with 25 pin fins of 9 mm2 cross section. (d) Filled with MEPCM–EG composite with 25 pin fins of 9 mm2 cross section.

The relationship between the enthalpy Hf and the temperature Tf without phase change is defined as:

Tf =

H f − Hre f cp f

(9)

At the interfaces between MEPCM–EG composite, pinfins, and shell of heat sink assembly, the coupled DirichletNeumann condition needs to be satisfied for continuity of heat transfer:

Tc = T f

(10)

−kc ∇ Tc = −k f ∇ T f

(11)

For the bottom base of heat sink assembly, the Neumann condition with heat flux q is applied due to the existence of heater:

−k f

∂ Tf =q ∂y

(12)

The adiabatic boundary conditions are applied for the other heat sink assembly surfaces:

∂ Tf =0 ∂n

(13)

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199

As shown in Fig. 2, a quarter of heat sink assembly is used for modeling so that symmetric conditions are applied on the front and right surfaces respectively:

∂ Tc ∂ Tc = 0 (right surface ), = 0 (front surface ) ∂x ∂y

(14)

The governing equations are solved with the 3D enthalpy-based lattice Boltzmann method through physical units and LBM units conversion by matching the Stefan number Ste, Fourier number Fo, thermal conductivity ratio ɛk , and density and specific heat ratio ε(ρ c p ) given as:

Ste =

c p c qL kc t , Fo = , k f hc (ρ c p )c L2

εk =

kc , kf

ε (ρ c p ) =

( ρ c p )c (ρ c p ) f

(15)

The detailed methodology of solving the above governing equations using LBM is presented in the Appendix with CUDA code validation for solid–liquid phase change and conjugate heat transfer. 3. Results and discussion As mentioned in the introduction section, the anti-leakage and low volume expansion/shrinkage advantages of MEPCM make it more appropriate for electronic thermal management over traditional PCMs. However, there still exists the low thermal conductivity issue for MEPCM. In order to ameliorate the heat transfer capability of MEPCM filled heat sink, pin-fins and expanded graphite are applied and compared for thermal performance optimization because of their intrinsically different heat transfer enhancement characteristics. Due to its direct contact with heat sink base, pin-fins have an obvious enhancement on the heat transfer rate in finned partial region of heat sink while expanded graphite could averagely ameliorate the heat transfer capability for the entire heat sink domain. To achieve an optimum heat transfer enhancement design for electronic heat sink assembly, the influence of pin-fin configurations on thermal performance of heat sink assembly with MEPCM under different melting temperature is firstly clarified. Then, the contribution of expanded graphite on accelerating the heat dissipation rate from electronic heat sink assembly is discussed. Finally, the MEPCM–EG composite with cascade melting temperature is studied in order to further improve the thermal management performance of electronic devices. Besides, the maximum temperature inside a side view or the whole heat sink assembly including the region of MEPCM–EG composite, copper pinfins, and copper heat sink shell is used to represent the thermal performance of electronic heat sink because the maximum temperature should be controlled under the critical temperature in order to guarantee the normal working of electronic devices. 3.1. The effects of pin-fin configuration and MEPCM melting temperature The pin-fin configuration plays an essential role on its heat transfer enhancement performance for electronic thermal management. When the length of pin-fins is prolonged, the heat transfer depth and the heat transfer area between pin-fins and MEPCM are increased which are beneficial for ameliorating the heat dissipation rate from electronic heat sink. However, the conduction heat transfer resistance is simultaneously increased with the increment of pin-fin lengths. On the other hand, using pin-fins with a larger cross-section area could reduce their conduction heat transfer resistance to some extent. Unfortunately, the increment of pin-fin cross-section area causes a decreasing volume fraction of MEPCM in heat sink assembly under which circumstance the latent heat available for electronic thermal management is reduced.

5

Confronting the above tradeoff effects, the influence of pin-fin configurations on electronic thermal management with MEPCM is discussed and optimized in this section in terms of considering the critical temperature of electronic devices. To clearly demonstrate the effectiveness of pin-fins for electronic thermal management, the thermal performance of MEPCM filled heat sink assembly without pin-fins (Fig. 2(a)) is investigated as a reference for comparing with MEPCM filled heat sink using 100 pin-fins of 1 mm2 cross section (Fig. 2(b)) and that using 25 pin-fins of 9 mm2 cross section (Fig. 2(c)). In Fig. 3, the transient temperature contours of MEPCM filled heat sink without pin-fins at melting temperature Tm = 45 ◦ C and heat flux of q = 6 kW/m2 are presented from different sideviews. As shown in Fig. 3(a) for the x–z surface at y = 31mm and t = 450 s, the temperature inside the heat sink base is nearly uniform because of the high thermal conductivity of copper. Nevertheless, due to the low thermal conductivity of MEPCM, the heat transfer rate from heat sink base to MEPCM is highly hindered, and a large temperature gradient between heat sink surface and MEPCM is formed. As a consequence, the maximum temperature inside this sideview reaches 57.23 ◦ C at t = 450 s. Similarly, as displayed in Fig. 3(b) for the x–y surface at z = 9.5 mm and t = 450 s, it is observed that the temperature of MEPCM is slowly increased and still lower than its melting temperature of Tm = 45 ◦ C due to the high MEPCM thermal resistance even though the maximum temperature inside the heat sink shell is already 54.19 ◦ C. From the above discussion, it is obvious that the low thermal conductivity of MEPCM is indeed a major serious issue which impedes the heat transfer in electronic heat sink and affects the corresponding device working efficiency. As a comparison, the transient temperature contours of MEPCM filled heat sink with 100 pin-fins of 1 mm2 cross section and fin height of L = 15 mm at Tm = 45 ◦ C and heat flux of q = 6 kW/m2 are displayed in Fig. 4. As shown in Fig. 4(a) for the x–z surface at y = 31 mm and t = 450 s, with the existence of copper pin-fins, the heat transfer depth is highly extended so that heat dissipation rate from heat sink base into the MEPCM region is obviously accelerated. Under this circumstance, the maximum temperature inside heat sink base of this sideview is 52.08 ◦ C which is actually 5.15 ◦ C lower than that heat sink without pin-fins. For the x–y surface at z = 9.5 mm and t = 450 s as shown in Fig. 4(b), it is clearly observed that the temperature of 100 copper pin-fins is higher than that of the surrounding MEPCM which actually contributes to the increasing heat transfer rate from heat sink base into deep region of MEPCM. The maximum temperature in this sideview is 50.55 ◦ C which is 3.64 ◦ C below that case without using pin-fins. Besides, to clarify the influences of pin-fin numbers and cross-section areas on heat transfer in electronic heat sink, the transient temperature contours of MEPCM filled heat sink with 25 pin-fins of 9 mm2 cross section and fin height of L = 15 mm at Tm = 45 ◦ C and heat flux of q = 6 kW/m2 are shown in Fig. 5. Compared with the heat sink using 100 pin-fins of 1 mm2 cross section in Fig. 4, the number of pin-fins is reduced to 25 while the cross section area of each fin is increased up to 9 mm2 . As shown in Fig. 5(a) for x–z surface at y = 31 mm and = 450 s, the temperature inside each pin-fin is almost uniform due to its decreased condution thermal resistance with enlarged pin-fin cross section. Based on this, the corresponding maximum temperature inside heat sink base of this sideview is 51.22 ◦ C which is actually a little bit lower than that case with 100 pin-fins of 1 mm2 cross section. Furthermore, the result in Fig. 5(b) indicates that the temperature inside the pin-fins on the x–y surface at z = 9.5 mm and t = 450 s is much higher than MEPCM temperature in comparison with the result in Fig. 4(b) which further demonstrates the enhanced conduction heat transfer inside pin-fins by enlarging their cross sections. The maximum temperature inside heat sink shell is 50.21 ◦ C which is also lower than that in heat sink with 100 pin-fins of 1 mm2 cross section. As

