Thermal Control of Protruding Electronic Components with PCM: A Parametric Study

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Numerical Heat Transfer, Part A:

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Thermal Control of Protruding Electronic Components with PCM: A Parametric

Study

Mustapha Faraji

, Hamid El Qarnia

& Juan Carlos Ramos

a

Cadi Ayyad University, Faculty of Sciences Semlalia, Department of Physics , Fluids Mechanic and Energetic Laboratory , Marrakesh, Morocco

b

TECNUN—University of Navarra, Department of Mechanical Engineering , Thermal and Fluids Engineering Division , San Sebastian, Spain

Published online: 23 Oct 2009.

To cite this article: Mustapha Faraji , Hamid El Qarnia & Juan Carlos Ramos (2009) Thermal Control of Protruding Electronic Components with PCM: A Parametric Study, Numerical Heat Transfer, Part A: Applications: An International Journal of Computation and Methodology, 56:7, 579-603, DOI:

10.1080/10407780903323611

To link to this article: http://dx.doi.org/10.1080/10407780903323611

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THERMAL CONTROL OF PROTRUDING ELECTRONIC COMPONENTS WITH PCM: A PARAMETRIC STUDY

Mustapha Faraji

, Hamid El Qarnia

, and Juan Carlos Ramos

1

Cadi Ayyad University, Faculty of Sciences Semlalia, Department of Physics, Fluids Mechanic and Energetic Laboratory, Marrakesh, Morocco

2

TECNUN—University of Navarra, Department of Mechanical Engineering, Thermal and Fluids Engineering Division, San Sebastian, Spain

This work deals with the melting and natural convection in a rectangular enclosure heated from three discrete protruding electronic components (heat sources) mounted on a conduct- ing vertical plate (substrate). The heat sources generate heat at a constant and uniform volumetric rate. A part of the power generated in the heat sources is dissipated to the phase change material (PCM,n-eicosane with a melting temperature Tm¼36C) that filled the enclosure. To investigate the thermal behavior of the proposed heat sink, a mathematical model, based on the mass, momentum, and energy conservation equations was developed.

The model has been verified and then validated comparing the melting front with available experimental results. Numerical investigations have been conducted in order to examine the effects of the electronic components thickness and the plate thermal diffusivity on the maximum temperature of electronic components. The percentage contribution of plate heat conduction on the total removed heat and temperature profile in the plate have also been analyzed. Correlations for the nondimensional secured working time (time to reach the threshold temperature, Tcr¼75C) and its corresponding melt fraction were derived.

I. INTRODUCTION

Rapid advances in technologies in the last decade have brought new challenges in thermal sciences and fluid mechanics. For instance, thermal management of elec- tronic packaging is critical as we approach the submicron and even nanoscale feature sizes of the chip. The power densities generated in future miniaturized electronic devices are expected to exceed 100 W=cm

[1]. Therefore, advanced thermal control technologies are required to meet the thermal requirements of the high power density components. Some of the technologies under consideration are high thermal conduc- tivity materials, passive two-phase devices, such as micro heat pipes embedded in electronic packages and thermo-electric coolers for active cooling. All of these tech- nologies have limitations on how effectively they keep high power density electronic components below their upper temperature limit, or how easily they integrate with heat dissipating components [1]. Garimella [2] considers recent advances in a number

Received 20 February 2009; accepted 24 August 2009.

Address correspondence to Hamid El Qarnia, Cadi Ayyad University, Faculty of Sciences Semlalia, Department of Physics, Fluids Mechanic and Energetic Laboratory, P.O. 2390, Marrakesh, Morocco. E-mail: elqarnia@ucam.ac.ma

Copyright#Taylor & Francis Group, LLC ISSN: 1040-7782 print=1521-0634 online DOI: 10.1080/10407780903323611

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of novel high-performance cooling techniques for emerging electronics applications, and presents various thermal management technologies: micro channel transport and micro pumps, jet impingement, miniature flat heat pipes, transient phase change energy storage systems, and piezoelectric fans. Application of the melting process of phase change material (PCM) in the cooling management of electronic components as an alternative cooling method for various applications, such as spacecraft and avionics thermal control, personal computing and communication equipment, laptops, electronic equipments, mobile phones, etc., has received significant attention in recent years. The concept is simple as it uses specific cavities filled with PCMs as the heat sink. The use of PCM as a storage medium can reduce the size of the cooling system, due to relative high energy storage density of the PCM. During the working period, electronic components dissipate heat through their exposed faces and the solid PCM continually melts as heat sources continually dissipate heat. The heat stored in the PCM is naturally rejected to the ambient and the melted PCM resoli- difies during the stop periods. As the melted PCM resolidifies it can be used in the next cycles. The use of PCM as a heat storage medium in electronics cooling has been the object of different studies. Shanmugasundaran et al. [3] present a design optimization of thermal management of high heat flux sources using PCM. They showed that this method may be a promising alternative of periodically operating

NOMENCLATURE

b, C constant values in porosity function, Eqs. (3) and (4)

cp specific heat at constant pressure, J kg¹K¹

f liquid fraction g gravity, m s²

h specific enthalpy, J kg¹ Hm height of the PCM enclosure, m k thermal conductivity, W m¹K¹ lo characteristic length, m,

lo¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LmHm3XcLc

p

lEC represents the electronic component volume, m,l_EC² ¼XcLc

Lc electronic component height, m Le space between two successive

sources, m

Lh position of the bottom electronic component, m

Lm width of the PCM enclosure, m Nu average heat source Nusselt number p constant pressure, Pa

