Numerical study of free convection dominated melting in an isolated cavity heated by three protruding electronic components

Numerical Study of Free Convection Dominated Melting in an Isolated Cavity Heated by Three

Protruding Electronic Components

Mustapha Faraji and Hamid El Qarnia

Abstract—This paper presents the results of a numerical study of the melting and natural convection in a rectangular enclosure heated with three discrete protruding electronic components (heat sources) mounted on a conducting vertical plate. The heat sources generate heat at a constant and uniform volumetric rate. A part of the power generated in the heat sources is dissipated in phase change material (PCM, n-eicosane with melting temperature, T

m

= 36 °C) that filled the enclosure. The advantage of using this cooling strategy is that the PCMs are able to absorb a high amount of heat generated by electronic components without activating the fan. To investigate the thermal behavior of the proposed cooling system, a mathematical model, based on the mass, momentum, and energy conservation equations, was developed. The governing equations are next discretized using a finite volume method in a staggered mesh, and a pressure correction equation method is employed for the pressure–velocity coupling. The energy conservation equation for the PCM is solved using the enthalpy method. The solid regions (substrate and heat sources) are treated as fluid regions with infinite viscosity.

A parametric study was conducted in order to optimize the thermal performance of the heat sink. The optimization involves determination of the key parameter values that maximize the time required by the electronic component to reach the critical temperature (T < T

_cr

).

Index Terms—Passive cooling, phase change material, protrud- ing electronic component, thermal control.

Nomenclature

b, C Constant values (9).

c p Specific heat (J·kg ^{− 1} ·K ^{− 1} ).

f Liquid fraction.

g Gravity (m·s ^{− 2} ).

h Enthalpy (J · kg ^{− 1} ).

h c Local heat transfer coefficient (W · m ⁻² · K ⁻¹ ).

H m Height of the PCM enclosure (m).

k Thermal conductivity (W·m ⁻¹ ).

l _o Characteristic length (m), l _o = √

L _m H _m − 3 X _c L _c . l _EC Constant length (m), l _EC = √

X _c L _c .

Manuscript received November 5, 2008; revised February 23, 2009 and July 7, 2009. First version published November 10, 2009; current version published March 10, 2010. Recommended for publication by Associate Editor P. Dutta upon evaluation of reviewers’ comments.

The authors are with the Department of Physics, Fluid Mechanics and Energetics Laboratory, Cadi Ayyad University, Marrakesh 40000, Morocco (e-mail: farajimustapha@yahoo.fr; elqarnia@ucam.ac.ma).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCAPT.2009.2029086

L c Electronic component height (m).

L e Space between two consecutive electronic components (m).

L h Position of the bottom electronic component (m).

L m Width of the PCM enclosure (m).

Nu Average heat source Nusselt number, Nu = h l o /k m,l . p Constant pressure (Pa).

Q Heat generation per unit length of heater (W·m ⁻¹ ).

q Local heat flux density (W·m ⁻² ).

¯

q" Non-dimensional local heat flux density, q /(3Q / l _o ).

T Temperature (°C).

t Time (s).

X c , X s Electronic component, plate thicknesses (m).

Subscripts

1, 2, 3 bottom, median, and upper electronic components, re- spectively.

cr Critical value.

c, EC Electronic component.

f Fluid.

l Liquid.

m, l Melting, liquid PCM.

s Substrate, solid.

Greek symbols

α Thermal diffusivity (m ² ·s ⁻¹ ), α = k/ρc.

β Thermal expansion factor (K ⁻¹ ).

τ _max Maximum dimensionless working time, τ _max = α _m,l t _max /l ² _o .

δ i δ Kronecker symbols, distance (m).

µ Dynamic viscosity (kg·m ^{− 1} ·s ^{− 1} ).

ν Kinematic viscosity (m ² ·s ^{− 1} ).

H f Latent heat (J·kg ^{− 1} ).

t Time step (s).

ρ Density (kg · m ^{− 3} ).

ψ Stream function, ψ = (1/α m,l )

udy − vdx.

I. Introduction

C OOLING needs of electronic devices have surpassed the capabilities of conventional methodologies such as natural convection or forced convection (fan-blown air cool- ing). The fundamental problem of interest for the cooling of electronic equipment is maintaining safe temperatures in the electronic devices [1]. Considering the effects of tem-

1051-8215/$26.00 c 2010 IEEE

perature on the reliability of the electronic components, the thermal design must be able to keep the working temperature of such devices below their respective allowable maximum temperatures (generally ranging from 75 °C to 120 °C) at all times during normal operation. Lin et al. [2] intended to improve the performance of cooling air flow fan to enhance the heat dissipating ability for a heat sink assembly. Their experimental results showed some promising findings for the thermal management of notebook and desktop computers.