6

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199

Fig. 3. Transient temperature of MEPCM filled heat sink assembly without pin-fins at melting temperature Tm = 45 ◦ C and heat flux q = 6 kW/m2 , (a) Sideview of the x–z surface at y = 31 mm and t = 450 s. (b) Sideview of the x–y surface at z = 9.5 mm and t = 450 s. (c) Sideview of the x–z surface at y = 31 mm and t = 1350 s (d) Sideview of the x–y surface at z = 9.5 mm and t = 1350 s.

the working time of electronic devices evolves, the MEPCM filled in heat sink continues to be melted due to the heat dissipated from the base heater. The temperature contours of MEPCM filled heat sink at t = 1350 s without pin-fins, with 100 pin-fins of 1 mm2 cross section and fin height of L = 15 mm, and with 25 pin-fins of 9 mm2 cross section and fin height of L = 15 mm are plotted in Figs. 3(c), 4(c), and 5(c) for the x–z surface at y = 31 mm as well as Figs. 3(d), 4(d), and 5(d) for the x–y surface at z = 9.5 mm. For the x–z surface at y = 31 mm, the results show that the maximum temperatures inside the aforementioned three heat sinks are 98.94 ◦ C, 93.82 ◦ C, and 93.07 ◦ C, respectively. Besides, the maximum temperatures for the x–y surface at z = 9.5 mm are 96.36 ◦ C, 91.52 ◦ C, and 92.21 ◦ C respectively. The above results further demonstrate that the thermal performance of MEPCM filled heat sink could be improved by adding high thermal conductivity pin-fins for long operational time even until its maximum temperature reaches to more than 90.0 ◦ C. After successful demonstration of pin-fin effectiveness on enhancing the thermal management performance of electronic devices, the maximum temperature Tmax and average liquid fraction fl, ave of MEPCM filled heat sink at melting temperature Tm = 45 ◦ C and heat flux of q = 6 kW/m2 are plotted in Fig. 6 with respect to different pin-fin configurations in order to optimize their thermal performances. As shown in Fig. 6(a), the maximum temperature Tmax of heat sink assembly is presented versus pin-fin num-

bers and cross section areas. It could be obviously found that the maximum temperature Tmax of heat sink without pin-fins increases more dramatically than that of heat sink using pin-fins. For the heat sink with 25 pin-fins of cross section area varied from 1 mm2 to 49 mm2 , at an early stage, the increasing rate of maximum temperature Tmax decreases for the heat sink using pin-fins with larger cross section due to the decreased conduction heat transfer resistance. For instance, at the time of t = 360 s, the heat sink with 25 pin-fins of 49 mm2 cross section has the lowest Tmax = 47.72 ◦ C. However, as the working time of electronic heat sink continutes, several intersections between maximum temperature Tmax curves are observed in Fig. 6(a). The maximum temperature Tmax of heat sink using 25 pin-fins of 49 mm2 cross section is 51.9 ◦ C at t = 510 s which is 0.72 ◦ C higher than that of heat sink with 25 pinfins of 25 mm2 cross section. Furthermore, at the time of t = 570 s , the maximum temperature Tmax of heat sink with 25 pin-fins of 9 mm2 cross section is even 0.21 °C lower than that of heat sink with 25 pin-fins of 49 mm2 cross section. According to the average liquid fraction fl, ave as displayed in Fig. 6(b), when the cross section areas of pin-fins are enlarged, the melting speed of MEPCM in heat sink becomes faster due to the decreased MEPCM volume fraction with less latent heat energy storage capacity for electronic thermal management. As a consequence, the maximum temperature Tmax of heat sink for pin-fins with larger cross section becomes higher than that of heat sink using pin-fins with smaller