Q⁰ power per unit length, W m¹ s heat source peripheral distance, m T temperature, K

t time, s

u, v x, yvelocity, m s¹ U,V dimensionless velocities:_a^u

m;l=lo; _a^v

m;l=lo

Xc,Xs electronic component and plate thicknesses, m

x,y Cartesian coordinates, m Subscripts

1, 2, 3 refers to the bottom, central, and upper electronic components, respectively cr critical value

c,EC electronic component

f fluid

l liquid

m– m,l melting–liquid PCM, respectively max maximum value

o initial

s substrate

t total

a thermal diffusivity (m²s¹),a¼_qc^k b expansion factor, K¹

di,d kronecker symbols, distance, m DHf latent heat, J kg¹

Dt time step, s

g perpendicular to the hot wall=liquid PCM interfaces, m

m dynamic viscosity, kg m s¹ n kinematic viscosity, m²s¹ q density, kg m³

s dimensionless time,s¼^a_l^m;l2 o t w stream function,w¼_a¹

m;l

Rudyvdx

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components due to relative high removal capabilities of PCMs compared with liquid and air options [3]. Pal and Joshi [4] studied computationally and experimentally melting of a PCM, n-triacontane, in a side uniformly heated (isothermal wall) tall enclosure of aspect ratio of 10 for airspace applications. Alawadhi and Amon [5]

investigated the effectiveness of a thermal control unit (TCU) made of PCM and aluminum fins for portable electronic devices. Yin et al. [6] prepared a composite PCM with excellent thermal storage performance, and its heat storage period and heat release period are shortened by 65.3% and 26.2%, respectively, compared with that of the paraffin. Their study shows that applying the composite PCM to an elec- tronic device’s heat sink can effectively improve the performance of resisting the shock of high heat flux and warrants the reliability and operating stability of elec- tronic and electrical equipments. The experimental results show that the apparent heat transfer coefficients of the experimental heat sink with the PCM are 1.36–2.98 times higher than the heat sink without the PCM. Nayak et al. [7] numeri- cally studied the performance of heat sinks with various volume fractions of thermal conductivity enhancers (TCEs) for different configurations. The time variations of heat source temperature, melt fraction and dimensionless temperature difference within the PCM were analyzed. Results illustrate significant effect of the thermal conductivity enhancer on the performance of heat sinks. Gurrum et al. [8] studied the cooling management of pulsed electronics using metallic PCMs. PCM effective- ness and temperature reductions as a function of chip thickness, channel width and the power dissipated are reported. Temperature reductions up to 25

C can be rea- lized with a combination of metallic spreader and PCM. Pal et al. [9] analyzed the passive cooling of a plastic quad flat package by introducing a phase change material under the substrate. The effect of the thermal conductivity of the substrate was reported. It was found that temperature level decreases with an increase in the ther- mal conductivity. For lower thermal conductivity of the board, the melt region was found to be localized near the package footprint, while for higher board conduc- tivity, the melt region extends along the board. Sasaguchi et al. [10] performed a numerical study to examine the feasibility of using the melting process for cooling of heated surfaces. It was found that natural convection in the melt region kept the temperature of the surface almost constant for a long period depending on the surface heating conditions. Krishnan et al. [11] proposed a hybrid heat sink, which consists of a plate fin heat sink with the tip immersed in a rectangular enclosure filled with a phase change material (PCM). A one-dimensional fin equation is formulated which accounts for the simultaneous convective heat transfer from the finned surface and melting of the PCM at the tip. The influence of the location, amount, and type of PCM, as well as the fin thickness on the thermal performance of the hybrid heat sink, is investigated. Simple guidelines are developed for preliminary design of the heat sink. Recently, cooling management of microprocessors (CPU) was also investigated by Faraji and El Qarnia [12] using a hybrid heat sink which consists of a rectangular cavity filled with PCM (SunTech P111) attached to a conventional aluminum fins with a constant heat flux density imposed to the base of the microprocessor. Numeri- cal simulations were conducted to optimize the thermal performance of the heat sink in order to maximize the working time of the microprocessor. The obtained optimal configuration was subject to the cyclic operating mode (charging, fan off=dischar- ging, fan on); the established periodic mode was reached after three cycles of work-

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ing. An experimental study of melting and natural convection heat transfer in an enclosure with three discrete protruding heat sources at a constant rate attached on one of its vertical wall was conducted by Ju et al. [13]. The horizontal walls were adiabatic. Their results were compared with those of Keyhani et al. [14]. This com- parison shows that the rise in the temperature of the heat sources can be reduced to 50–70% by using PCM melting and natural convection instead of natural convec- tion of ethylene glycol. In another experimental study conducted by Jianhua et al.

[15], the melting process of n-octadecane in a rectangular cavity with three discrete protruding heat sources, placed on the lower side of the enclosure has been studied.

The effects of the Stefan number, the sub-cooling, and aspect ratio on the melting process were analyzed. It may be observed from the above discussion that a substan- tial amount of research has been reported on the thermal management of electronics using PCM with metal fins and flush heat sources under boundary conditions of uni- form heat flux or temperature. No numerical studies have been developed to examine the problem of thermal control of protruding electronic components with volumetric heat generation. This article presents a numerical model to perform a parametric study of an electronics cooling configuration previously analyzed experimentally by Ju et al. [13]. The problem studied corresponds to the melting natural convection inside a vertical rectangular enclosure with protruding heat sources attached on one of the vertical walls. This article improves the analysis performed in reference [13] by considering uniform volumetric heat generation inside the heat sources instead of uniform heat flux on their back faces. The objectives of the current study are as follows. 1) To develop a mathematical model to study the thermal behavior of the proposed system, and 2) to validate it and to examine the effects of the heat sources thickness and substrate thermal diffusivity on the maximum temperature of the heat sources, on the percentage contribution of substrate heat conduction to the total removed heat, and on the temperatures profile in the substrate. In addition, correla- tions are developed for the secured working time and the corresponding realized melt fraction.