Application of melting process of phase change materials (PCMs) in the cooling management system for computer elec- tronic components has received significant attention in recent years and may be a promising alternative cooling management system for electronic devices. Several investigations were conducted on cooling electronic devices using PCMs. Faraji and El Qarnia [3] and [4] analyzed the active/passive cooling of microprocessors using a hybrid heat sink. This consists of a rectangular cavity filled with PCM (SunTech P111) attached to a conventional aluminum fan with a constant and uniform heat flux density at the base of the microprocessor. Numerical investigations were conducted in order to optimize the thermal performance of the heat sink by maximizing the safe working time (time required by the heat sink before reaching the al- lowable maximum temperature). The obtained optimal config- uration was next subject to the cyclic operating mode (charge, fan “off” and discharge, fan “on”), and the established periodic mode was reached after three cycles of working. The cooling of mobile electronic devices, such as personal digital assistants (PDAs), using a heat storage unit (HSU) filled with n-eicosane, was also experimentally studied by Tan and Tso [5]. The PCMs absorb the heat dissipated by the chips and can maintain their temperatures below the allowable service temperature of 50 °C, for 2 h of transient operations of the PDA. Zhang et al. [6] experimentally investigated the melting process of n-octadecane that is discretely heated by heat sources. At a constant rate, from one side of an enclosure, the other surfaces were adiabatic. One of the important experimental results shows that, compared to natural cooling by air, the temperature rise of a component could be reduced as much as 50%

when PCM melting is used for cooling [6]. Hodes et al. [7]

experimentally studied the feasibility of transient thermal management of handsets using (PCMs). They also conducted a numerical investigation in order to compute the transient and steady-state heat transfer rates by natural convection and radiation from the handset to the environment. They ana- lyzed the effects of PCM, power supplied to the handset, and handset orientation in time required by the handset to reach the maximum temperature and recovery time. Krishnan et al. [8] proposed a hybrid heat sink, which consists of a plate fan heat sink with the tip immersed in a rectangular enclosure filled with a PCM. To the author’s knowledge, numerical study of the PCM-based heat sink using for cooling protruding substrate mounted electronic devices has not been examined up to now. The present paper considers the problem of the natural convection dominated melting in a rectangular enclosure, which is heated with three protruding heat sources generating constant and uniform volumetric power. The proposed problem is numerically studied. The objective of this investigation is

Fig. 1. Schematic view of the physical model.

to analyze the thermal and dynamic behaviors of the proposed cooling system and examine the effects of the volumetric power, positions, and thicknesses of the heat sources on the maximum temperature, the percentage contribution of sub- strate heat conduction on the total removed heat, and substrate temperature profile. A correlation is developed for the secured working time and its corresponding liquid fraction.

II. Analysis and Modeling

A. Physical Model

A schematic view of the physical model studied in this paper is shown in Fig. 1. It consists of a rectangular enclosure, with height H _m and length L _m containing a PCM. The left wall of the cavity consists of a thin substrate layer with thickness X _s , and supporting three electronic components (EC), which are simulated by three identical rectangular heat sources (EC) i=1,2,3 . Each protruding heat source has thickness X c and height L c . The distance between two successive heat sources is L e and the position of the lower heat source is L h . Power per unit length Q is internally generated within every heat source, at a constant and uniform magnitude. The heat sources dissipate heat to the PCM through their exposed faces. The substrate participates by absorbing, diffusing, and transferring the heat to the PCM. The particularity of the present cooling strategy is that no fan acts until all the PCMs melt or if one of the three electronic components reaches the critical temperature, T cr .

It can be noticed that, in most recent desktop computers

assembly motherboards, there are three electronic components

directly related to the generation of heat: CPU, onboard AGP

controller, and Chipset. These components contribute to the

greatest portion of the amount of the total heat generated

within the computer central unit case. In addition, a wide

range of computer cases is made in Tower position. These

are the reasons and motivations behind choosing the current

configuration (three protruding electronic components attached

to a vertical substrate).

B. Governing Equations

The flow is assumed to be 2-D, Newtonian, laminar, and incompressible. The thermal properties of materials are con- stant in the temperature range under study. Thermal properties of the PCM may be different from one phase to the other.

In this paper, the thermal properties of both phases are equal, except the thermal conductivities. The phase change is isotherm and the PCM is assumed initially solid at its melting temperature, T o = T m . The Boussinesq approximation is used and a reference temperature is selected equal to the melting temperature, T _ref = T m . The density of the PCM is taken as reference density, ρ _ref = ρ m . The equations governing heat transfer and flow in the studied configuration are as follows:

Continuity equation

∂(ρu)

∂x + ∂(ρv)

∂y = 0. (1)

Momentum equation for x and y direction

∂

∂t (ρu) + ∂

∂x (ρuu) + ∂

∂y (ρvu)

= − ∂p

∂x + ∂

∂x

µ ∂u

∂x

+ ∂

∂y

µ ∂u

∂y

+ S u (2)

∂

∂t (ρv) + ∂

∂x (ρuv) + ∂

∂y (ρvv)

= − ∂p

∂y + ∂

∂x

µ ∂v

∂x

+ ∂

∂y

µ ∂v

∂y

+ S v . (3) Energy transport equation

∂

∂t (ρh) + ∂

∂x (ρuh) + ∂

∂y (ρvh)

= ∂

∂x k

c _p

∂h

∂x

+ ∂

∂y k

c _p

∂h

∂y

+ S T (4)

where h = _T

T

m

c p dT +h(T _m ) S _u = − C (1 − f)²

(f ³ +b) u, S v = − C (1 − f )²

(f ³ + b) v + ρ _ref gβ(T − T m ) S T = δ ₁

− (1 − δ ₂ )ρH f

∂f

∂t + δ ₂ Q X c · L c

. (5)

S u and S _v are source terms used for velocity suppression in the solid regions (solid PCM, substrate, and heat sources).

One of the common models for velocity suppression is to introduce a Darcy-like term [10] (C = 10 ²⁵ kg·m ⁻³ ·s ⁻¹ and b = 0.005 are used). The same full set of governing equations throughout the entire enclosure governs conjugate heat transfer in both the liquid and solid regions while taking a large value of the viscosity in solid regions. The conductivity and the step function δ are set as follows:

δ ₁ =

0, for the substrate

1, for electronic components and PCM (6) δ ₂ =

0, for PCM

1, for electronic components k =

⎧ ⎨

⎩

k m , for PCM k s , for substrate

k c , for electronic components.