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199

7

Fig. 4. Transient temperature of MEPCM filled heat sink assembly with 100 pin-fins of 1 mm2 cross section and fin height of L = 15 mm at melting temperature Tm = 45◦ C and heat flux q = 6 kW/m2 . (a) Sideview of the x–z surface at y = 31 mm and t = 450 s. (b) Sideview of the x–y surface at z = 9.5mm and t = 450 s. (c) Sideview of the x–z surface at y = 31 mm and t = 1350 s. (d) Sideview of the x–y surface at z = 9.5 mm and t = 1350 s.

cross section as the device working time evolves. A similar phenomenon is also observed when 10 0–40 0 pin-fins at fixed 1 mm2 cross section are used for electronic heat sink. By using a large number of pin-fins, the maximum temperature Tmax of heat sink with 400 pin-fins is lowest at the early working stage because of the increased number of high-speed heat transfer channels. Nevertheless, with the continuous melting of MEPCM in heat sink, its maximum temperature Tmax is finally higher than that of the heat sink with 100 or 225 pin-fins due to the reduced latent heat energy storage capability as represented by the accelerated increasing rate of average liquid fraction fl, ave in Fig. 6(b). From the enlarged plot in Fig. 6(a), the optimum thermal management performance is achieved for the heat sink with 100 pin-fins of 1 mm2 cross section or 25 pin-fins of 9 mm2 cross section after the working time of t = 810 s. Besides, the influences of pin-fin height on thermal management of MEPCM filled heat sink is presented in Fig. 6(c) with 100 pin-fins of 1 mm2 cross section and 25 pin-fins of 9 mm2 cross section. The maximum temperature Tmax of heat sink using higher pin-fins increases more slowly at early stage, and it means that the increased heat transfer depth is superior to the increased conduction thermal resistance when higher pin-fins are used. Hence, the heat dissipation rate from heat sink base to MEPCM is actually enhanced with the increasing pin-fin height. However, the corresponding MEPCM volume fraction is also decreased with the increment of pin-fin height making the maximum temperature Tmax

increase seriously at a later time stage when the heat absorption through solid–liquid phase change is ceased. The efficiency and mechanism of electronic thermal management using MEPCM highly depends on its latent heat thermal energy storage capacity and solid–liquid phase change rate under which circumstance the melting temperature of MEPCM becomes one of the significant factors. The influence of MEPCM melting temperature on electronic heat sink assembly with 25 pin-fins of 9 mm2 cross section and fin height of L = 21 mm with respect to different heat flux q is investigated and presented in Fig. 7. Due to the nearly constant solid–liquid phase change temperature of MEPCM, the maximum temperature Tmax of heat sink increases much more slowly onces it reaches the value of MEPCM melting temperature as shown in Fig. 7(a). At low heat flux of q = 4.5 kW/m2 , the maximum temperature Tmax of heat sink using MEPCM with Tm = 45 ◦ C is always lower than that of MEPCM filled heat sink at Tm = 60 ◦ C. Actually, as displayed in Fig. 7(b), the MEPCM with Tm = 60 ◦ C in heat sink is not fully melted even until time t = 1500 s which means that partial latent heat for electronic thermal management is wasted. As a result, its maximum temperature Tmax is higher than that of heat sink using MEPCM of Tm = 45 ◦ C. When the heat flux q increases to 6 kW/m2 or 7.5 kW/m2 , the maximum temperature Tmax of heat sink with a lower MEPCM melting temperature increases more slowly at an early stage. However, as the electronic device continues working, the MEPCM with

8

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199

Fig. 5. Transient temperature of MEPCM filled heat sink assembly with 25 pin-fins of 9 mm2 cross section and fin height of L = 15 mm at melting temperature Tm = 45◦ C and heat flux q = 6 kW/m2 . (a) Sideview of the x–z surface at y = 31 mm and t = 450 s. (b) Sideview of the x–y surface at z = 9.5 mm and t = 450 s. (c) Sideview of the x–z surface at y = 31 mm and t = 1350 s. (d) Sideview of the x–y surface at z = 9.5 mm and t = 1350 s.

Tm = 45 ◦ C melts faster than the MEPCM with Tm = 60 ◦ C. Hence, after the MEPCM of Tm = 45 ◦ C is almost melted with weaker ability of absorbing heat through solid–liquid phase change, its maximum temperature Tmax in heat sink assembly increases more rapidly and achieves the same value with that of heat sink using MEPCM of Tm = 60 ◦ C. In Fig. 7, it could be found that the maximum temperature Tmax of heat sink with Tm = 45 ◦ C is never higher than that of heat sink using MEPCM at Tm = 60 ◦ C. The result indicates that MEPCM with relatively low melting temperature needs to be used when the critical temperature of electronic device is low. Nevertheless, it should be pointed out that the solidification process of MEPCM with lower melting temperature is slow which hinders the recycling period of controlling the electronic device temperature using MEPCM. Therefore, appropriate MEPCM melting temperature should be carefully chosen by simultaneously considering the critical temperature of electronic device and its working period for recharging the latent heat of MEPCM through solidification process. 3.2. Heat transfer enhancement with expanded graphite The expanded graphite is also commonly used to improve the thermal performance of PCMs due to its interconnected conductive heat transfer channels. Accompanying with the anti-leakage advantage of MEPCM, the MEPCM–EG composite becomes a potential candidate with large latent heat and enhanced thermal con-

