Numerical study of melting in an enclosure with discrete protruding heat sources

Numerical study of melting in an enclosure with discrete protruding heat sources

Mustapha Faraji, Hamid El Qarnia

^*

Cadi Ayyad University, Faculty of Sciences Semlalia, Physics Department, Fluids Mechanic and Energetic Laboratory, P.O. Box 2390, Marrakesh, Morocco

a r t i c l e i n f o

Article history:

Received 26 September 2008 Accepted 18 August 2009 Available online 22 August 2009

Keywords:

Phase change material Melting

Heat transfer Protruding heat source Electronic cooling Heat storage

a b s t r a c t

In order to explore the capability of a solid–liquid phase change material (PCM) for cooling electronic or heat storage applications, melting of a PCM in a vertical rectangular enclosure was studied. Three protruding generating heat sources are attached on one of the vertical walls of the enclosure, and generating heat at a constant and uniform volumetric rate. The horizontal walls are adiabatic. The power generated in heat sources is dissipated in PCM (n- eicosane with the melting temperature,Tm= 36°C) that filled the rectangular enclosure.

The advantage of using PCM is that it is able to absorb high amount of heat generated by heat sources due to its relatively high energy density. To investigate the thermal behav- iour and thermal performance of the proposed system, a mathematical model based on the mass, momentum and energy conservation equations was developed. The governing equa- tions are next discretised using a control volume approach in a staggered mesh and a pres- sure correction equation method is employed for the pressure–velocity coupling. The PCM energy equation is solved using the enthalpy method. The solid regions (wall and heat sources) are treated as fluid regions with infinite viscosity and the thermal coupling between solid and fluid regions is taken into account using the harmonic mean of the ther- mal conductivity method. The dimensionless independent parameters that govern the thermal behaviour of the system were next identified. After validating the proposed math- ematical model against experimental data, a numerical investigation was next conducted in order to examine the thermal behaviour of the system by analyzing the flow structure and the heat transfer during the melting process, for a given values of governing parameters.

Ó2009 Elsevier Inc. All rights reserved.

1. Introduction

The miniaturization of electronic components and the growing of the heat generated within them due to a technology progress in electronic industry results for high heat dissipation. Therefore cooling management strategies must be consid- ered to remove the heat from electronic components in order to overcome the superheating problems and ensure reliable operation of electronic systems. Several works on cooling strategies using air or liquid as cooling fluids were conducted.

Kuhn and Oosthuizen

[1]

simulated three-dimensional flow and heat transfer for one or two discrete flush-mounted heat sources mounted to one wall of a vertical cavity with negligible substrate conduction. The opposing wall was chilled, while the top and bottom surfaces were assumed to be adiabatic. For certain conditions considered in their study, the authors show that the effect of the heater location was shown to be small and buoyancy-driven fluid motion was reduced and heat transfer was dominated by conduction across the fluid. They also show that the three-dimensional flow near the vertical edges of the

0307-904X/$ - see front matterÓ2009 Elsevier Inc. All rights reserved.

doi:10.1016/j.apm.2009.08.012

* Corresponding author. Tel.: +212 66 35 00 16; fax: +212 24 43 74 10.

E-mail address:elqarnia@ucam.ac.ma(H. El Qarnia).

Contents lists available atScienceDirect

Applied Mathematical Modelling

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a p m

Nomenclature

A aspect ratio, l/w B porosity function

c

p

specific heat a constant pressure (J kg

¹

K

¹

) e thickness (m)

f liquid fraction g gravity (m s

²

)

h volumetric enthalpy (J m

³

) or heat transfer coefficient (W/m

²

K)

H height (m)

k

m

PCM thermal conductivity (W m

¹

K

¹

)

k solid (wall or heat sources) thermal conductivity (W m

¹

K

¹

) K dimensionless thermal conductivity, k/k

m,l

l

o

, l

co

characteristic lengths (m): l

²_o¼

l w 3l

²_co

and l

²_co¼

l

c

e

c

l height of the enclosure (m) Nu Nusselt number

¼^{h l}_k ^o

p pressure (Pa)

m;l

Pr Prandtl number

¼_a^mm;l

Q

⁰

heat generation per unit length of the heat source (W m

m;l ¹

) = q

⁰⁰⁰

l

c

e

c

q

⁰

dimensionless heat flux transferred from the left interface to the PCM q

⁰⁰⁰

volumetric heat generation (W m

³

)

q

⁰⁰

dimensionless local heat flux density Ra Rayleigh number

¼^gbl

3 oDTref

tm;lam;l

s peripheral distance at the interface solid (wall or heat sources/PCM liquid)

S source term

Ste Stefan number

¼

Ste

¼^ðcpÞm;l_DH^DTref

T temperature (K)

t time (s)

x, y cartesian coordinates (m) U, V dimensionless velocity w width of the enclosure (m) Subscripts

cr critical value c heat sources

f fluid

i, j control volume location k kth time iteration l liquid, local

m melt, PCM

max maximum value

o initial or characteristic length out outlet

p constant pressure s wall, solid Greek symbols

a thermal diffusivity (m

²

s

¹

)

a dimensionless thermal diffusivity

¼_a^a

m;l

b

volumetric thermal expansion factor of PCM liquid (K

¹

)

d1;2;d

Kronecker symbols, distance (Fig. 1)

D H latent heat (J kg

¹

)

D x, D y space steps in x and y directions, respectively D s time step

u dimensionless flow mass

c space between two consecutive heat sources (m), (Fig. 1)

g dimensionless distance perpendicular to the wall–heat sources/liquid PCM interface

l dynamic viscosity (kg m s

¹

)

m cinematic viscosity (m

²

s

¹

)

h

dimensionless temperature

q density (kg/m

³

)

heater increased local convection coefficients, causing average Nusselt numbers for the heaters to exceed predictions based on two-dimensional model. Heindel et al.