(7)

The thermal conductivity k _m of the PCM and the thermal conductivity at interfaces k i are expressed as follows:

k m = f k m,l + (1 − f ) k m,s (8) k i = k ₊ k ₋ (δ ₊ + δ ₋ )

k ₊ δ ₋ + k ₋ δ ₊ (9) where δ ₊ and δ ₋ are distances separating the interface to the first neighboring nodes, ‘+’ and ‘−’. k ₊ and k ₋ are the thermal conductivities at nodes ‘+’ and ‘−’, respectively.

C. Boundary Conditions

At the interfaces between two different materials (1) and (2) (substrate, PCM, or heat sources)

k ₁ ∂T

∂η

_interface = k ₂ ∂T

∂η

interface

, T ₁ = T ₂ (where η ⊥ interface).

(10) At the adiabatic walls

∂T

∂η

wall

= 0. (11)

No slip and nonpermeability at the solid interfaces and walls

u = v = 0. (12)

D. Initial Conditions

u(x, y, t = 0) = v(x, y, t = 0) = f (x, y, t = 0) = 0

T (x, y, t = 0) = T m . (13)

The discretized equations are obtained by integrating the governing equations in a staggered mesh, with M nodes in the (x) direction and N nodes in the (y) direction, using a finite volume method developed by Patankar [11]. The power law scheme is used to evaluate 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 p φ p = a E φ E + a S φ S + a W φ W + a N φ N + S (14) where φ is a general variable (φ = u, v, h ). P, E, S, W, and N, are the center, east, south, west, and north nodes of the control volume, respectively. An expression for coefficients appearing in 14 may be found in [11]. Here, S is the source term and it includes the value φ _P ^old , at the previous time step.

It should be noted that the energy equation for the PCM is formultaed using the enthalpy fixed-grid technique [10]. The central feature of this technique is the source term S for the energy equation

S = ρH _f (f ^old − f ⁱ ) xy

t + ρ xy

t h ^old _p . (15) The first term on the right-hand side of (15) 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 vicinity of the melting front. Its value is determined

TABLE I

Sensitivity of the Results for ( a ) Varying Grid, t = 20 s and ( b ) Time Step, M × N = 60 × 80 (a)

M × N Maximum working time, t

max

(s) Deviation (%) Maximum liquid fraction, f

t,max

Deviation (%)

40 × 60 9200 – 0.895 –

60 × 60 10 300 11.95 0.947 5.81

60 × 80 10 960 6.31 0.982 3.69

80 × 120 10 980 0.18 0.983 0.10

(b)

Time step (s) Maximum working time, t

max

(s) Deviation (%) Maximum liquid fraction, f

t,max

Deviation (%)

60 8760 – 0.791 –

30 10 060 14.84 0.885 11.88

20 10 960 8.94 0.982 10.96

5 10 965 0.041 0.987 0.509

iteratively from the solution of the enthalpy equation. Hence, after the (i + 1) ^th numerical solution of the energy equation over the entire computational domain (14) with φ = h, it may be written as

a _p h _p = a _E h _E + a _S h _S + a _W h _W +

a _N h _N + ρH _f (f ^old − f ⁱ ) xy

t + ρ xy

t h ^old _p . (16) If the phase change is occurring at (P) ^th node, i.e., 0 < f <

1, then the (i + 1) ^th estimate of the liquid fraction needs to be updated such that the left hand side of (16) is zero, (h p = 0);

that is

0 = a E h E + a S h S + a W h W + a N h N + ρH f (f ^old − f ⁱ⁺¹ ) xy

t + ρ xy

t h ^old _p . (17) Subtracting (17) from (16) yields the following update for the liquid fraction at nodes where the phase change is taking place:

f ⁱ⁺¹ = f ⁱ + ω t

ρH f xy a P h _P (18) where ω is a relaxation parameter. The liquid fraction update is applied at every node. To account for the fact that (18) is not appropriate at every node, the overshoot/undershoot correction

f = 0, f ≤ 0

f = 1, f ≥ 1 (19)

is used immediately after (18).

The resulting algebraic equations are solved, for every time step, using the Tri-Diagonal Matrix iterative method. The model was implemented by developing a personnal computer code in Fortran language. Typical execution times for the fine mesh runs exceed 9 h in a (CPU 2.6 GHz, 1 GB RAM) desk computer.

Numerical investigations were conducted to check the grid size and time step dependence results using different grid sizes and time steps. The power per length is Q = 30 W·m ^{− 1} (Ra = 2.508 × 10 ⁹ ), and the other parameters are given in Tables II and III. Results are shown in Table I(a) and (b). The analysis of such results shows that a nonuniform 60 × 80 grid and the time step of 20 s are sufficient to give accurate results.

The small time step of 5 s was used but, supplied a drastic CPU time, without giving appreciable accuracy in numerical

results. As can be seen, changing grid size from 60 × 80 to 80 × 120 leads to relative changes of the maximum working time, t _max , and its corresponding liquid fraction, f _t,max , of 0.18% and 0.1%, respectively. A fine grid size near solids was set to give more details for hydrodynamic and thermal behaviors near interfaces.