ductivity for electronic thermal management. The contribution of expanded graphite to the heat transfer enhancement on MEPCM filled heat sink assembly is clarified in this section according to various EG content. The thermal performance of MEPCM–EG composite filled heat sink at melting temperature Tm = 45 ◦ C and fin height L = 15 mm is investigated with respect to different pin-fin configurations and EG content in Fig. 8. To compare the different heat transfer enhancement mechanisms between using pinfins and EG, the maximum temperature Tmax of MEPCM–EG composite filled heat sink without pin-fins using 5% EG and MEPCM filled heat sink using 25 pin-fins of 9 mm2 cross section with pin-fin volume fraction of 4.63% are chosen to be the representative cases. As shown in Fig. 8(a), at relatively low heat flux of q = 4.5 kW/m2 , the maximum temperature Tmax of heat sink using 25 pin-fins is lower than that of heat sink using MEPCM–EG composite with 5% EG content at the time until t = 1410 s. From the average liquid fraction fl, ave displayed in Fig. 9(a), it is found that the MEPCM melting speed for heat sink using 25 pin-fins is much faster than that using 5% content EG because the pin-fins are directly contacted with the heat sink base which highly improves the corresponding heat dissipation rate. Due to the relatively slow heat dissipation rate and low MEPCM melting speed, the latent heat of MEPCM filled heat sink with 5% EG for controlling the temperature rise is not fully used until t = 1410 s making its maximum temperature Tmax higher than that of heat sink with 25 pin-fins. However, after the time t = 1410 s, the MEPCM inside

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199

9

Fig. 6. MEPCM filled heat sink at melting temperature Tm = 45 ◦ C and heat flux q = 6 kW/m2 with different pin-fin configurations versus time. (a) Maximum temperature Tmax of heat sink and its enlarged plot with different number of pin-fins with various cross section areas at fin height L = 15 mm. (b) Average liquid fraction fl, ave of heat sink with different number of pin-fins with various cross section areas at fin height L = 15 mm. (c) Maximum temperature Tmax of heat sink with different pin-fin heights and configurations.

Fig. 7. MEPCM filled heat sink assembly with 25 pin-fins of 9 mm2 cross section and fin height of L = 21 mm with respect to different melting temperature Tm and heat flux q. (a) Maximum temperature Tmax . (b) Average liquid fraction fl, ave .

10

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199

Fig. 8. Maximum temperature Tmax of MEPCM–EG composite (Tm = 45 ◦ C) filled heat sink and its enlarged plot versus different pin-fins at fin height of L = 15 mm and expanded graphite content with various heat flux q: (a) q = 4.5 kW/m2 , (b) q = 6 kW/m2 , (c) q = 7.5 kW/m2 .

the heat sink with 25 pin-fins is completely melted causing the heat sink to lose the capability of managing the temperature rise through solid–liquid phase change. Under this condition, the maximum temperature Tmax of heat sink using 5% EG becomes lower than that of heat sink with 25 pin-fins due to its remained latent

heat of MEPCM–EG composite and the averagely enhanced thermal conductivity throughout the entire domain. When the heat flux q is increased to 6.0 kW/m2 , the melting speed of MEPCM is accelerated compared with the case at heat flux of q = 4.5 kW/m2 as displayed in Fig. 9(a) and (b). Hence, the maximum temperature Tmax

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199

11

of heat sink with 25 pin-fins could only be lower than that of heat sink using MEPCM with 5% EG until the time of t = 1110 s which is reduced by 300 s due to the increment of heat flux from 4.5 kW/m2 to 6.0 kW/m2 . In addition, as the heat flux q is further increased to 7.5 kW/m2 , as shown in Figs. 8(c) and 9(c), the operating time for keeping the maximum temperature Tmax of heat sink using 25 pin-fins lower than that of heat sink using MEPCM with 5% EG is further decreased to t = 930 s because of the accelerated MEPCM melting rate. The aforementioned discussion indicates that pin-fins could minimize the temperature rise inside electronic heat sink at early operating stage because of its direct contact with heat sink base for enhanced heat transfer and the corresponding increased MEPCM melting rate for absorbing heat through solid–liquid phase change. Furthermore, as presented in Figs. 8 and 9, the increasing rate of maximum temperature Tmax in MEPCM–EG composite filled heat sink could be modulated in terms of pin-fin volume fraction and EG content. As previously discussed, there exists a tradeoff between accelerating the heat dissipation rate of heat sink by adding pin-fins or EG and its corresponding decreased latent heat energy storage capacity. Therefore, the pin-fin and EG volume fractions need to be seriously chosen according to the thermal requirement of electronic devices. As presented in Fig. 8(a), at heat flux of q = 4.5 kW/m2 , the heat sink with 25 pin-fins of 9 mm2 cross section and 10% EG has the lowest maximum temperature Tmax among all the cases until the time of t = 1170 s due to its excellent heat transfer capability by using pin-fins and EG simultaneously. Besides, as shown in Fig. 9(a), the corresponding melting speed of this heat sink is also highest with fully melted MEPCM at time t = 1170 s. Unfortunately, due to the cease of solid–liquid phase change, the maximum temperature Tmax of this heat sink dramatically increases to 77.78 ◦ C at time of t = 1500 s which is only lower than that of heat sink without using pin-fins and EG. Similarly, when the heat flux is increased to q = 6.0 kW/m2 and q = 7.5 kW/m2 , the maximum temperature Tmax of heat sink with 25 pin-fins and 10% EG is also maintained lowest until the MEPCM is completely melted at the time of t = 900 s and t = 750 s respectively. The aforementioned findings indicate that increasing the fractions of pin-fins and EG could enhance the heat dissipation rate and control the temperature rise of electronic heat sink at early stage before the solid–liquid phase change of MEPCM is completed. However, when the MEPCM is fully melted, the maximum temperature Tmax of heat sink exhibits dramatical increment due to the lack of heat absorption through solid–liquid phase change. Based on these phenomena, in order to efficiently limit the rapid temperature rise in short working time, the MEPCM–EG composite filled heat sink with high fractions of pin-fins and EG should be used especially for the electronic devices whose critical temperature is relatively low. On the other hand, for the electronic devices with higher critical working temperature, the fraction of pin-fins and EG in MEPCM–EG composite filled heat sink needs to be appropriately reduced in purpose of balancing its enhanced heat transfer capability and decreased latent heat energy storage capacity for thermal management. 3.3. Heat transfer enhancement with cascade MEPCM–EG composite

Fig. 9. Average liquid fraction fl, ave of MEPCM–EG composite (Tm = 45 ◦ C) filled heat sink versus different pin-fins at fin height of L = 15 mm and expanded graphite content with various heat flux q: (a) q = 4.5 kW/m2 , (b) q = 6 kW/m2 , (c) q = 7.5 kW/m2 .