II. ANALYSIS AND MODELING

Description of the Studied Configuration

Figure 1 illustrates the physical model and coordinate system. It consists of a rectangular enclosure containing a phase change material (PCM) with three identical discrete protruding heat sources, simulating electronic components, attached on the left wall. The height and thickness of each heat source are L

, and X

, respectively. The distance between two consecutive heat sources is L

, and the distance between the bottom enclosure wall and the bottom face of the lower heat source is L

. The height and width of the enclosure are H

and L

, respectively. The thickness of the plate (substrate) is X

. Each electronic compo- nent generates a constant and uniform volumetric power. It must be noted that all boundaries of the enclosure are adiabatic, and the particularity of the present cooling strategy is that there is no fan and that the device works until the complete melting of the PCM or if one of the heat sources reaches the critical temperature T

.

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Governing Equations

The flow is assumed to be two-dimensional, Newtonian, laminar, and incom- pressible. The thermophysical properties of the materials are constant at the temperature range under study. The density difference between solid and liquid phases is negligible, and the Boussinesq approximation is used. The PCM is assumed initially solid at its melting temperature T

and the phase change is isotherm.

The governing equations and boundary conditions, for the PCM, heat sources and substrate are as follows.

qðqUÞ

qt þ r ðquU CrUÞ ¼ S ð1Þ where C is the coefficient of diffusion and S is a source term associated to the variable U (u, v, or h). Therefore, the general equation consists of term of accumu- lation, a term of convection, a term of diffusion, and a source term. These quantities are summarized in Table 1.

h ¼ Z

Tm

c

dT þ hðT

Þ ð2Þ

where h is the specific enthalpy and

S

¼ C ð1 f Þ

ðf

þ bÞ u ð3Þ

Figure 1. The physical model.

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S

¼ C ð1 f Þ

ðf

þ bÞ v þ q

gbðT T

Þ ð4Þ

S

¼ d

ð1 d

ÞqDH

qf

qt þ d

Q

X

L

ð5Þ

Boundary Conditions

The interface conditions between two different materials (1 and 2) (substrate, PCM or electronic component) are set as follows.

k

qT qg

interface

¼ k

qT qg

interface

; T

¼ T

ð6Þ

with g? interface.

At the adiabatic walls

qT qg

wall

¼ 0 ð7Þ

No slip and nonpermeability at solid surfaces (the two components of velocity at all the solid surfaces are set to zero).

u ¼ v ¼ 0 ð8Þ

Initial Conditions

u ¼ v ¼ 0; f ¼ 0; T ¼ T

ð9Þ S

and S

are source terms used for the velocity suppression in the solid regions (solid PCM, substrate, electronic components). One of the common models for the velocity suppression is to introduce a Darcy-like term [16] (see expressions (3) and (4) with C ¼ 10

kg m

s

and b ¼ 5 10

). The same full set of governing equations throughout the entire enclosure governs conjugate heat transfer in both the liquid and solid regions with taking a large value of the viscosity for solid regions. The thermal conductivity, k, and the step function d are set as follows.

Table 1.Terms of the general equation

U C S

1 0 0

u m _qx^qpþSu

n m ^qp_qyþSv

h k=cp ST

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k ¼

k

for PCM k

for substrate k

for heat sources 8 <

:

d

¼ 1 for heat sources and PCM 0 for the substrate

d

¼ 1 for heat sources 0 for the PCM

ð10Þ

Thermophysical properties of the PCM are assumed constant in every phase, but may be different from a one phase to another. Thermal conductivity k

and specific heat (c

)

of PCM, and the thermal conductivity at interfaces k

are evaluated in term of the liquid fraction as follows.

k

¼ fk

þ ð1 f Þk

;

ðqc

Þ

¼ f ðqc

Þ

þ ð1 f Þðqc

Þ

; k

¼ k

k ðd

þ d Þ

k

d þ k d

ð11Þ

where d

and d

are distances separating the interface to the neighboring first nodes.

K

and K

are thermal conductivities at neighboring nodes þ and , respectively.

The discretized equations are obtained by integrating the governing equations in a staggered mesh, with M nodes in x direction and N nodes in y direction, using a finite-volume method developed by Patankar et al. [17]. The power law scheme is used for the evaluation of the total flux which combines convective and conductive terms. The SIMPLE routine is used to couple pressure and velocity equations. The general form of the discretized equations is given by the following expression.

a

/

¼ a

/

þ a

/

þ a

/

þ a

/

þ S ð12Þ

E, S, W, N, and P are the east, south, west, north, and central nodes of the control volume. Expressions for the coefficients in Eq. (12) may be found in reference [17].

Here, S is the source term and it includes the value /

at the previous time step.