III. Results and Discussion

This section presents the results of the numerical investi- gations of natural convection dominated melting of a PCM (n-eicosane, with a melting temperature, T m = 36 °C) in the rectangular enclosure shown in Fig. 1. During the melting process, the fan does not act and the heat sources are passively cooled by transferring a part of the heat generated to PCM;

the other part is stored in the solid walls (heat sources and substrate). Initially the solid PCM, heat sources, and substrate are at the temperature, T o = T m . The power per length Q ranges from 7.5 W·m ⁻¹ to 100 W·m ⁻¹ . This range covers the power generated by microprocessors of the first generation (1.6 GHz), second generation (2 GHz), and third generation (3.6 GHz). The thermo-physical and geometric parameters are summarized in Tables II and III.

The CPU temperature rises and reaches the limiting temper- ature, T cr ∼ 75 °C, since it is typically the highest operating temperature permissible for most chips to ensure reliability [9].

The time t cr required by the electronic components to reach this limit temperature depends on the geometric and operating parameters and thermo-physical properties of the PCM, heat sources, and substrate. In view of this, numerical simulations were conducted during the limiting time t _cr or until the liquid fraction f approaches 1. In this paper, the effects of the positions L _h and thickness X _c of the heat sources, and the heat generated per unit length Q (or Rayleigh number, Ra = g β l ³ _o (3Q

/ k m,l )/υ m,l α _m,l ) are examined. The range of each of these control parameters is given in Table IV.

A. General Trends of Flow and Heat Transfer

Examples of simulated temperature and flow patterns, for

power per unit length Q = 30 W · m ⁻¹ (Ra = 2.508 × 10 ⁹ ), at

different stages (initial, medium, and advanced stages of the

melting process), are shown in Figs. 2 and 3. The values of

the remaining parameters are summarized in Tables II and III.

TABLE II

Thermo Physical Properties [9], [12] and [13]

Modules (ceramics) ρ

c

= 3260 kg · m

⁻³

c

p

c

740 J · kg

⁻¹

k

c

= 170 W · m

⁻²

· K

⁻¹

T

cr

= 75 °C Plate (Al-substrate) ρ

s

= 3900 kg · m

⁻³

c

p

s

= 900 J · kg

⁻¹

k

s

= 19.7 W · m

⁻²

· K

⁻¹

PCM (n-eicosane) β = 8.5 × 10

⁻⁴

K

⁻¹

H

f

= 2.47 × 10

⁵

J · kg

⁻¹

T

m

= 36 °C

c

pm

= 2460 J · kg

⁻¹

k

m

= 0.1505 W · m

⁻²

· K

⁻¹

ρ

m

= 769 kg · m

⁻³

µ

m

= 4.15 × 10

⁻³

kg · m

⁻¹

· s

⁻¹

TABLE III

Geometric Parameters of the Cavity, the Plate, and the Components (m)

X

c

Lc L

e

L

h

X

s

L

m

H

m

0.003 0.015 0.01 0.03 0.005 0.035 0.123

TABLE IV

Range of the Control Parameters Used in This Paper

Parameters Range

Rayleigh number, Ra 6.27 × 10

⁸

(Q

= 7.5 W · m

⁻¹

) to 5.01 × 10

⁹

(Q

= 60 W · m

⁻¹

) Position, L

h

0.005 m to 0.05 m

Thickness, X

c

0.002 m to 0.012 m

The corresponding liquid fractions of these three stages are 0.04, 0.4 and 0.95, respectively. Fig. 3 displays the time variations of the mean temperatures of heat sources and liquid fraction. The analysis of such a figure shows that the temperature variation goes through three distinct regions. At early stages, it can be seen from Fig. 2 that the melting front, indicated by the isotherm 36 °C, is almost parallel to the substrate and chips faces, indicating that the substrate is already almost isothermal. Within every electronic component, isotherm contours are concentrically located; this configuration is typical to the volumetric heat generation sources. The flow inside the molten PCM 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 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 rate of changes. As a result, a decrease in the energy storage rate of each heat source and an increase in the heat transferred from heat sources to substrate and liquid PCM occur. Another important remark that can be made is that the liquid fraction increases linearly (Fig. 3, t < 1000 s), which corresponds to a constant exchanged heat flux at the melting front during this stage. The second stage starts when the liquid pockets, earlier

surrounding the electronic components, enlarge and combine.

As a result, natural convection establishes.

In the liquid PCM, the isotherms are distorted incessantly by the liquid motion as the flow becomes relatively strong and natural convection develops, affecting not only the melting rate, but also the shape of the remaining solid phase. Indeed, Fig. 2(b) shows that the rotating cells erode the remaining solid phase. The streamlines become relatively closed to the left wall and the melting front, which leads to a relatively fast flow at these boundaries. This intensifies the heat transfer between the liquid phase and its solid boundaries. Therefore, all the heat generated within the heat sources is approximately transmitted to the solid PCM. No sensible heat is stored in the heat sources and, hence, their corresponding mean temperatures reach a plateau region (Fig. 3) and remain constant throughout this stage (quasi-steady-state). The third stage starts when no solid PCM remains attached to the right wall, at the upper part of the enclosure. During this stage, the heat flux transmitted to the melting front decreases with time. As a result, the liquid fraction continues its increase with a decrease in the slope of the liquid fraction curve. At the same time, the temperature field stratifies within the cavity. The driven flow also vanishes and the liquid motion becomes almost weak at that location.

However, at the upper and lower parts of the cavity, rotating cells persist due to the horizontal temperature gradients at the interface left wall/liquid PCM. It should be noted that the rotation of the cells, at the core of the bottom region, causes an enhancement of the heat transfer between the liquid PCM and the left wall (lower part of the substrate and bottom electronic component). Consequently, isotherms are obviously inclined and the bottom heat source triumphs for the lowest temperature.