The MEPCM filled heat sink with constant melting temperature is proved to be effective for electronic thermal management by adding appropriate fraction of pin-fins and EG. Due to the existence of temperature gradient inside electronic heat sink, the melting speed of MEPCM is not uniform which inhibits the efficient usage of latent heat for controlling the temperature rise. Under this situation, the heat transfer in electronic heat sink using MEPCM–EG composite with cascade melting temperature deserves to be investigated in order to optimize the thermal performance of

12

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199

Fig. 10. Transient temperature and liquid fraction of MEPCM–EG composite ( = 5%) filled heat sink assembly with melting temperature Tm = 45 ◦ C and heat flux q = 6 kW/m2 for the x–z surface at y = 31 mm. (a) Temperature T at t = 450 s. (b) Liquid fraction fl at t = 450 s. (c) Temperature T at t = 1350 s. (d) Liquid fraction fl at t = 1350 s.

electronic devices. The transient temperature and liquid fraction of MEPCM–EG composite with 5% EG filled heat sink assembly at constant melting temperature Tm = 45 ◦ C and heat flux q = 6 kW/m2 for the x–z surface at y = 31 mm are presented in Fig. 10. As shown in Fig. 10(a) at time of t = 450 s, the temperature T inside heat sink base is seriously increased with maximum temperature Tmax to be more than 51.8 ◦ C. Besides, from the liquid fraction fl of MEPCM–EG composite at t = 450 s as displayed in Fig. 10(b), there is only a small amount of MEPCM being melted because the temperature of most MEPCM is still below its melting temperature Tm = 45 ◦ C at this stage. As a comparison, the transient temperature and liquid fraction of MEPCM–EG composite filled heat sink with cascade melting temperature which is Tm1 = 45 ◦ C in the bottom half region and Tm2 = 30 ◦ C in the top half region are shown in Fig. 11. In Fig. 11(a), in contrast with the temperature contours in Fig. 10(a), the temperature contours in the heat sink with cascade MEPCM–EG composite spread out with maximum temperature Tmax of 47.94 ◦ C which means that the temperature increasing rate is reduced by using MEPCM–EG composite with cascade melting temperature instead of that with constant melting temperature. As shown in Fig. 11(b), the liquid fraction of MEPCM–EG composite filled heat sink with cascade melting temperature indicates that a few amount of MEPCM in the top region is melted due to its relatively low melting temperature. Therefore, the temperature rise in this heat sink becomes slow because of the heat ab-

sorbed during the solid–liquid phase change of MEPCM in top half region. In addition, at the time of t = 1350 s, the maximum temperature Tmax of heat sink using MEPCM–EG with constant melting temperature in Fig. 10(c) is increased to 83.39 ◦ C with incompletely melted MEPCM as presented in Fig. 10(d). However, the maximum temperature Tmax of heat sink using cascade MEPCM–EG is only 76.71 ◦ C as presented in Fig. 11(c) because of its fully used latent heat for absorbing heat dissipated from the electronic heat sink base as displayed by Fig. 11(d). In order to further clarify the effectiveness of controlling the temperature rise in electronic heat sink using MEPCM–EG composite with cascaded melting temperature, the maximum temperature Tmax and average liquid fraction fl, ave for heat sink using MEPCM–EG composite with constant or cascade melting temperature are presented in Fig. 12 under different heat fluxes. At relatively low heat flux of q = 4.5 kW/m2 , as shown in Fig. 12(b), the melting speed of MEPCM–EG composite with cascaded melting temperature is faster than that of the uniform MEPCM–EG composite. Hence, the maximum temperature Tmax of heat sink using MEPCM–EG composite with cascade melting temperature is consistently lower than that of heat sink using uniform MEPCM–EG composite as presented in Fig. 12(a). On the other hand, when the heat flux q is increased to 7.5 kW/m2 , the maximum temperature Tmax of heat sink using MEPCM–EG composite with cascade melting temperature is lower than that of heat sink with uniform MEPCM–

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199

13

Fig. 11. Transient temperature and liquid fraction of MEPCM–EG composite ( = 5%) filled heat sink assembly with cascade melting temperature Tm1 = 45 ◦ C (bottom half region), Tm2 = 30 ◦ C (top half region), and heat flux q = 6 kW/m2 for the x–z surface at y = 31 mm, (a) Temperature T at t = 450 s. (b) Liquid fraction fl at t = 450 s. (c) Temperature T at t = 1350 s. (d) Liquid fraction fl at t = 1350 s.

Fig. 12. MEPCM–EG composite ( = 5%) filled heat sink assembly with constant or cascade melting temperature at different heat flux q. (a) Maximum temperature Tmax . (b) Average liquid fraction fl, ave .