[2]

conducted calculations for 3 3 array of discrete heat sources mounted flush to one wall of a rectangular enclosure with negligible substrate conduction. Isoflux conditions were prescribed at each of the heaters and the opposing chilled wall was isothermal, while the remaining surfaces were assumed to be adiabatic. The com- puted temperature field was used to determine local and average Nusselt numbers for the heaters. The same authors

[3]

con- ducted numerical investigations for a vertical array of three flush-mounted strip heaters with substrate conduction.

Protruding heat sources have been also studied by several investigators. For more details, the lecturer can to refer to the book of Incropera

[4]. Recently, Desrayaud et al.[5]

analyzed a steady, two-dimensional laminar natural convection in a sys- tem of parallel vertical channels with a single protruding heat module mounted mid-height on a substrate of finite-thick- ness. A parametric study was conducted by varying the thermal conductivity and the thickness of the substrate and the width of the module. The results obtained showed that a variation in the substrate conductivity considerably affects the module temperature while the flow structure is only slightly altered. It is also demonstrated that streamwise conduction through the substrate is an important cooling mechanism and must be accounted for. Rao and Narasimham

[6]

has con- ducted a numerical investigation to examine the conjugate mixed convection arising from protruding heat generating ribs attached to substrates forming channel walls. The maximum dimensionless temperature is correlated in terms of pure nat- ural convection and forced convection inlet velocity asymptotes. For the parameter values considered in their study, the heat transferred to the working fluid via substrate heat conduction is found to account for 41–47% of the heat removal from the ribs. Other investigations on natural convection cooling, using ethylene glycol as fluid, for discrete heat sources attached to one vertical wall of rectangular enclosures were carried out by Keyhani et al.

[7]

and Ju and Chen

[8].

The experimental results obtained by Kelleher et al.

[9]

and Turner and Flack

[10]

on air natural convection heat transfer in rectangular enclosures equipped with heat sources, simulating the air natural convection cooling of chips attached on sub- strates, shows, however, the limitation of the cooling capacity of air natural convection. A promising alternative for period- ically or intermittently operated electronic devices is a solid–liquid phase change cooling. This is justified by the fact that the phase change materials (PCMs) have a high energy storage density and a relatively high removal heat compared to liquids and air. The fundamental problem of interest for the cooling of electronic equipment is maintaining safe temperatures in the electronic devices. The use of PCM as a storage medium can reduce the size of the cooling system and allows the system its continuous cooling capacity. During the working period, electronic components dissipate heat through their exposed area and the solid PCM continually melts as heat sources continually dissipate heat. Such a conception would sustain its dissipa- tion capability as long as molten PCM would not become overheated. The heat stored in the PCM is naturally rejected to the ambient and the melted PCM re-solidifies during the stop periods. As the melted PCM re-solidifies it can be used in the next cycles. Zhang et al.

[11]

have conducted an experimental study of melting of PCM (n-octadecane) inside a rectangular cavity heated by flush-mounted heat sources on one of its vertical wall at constant and uniform heat generation. The horizontal walls are adiabatic. The results obtained show that the cooling of heat sources using PCM melting natural convection leads to a drop of the mean temperatures of heat sources as much as 50% compared to air natural convection cooling. Bruno et al.

[12]

developed a (2D) mathematical model simulating the thermal behaviour of a rectangular enclosure similar to the one studied by Zhang et al.

[11,13]. Their study finds applications in the design of heat storage units and cooling of electronic

equipments. The results show that enclosures with high aspect ratio (>4), to better control temperature heat sources and offer relatively extended melting durations. In another experimental study conducted by Jianhua et al.

[14], the melting pro-

cess 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. An experimental study of melting 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.

[15]. The horizontal walls were

adiabatic. Their results were compared with those of Keyhani et al.

[7]

and Ju and Chen

[8]. This comparison shows that the

rise in the temperature of the heat sources can be reduced to 50–70% by using PCM melting natural convection instead of natural convection of ethylene glycol. Another comparison with the results obtained by Zhang et al.

[11]

shows that the sur- face temperatures of protruding heat sources were lower than those of flush-mounted heaters.

The above literature survey shows clearly that the previous studies relating to the PCM electronic cooling were carried out experimentally, except that of Bruno

[12]. Although in this latest work a numerical study of melting natural convection in a

rectangular enclosure has been conducted, only flush-mounted heat sources were considered. Melting natural convection in a vertical rectangular enclosure with protruding heat sources attached on one of the vertical walls has been the subject of the experimental study only

[15]. Further, only a heat flux, at a constant rate, was imposed on the back face of the heat sources.

The present study overcomes this limitation by considering the problem of melting natural convection in a rectangular enclosure heated with three protruding heat sources with a constant and uniform volumetric heat generation. The proposed problem is numerically studied. The objectives of the present study are (1) to develop a mathematical model and validate it with experimental data and (2) to examine the thermal behaviour of the proposed thermal system by analyzing the flow structure and the heat transfer during the melting process, for a given governing parameters.

s dimensionless time

w

stream function,

w¼R

UdY VdX

2. Statement of the problem and solution

The schematic view of the physical model considered in the present study is given in

Fig. 1. The physical model consists of

a rectangular enclosure containing a phase change material (PCM) with three discrete protruding heat sources attached on one of its walls, each of them generates a constant and uniform volumetric heat generation, q

⁰⁰⁰

. The height and thickness of each heat source are l

c

and e

c

, respectively. The distance between two consecutive heat sources is c and the distance between the bottom enclosure wall and the bottom face of the lower heat source is,

d. The height and width of the rectangular enclo-

sure are, l and w, respectively. The thickness of the wall is, e. The thermal conductivity, k

_c

, specific heat at constant pressure, c

p;c

, and density, q

c

of the heat sources are different from those of the wall, k

s

, c

p,s

and q

s

. Identical heat sources and volu- metric heat generations are considered in this study.