B. Parametric Study

1) Effect of the Rayleigh Number, Ra: Fig. 4 displays

the time wise variations of the mean temperatures of the

electronic components and the total PCM melt fraction, f t ,

for various Rayleigh numbers, R a . It should be noted that the

Rayleigh number, Ra, varies in the range indicated above,

in Table IV, and its variation is due to that of the power

per unit length Q when the latter varies from 7.5 W · m ⁻¹

to 60 W · m ⁻¹ . Analysis of Fig. 4 shows that, for this range

of Rayleigh numbers, the temperature variations go to three

distinct regions. In the first stage, pure conduction prevails

near the chips, and the corresponding temperature variations

are nearly linear with time. During the second stage (f t > 0.1),

natural convection develops, and almost all heat dissipated by

the heat sources is transferred to the melting front. The heat

transfer is quasi-steady-state (for example, for Ra = 6.27×10 ⁸ ,

the curves reach a temporary plateau region limited by the

interval [t s1 , t _s2 ]) and the mean temperatures of heat sources

remain nearly constant during this interval. Fig. 4 shows, also,

that the duration of the quasi-steady-state region and the time

to reach the plateau region, t _s1 , decrease with the increasing

Q . Furthermore, the quasi-steady-state temperature increases

sharply with Ra. The last stage starts when the liquid fraction

reaches a value of almost 0.58. During this stage, the electronic

component temperature T ₃ and the melt fraction f _t increase

Fig. 2. (a) Isotherms (b) Streamlines Ra = 2.508 × 10

⁹

(Q

= 30 W · m

⁻¹

).

with a decrease of the slope of the liquid fraction curve. It can also be seen that the critical temperature was reached faster for highly powered chips, because the PCM is rapidly melted.

For example, after 4500 s, the liquid fraction f t was found to be 0.14 and 0.87, for Ra = 6.27 × 10 ⁸ and 5.01 × 10 ⁹ , respectively.

Fig. 5 illustrates the streamline and isotherm plots describ- ing the flow and temperature fields for two values of Rayleigh number, Ra = 2.508 × 10 ⁹ and 5.01 × 10 ⁹ , at t = 2800 s.

All other parameters are kept constant and equal to their reference values in Tables II and III. As it can be expected, the two-dimensionality of the temperature field is evident and the temperature distribution is nonuniform. Natural convection flow is clockwise, upward near the hot walls (heat sources and

Fig. 3. Temporal variations of the average temperatures of heat sources, maximum temperature, and liquid fraction. Ra = 2.508 × 10

⁹

(Q

= 30 W · m

⁻¹

).

Fig. 4. Temporal variations of the average electronic components tempera- tures and the melt fraction f

t

.

substrate) and downward in the vicinity of the solid–liquid interface (cold wall). Smaller PCM liquid cavity is obtained for a low Rayleigh number, Ra = 2.508 × 10 ⁹ . The flow is multicellular and the maximum value of the stream function

| ψ _max | increases with the increase in the Rayleigh number due to the increase in the driven temperature difference between the left hot wall and the melt front, (T ) max = T _max − T m . For example, driving temperature differences were found to be around 9.7 °C and 21 °C, while the total liquid fraction f t

values were 0.21 and 0.62 for Ra = 2.508×10 ⁹ and 5.01×10 ⁹ ,

respectively. Fig. 5 also shows that for Ra = 5.01 × 10 ⁹ , the

temperature field rapidly stratifies on the top region of the

cavity, whereas for a low Rayleigh number, isotherms remain

inclined. The upper electronic component achieves the highest

temperature for the two cases but it is colder for the lower

Rayleigh number (T ₃ = 57 °C for Ra = 5.01 × 10 ⁹ and

T ₃ = 45.3 °C for Ra = 2.508 × 10 ⁹ ). It can be seen that

the cells are trapped and distorted by the corners of the heat

Fig. 5. Isotherm and streamline contours at t = 2800 s.

Fig. 6. Substrate temperature profile dependency on the Rayleigh number, Ra.

sources, and their number is reduced with an increase in the Rayleigh number.

Fig. 6 depicts the effect of the heat source power level on the temperature profile in the substrate. The curves represent the temperature at mid-width of the substrate. The analysis of such a figure shows that the substrate is overly isotherm for a low Rayleigh number, Ra = 6.27 × 10 ⁸ (Q = 7.5 W·m ⁻¹ ).

This arises because the increase of the heat source power leads to an increase in the heat transferred by conduction through the substrate, and then an increase in the thermal gradient.

The overall temperature difference which can be found in the substrate for the low power level (Q = 7.5 W·m ^{− 1} ) is less than 1.5 °C. With the increase in power level (Ra = 5.01 × 10 ⁹ , Q = 60 W · m ⁻¹ ), natural convection intensifies and the convection currents force the liquid PCM to extract more heat from the lowest electronic component and redirect that heat to the top regions of the cavity. As a result, the temperature of the

Fig. 7. Temporal variations of the heat sources average Nusselt number, for Ra = 2.508 × 10

⁹

(Q

= 30 W · m

⁻¹

).

electronic component located at the top of the cavity rises. In this case, Fig. 6 shows that the substrate is not isothermal and the temperature difference within the substrate exceeds 12 °C at t = 2800 s, and the mean temperature of the chips (EC) 3 , increases faster toward the limit temperature.