14

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199

EG composite at early stage as displayed in Fig. 12(a). Nevertheless, after the MEPCM–EG composite with cascade melting temperature is fully melted at the time of t = 1230 s, the maximum temperature Tmax in the corresponding heat sink without latent heat increases more dramatically, and it finally approaches to the maximum temperature of heat sink with uniform MEPCM–EG composite. Furthermore, for the intermittent working electronic devices, the latent heat of MEPCM–EG composite should be recharged during the ceasing time under natural convection with environment in order to manage the thermal issue when the electronic device restarts. For the solidification process, the bottom region of cascade MEPCM–EG composite filled heat sink could be firstly recharged due to its relatively high melting temperature so that its latent heat could be efficiently recovered for thermal management. The above detailed discussion demonstrates that filling the electronic heat sink by MEPCM–EG composite with cascade melting temperature could not only decrease the temperature rise rate through more efficient MEPCM solid–liquid phase change in the top region but also efficiently recharge the MEPCM in the bottom region during solidification process. 4. Conclusions Due to the advantages of low volume expansion/shrinkage and anti-leakage characteristics, MEPCM exhibits excellent performance over traditional PCMs in electronic thermal management when efficient heat transfer enhancement technologies are applied. However, there exists several tradeoff effects between enhancing the heat transfer rate of MEPCM and its reduced latent heat energy storage capacity. In this paper, thermal management of electronic devices using cascade MEPCM–EG composite filled heat sink with pin-fins is investigated in detail through 3D enthalpy-based lattice Boltzmann modeling. The following conclusions could be drawn from the current investigation: 1) The heat transfer depth in electronic heat sink could be enhanced by increasing the pin-fin lengths. Nevertheless, the conduction thermal resistance of prolonged pin-fins is simultaneously increased. Under this situation, enlarging the cross section area of pin-fins reduces their conduction thermal resistance to some extent. Unfortunately, the latent heat energy storage capacity of MEPCM-filled heat sink decreases with the increment of pin-fin length and pin-fin cross section which limits the operating time of controlling the electronic temperature through solid–liquid phase change. Hence, the pin-fin arrays need to be carefully designed with appropriate fin number and configurations depending on the thermal requirement of electronic devices in order to balance the increased heat transfer capability and the decreased latent heat energy storage capacity of heat sink. For the electronic device with relatively low critical temperature, the number, length, and cross section of pin-fins could be increased in order to improve the heat dissipation rate at early operating stage. Otherwise, the volume fraction of pin-fins should be reduced to extend the operating time of limiting the temperature rise in electronic devices through solid–liquid phase change. 2) Due to the different heat transfer enhancement mechanism of pin-fin arrays and expanded graphite, the volume fraction of pin-fins and EG needs to be carefully chosen with respect to the thermal requirement of electronic devices. At the early working stage of electronic devices, pin-fins are found to be more effective for limiting the temperature rise with higher heat dissipation rate because of its direct contact with heat sink base. However, as the working time of electronic device evolves, the MEPCM–EG composite exhibits better thermal performance over pin-fin based MEPCM due to the averagely enhanced ther-

mal conductivity by adding EG into MEPCM to form interconnected heat transfer channels. Based on this, the pin-fin array is recommended to be used in the electronic heat sink with relatively low critical temperature while EG is beneficial for electronic heat sink with high critical temperature and long operating time. 3) The MEPCM–EG composite with cascade melting temperature is presented and investigated in the current work under different heat flux conditions. The results indicate that the cascade MEPCM–EG composite could enhance the heat absorption through solid–liquid phase change in the top half region of heat sink due to the relatively low melting temperature in this specific region. In addition, with the relatively high MEPCM melting temperature in the bottom half region of heat sink, the recharging process for MEPCM latent heat during the cease working period of electronic devices through solidification process is also accelerated compared with the heat sink using uniform MEPCM. Declaration of Competing Interest None Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 51806168 and No. 51776145) and the China Postdoctoral Science Foundation (No. 2017M623169). Appendix Lattice Boltzmann method: The D3Q7 enthalpy-based multiple-relaxation-time (MRT) lattice Boltzmann method developed by Li et al. [38] is used in the current work to model the solid–liquid phase change and conjugate heat transfer during thermal management of electronic devices with MEPCM–EG composite and pin-fins. By using the MRT scheme instead of single-relaxation-time (SRT) scheme for collision, the numerical stability is highly improved. The collision of distribution function in MRT model is completed in the momentum space as:

m(x, t + t ) = m(x, t ) − S [m(x, t ) − meq (x, t )]

(A1)

The distribution function m in momentum space is computed from the distribution function in velocity space gi as:

m = M (g0 , g1 , g2 , g3 , g4 , g5 , g6 )T

(A2)

where M is the transformation matrix given by−:

M=

(A3)

The equilibrium distribution function in momentum space meq (x, t) is expressed as:



meq (x, t ) = h,

T

c p T ux c p T uy c p T uz , , , 6h − 21ωT c p re f T , 0, 0 c c c

(A4) ux , uy , and uz are velocities in the three Cartesian coordinate directions, and ωT is a constant which is set to be 0.25 in the current work. c is the lattice speed, and S is the relaxation matrix defined by:

S = diag(σ0 , σ1 , σ2 , σ3 , σ4 , σ5 , σ6 )

(A5)

where σ0 = 1, σ1 = σ2 = σ3 = τ , σ4 = σ5 = σ6 = 2 − τ , and τ is the relaxation time given by: 1



τ=

4ρ k 4 ρ cfp re f re f

kc re f c p re f

1

+ 0.5, PCM − EG Composite

+ 0.5, Copper f ins and heat sink shell

(A6)

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199

15

CUDA implementation for lattice Boltzmann method is presented in detail in our previous work [40]. Conjugate heat transfer: The developed 3D CUDA Fortran Code is firstly calibrated by one-dimensional conjugate heat transfer without solid–liquid phase change in infinite two regions. Originally, the temperature T is set to be 1 in the region A at x > 0 while its value is set to be 0 in the region B at x < 0. The analytical solution for this problem is as follows:

T A (x, t ) =

× 1+

1+





1

(ρ C p )

B B k /

(ρ C p )

B B k /

(ρCp )A kA

(ρ C p )

A A k er f



x

2 kA t/(ρC p )A

T B (x, t ) =

1+



1

(ρCp )B kB /(ρCp )A kA

Fig. A1. Calibration of CUDA code for conjugate heat transfer.