2.1. Governing equations

The physical properties of the materials are constant at the temperature range under study. The density difference be- tween solid and liquid phases is negligible. The Boussinesq approximation was used. The phase change is isothermal and the PCM is initially solid at its melting temperature, T

o

= T

m

. The reference temperature is equal to the melting temperature T

m

. The density of the PCM is taken as a reference density. The thermal properties of materials are given in

Table 4. The flow

in the liquid phase of the PCM is assumed to be laminar, incompressible and Newtonian. The heat transfer and fluid flow are assumed 2D.

Based on the above mentioned assumptions, the non-dimensional form of the governing equations for the PCM, heat sources and substrate can be written as follows:

Continuity equation

@U

@X þ @V

@Y ¼ 0 ð1Þ

Momentum equations

@U

@ s ^þ

@UU

@X þ @VU

@Y ¼ @P

@X þ Pr @

@X

@U

@X

þ Pr @

@Y

@U

@Y

þ S

u

ð2Þ

@V

@ s ^þ

@UV

@X þ @VV

@Y ¼ @P

@Y þ Pr @

@X

@V

@X

þ Pr @

@Y

@V

@Y

þ S

_v

ð3Þ

Energy equation

@h

@ s ^þ

@Uh

@X þ @Vh

@Y ¼ @

@X a ^@h

@X

þ @

@Y a ^@h

@Y

þ S

T

ð4Þ

With

S

u

¼ BU ð5Þ

S

_v

¼ BV þ RaPr h ð6Þ

l l

γ

δ e

e

c c

PCM

0

w

Fig. 1.The schematic view of the physical model.

S

T

¼ d

1

ðd

2

1Þ 1 Ste

@f

@ s ^þ

d

2

3E

c

L

c

ð7Þ

d

1

¼ 1 for heat sources and PCM 0 for the substrate

; d

2

¼ 1 for heat sources 0 for the PCM

; B ¼ C ð1 fÞ

²

b þ f

³

ð8Þ

The above dimensionless governing equations, and the following boundary and initial conditions are obtained using the following dimensionless independent and dependant variables and parameters:

X ¼ x l

o

; Y ¼ y l

o

; s ¼ a

m;l

t l

²_o

U ¼ u

a

m;l

=l

o

; V ¼ v a

m;l

=l

o

; D T

ref

¼ 3Q

⁰

k

m;l

; h ¼ T T

m

D T

ref

ð9Þ

Ra ¼ g b l

³_o

D T

ref

t

m;l

a

m;l

Ste ¼ ðc

p

Þ

_m;l

D T

ref

D H ; Pr ¼ t a

m;l

; P ¼ p

q ð a

m;l

=l

o

Þ

²

The characteristic length l

o¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lw 3l

c

e

c

p

represents the mass of a PCM, which is kept constant in the present study.

S

u

and S

v

are the source terms used for the velocity suppression in the solid regions (solid PCM, wall, heat sources). One of the common models for the velocity suppression is to introduce a Darcy-like term as described in

[16]. The function

B must be large where f is equal to zero and must go to zero as f go to one. A function based on the Carman–Koseny relation for a porous medium as described in

[16]

was employed in this study, with C = 10

²⁵

and b = 0.005. This numerical artefact can be viewed as a way of modelling the transition zone between the solid and liquid phases. The same full set of governing equa- tions throughout the entire enclosure govern temperature and velocity distribution in both the liquid and solid regions with taking a large value of the viscosity for solid regions.

2.2. Boundary conditions Adiabatic walls:

@ h

@ g

wall

¼ 0; g ? wall ð10Þ

Interface wall/ith heat source (X = 0, i = 1, 2, 3):

h

c

¼ h

s

and K

c

@h

@ X

_X¼0

¼ K

s

@h

@X

_X¼0

; D þ ði 1Þð C þ L

c

Þ 6 Y 6 D þ ði 1Þ C þ iL

c

ð11Þ

Interface wall–PCM:

h

s

¼ h

m

and K

s

@h

@X

_X¼0

¼ K

m

@h

@X

_X¼0

ð12Þ

Interface heat sources–PCM:

h

c

¼ h

m

and K

c

@h

@ g

_c

¼ K

m

@h

@ g

_m

ð13Þ

( g is the distance measured normal to the interface heat source PCM) Wall:

U ¼ V ¼ 0 ð14Þ

Initial conditions:

h ¼ U ¼ V ¼ 0 ð15Þ

The dimensionless PCM thermal conductivity and thermal diffusivity at each location are calculated as follows:

K

m

¼ f þ ð1 fÞK

m;s

and a

m

¼ f þ ð1 f Þ a

m;s

ð16Þ

The dimensionless thermal conductivities at the interfaces are obtained by the harmonic mean method:

K

i

¼ K

þ

K ðD

þ

þ D Þ K

þ

D þ K D

þ

ð17Þ Signs ‘+’ and ‘’ refer to the first right (top) and left (bottom) nodes, from the interface respectively.