In order to calculate the heat flux from the surface of each heat source to the PCM, using the maximum temperature of the heat sources T _max , the following expression is used for the local heat flux density:

q = h _c (T _max − T _m ) (20) where h c is the local heat transfer coefficient. The average Nusselt number for each heat source is expressed as follows:

Nu ¯ _1,2,3 = − 1

(L _c + 2 X _c ) (T max − T _m )

×

L

_c

+2 X

_c

0

k _i ∂T

∂η ds i

s

_1,2,3

(21) where s _i is the heat source periphery distance (0 ≤ si ≤ L _c + 2X c ), and k i is the thermal conductivity at the interface.

Fig. 7 displays the temporal variations of the average Nusselt number of each heat source. Analysis of such a figure shows that the mean heat source Nusselt number decreases, increases, and reaches a plateau as melting progresses. The presence of this plateau in this figure shows that all the heat generated is absorbed by the melting front. Also, it can be de- duced from the analysis of such a figure that the average Nus- selt number, Nu, has a variation which is the opposite of that of T _max (which is, in general, close to the average temperature of the upper heat source). The variation of Nu is not, however, proportional to 1/T _max . Indeed, analysis of (21) shows that the average Nusselt number Nu is proportional to the product of 1/T _max and L

c

+2X

c

0 [k i (∂T /∂η)ds i ] s . This latest term refers to

the heat flux dissipated by the three faces of each heat source

to the liquid PCM, and it is not constant. This is because a

part of the heat generated in the heat source is stored in itself

and the other part is conducted through the substrate.

Fig. 8. Local temperature and non-dimensional local heat flux density dis- tributions at the left hot wall (interface heat sources substrate/liquid PCM), for Ra = 2.508 × 10

⁹

(Q

= 30 W · m

⁻¹

).

Local temperature distributions at the left solid-liquid inter- face were depicted in Fig. 8, at various times. Moving along the interface, from the bottom to the top of the enclosure, the temperature oscillates. A local minimum near the upper east corner (D) of the bottom heat source is observed. The reason behind this is that the corner which corresponds to the thin location of the developing thermal boundary layer is crowded by the ascending liquid PCM. Another observation that can be made is the uniformity of the temperature in the heat sources regions, which is due to their relatively high thermal conductivity. Analysis of Fig. 8 shows that the temperature, at any location, increases with time. At quasi-steady-state, the temperature at any location remains, approximately, constant.

This is shown by the closer curves.

Interesting heat transfer trends are also revealed when the distribution of the local heat flux density along the left interface between electronic component plate and liquid PCM, at the selected instants, is examined. As depicted in Fig. 8, at power level Q = 30 W·m ⁻¹ (Ra = 2.508 × 10 ⁹ ), the distribution is highly nonuniform and presents local maxima and minima. At early stages (t ∼ 320 s), because the melting process occurs only in the vicinity of the modules and in the regions between them, a weak heat transfer is manifested at lower and upper regions of the plate (A–B), (M–N). The reason for this is that at earlier stages, due to thermal inertia of the plate, heat is not sufficiently conducted from the modules to the upper and lower-most plate regions to sustain high plate temperature. The local maxima and minima that occurred near electronic components corners are associated to boundary layer thinning (C, D, G, H, K, L) and stagnant fluid in space between modules. Maximum heat flux occurs at the right faces of each module (CD, GH, KL) since this surface is parallel to the main stream direction. The interfacial heat flux density decays along heat sources’ right faces. In the regions between heat sources, for which the fluid is quasi-stagnant, heat flux density reaches its minimum value.

2) Effect of the Position, L h : Because the liquid PCM transfers heat at the melting front while descending, lower temperatures are observed at the bottom of the cavity. There-

Fig. 9. Effect of L

h

on the temporal variations of the mean temperature of the upper electronic component (EC

3

), for Ra = 5.01 × 10

⁹

.

fore, more heat is exchanged between the lower electronic component and liquid PCM, which explains the lower tem- perature of (EC 1 ). In ascending direction, however, the liquid becomes more and more hot, which reduces the heat transfer between the electronic components and liquid PCM. This is the reason behind the increase in the temperature of the upper electronic component placed at the top of the cavity (EC ₃ ). Analysis of Figs. 9 and 10 showed that the more the heat sources are positioned and assembled in the bottom of the cavity, the better they are cooled and their temperatures are reduced. It should be noted that the rea- son behind the criss-crossing of the temperature curves in Fig. 10 is that, as heat sources ascend, by increasing the distance L h , the local temperatures maxima move in the ascendant direction, which leads to the cooling of the lower substrate portion (0 < y < L h ) and heating of the upper one (L h +2L e +3L c < y < H m ). Also, Fig. 9 shows that, while reducing the chip position L h from 0.05 m to 0.005 m, the secured working time, defined as the time taken by electronic components to reach the critical temperature, T ∼ T cr , is improved from t cr = 3400 s to t cr = 5400 s. Consequently, when heat sources are positioned near the bottom location of the PCM enclosure, the secured working duration can be relatively extended.

3) Effect of the Module Thickness, X _c : In this paper, the volumes of the PCM and heat sources were kept constant.

Their values are represented by the two constant lengths, l ₀ , and, l EC , respectively, with

l _o =

H _m L _m − 3X c L _c = 0.06 m l EC =

X c L c = 6.7 × 10 ⁻³ m. (22) In order to maintain the quantity l EC constant when varying the electronic component thickness X c , the height L c must be adjusted using the relationship L c = l ² _EC /X c . The thickness X c

ranges from 0.002 m to 0.01 m, corresponding to dimensions frequently used in electronics [9].