The energy equation for PCM is solved using the enthalpy fixed-grid technique developed by Voller et al. [16]. The central feature of a such technique is the source term S for the energy equation

S ¼ qDH

ðf

f

Þ DxDy

Dt þ q DxDy

Dt h

ð13Þ

The first term on the right-hand side of Eq. (13) keeps track of latent heat evolution, and its driving element is the local liquid fraction f. This fraction takes the values of 1 in fully liquid regions, 0 in fully solid regions, and lies in the interval ]0,1[ in the vicin- ity of the melting front. In the numerical implementation, its value is determined iteratively from the solution of the enthalpy equation. Hence, after the (i þ 1)th numerical solution of the energy equation over the entire computational domain,

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Eq. (12), for / ¼ h may be written as a

h

¼ a

h

þ a

h

þ a

h

þ a

h

þ qDH

ðf

f

Þ DxDy

Dt þ q DxDy

Dt h

ð14Þ If the phase change is occurring about the (p)th node, i.e., 0 <f < 1, then the (i)

estimate of the liquid fraction needs to be updated such that the left-hand side of Eq. (14) is zero (h

¼ 0); that is,

0 ¼ a

h

þ a

h

þ a

h

þ a

h

þ qDH

ðf

f

Þ DxDy

Dt þ q DxDy

Dt h

ð15Þ Subtracting Eq. (15) from Eq. (14) yields the following update for the liquid fraction at nodes where the phase change is taking place.

f

¼ f

þ x Dt

qDH

DxDy a

h

ð16Þ

where x is a relaxation parameter. The liquid fraction update is applied at every node. To account for the fact that Eq. (16) is not appropriate at every node, the overshoot=undershoot correction is

f ¼ 0: f

0

f ¼ 1: f

1 ð17Þ

is used immediately after Eq. (16).

The iterative solution continues until convergence of the flow and energy fields at every time step is reached. Convergence is declared when the following criteria relating to mass and energy balances, e

, e

, defined as follows, are smaller than 10

and 10

, respectively.

e

¼ max j u

ði; jÞ u

ði; jÞ j ð18Þ

e

¼ 1 ðQ

þ Q

þ Q

þ Q

ð19Þ

where u

and u

are the inlet and outlet mass flow per unit length.

u

ði; jÞ ¼ Uði; jÞDY þ Vði; jÞDX

u

ði; jÞ ¼ Uði þ 1; jÞDY þ Vði; j þ 1ÞDX

1 i M þ 1; 1 j N þ 1 ð20Þ

Sensible heat stored in the wall, liquid PCM, heat sources, and the latent heat stored in the liquid PCM are defined as follows.

Q

¼ P

substrate

q

ðh h

ÞDxDy=Dt

3Q

ð21Þ

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Q

¼ P

liquid PCM

qðh h

ÞDxDy=Dt

3Q

ð22Þ

Q

¼ P

heat source1;2;3

q

ðh h

ÞDxDy=Dt

3Q

ð23Þ

Q

¼ P

PCM

qDH

DxDy

3Q

ð24Þ

Validation

The model was used for validation against experimental data obtained by Ju et al. [13] for a particular configuration of protruding heat sources similar to the present system. This configuration consists of an insulated rectangular enclosure of height H

¼ 9.0 cm, width L

¼ 6.0 cm, and a depth of 6.0 cm. The left wall is made with Plexiglas material and has a thickness of X

¼ 2 cm. It supports three protruding heat sources of height L

¼ 1.5 cm and thickness X

¼ 0.9 cm. The heat flux density at the base of each heat source is equal to 900 W=m

. The lower heat source is placed at a distance L

¼ 0.75 cm from the bottom wall, and the distance between two consecutive heat sources is L

¼ 1.5 cm. Initially, the cavity is filled with a solid PCM (n-octadecane) at its melting temperature T

¼ 28

C. The thermal properties of this PCM are shown in Table 2.

Air layer of 1 cm thickness was provided for PCM expansion during the melt- ing. The numerical code was adjusted to setup conditions using a grid size of 40 60 and a time step of 20 s, and numerical results are compared with experimental data.

It can be seen, taking into account the complexity of the phenomenon, that there is a satisfactory agreement between the present and published experimental data, as shown in Figure 2. The concordance is better at t ¼ 25 min and t ¼ 50 min, except at the lower and upper heat sources that seem to furnish less heat to the PCM in comparison with what predicts the mathematical model. Other reasons are also behind to this slightly difference in melting front positions: the surrounding heat lost and the remove of the insulation during photographing the melting front.

At t ¼ 95 min and t ¼ 110 min the agreement is good but a deviation at the top por- tion was observed. In fact, the melted PCM expands during the melting process and the hot liquid exits over and goes up the free surface of solid PCM and accelerates its melting. This explains the deviation to the right of the experimental melting front.

Table 2.Thermophysical properties ofn-octadecane [18, 19]

Tm

(K)

km,s

(W=m K) km,l

(W=m K)

(cp)m,s

(J=kg K)

(cp)m,l

(J=kg K) qm

(kg=m³) m (kg=m s)

b (K¹)

DH (J=kg) 301.16 0.38 0.15 1,891 2,251 771.2 3.610³ 9.110⁴ 2.43410⁵

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A part of liquid PCM exits the computational domain and breaks the adiabatic condition in the north boundary.

III. RESULTS AND DISCUSSION Range of Parameters

In this study, the volumes of PCM and heat sources were kept constant, and their values are represented by the two constant characteristics lengths l

and l

, respectively,

l

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H

L

3X

L

p ¼ 0:06 m; l

¼ ffiffiffiffiffiffiffiffiffiffiffi X

L

p ¼ 6:7 10

m

During the parametric study, numerical investigations are driven with varying para- meters values from the reference values of the baseline case given in Table 3. The power per unit length delivered by each heat source is kept constant, Q

¼ 60 W=m [20]. In order to kept constant the mass of PCM (constant value of l

), the length l

must be kept constant when varying the electronic component thickness X

So, the height L

must be adjusted using the relationship: L

¼ l

=X

.