The governing parameters of the problem are:

E ¼ e l

o

; E

c

¼ e

c

l

o

; L

c

¼ l

c

l

o

; D ¼ d l

o

; C ¼ c

l

o

; L ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 1 þ 3 l

co

l

o 2

! v u

u t ; W ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ 3ð

^lco_l

o

Þ

²

Þ A s

A ¼ l

w ; K

m;s

¼ k

m;s

k

m;l

; K

s

¼ k

s

k

m;l

; K

c

¼ k

c

k

m;l

; a

m;s

¼ a

m;s

a

m;l

; a

s

¼ a

s

a

m;l

; a

c

¼ a

c

a

m;l

; Ra; Ste; Pr ð18Þ

2.3. Numerical procedure

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 control volume approach 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 energy equation for PCM is solved using the enthalpy method devel- oped by Voller

[18]. The source term,ðST¼ _Ste^{1 @f}_@sÞ, is the central feature of this technique for the energy equation (Eq. 4),

which 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. In the numerical implementation, its value is determined iteratively from the solution of the energy equation. A tri-diagonal matrix iterative method is used to solve for U, V and

h. The iterative solution continues until convergence of the flow and energy

fields at every time step, is reached. Convergence is declared when the following criterions relating to mass and energy bal- ances, e

m;

e

T

, defined as follows, are smaller than 10

⁸

and 10

²

, respectively.

e

m

¼ Maxj u

_in

ði; jÞ u

_out

ði; jÞj ð19Þ

e

T

¼ 1 ðQ

sen;s

þ Q

sen;c

þ Q

sen;l

þ Q

lat

Þ ð20Þ

where u

_in

and u

_out

are the inlet and outlet mass flow:

u

_in

ði; jÞ ¼ ðUði; jÞ D Y þ Vði; jÞ D XÞ

u

_out

ði; jÞ ¼ ðUði þ 1; jÞ D Y þ Vði; j þ 1Þ D XÞ

1 6 i 6 M þ 1; 1 6 j 6 N þ 1 ð21Þ

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

Q

sen;s

¼ X

substrate

R

s

D X D Yðh

^kþ1

h

^k

Þ =d s ; Q

sen;l

¼ X

liquid PCM

D X D Yðh

^kþ1

h

^k

Þ =d s

Q

sen;c

¼ X

heat source1;2;3

R

c

D X D Yðh

^kþ1

h

^k

Þ =d s ð22Þ

Q

lat

¼ X

PCM

1 Ste

@f

@ s ^{D X D Y; where R}

^s

^¼ q

_s

c

p;s

q

_l

c

p;l

; R

c

¼ q

_c

c

p;c

q

_l

c

p;l

ð23Þ

The local heat transfer density at the left wall–heat sources–PCM interface is expressed as follows:

q

⁰⁰_i

¼ k

i

@T

@ g

_i

ð24Þ

where the distance g is measured normal to the interface. The dimensionless heat flux density is expressed as:

q

⁰⁰_i

¼ K

i

@h

@ g

i

ð25Þ g is the dimensionless distance measured normal to the interface

Average Nusselt numbers. In order to calculate the heat flux dissipated from the surface of each heat source to the PCM, using the maximum temperature, the following expression is used for the local heat flux density at the heat source surface:

q

⁰⁰_j

¼ h

j

ðT

max

T

m

Þ ð26Þ

where h

j

is the local heat transfer coefficient based on the maximum temperature, T

max

.

Using the dimensionless quantity defined above, the average Nusselt number for each heat source is expressed as follow:

Nu ¼ hl

o

k

m;l

¼ 1 Sh

max

Z

S 0

K @h

@ g ^{d s}

i

ð27Þ where S

¼^lcþ2e_l ^c

o

2.4. Grid independence study

Numerical investigations were conducted to check the grid size and time step dependence results. The results are shown in

Table 1a and b andFig. 2.

The analysis of the obtained results in

Table 1a and b shows that a non uniform 60

80 grid and the time step, D s

¼

4:42 10

⁴

(20 s), were found sufficient to give accurate results. Other small time steps were used but, supply a drastic CPU time, without giving appreciable accuracy in numerical results. The effects grid size and time step on the sensitivity of the results are shown in

Table 1

and

Fig. 2.Table 1a andFig. 2a show the effects of three grid sizes: 40x60, 60x80 and

80 100, for a dimensionless time step, D s = 4.42 10

⁴

(20 s). As it can be seen, changing grid size for 60 80 to 80 100 leads to a relative change of liquid fraction and maximum dimensionless temperature of 0.55 and 0.13%, respec- tively. The melting fronts are practically confused for these two grid sizes.

Table 1b shows that using a dimensionless time

step of D s = 4.42 10

⁴

leads to relative changes of liquid fraction and dimensionless maximum temperature of 0.17% and 0.09%, respectively.

Fig. 2b shows that the melting front are approximately confused for,

D s

⁶

4:42 10

⁴

. A fine grid size near solids was set to give more details for hydrodynamic behaviour near interfaces. It should be noted that this grid refine- ment is available only for the present configuration with a fixed aspect ratio at A = 4. Certainly, other grids size will be

Table 1

Sensitivity of the results for varying grid (a) and time step (b).

MN f Deviation (%) hmax Deviation (%)

(a)

4060 0.3290 – 0.02156 –

6080 0.3507 6.6 0.02275 5.52

80100 0.3527 0.55 0.02278 0.13

Time step,Ds f Deviation (%) hmax Deviation (%)

(b)

1.3210³(60 s) 0.3931 – 0.02374 –

7.7310⁴(35 s) 0.3622 7.86 0.02291 3.50

4.4210⁴(20 s) 0.3508 3.14 0.02275 0.70

2.2110⁴(10 s) 0.3502 0.17 0.02273 0.09

Fig. 2.Effect of grid size, (a) and time step, (b) on the melting front position,s= 0.058.

adopted for other aspect ratios. Typical execution times for the retained mesh runs were around 9 h on a 2.2 GHz Pentium 4 computer.