Fig. 11 shows the effect of the heat source thickness Xc

on the temperature distribution within the substrate at the

Fig. 10. Effect of L

h

on the temperature profile within the substrate at t = 3800 s, x = X

s

/2, for Ra = 5.01 × 10

⁹

.

Fig. 11. Substrate temperature profile dependency on the heat sources thick- ness X

c

, at t = 3800 s and x = X

s

/2, for Ra = 5.01 × 10

⁹

.

location x = X s /2. It can be seen from this figure that the plate and heat sources are well cooled for higher heat source thinness X c . As stated above, the same reason is behind the criss-crossing temperature curves in Fig. 11. It can also be seen that an increase in the heat sources’ thickness X _c leads to a reduction in the overall temperature difference within the substrate (T s,max is equal to 24 °C and 12 °C for X c = 0.002 m and 0.012 m, respectively).

In order to quantify the contribution of all parts of the left hot wall in terms of heat transfer, the percentages of heat transferred from the plate and through the exposed faces of the heat sources (= [Q (plate portion or heat source faces) /3Q ] × 100) to the core flow, as a function of the modules thicknesses X c , are calculated and summarized in Table V. Data analysis leads us to conclude that for all values of the thickness X c

considered in this paper, the plate portions between modules and the upper plate portion transfer less heat to the core flow in comparison to the amount of heat delivered by the bottom plate portion. Analysis of Table V also shows that an increase

in the heat source thickness X _c contributes to the increase in the heat transfer rate from the upper substrate portion.

This is due, in part, to the increase of the heat exchange surface of the upper substrate portion. Table V also shows that the increase of X c leads to a decrease in the heat transfer rate from the plate portions between modules. Indeed, as X c increases, the height of each heat source, and the space between heat sources, decrease. The trapped fluid becomes more and more stagnant in the micro cavity (space between two heat sources), and the heat transfer rate, which occurs by conduction in these locations, diminishes. As was observed in Fig. 11, the high thermal conductivity of the substrate, relative to that of the PCM, allows for thermal spreading to occur through the substrate. For all values of X _c , no less than 35% of the heat generated in modules is transferred through the back face of the modules to the plate and convected to the liquid PCM via the exposed plate faces (Table V). The percentage of the generated heat which is transferred to the liquid PCM through the exposed faces of the heat sources decreases with the increase of X c . Indeed, varying the module thickness X c from 0.002 m to 0.012 m leads to a decrease of the height of the vertical faces of the modules, and then in the vertical heat transfer surface. This explains, in part, the reduction of the heat transfer from the module faces, which varies from 59.6% to 48.71%. The plate contribution, however, is improved from 35.62% to 50.45%. It can also be deduced that the increase in the X _c , from 0.002 m to 0.012 m enhances the total heat absorbed by the PCM from 95.22% to 99.16%. Further increase in the module thickness, X _c (X c > 0.012 m), leads to the slight improvement in a heat transfer.

Fig. 12(a) shows the influence of the module thickness X c

on the thermal performance of the PCM based heat sink.

The analysis of such a figure shows that the increase of the thickness X c in the range of 0.002–0.012 m implies an increase on the operating time of the modules. Indeed, as stated above, the increase of the module thickness causes an enhancement of the total heat transfer, and so a decrease in the temporal change rate of the modules temperature. Therefore, the heat sources take more time to reach the limit temperature, which extends the working time of the heat sources without using any fan.

As mentioned above, the increase of the heat source thickness entails an enhancement of the heat rate dissipated from the left hot solid/liquid interfaces and, consequently, an increase on the mass of the liquid phase of the PCM. More of an increase in X _c has a slight effect on the secured operating timeτ max and liquid fraction f _t,max as was also concluded above.

In general, electronic components engineers are interested in the maximum temperature at the chip. The reliability of electronic devices depends on this maximum temperature. The chip temperature increases transiently from 0 to T cr . When the chip temperature approaches the threshold value (T ∼ 75 °C) the device must cease working. The proposed concept based on the PCM heat storage reservoir must work intermittently. The heat stored in the PCM is naturally rejected to the ambient and the melted PCM resolidifies during the stop periods.

As the melted PCM resolidifies, it can be used in the next

cycles. The maximum dimensionless working time τ _max and

TABLE V

Plate and Modules Heat Transfer Contribution for Various Heat Sources Thicknesses During Steady-State

X

c

(m) % Heat generating in modules

0.002 17 20.3 7.23 22.3 7.32 17 4.07

0.003 18.74 18.63 7.24 19.85 6.78 17.30 8.50

0.0035 19.13 17.8 7.28 20.01 6.68 16.2 11.25

0.008 21.07 18.2 6.01 17.12 4.01 15.4 16.8

0.012 23.4 18.64 5.97 15.03 3.23 15.04 17.85

Fig. 12. Maximum electronic component dimensionless working time τ

max

and liquid fraction f

t,max

as a function of the ratio X

c

/X

c,ref

(a) and Parity plot between numerical and correlation values (b) (X

c,ref

= 0.003 m).

the total melt fraction f _t,max achieved at the end of the charging process (T = T cr ) vary approximately linearly on a logarithmic scale with respect to the module thickness X c over the range 0.002–0.012 m. The following numerical correlations were derived:

τ _max = 0.1035 X _c

X _c,ref 0.0941

, f _t,max = 0.87 X _c

X _c,ref 0.117

. (23)

The parity plot [Fig. 12(b)] shows a good agreement of the above correlations with the numerical solution. Maximum deviation was found to be less than 2%. Correlation (23) can be used in the design of the PCM-based heat sinks to predict the modules thickness X _c and liquid fraction f _t,max for a given secured working time τ _max .