Figure 2. Comparison between numerical predicted melting fronts (solid lines) and photographed melting fronts obtained by Ju et al. [13] (dashed lines) att1¼25 min,t2¼50 min,t3¼95 min, and t4¼110 min.

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Values given in Tables 3–5 correspond to dimensions and thermophysical properties frequently used in electronics [20, 21].

Numerical investigations were conducted to check the grid size and time step dependence results using different grid sizes and time steps. The results are shown in Tables 6a and 6b. The analysis of the obtained results shows that a nonuniform 60 80 grids and the time step 20 s were found sufficient to give accurate results.

The small time step of 10 s was used but supply a drastic CPU time without giving appreciable accuracy in numerical results. As it can be seen, changing the grid size from 60 80 to 80 100 leads to a relative change of liquid fraction of 0.55%. A fine grid size near solids was set to give more details for hydrodynamic behaviors near interfaces. For every time step, the resulting algebraic equations are solved using the tridiagonal matrix. The duration of a typical simulation exceeds 9 h in a (CPU 2.6 GHz, 1 Gb RAM) desk computer.

General Trends of Flow Pattern and Heat Transfer

Figure 3 displays the timewise variations of the mean temperatures of the heat sources and liquid fraction. Analysis of Figure 3 shows that the temperature vari- ation goes through three distinct regions: beginning, plateau region, and completion of melting process. At the earlier stage, pure conduction prevails during the melting

Table 3a. Thermophysical properties of modules, substrate, and PCM

Modules (Ceramics) Plate (Al-substrate) PCM (n-eicosane)

b¼8.510⁴K¹

qc¼3,260 kg=m³ DHf¼2.4710⁵J=kg

(cp)c¼740 J=kg qs¼3,900 kg=m³ Tm¼36C

kc¼170 W=m K (cp)s¼900 J=kg (cp)m¼ 2,460 J=kg

Tcr¼75C ks¼19.7 W=m K km¼0.1505 W=m K

qm¼769 kg=m³ mm¼4.1510³kg=m s

Table 3b. Geometrical dimensions corresponding to the reference case

Xc(m) Lc(m) Le(m) Lh(m) Hm(m) Xs(m) Lm(m)

0.003 0.015 0.01 0.03 0.121 0.005 0.035

Table 4.Typical values of thermophysical properties of substrate used in electronic packaging Plate material ks(W=m K) qs(kg=m³) Cp,s(J=kg K) as(m²=s)

Plexiglas 0.19 1,188 1,445 0.1110⁶

Mica 2.4 2,600 800 1.1510⁶

Al-substrate 19.7 3,900 900 5.6110⁶

Boron nitride 30 1,900 1,610 9.8110⁶

AL-Si C 20 3,000 500 1.3310⁵

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process, near the heat sources. The mean temperatures of heat sources increase with a decrease in their corresponding change rates. This corresponds to a decrease in their energy storage and an increase in the extracted heat from heat sources to sub- strate and liquid PCM. Another important remark that can be made is that the liquid fraction increases linearly, which corresponds to a constant exchanged heat flux at the melting front, during this stage. With the melting progress, f 0.1, liquid pockets surrounding the chips enlarge and combine and natural convection estab- lishes, what leads to a slightly chips temperature drop due to the impact of cold cur- rent flowing from the melting front. Similar results are reported by Zhang et al. [22].

During the second stage (plateau region), natural convection develops and intensifies the heat transfer between the liquid phase and its solid boundaries (heat sources, sub- strates, and solid PCM). Therefore, all of the heat generated within the heat sources is practically transmitted to the solid PCM. No sensible heat is stored in the heat sources and, hence, their corresponding means temperatures remain constant throughout this stage (quasi-steady state). During the third stage, heat flux trans- mitted to the liquid solid interface decreases with time which explains the decays of the rate of change in the liquid fraction. The upper heat source has the highest temperature and the bottom heat source triumphs for the lowest temperature.

Figure 4 presents examples of the simulated phase distributions, flow pattern, and isotherms for power per unit length, Q

¼ 60 W=m (corresponding to the

Table 5. Range of parameters used in the present study (Xc,ref¼ 0.003 m,as,ref¼5.6110⁶m²=s)

Parameters Range

as=as,ref 0.02–2.38

Xc=Xc,ref 0.67–4

Table 6a. Sensitivity of the results for varying grid

MN f Deviation (%)

4060 0.3290 –

6080 0.3507 6.6

80100 0.3527 0.55

Dt¼20 s.

Table 6b. Sensitivity of the results for time step

Time step f Deviation (%)

60 s 0.3931 –

35 s 0.3622 7.86

20 s 0.3507 3.14

10 s 0.3502 0.17

MN¼6080.

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reference case). At early stages, the melting front, indicated by the isotherm 36

C, is almost parallel to the substrate and chips faces, indicating that the substrate is almost isothermal. Temperature contours within every heat source reflect typically the heat generation. The heat generated by the modules has two distinct paths.

Figure 3.Timewise variation of average heat sources temperature, maximum temperature, and liquid fraction,tmax¼4,570 s.

Figure 4. (a) Temporalwise evolution of the isotherms, and (b) streamlines,tmax¼4,570 s.

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It reaches the surface by conduction and convects away by the melted PCM stream or it spread in the substrate plate (PC Board). Since the plate and modules have finite thermal conductivity ratios, (k

=k

¼ 131 and k

=k

¼ 1,130), conjugate conduc- tion coupled to the fluid flow and phase change is included in the heat transfer model.