3. Validation

In order to carry out numerical investigations for the system presented in this study, a personal computer program was implemented in Fortran language. Before conducting such numerical investigations, the computer program was first used for validation against experimental data obtained by Gau et al.

[19], for the two-dimensional problem of melting of Gallium with

the presence of natural convection in a rectangular enclosure. The numerical code was also validated with the experimental data obtained by Ju et al.

[15]

related to the melting of n-octadecane inside a rectangular cavity heated by three discrete protruding heat sources attached on one of the vertical walls.

For the first validation, the dimension of the cavity used in the experiments are l = 6.35 cm and w = 8.89 cm. The left hot wall and the right cold wall are maintained at temperatures, T

H

= 311.5 K and T

C

= 301.45 K, respectively, and it was filled by solid Gallium initially at temperature T

_C

. The horizontal walls are insulated.

The duration of the melting of the Gallium takes 20 min. The numerical code was readapted to the setup conditions.

Thermo physical properties of Gallium used in the experiment setup are summarized in

Table 2.

A comparison between numerical and experimental melting fronts is sketched in

Fig. 3. The accordance was found suf-

ficient between experimental and numerical data. The discordance observed may be explained by the difficulty encountered by Gau et al.

[19]

to ensure the stability in vertical walls temperatures during the experiment. The same remarks were made by other authors when validating their numerical code.

For the second validation, the experimental tests of Ju et al.

[15]

have been used therefore to check the predictions of the present numerical model for a particular configuration of protruding heat sources as displayed in

Fig. 4. This configuration

consists of an insulated rectangular enclosure of height, l = 9.0 cm, and width, w = 6.0 cm, the left wall is made with Plexiglas material and have a thickness of, e = 2 cm. It supports three discrete protruding heat sources of height, l

_c

= 1.5 cm, and thick- ness, e

c

= 0.9 cm. The heat flux density delivered by each heat source equals to 900 W/m

²

.

Table 2

Thermo physical properties of Gallium[20].

Tm(K) k(W/m K) cp(J/kg K) qm(kg/m³) l(kg/m s) b(K¹) DH(J/kg)

302.93 32 381.5 6093 1.8110³ 1.210⁴ 0.801610⁵

Fig. 3.Numerical and experimental melting front positions.

The lower heat source is placed at a distance

d

= 0.75 cm to the bottom wall and the distance between tow consecutive heat sources equals to c = 1.5 cm. Initially, the cavity is filled with a solid PCM (n-octadecane) at its melting temperature, T

m

= 28

°C. An air layer of 1 cm thickness was provided for PCM expansion during the melting. Thermo-physical properties

of the PCM are summarized in

Table 3.

The numerical code was adjusted to setup conditions and, after a grid refinement (aspect ratio A = 1.5), the computer pro- gram 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

Fig. 5. The concor-

dance is better at t = 25 min and t = 50 min except at the lower and upper heat sources that seem furnishing less heat to the PCM in comparison with what predicts the mathematical model.

Other causes are also behind this slightly difference in melting front positions: the surrounding heat lost and the remove of the insulation during photographing the melting front. At t = 95 min and t = 110 min the agreement is well but a deviation at the top portion was clearly observed. Indeed, an amount of melted n-octadecane 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. Many experiments related to the melting of n-octadecane encountered this phe- nomenon

[12,13]. A part of liquid PCM goes out of computational domain and breaks the adiabatic condition in the north

boundary.

l

w

γ l

δ e

e

c c

air

n-octadecane

Fig. 4.Experimental setup of Ju et al.[15].

Table 3

Thermo physical properties of n-octadecane[21].

Tm(K) km,s(W/m K) km,l(W/m K) ðcpÞ_m;s(J/kg K) ðcpÞ_m;l(J/kg K) qm(kg/m³) l(kg/m s) b(K¹) DH(J kg)

301.3 0.38 0.15 1891 2251 771.2 3.610³ 9.110⁴ 2.434105

Fig. 5.A comparison between numerical predicted melting fronts (solid line) and photographed melting fronts obtained by Ju et al.[15](dashed lines) at t1= 25 min,t2= 50 min,t3= 95 min andt4= 110 min.

4. Results and discussion

Numerical investigations were carried out to examine the thermal behaviour of the proposed system by analyzing the thermal and flow fields evolution with time for a given values of the governing parameters. The PCM, wall and heat sources used for the numerical simulations are the n-eicosane, Alumina substrate and Aluminum Ceramics, respectively. Their cor- responding thermal properties are given in

Table 4. The geometrical parameters of the enclosure, heat sources and wall are

displayed in

Table 5. The corresponding values of the governing parameters are displayed inTable 6, and have been calcu-

lated using

Tables 4 and 5. The heat generating per unit of length within each heat source,

Q’, is equal to 20 W/m. The char- acteristic length, l

o

, (mass of PCM) is equal to 0.06 m.

Initially the PCM is solid at the temperature, T

o

= T

m

= 36

°C. The heat generation supplied to the heat sources is equal to

that generated within the actual Pentium micro processors (CPU) Centrino Duo 4 (2.8 Ghz). During the melting process, no fan was used. One part of the heat generated is stored in substrate and electronic components; the other part is transported by conduction and natural convection to the PCM. The heat sources temperatures rise and reach a limit temperature (70

°C)

since it is typically the highest operating temperature permissible for most chips to ensure reliability

[24]. The time,

t

cr

, re- quired for the electronic components to reach this limit temperature depends on the governing parameters of the heat sink.