2 c pc c p f c pc + c p f

(A7)

After the collision step, the distribution function gi in velocity space is computed according to the inverse transformation:

gi (x, t + t ) = M −1 m(x, t + t )

(A8)

Then the streaming process is carried out in velocity space as:

gi (x + ei t, t + t ) = gi (x, t + t )

(A9)

After tackling the thermal boundary conditions, the enthalpy H is calculated as:

H=

6 

gi



x

2 kB t/(ρC p )B (A12)

The reference density ρ ref is set to be 1, and the reference specific heat cp ref is defined by the harmonic mean as:

c pre f =

er f c −

(A11)

(A10)

i=0

Hence, the corresponding PCM–EG composite temperature Tc and temperature of heat sink shell with pin-fins Tf could be achieved by using Eq. (7) and Eq. (9) respectively. The technique of

In the current calibration, MEPCM is chosen to be the material in the region B while copper is used as the material in region A. As shown in Fig. A1, the results from lattice Boltzmann method using grid size of x = 0.01 m agree well with the analytical solutions at different physical times. Similarly, the 3D CUDA Fortran Code is also validated for the y coordinate direction and z coordinate direction for conjugate heat transfer respectively. Based on the above calibrations, the current LBM CUDA code is demonstrated to be capable of simulating the conjugate heat transfer between PCM–EG composite, heat sink copper shells, and copper pin-fins. 3D solid–liquid phase change with natural convection: In order to verify the current CUDA Fortran code for 3D solid–liquid phase change heat transfer, the current results are compared with the Case 1 (Ra = 25, 0 0 0, P r = 0.02) and Case 2 (Ra = 25, 0 0 0, P r = 10) for solid–liquid phase change with natural convection in the Section 3.3 of reference by Li et al. [38] with the same grid size. The fluid flow is simulated using D3Q19 LBM model which is not presented in this paper for simplicity. As displayed in Fig. A2, the average Nusselt number Nu along the left wall and the average solid–liquid fraction fl are presented, and it could be found that the results computed from the present CUDA Fortran code are highly consistent with those in the reference.

Fig. A2. Calibration of CUDA code for 3D solid–liquid phase change.

16

Q. Ren, P. Guo and J. Zhu / International Journal of Heat and Mass Transfer 149 (2020) 119199

References [1] Z. Ling, et al., Review on thermal management systems using phase change materials for electronic components, Li-ion batteries and photovoltaic modules, Renew. Sustain. Energy Rev. 31 (2014) 427–438. [2] R. Kalbasi, M. Afrand, J. Alsarraf, M.-D. Tran, Studies on optimum fins number in PCM-based heat sinks, Energy 171 (2019) 1088–1099. [3] P. Ping, R. Peng, D. Kong, G. Chen, J. Wen, Investigation on thermal management performance of PCM-fin structure for Li-ion battery module in high-temperature environment, Energy Convers. Manag. 176 (2018) 131–146. [4] R. Pakrouh, M.J. Hosseini, A.A. Ranjbar, R. Bahrampoury, A numerical method for PCM-based pin fin heat sinks optimization, Energy Convers. Manag. 103 (2015) 542–552. [5] M.J. Ashraf, H.M. Ali, H. Usman, A. Arshad, Experimental passive electronics cooling: parametric investigation of pin-fin geometries and efficient phase change materials, Int. J. Heat Mass Transf. 115 (2017) 251–263. [6] H.M. Ali, et al., Thermal management of electronics: an experimental analysis of triangular, rectangular and circular pin-fin heat sinks for various PCMs, Int. J. Heat Mass Transf. 123 (2018) 272–284. [7] X.-H. Yang, S.-C. Tan, Z.-Z. He, Y.-X. Zhou, J. Liu, Evaluation and optimization of low melting point metal PCM heat sink against ultra-high thermal shock, Appl. Therm. Eng. 119 (2017) 34–41. [8] Q. Ren, C.L. Chan, GPU accelerated numerical study of PCM melting process in an enclosure with internal fins using lattice Boltzmann method, Int. J. Heat Mass Transf. 100 (2016) 522–535. [9] Q. Ren, H. Xu, Z. Luo, PCM charging process accelerated with combination of optimized triangle fins and nanoparticles, Int. J. Therm. Sci. 140 (2019) 466–479. [10] W.G. Alshaer, S.A. Nada, M.A. Rady, E.P.D. Barrio, A. Sommier, Thermal management of electronic devices using carbon foam and PCM/nano-composite, Int. J. Therm. Sci. 89 (2015) 79–86. [11] W.G. Alshaer, S.A. Nada, M.A. Rady, C.L. Bot, E.P.D. Barrio, Numerical investigation of using carbon foam/PCM/Nano carbon tubes composites in thermal management of electronic equipment, Energy Convers. Manag. 89 (2015) 873–884. [12] Z.-Q. Zhu, Y.-K. Huang, N. Hu, Y. Zeng, L.-W. Fan, Transient performance of a PCM-based heat sink with a partially filled metal foam: effects of the filling height ratio, Appl. Therm. Eng. 128 (2018) 966–972. [13] Q. Ren, Y.-L. He, K.-Z. Su, C.L. Chan, Investigation of the effect of metal foam characteristic on the PCM melting performance in a latent heat thermal energy storage unit by pore-scale lattice Boltzmann modeling, Numer. Heat Transf., A: Appl. 72 (2017) 745–764. [14] Q. Ren, F. Meng, P. Guo, A comparative study of PCM melting process in a heat pipe-assisted LHTES unit enhanced with nanoparticles and metal foams by immersed boundary-lattice Boltzmann method at pore-scale, Int. J. Heat Mass Transf. 121 (2018) 1214–1228. [15] J. Krishna, P.S. Kishore, A. Brusly Solomon, Heat pipe with nano enhanced-PCM for electronic cooling application, Exp. Therm. Fluid Sci. 81 (2017) 84–92. [16] H. Behi, M. Ghanbarpour, M. Behi, Investigation of PCM-assisted heat pipe for electronic cooling, Appl. Therm. Eng. 127 (2017) 1132–1142. [17] Z.Y. Jiang, Z.G. Qu, Lithium-ion battery thermal management using heat pipe and phase change material during discharge–charge cycle: a comprehensive numerical study, Appl. Energy 242 (2019) 378–392. [18] Q. Ren, Enhancement of nanoparticle-phase change material melting performance using a sinusoidal heat pipe, Energy Convers. Manag. 180 (2019) 784–795. [19] W. Wu, G. Zhang, X. Ke, X. Yang, Z. Wang, C. Liu, Preparation and thermal conductivity enhancement of composite phase change materials for electronic thermal management, Energy Convers. Manag. 101 (2015) 278–284. [20] Z. Huang, Z. Luo, X. Gao, X. Fang, Y. Fang, Z. Zhang, Preparation and thermal property analysis of Wood’s alloy/expanded graphite composite as highly conductive form-stable phase change material for electronic thermal management, Appl. Therm. Eng. 122 (2017) 322–329.