The flow inside the PCM molten layer is weak and does not have a significant impact on the melting process. Thus, the liquid layer presents mainly an additional resistance to the heat transfer from the heated walls (heat sources and substrate plate) to the solid PCM. At this earlier stage, pure conduction prevails during the melting process, near the heat sources and the plate. The mean temperatures of heat sources increase, with a decrease in their corresponding change rates. This corre- sponds to a decrease in their energy storage rates and hence, an increase in the trans- ferred heat from heat sources to substrate and liquid PCM. Another important remark that can be made is that the liquid fraction increases linearly, as illustrated in Figure 3 (t=t

< 0.62), which corresponds to a constant transferred heat flux at the melting front during this stage. As time progresses, the liquid pockets early surrounding the electronic components enlarge and combine and natural convection establishes.

In the liquid PCM, isotherms are distorted incessantly by the liquid motion and the flow becomes relatively strong and natural convection develops, affecting not only the melting rate but also the shape of the remaining solid phase. In fact, Figure 4 shows that the rotating cells erode the remaining solid phase at this second stage. The streamlines become relatively closed to the left wall and melting front, which leads to a relatively fast flows at these boundaries. This intensifies the heat transfer between the liquid phase and its solid boundaries (heat sources, substrate, and solid PCM). Therefore, all of the heat generated within the heat sources is prac- tically transmitted to the solid PCM. No sensible heat is stored in the heat sources and, hence, their corresponding means temperatures reach a plateau region (Figure 3) and remain constant during this stage (quasi-steady state).

The third stage starts when there is no solid PCM around the upper east corner of the enclosure; the melt front moves down. The heat flux transmitted at the liquid=

solid interface decreases with time, which explains the decrease of the change rate in the liquid fraction. At the same time, the temperature field stratifies at the upper region of the cavity. In the center of the cavity, between lower and upper heat sources, the temperature field is homogenized; the driven flow vanishes and the liquid remains pseudostagnant in this location. Upper to the top heat source, the liquid is super heated; whereas, for the core of the bottom region, the rotation of the cells in the lower zones strengths the fluid to absorb more heat from substrate and the bottom electronic component faces and rejects it to the remaining melting front. Consequently, isotherms are obviously inclined and the bottom heat source triumphs for the lowest temperature.

Parametric Study

Effect of modules thickness, X

/X

. Figure 5 displays the isotherms and flow field evolutions with time for various modules thickness X

. The other para- meters are kept constant and equal to their reference values, shown in Table 3.

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The position and the shape of the melting front (isotherm 36

C) depend on the mod- ules thickness X

. Indeed, Figure 5 shows that isotherms and velocity field are altered by the chip protrusion. Another important remark that can be made is that at early

Figure 5. Isotherms and velocity fields for various modules thickness at, (a)t¼800 s, (b)t¼3,800 s, and (c) flow details around heat sources (zooming from Figure 5,Xc=Xc,ref¼4 andt¼3,800 s).

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stages (Figure 5a, t ¼ 800 s), the liquid pockets early surrounding the electronic components enlarge and combine, and natural convection establishes at the upper portion of the cavity for lower thicknesses (X

=X

1). For the highest values of X

(X

=X

> 1) natural convection establishes only around heat sources. One counterrotating cell appears at the top faces of the lower and central heat sources.

The thermal field, in these locations, is similar to the case of the cavity heated from below, which is characterized by the apparition of wall jet and the plume structure (Rayleigh - Be´nard cells - Figure 5a, X

=X

¼ 4). The velocity field is found per- fectly parabolic within the developing boundary layer in the vicinity of all solid walls.

As time progresses, the flow become relatively strong and natural convection devel- ops. Consequently, isotherms become incessantly distorted by the liquid motion. For low modules thicknesses (X

=X

1), Figure 5 shows clearly that at t ¼ 3,800 s no solid PCM remains at the top part of the cavity.

For highest values of thickness (Figure 5, X

=X

¼ 4), however, one isolated solid PCM bloc remains near the upper electronic component and important thermal gradients occur near the upper heat source due to the existence of this nonmelted solid PCM bloc that plays a role of heat sink (cold source). It should be noticed that, in the case of higher thickness X

for which a solid PCM bloc remains at the upper part of the cavity, the streamlines become relatively closed to the top heat source faces. This leads to a relatively fast flow at these boundaries and an enhancement of the heat transfer between liquid PCM and the top heat source.

Figure 5. Continued.

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Figure 6 shows the effect of the heat source thickness X

on the temperature distribution within the substrate. The other parameters are kept constant and equal to their reference values, shown in Table 3. It can be seen from this figure that with the increase of the thickness X

the plate and heat sources are well cooled. When the heat sources are profoundly inserted in the PCM with increasing X

, heat transfer through the exposed plate portion and heat sources faces is enhanced, which allows to a reduction of heat sources temperatures. It can also be seen that increasing the thickness of heat sources X

leads to a reduction in the maximum temperature dif- ference that can be found in the substrate. This difference equals 21

C and 11

C for, X

=X

¼ 0.67 and 4, respectively.