In view of this, numerical simulations were conducted during the limiting time, t

cr

, or until the melt fraction, f, approaches 1.

4.1. Streamlines and isotherms plots

The streamlines and isotherms plots describing the flow and temperature fields are presented in

Figs. 6a and b, at differ-

ent times. The analysis of these figures shows that natural convection starts to develop near the top east corners of heat sources (distortion of isotherms) at s = 0.042, while conduction still prevails near vertical substrate surfaces (vertical iso- therms). The flow is clockwise, upward near the heat sources and downward in the vicinity of the melt front interface. It starts multi-cellular because of a very slender vertical liquid layer; this behaviour is frequently encountered in the tall con- vective cavity. Seven separated convection cells are formed in the melt region with one convection cell near each vertical face of heat sources. The upper and lower cells are located above and under the top and bottom heat sources, respectively.

The solid–liquid interface move away uniformly from the three vertical faces of protruding heat sources, and the melt re- gions in the vicinity of the heat sources developed independently with approximately the same shapes. As a result, the inter- face starts exhibiting a distortional shape near the top east corners of heat sources while it remains uniform in the lower portions. The streamlines are deformed by the corners of the protruding chips and the cells are partially trapped in the re- gions between the heat sources. This trend is only observed for the complex case of protruding heat sources and was absent in the case of flush-mounted heat sources. As time progresses

ð

s

>

0:042Þ, the cells formed earlier grow in volume and push out away from heat sources. They combine together and form a bicellular flow

ð

s

¼

0:096Þ. As time progresses, buoyancy

Table 5

Geometrical parameters[24–26].

l(m) w(m) e(m) ec(m) lc(m) c(m) d(m)

0.122 0.03055 0.005 0.003 0.015 0.01 0.03

Table 4

Thermo physical properties of PCM, wall and heat sources[20–23].

PCM (n-eicosane) Substrate (Alumina substrate) Heat source (Alumina ceramic)

km=km,l=km,s= 0.1505 W/m K ks= 19.7 W/m K kc= 40 W/m K

qm= 769.0 kg/m³ qs= 3900 kg/m³ qc= 3800 kg/m³

ðcpÞm;l¼ ðcpÞm;s¼2460 J=kg (cp)s= 900 J/kg (cp)c= 770 J/kg

l= 4.1510³m²/s b= 8.0510⁴K¹ DH= 2.4710⁵J/kg Tm= 36°C

Table 6

Governing parameters.

E Ec Lc D C A L W –

0.083 0.05 0.25 0.5 0.167 4.0 2.03 0.51 –

Km,s Ks Kc am;s as ac Ra Ste Pr

1.0 130.9 265.78 1.0 70.56 171.85 1.67310⁹ 3.97 67.83

driven flows become more vigorous in the upper region of the melt, and natural convection effects intensifies. This explains the more rapid advances of the melting front at the top part of the enclosure. Indeed, as the melt moves upward adjacent to the heated wall, it gains heat and reaches its maximum temperature near the top heat source. The heated liquid turns,

τ=7.4 10 ψ =0.00

max x ^-5

0.7 0.7

0.7

1.6

τ=0.042 ψ =1.6

max 0.7

1.72.7 3.7

4.7 5.7 6.6

τ=0.096 ψ =6.6

max 0.7

2.7 3.7

1.7 4.7 6.7

5.7 6.7 7.7

8.0 5.7

τ=0.152 ψ =8.0

max

0.71.700 2.70 4.70 4.85

3.70 4.70

3.70 1.70

τ=0.197 ψ = 4.85

max

0.70 1.70

3.702.70 4.70 5.50

0.70 1.70

2.70

τ=0.230 ψ =5.5

max 1.700.70

2.70 3.704.16

0.70 1.70 2.70

ψ = 4.16 τ=0.275 max

0.70 1.70 2.70 3.53

0.70 1.70

1.70 2.70

ψ =3.53 τ=0.296 max

3.25

3.25

1.82 1.46

1.46 ψ =3.25

τ=0.320^max

0.70 0.70

(a)

Fig. 6.(a) Streamlines plot; (b) Isotherm plots.

deflecting away from the heated wall toward the solid–liquid interface. Therefore, significantly heat is transferred to the so- lid PCM and more melting occurs in this region. As the melt flows downward along the solid–liquid interface, it loses heat

0.001

0.0000.001

0.001

τ=7.4 10 θ =0.001

max

x ^-5

0.0000.015

0.021 0.020

0.005 0.010

τ=0.042

θ=0.021_max ^0.

000

0.005

0.010 0.015

0.020

0.022

τ=0.096 θ =0.022

max

0.000

0.005 0.010

0.015 0.020

0.022

τ=0.152 θ =0.022_max

0.000 0.005 0. 010 0.015

0. 020 0. 025

0.028

τ=0.197 θ =0.028

_max

0.0.000005 0. 010

0. 015 0.0200. 025

0.030 0.035 0. 040 0.045

τ=0.275 θ =0.045

_max

0.000 0. 010

0. 015 0. 020 0. 0250. 030 0. 035

0.040 0.045

0. 050 0. 055

0. 060

0.064

0.005

τ=0.296

θ =0.064_max

0.0.000005 0. 010

0. 015 0. 020 0. 025

0. 030 0. 035

0.03

7

τ=0.230 θ =0.037

max

0.077

0. 070

0.065

0. 060

0.055 0. 050

0.045 0. 040 0. 035

0. 030 0.025

0. 020 0.015

0. 010

τ=0.320 θ =0.077_max

(a) (b)