[21] T. Xu, Y. Li, J. Chen, H. Wu, X. Zhou, Z. Zhang, Improving thermal management of electronic apparatus with paraffin (PA)/expanded graphite (EG)/graphene (GN) composite material, Appl. Therm. Eng. 140 (2018) 13–22. [22] C. Kinkelin, et al., Theoretical and experimental study of a thermal damper based on a CNT/PCM composite structure for transient electronic cooling, Energy Convers. Manag. 142 (2017) 257–271. [23] A. Farzanehnia, M. Khatibi, M. Sardarabadi, M. Passandideh-Fard, Experimental investigation of multiwall carbon nanotube/paraffin based heat sink for electronic device thermal management, Energy Convers. Manag. 179 (2019) 314–325. [24] L.-W. Fan, Z.-Q. Zhu, Y. Zeng, Y.-Q. Xiao, X.-L. Liu, Y.-Y. Wu, Q. Ding, Z.-T. Yu, K.-F. Cen, Transient performance of a PCM-based heat sink with high aspect-ratio carbon nanofillers, Appl. Therm. Eng. 75 (2015) 532–540. [25] Z. Yang, et al., Thermally conductive, dielectric PCM-boron nitride nanosheet composites for efficient electronic system thermal management, Nanoscale 8 (2016) 19326–19333. [26] C.Y. Zhao, G.H. Zhang, Review on microencapsulated phase change materials (MEPCMs): fabrication, characterization, and applications, 15 (2011) 3813– 3832. [27] J. Giro-Paloma, M. Martinez, L.F. Cabeza, A.I. Fernandez, Types, methods, techniques, and applications for microencapsulated phase change materials (MPCM): a review, Renew. Sustain. Energy Rev. 53 (2016) 1059–1075. [28] W. Li, H. Wan, H. Lou, Y. Fu, F. Qin, G. He, Enhanced thermal management with microencapsulated phase change material particles infiltrated in cellular metal foam, Energy 127 (2017) 671–679. [29] W.Q. Li, H. Wan, T.T. Jing, Y.B. Li, P.J. Liu, G.Q. He, F. Qin, Microencapsulated phase change material (MEPCM) saturated in metal foam as an efficient hybrid PCM for passive thermal management: a numerical and experimental study, Appl. Therm. Eng. 146 (2019) 413–421. [30] B. Praveen, S. Suresh, Thermal performance of micro-encapsulated PCM with LMA thermal percolation in TES based heat sink application, Energy Convers. Manag. 185 (2019) 75–86. [31] T. Wang, S. Wang, W. Wu, Experimental study on effective thermal conductivity of microcapsules based phase change composites, Int. J. Heat Mass Transf. 109 (2017) 930–937. [32] S. Chen, G.D. Doolen, Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech. 30 (1998) 329–364. [33] C.K. Aidun, J.R. Clausen, Lattice-Boltzmann method for complex flows, Annu. Rev. Fluid Mech. 42 (2010) 439–472. [34] A. Xu, W. Shyy, T. Zhao, Lattice Boltzmann modeling of transport phenomena in fuel cells and flow batteries, Acta Mech. Sin. 33 (3) (2017) 555–574. [35] R.Z. Huang, H.Y. Wu, P. Cheng, A new lattice Boltzmann model for solid-liquid phase change, Int. J. Heat Mass Transf. 59 (2013) 295–301. [36] R.Z. Huang, H.Y. Wu, Phase interface effects in the total enthalpy-based lattice Boltzmann model for solid-liquid phase change, J. Comput. Phys. 294 (2015) 346–362. [37] Dong Li, Qinlong Ren, Zi-Xiang Tong, Ya-Ling He, Lattice Boltzmann models for axisymmetric solid-liquid phase change, Int. J. Heat Mass Transf. 112 (2017) 795–804. [38] D. Li, Z.-X. Tong, Q. Ren, Y.-L. He, W.-Q. Tao, Three-dimensional lattice Boltzmann models for solid-liquid phase change, Int. J. Heat Mass Transf. 115 (2017) 1334–1347. [39] Q. Ren, C.L. Chan, Numerical study of double-diffusive convection in a vertical cavity with Soret and Dufour effects by lattice Boltzmann method on GPU, Int. J. Heat Mass Transf. 93 (2016) 538–553. [40] Q. Ren, C.L. Chan, Natural convection with an array of solid obstacles in an enclosure by lattice Boltzmann method on a CUDA computation platform, Int. J. Heat Mass Transf. 93 (2016) 273–285. [41] A. Xu, L. Shi, T.S. Zhao, Accelerated lattice Boltzmann simulation using GPU and OpenACC with data management, Int. J. Heat Mass Transf. 109 (2017) 577–588. [42] Q. Ren, F. Meng, C.L. Chan, Cell transport and suspension in high conductivity electrothermal flow with negative dielectrophoresis by immersed boundary-lattice Boltzmann method, Int. J. Heat Mass Transf. 128 (2019) 1229–1244.