Table 7 summarizes the percentage of the mean heat transferred from various plate portions and exposed heat sources faces to the core flow for various thicknesses X

. The other parameters are kept constant and equal to the reference values, shown in Table 3. Data analysis concludes that whatever the thickness X

is the amount of

Table 7.Plate and modules heat transfer contribution for various heat sources thickness att¼2000 s (the plateau region)

Xc=Xc,ref % Heat generating in modules

0.67 17 20.3 7.23 22.3 7.32 17 4.07

1 18.74 18.3 7.24 21.73 6.78 16.3 9.02

1.2 19.13 17.8 7.28 20.01 6.68 16.2 11.25

2.6 21.07 18.2 6.01 17.12 4.01 15.4 16.8

4 23.4 18.64 5.97 15.03 3.23 15.04 17.85

Figure 6. Substrate temperature profile dependency on the heat sources thicknessXcatt¼4,300 s.

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heat delivered by the bottom plate portion is higher than those delivered by other portions. The reason is that the increase of X

leads to an overheating of liquid in the microcavity (space between two consecutive heat sources), and the heat transfer occurs, at priory, by conduction in these locations. Indeed, zooming around heat sources (see Figure 5c), reveals a relatively weak liquid motion in the microcavities for X

¼ 0.012 m. It should be noted that the increasing of thickness X

leads to an increase on the length of the upper substrate portion (the spaces between heat sources L

and relative position L

are kept constant). This leads, consequently, to an enhancement of heat transfer at this location. Whatever the X

is, no less than 35% of heat generated in modules is transferred through the back faces of the mod- ules to the plate and convected to liquid PCM via the exposed plate faces (Table 7).

The percentage of the generated heat which enters to the fluid through the modules exposed faces decreases with the increase of X

. In fact, varying the module thickness ratio X

=X

from 0.67 to 4, leads to a reduction of the percentage of the heat trans- ferred from the module faces from around 59.6% to 48.71%, but the plate contri- bution is improved from 35.62% to 50.45%. It can also be deduced that the increase in the ratio X

=X

from 0.67 to 4, enhances total heat absorbed by the PCM from 95.22% to 99.16%. Further increase in X

(X

=X

> 4) leads to a slight improvement in the heat transfer. The vertical heat source face furnishes less heat transfer when it is closer to the right insulated boundary of the PCM cavity.

The above observations are confirmed by analyzing Figure 7, which displays the variations of the heat sources average Nusselt numbers Nu and the maximum temperature difference T

T

reached at the plateau region, with the compo- nents thickness ratio X

=X

. The other parameters are kept constant and equal to the reference values shown in Table 3.

Figure 7. Average heat sources Nusselt numbers and maximum temperature dependency on the thickness Xc,t¼2,000 s.

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The average Nusselt number Nu based on the maximum temperature T

is defined as follows.

Nu

¼ 1

ðL

þ 2X

ÞðT

T

Þ

Z

0

k

qT qg ds

s;1;2;3

ð25Þ

It can be seen that the heat sources average Nusselt number Nu

increase with the increase in the thickness ratio X

=X

for all heat sources and reaches a maximum value around X

=X

¼ 1.2, and next, decreases with further increase in the thick- ness ratio X

=X

. The top electronic component is characterized by the lower Nuselt number, for X

=X

< 3.2, and it is rapidly heated and reaches the maximum temperature in comparison with the other components. For the highest thickness ratio, the central heat source registers the lowest Nusselt number, which explains its rapid overheating in comparison with the other heat sources.

Figure 8a shows the influence of the module thickness ratio X

=X

on the maximum heat sources dimensionless working time s

and its corresponding liquid fraction f

. The analysis of this figure shows that the increase of the thickness ratio X

=X

from 0.67 to 4, leads to an increase on the operating time of the mod- ules. In fact, the increase of the modules thickness enhances the heat rate dissipated by the electronic components as stated above, which leads to a decrease in the tem- poral change rate of the modules temperatures. Therefore, the time required by heat sources to reach the limit temperature increases, and the temperature remains almost constant during an important time period without using a fan. The enhancement of the heat transferred by heat sources through their faces and via substrate to the core flow leads to an increase on the mass of the liquid phase of the PCM. An increase in the module thickness ratio has a slight effect on the operating time s

and its cor- responding liquid fraction f

. The dimensionless working time s

and the total melt fraction f

achieved at the end of the melting process (T ¼ T

) vary as the following numerical correlations with respect to the module thickness ratio X

=X

over the range 0.67 to 4.

s

¼ 0:12243 0:02113 X

X

f

¼ 0:99107 0:12307 X

X

ð26Þ

The parity plot (Figure 8b) shows good agreement for the above correlations with the numerical solution. Maximum deviation was found less than 1%.

Effect of substrate thermal diffusivity ratio, a

/a

. The substrate ther- mal diffusivity range considered herein is given in Tables 4 and 5, and corresponds to frequently used plates in electronic assembly [18, 20, 21], such as plexiglas, alumina, epoxy glass, and ceramics composite substrates.

Figure 9 displays the effect of the ratio a

=a

(where a

¼ 5.61 10

m

s

) on the time variations of the maximum temperature of the electronic components.

The other parameters are kept constant and equal to the reference values shown in

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Table 3. Curves illustrating the mean temperatures of electronic components present the same trends and are not reported in this figure for brevity. As shown in Figure 9, for a

=a

1.75, the higher the ratio a

=a

is the smoother the change of the mean chip temperature. In fact, when a

=a

ranges from 0.02 to 1.75, the ability of the substrate to conduct heat relative to the stored thermal energy increases.

Figure 8. (a) Electronic component dimensionless working timesmaxand its corresponding liquid fraction ft,maxas a function of the ratio Xc=Xc,refand (b) the parity plot comparisons between numerical and correlation values.