Fig. 6.(continued)

and directly contributes to the melting of the solid and hence more cooling of the bottom heat source occurs. Thereby, its capability to melt the solid is gradually decreased to the minimum at the bottom of the cavity. As the heating continues, the melt cavity expands and buoyancy-driven convection intensifies and establishes itself in the entire melt region. The flow structure changes next and four convection cells formed at the core of the melting region

ð

s

¼

0:152Þ. The melting front reaches the right wall with a relatively accentuated distortion in its shape. This position offers a maximum heat transfer area (the active cold wall). Therefore, relatively more heat is exchanged at the PCM solid–liquid interface. The maximum strength of the convection current i.e.

w_max

is also reached

ðw_max

8:00Þ. At the same time, the two central cells combine while the other cells remain at their locations, at s = 0.197. For s > 0.23, the core flow splits into two independent clockwise cells sep- arated by a stagnant melting region. The top cell grows while the bottom decreases in size with elapsed time. This hydro- dynamic behaviour is observed throughout the melting process. Another important observation that can be made is that the flow intensity diminishes with time.

The isotherm plots are displayed in

Fig. 6b at various times. At early stages, the heat transfer in the melt zone is domi-

nated by conduction, and the isotherms appear to be uniformly parallel to the substrate and right faces of the heat sources.

As to be expected, when time progresses, the two dimensionality of the temperature field is evident. The temperature dis- tribution inside any heat source is typical to the case where volumetric heat generation is present, i.e., closed isotherms. The homogenization of the heat sources temperatures is due to the large values of their dimensionless thermal conductivity (K

c

= 265.78). The maximum temperature, in general, occurs at the top heat source when natural convection establishes.

The heat transfer is characterized by a constriction of the isotherms in the vicinity of the vertical solid hot and cold walls, as illustrated in

Fig. 6b, which mean that the flow comes in the boundary layer regime, at quasi-steady state period ð0:0426

s

⁶

0:152Þ. At steady state period, natural convection prevails in the core liquid region and the convection current forces the fluid to extract more heat from heat sources and redirect it to the upper regions of the cavity. The temperature is homogenized in central liquid zone and a higher thermal gradient is found in the vicinity of the left and right active walls.

The maximum driven temperature difference between the left wall and the melt front interface increases during the melting process and reaches a constant value,

hmax¼

0:022

ðTmax

T

m¼

8:6

CÞ.

The maximum temperature difference, D

hsubstrate

, found in substrate exceeds 0.013 (T

max

T

m

= 5

°C), which shows that

the substrate is not isotherm. This can be explain by the low value of its relatively dimensionless thermal conductivity, K

_s

= 130.9. For s > 0.197, the melting front intercepts the right wall at a location which moves down. The descendant dis- placement of this location reduces the heat transfer area at the PCM solid–liquid interface. This forces the heated ascendant liquid to rest at the upper region of the cavity (between the top adiabatic wall and a part of the right adiabatic wall). In this upper region, the temperature field stratifies, which leads to a negligible temperature gradients. At the bottom region, how- ever, higher thermal gradients prevail in the vicinity of the left hot wall and the right melting front interface.

Fig. 7.Time wise variation of the average temperature heat sources, dimensionless maximal temperature and liquid fraction.

4.2. Dimensionless average temperature heat sources and liquid fraction

Fig. 7

displays the time wise variations of the dimensionless mean temperatures of the heat sources and liquid fraction.

The analysis of such figure shows that the temperature and liquid fraction variations go through three distinct regions. At the earlier stage, pure conduction prevails during the melting process, near the heat sources. The dimensionless mean temper- atures 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 ejected heat from heat sources to substrate 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.

During the second stage

ð0:066

s

⁶

0:152Þ, the natural convection develops and 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, substrates and solid PCM). Therefore, all 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 dimensionless means temperatures remain constants throughout this stage (quasi-steady state). The third stage starts when the location of the intersection between the melting front and right wall, moves down

ð

s

P

0:197Þ. The heat flux transmitted to the liquid-solid interface decreases with time which explains the decays of the rate of change in the liquid fraction. As explained earlier, the temperature field stratifies at the upper region of the cavity,

Fig. 8.Dimensionless temperature at the left solid–liquid interface: versus interface distance (a) and time (b).

whereas, for the core of the bottom region, isotherms are obviously inclined and the bottom heat source triumph for the low- est temperature.

4.3. Dimensionless left solid–liquid interface temperature

The dimensionless temperature at the left solid–liquid interface is depicted in

Fig. 8a, at various time. With moving along

the interface, from the bottom to the top of the enclosure, the temperature, in general, increases. A local minima near the upper east corner (D) of the bottom heat source is observed.

The reason behind the temperature drop at that corner is that it is crowded by the PCM liquid as it moves up, and cor- responds to the thinning location of the formed thermal boundary layer. Another observation that can be made is the homogenization temperature in the heat sources regions, which is due to their relatively high thermal conductivity. The

Fig. 10.Dimensionless local heat flux density distribution along the left interface between heat sources-wall and liquid PCM.

0 0.05 0.1 0.15

τ

0.2 0.25 0.3

0 50 100 150 200 250 300

Nu1 Nu2 Nu3 Nu

_

_ _ _

Fig. 9.Temporal evolution of the heat sources average Nusselt numbers,Nu. (1: bottom heat source, 2: central heat source, 3: upper heat source).

analysis of

Fig. 8b shows that the dimensionless temperature, at any location, increases with time (this may be also seen in Fig. 8b). At quasi-steady stateð0:066

s

⁶

0:152Þ, the temperature at any location remains, approximately, constant. This is showed by the closer temperature curves (plateau) that appear in

Fig. 8b.