IV. Conclusion

In this paper, melting of a PCM in a rectangular cavity heated with three protruding heat sources mounted on a vertical conducting plate has been studied numerically. It has been shown that PCMs can be used to absorb tran- sient power dissipation by electronics. The results show the following.

1) At the beginning of the melting process, the liquid PCM region is predominated by conduction.

2) The flow is clockwise, multicellular, and the streamlines are distorted by the protrusion.

3) Heat from the module is removed in part by natural convection liquid PCM while the rest is conducted to the wall before being dissipated in the PCM cavity.

4) During the beginning and the plateau regions, liquid fraction increases linearly with time.

5) Heat sources temperature increases approximately lin- early at the beginning stage before reaching the steady- state.

6) The maximum temperature is located in the central heat source when conduction prevails, but when convection develops the upper heat source registers the maximal temperature.

7) The highest heat transfer rates are observed for the bottom heat source.

8) Critical temperature was reached faster for high powered chips because the PCM is rapidly melted.

9) It is recommended to set up the electronic components at the bottom of the cavity to improve their working time.

10) Within the same geometry, the secured working time and the melt fraction depend on the heat sources thickness.

A correlation is suggested for the maximum working time and the corresponding melt fraction.

11) The approach developed herein can be used in the design

of PCM-based cooling systems.

References

[1] Y. Jaluria, “Heat transfer,” in Design and Optimization of Thermal Systems, 2nd ed. Boca Raton, FL: CRC Press, 2008, ch. 3, pp. 133–

148.

[2] S. C. Lin and C. L. Huang, “The study of a small centrifugal fan for notebook computers,” J. Chin. Soc. Mech. Eng., vol. 22, no. 5, pp. 421–

431, 2001.

[3] M. Faraji and H. El Qarnia, “Optimisation d’un système de stockage d’énergie par chaleur latente de fusion: application au refroidissement d’un composant électronique,” in Proc. Congr´es Fran¸cais Thermique (JITH), vol. 2. Albi, France, 2007, pp. 141–145.

[4] M. Faraji and H. El Qarnia, “Numerical optimization of a thermal performance of a phase change material based heat sink,” Int. J. Heat Technol., vol. 26, no. 2, pp. 17–24, 2008.

[5] F. L. Tan and C. P. Tso, “Cooling of mobile electronic devices us- ing phase change materials,” Appl. Thermal Eng., vol. 24, nos. 2–3, pp. 159–169, Feb. 2004.

[6] Y. Zhang, Z. Chen, Q. Wang, and Q. Wu, “Melting in an enclosure with discrete heating at a constant rate,” Exp. Thermal Fluid Sci., vol. 6, no. 2, pp. 196–201, Feb. 1993.

[7] M. Hodes, R. D. Weinstein, S. J. Pence, J. M. Piccini, L. Manzione, and C. Chen, “Transient thermal management of a handset using phase change material (PCM),” Amer. Soc. Mech. Eng. J. Electr. Packag., vol. 124, no. 4, pp. 419–426, Dec. 2002.

[8] S. Krishnan, S. V. Garimella, and S. S. Kang, “A novel hybrid heat sink using phase change materials for transient thermal management of electronics,” IEEE Trans. Compon. Packag. Manufact. Technol., vol. 28, no. 2, pp. 281–289, Jun. 2005.

[9] B. Joiner and S. Neelakantan, “Integrated circuit package types and thermal characteristics,” Electron. Cool, vol. 12, no. 1, pp. 10–17, 2006.

[10] V. R. Voller, M. Cross, and N. C. Markatos, “An enthalpy method for convection/diffusion phase change,” Int. J. Numer. Methods Eng., vol.

24, no. 1, pp. 271–284, 1987.

[11] S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Washington D.C.: Hemisphere, 1980.

[12] E. Dorre and H. Hubner, “Alumina,” in Alumina: Processing, Properties and Applications, Berlin, Germany: Springer-Verlag, 1984, ch. 6, pp.

180–201.

[13] W. R. Humphries and E. I. Griggs, “A design handbook for phase change thermal control and energy storage devices,” NASA Technical Paper 1074, NASA Scientific Tech. Info. Office, Hampton, VA, 1977.

Mustapha Faraji received the Engineer M.E. degree in industrial processes from the National School of Mineral Industry, Rabat, Morocco, in 1995. He is currently pursuing the Ph.D. degree in ther- mal and fluid mechanics from the Faculty of Sci- ences Semlalia, Cadi Ayyad University, Marrakesh, Morocco.

In 1996, he joined the Technological Training Institute, Marrakesh, Morocco, as a Training Officer in computer sciences. His current research interests include heat transfer during the melting and solidifi- cation of phase change materials and their applications to electronics cooling.

Hamid El Qarnia received the License degree in physics and the D.E.S. degree in energetics from Cadi Ayyad University, Marrakesh, Morocco, in 1985 and 1988, respectively. He received the Ph.D. degree in mechanical engineering from the University of Sherbrooke, QC, Canada, in 1999.

In 1989, he joined the Faculty of Sciences

Semlalia, Cadi Ayyad University, Marrakesh,

Morocco, where he is currently a Professor in the

Department of Physics and is a Member of the

Fluid Mechanics and Energetics Laboratory. His

current research interests include the area of heat transfer—in particular,

heat exchange, heat storage, liquid–solid phase change processes and their

applications for the thermal comfort and cooling management of electronic

components, and thermal systems optimization.