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Because for this range, the thermal conductivity of the substrate increases as a

=a

increases, the substrate thermal resistance decreases, and the heat transferred from heat sources toward substrate enhances. Consequently, the stored sensible energy in heat sources decreases. This is the reason behind the smoother change of the mean chip temperature. For the highest values of a

=a

(a

=a

¼ 2.37), the ability of the substrate to conduct heat relative to the stored thermal energy is the highest.

However, heat sources transfer less heat to the substrate, in comparison to the case of a

=a

¼ 1.75, due to the relative lower value of the substrate thermal conductivity (k ¼ 30 and 20 W=m K for a

=a

¼ 1.75 and 2.37, respectively). This can be explained by the decrease of the secured working time when the ratio a

=a

¼ 2.37, as indicated later in Figure 12. It can be noticed that the optimal value of the ratio a

=a

corresponding to the best cooling of electronic components with a minimum melted PCM required, during the quasi-steady state, is estimated to be about 1.75.

The corresponding maximum temperature difference T

T

is almost 19.02

C.

These results can also be checked by the analysis of Figure 10, which illustrates the substrate temperature profile at location x ¼ X

=2. The analysis of this figure shows that the substrate contributes more to heat removal toward the PCM as the substrate thermal diffusivity ratio a

=a

increases (a

=a

1.75), which explains the reduced maximum temperature for this case. Also, the higher the ratio a

=a

is the more the substrate is isothermal, and the more the heat sources lose their thermal identity. It is important to note the apparition of a relatively important thermal gra- dients within the substrate for small values of a

=a

(case with insulated-none con- ductive plates, a

=a

< 0.21). In this case, a relatively high temperature was reached near the electronic components junctions which can create a serious split problem (crack of the motherboard).

Figure 9.Time variation of the electronic components maximum temperature (solid lines) and liquid fraction (dashed lines) for various substrate thermal diffusivity ratioas=as,ref.

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The solid–liquid interface shape and position, for different substrate thermal diffusivity ratios a

=a

are depicted in Figure 11. At lower thermal diffusivity ratios the plate doesn’t work and it is considered as an insulator, heat is removed only through the chip faces and low heat transfer rate is driven from the substrate to the PCM. This leads to a rapid overheating of electronic components. The melting is manifested, particularly, near the exposed faces of all chips; this explains the shape of the melt front and the presence of the melt pockets surrounding the chips (similar results are reported in reference [9]). As the ratio a

=a

increases (a

=a

1), the melting zones combine and become larger. The melt sprawls rigorously along the hot wall, appearing to be a situation similar to the melting inside an isothermal wall.

However, for further increase in a

=a

(a

=a

¼ 1.75), the effect of the substrate thermal diffusivity on the melt front position becomes less important, as illustrated in Figure 11.

Figure 12 displays the variations of the dimensionless working time s

made by the electronic components to reach the critical temperature (T

75

C) for dif- ferent values of the ratio a

=a

. Analysis of this figure shows that the increase of the substrate thermal diffusivity ratio a

=a

has a substantial impact on the elec- tronic components secured working time s

. Indeed, when the ratio a

=a

varies from 0.08 to 1, the dimensionless working time s

improves while the passing of s

¼ 0.024 (t

¼ 1,086 s) to s

¼ 0.102 (t

¼ 4,616 s). But from a

=a

> 1, 1, the increase of s

occurs with a relatively lower rate of change. For higher ratios, a

=a

1.75, the dimensionless working time s

decreases. The analysis of Figure 12 clearly shows that the maximum dimensionless working time (s

¼ 0.1055, t

¼ 4,750 s) corresponds to the substrate thermal diffusivity ratio

Figure 10. Substrate temperature profile for various substrate thermal diffusivity ratioas=as,ref,t¼1,800 s, x¼Xs=2.

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Figure 11. Effect of the substrate thermal diffusivity ratioas=as,refon the melt front shape and position, t¼2,000 s.

Figure 12.Dimensionless working timesmaxand the corresponding total liquid fractionfmaxas a function of the ratioas=as,ref.

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a

=a

¼ 1.75 (boron nitride). Because this dimensionless working time varies with only 3% when the value of a

=a

varies from 1 to 1.75, it’s practical to select the substrate for which a

=a

¼ 1 (al-substrate) due to its relative low cost.

Analysis of Figure 12 also shows an increase of the melted PCM volume with the ratio a

=a

for a

=a

1.75. For a

=a

¼ 2.37, as stated above, relatively lower heat transfer is extracted from solid walls (heat sources and substrate) toward the PCM. According to this, lower total melt fraction is obtained in comparison to the case of a

=a

¼ 1.75.

VI. CONCLUSION

In the current study, the melting of a phase change material (PCM) in a rectangular cavity heated with three protruding heat sources mounted on a vertical conducting plate has been studied numerically. Numerical investigations were carried out to study the effects of the heat sources thickness and substrate thermal diffusivity on the thermal performance of the cooling PCM-heat sink. Two correla- tions have been developed for the maximum working time and the corresponding melt fraction in term of the module thickness ratio. The results also showed the existence of an optimal substrate thermal diffusivity ratio that maximizes the work- ing time s

required by the chips to reach the critical temperature (T

). The maximum duration of the plateau region, corresponding to the stable values of chips temperatures, was obtained for this optimal value of the substrate thermal diffusivity ratio. A very important variation in temperature gradient was found in the substrate layer near the electronics junction for the lowest thermal diffusivity ratios. This can create a serious splits (crack) problem. In future numerical simula- tions, the effects of other control parameters will be investigated in order to find the optimal values that maximized the thermal performance of the PCM-heat sink.

The approach developed herein can be used in the design of PCM-based cooling systems.

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