4.4. Average heat sources Nusselt numbers

The temporal variation of the average Nusselt numbers, based on the maximum temperature of heat sources are shown in

Fig. 9. One of the observations that can be deduced from the analysis of such figure is that,

Nu, has a variation which is oppo- site of,

hmax

(which is, in general, close to the average temperature of the upper heat source). The variation of Nu is not, how- ever, proportional to 1=h

max

. Indeed, the analysis of the expression (27) shows that the average Nusselt number Nu is proportional to the product of 1=h

max

and

RS

0½

K

_@^@h_g

d s

_i

. This latter term refers to the heat flux dissipated by the three faces

Fig. 11.Temporal evolution of the dimensionless heat flux density at the left interface (a), and the dimensionless heat flux, storage energy rate of the wall and heat sources (b).

of the heat source to the PCM liquid, and 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.

4.5. Dimensionless heat flux density at the left solid–liquid interface

The distribution of the dimensionless heat flux density along the left interface between heat sources-wall and liquid PCM is depicted in

Fig. 10, at various timeð06

s

⁶

0:296Þ. As it can be seen, at this Rayleigh number (Ra = 1.67 10

⁹

), the distribution is highly non uniform and presents local maxima and minima. At earlier stages, heat is transferred by pure conduction and, as can be expected, the profile of the dimensionless heat flux density is symmetric. As it can be seen in

Fig. 6b, high temperature gradients at the interface wall–PCM near the heat sources are observed. This explains

the heat flux density profile at these locations. The melting process occurs only in the vicinity of the heat sources and in the regions between them, and a relatively lower heat transfer is manifested at the lower region (A–B) and at the upper region (M–N). The reason of this is that, at earlier stages, heat is not sufficiently conducted from the heat sources to the most upper and lower wall regions to sustain large wall temperature. When natural convection develops, local maxima and minima occurred near at protrusion corners (C, D, G, H, K, L) and are associated to boundary layer thicknesses. The dimensionless interfacial heat flux density decays along the right face of each heat source due to the formed boundary layer. In the regions between heat sources, the fluid is stagnant and the heat flux density reaches its minimum value near the corners (B, E, F, I, J, M). Depending on the flow structure, the dimensionless heat flux density may increases or decreases with time.

Fig. 11a displays the temporal variation of the dimensionless heat flux density,

q

⁰⁰

, at some interfacial locations and the dimensionless heat flux, q

⁰

, transferred from the left interface to the PCM. As it can be deducted from the analysis of such figure, at earlier stages, when conduction prevails heat sources in the melt, the temperature difference between the left wall and the PCM layer increases causing an increase in the local dimensionless heat flux density and the dimensionless heat flux;

the melt cavity expands close to the right vertical face of heat sources and, later, close to the rest of the left walls. Next, the thermal resistance across the liquid layer increases, which results, in general, in a decrease in the magnitude of the local dimensionless heat flux density on the left solid–liquid interface. It should be noted that during that first stage of the melting process, one amount of the heat generated in heat sources is transferred to the PCM layer in contact with the right vertical faces of heat sources. The other part of that heat is transmitted by conduction to wall to increase itself temperature. The heat flows weakly to the PCM layers in the vicinity of the wall because of the wall thermal inertia. This behaviour explains the negligible interface wall–PCM local dimensionless heat flux density at the beginning of the melting process. When buoyancy driven flow appears in the liquid zones located in front of the vertical faces of heat sources, the local dimensionless heat flux density stops decreasing in these locations and then slightly increases. When natural convection establishes, quasi-steady state period is reached, and all the heat generated in heat sources is practically absorbed by the melting front (the temper- atures of heat sources-wall remain constant). The local dimensionless heat flux density reaches a constant value and the curves become closer to horizontal solid lines. During that stage, the dimensionless heat flux density also remains constant.

After this stage

ð

s

P

0:15Þ, the dimensionless local heat flux density increases, decreases or remains constant before decreas- ing, depending on the flow structure. At the same time, the dimensionless heat flux slightly increases and decreases next until the end of the melting process. The reason behind this decay is due to the increase in the storage energy rate of the wall when the melting process approaches its end (see

Fig. 11b).

5. Conclusion

A 2D mathematical model based on mass, momentum and energy conservation equations was developed for a conjugate conduction-natural convection dominated melting within a rectangular cavity with three volumetric protruding heat sources mounted to one of the vertical walls. The resulting equations were next integrated using the control volume ap- proach. The algebraic equations were solved by an iterative method (TDMA). Numerical investigations were carried out to examine the thermal trends of the proposed phase change material based heat sink. The results show that:

– At the beginning of the melting process, the bottom PCM region is predominated by conduction;

– the flow is clockwise multi-cellular and the streamlines are deformed by the protrusion;

– the flow is broken into tow separate zones when the melting front touch the right adiabatic walls;

– heat from the modules is removed in part by natural convection liquid PCM while the rest is conducted to the wall before being dissipated in PCM cavity;

– the liquid fraction increases linearly but will be late at the end of the process;

– at earlier stages, temperatures of heat sources increase practically linearly;

– the maximum temperature situates in central heat source when conduction prevails, but when convection develops the upper heat source registers this maximum temperature;

– wall is not isotherm, this leads to conclude that it is inappropriate to decouple conductive heat transfer within both the wall and the components and natural convection in the cavity;

– the highest heat transfer rates are observed for the bottom heat source;

– the mean heat source Nusselt number decreases and reaches a plateau as melting progress showing that all heat gener- ated is absorbed by the melting front and heat sources temperature is stationary;

– the model can also be used for thermal design for other geometric PCM-based heat sink.

In the future numerical simulations, the effect of the control parameters will be investigated in order to find their optimal values that maximize the thermal performance of the PCM-heat sink.

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