Thermal analysis of a phase change material based heat sink for cooling protruding electronic chips

Received: December 2008 H. EL Qarnia: Professor

www.springerlink.com

DOI: 10.1007/s11630-009-0268-1 Article ID: 1003-2169(2009)03-0268-08

Thermal Analysis of a Phase Change Material Based Heat Sink for Cooling Protruding Electronic Chips

Mustapha FARAJI

, Hamid EL QARNIA

and El Khadir LAKHAL

1. Fluids Mechanic and Energetic Laboratory,

2. Automatic, Environment and Transfer Process Laboratory, Cadi Ayyad University, Faculty of Sciences Semlalia, De- partment of Physics P. O. Box 2390, Marrakech, Morocco

This work aims to numerically study the melting natural convection in a rectangular enclosure heated by three discreet protruding electronic chips. The heat sources generate heat at a constant and uniform volumetric rate. A part of the power generated in the heat sources is dissipated to a phase change material (PCM, n-eicosane with melting temperature, Tm = 36

). Numerical investigations were carried out in order to examine the effects of the plate thickness on the maximum temperature of electronic components, the percentage contribution of plate heat conduction on the total removed heat and temperature profiles in the plate. Correlations for the dimensionless se- cured working time (time to reach the threshold temperature, Tcr = 75

) and the corresponding liquid fraction were derived.

Keywords: Phase change material, protruding electronic component, natural convection

Introduction

Thermal management of electronic packaging is critical as we approach the sub-micron and even nano-scale sizes on a chip. Garimella [1] considered recent advances in a number of novel high-performance cooling techniques for emerging electronics applications. Application of melting process of phase change material (PCM) to the cooling management of electronic components as an al- ternative cooling method for various applications such as spacecraft and power electronic equipment has received significant attention in recent years. This may be a prom- ising alternative due to relative high removal capabilities of PCMs compared with liquid and air options [2].

O’Connor and Weber [3] measured the thermal perform- ance of the PCM heat sink. Abhat [4] performed a com- bined experimental and computational study of a thermal control unit, where a PCM was incorporated in hexago-

nal honeycomb cells aligned parallel to the heating sur- face. Alawadhi and Amon [5] investigated the effective- ness of a thermal control unit (TCU) made of PCM and aluminium fins for portable electronic devices. Nayak et al. [6] numerically studied the performance of a PCM based heat sink with various volume fractions of thermal conductivity enhancers (TCEs) for different configura- tions. Results illustrate a significant effect of the thermal conductivity enhancer on the performance of heat sinks.

Cooling management of microprocessors (CPU) was also

investigated by Faraji and El Qarnia [7] using a hybrid

heat sink consisting of a rectangular cavity filled with a

PCM (SunTech P111) attached to the conventional alu-

minium fins with a density of heat flux imposed to the

base of the microprocessor. Zhang et al. [8] experimen-

tally investigated the melting process of n-octadecane

that is discretely heated by flush mounted heat sources at

a constant rate. One of the important results shows that,

Nomenclature x, y Cartesian coordinates (m) c

specific heat (J kg

K

) Subscripts

f liquid fraction, plate thickness (m) 1, 2, 3 refer to the bottom, middle and the upper electronic components, respectively.

g gravity (m s

) cr critical value

Gr

Grashoff number, Eq. (20) c, EC electronic component H enthalpy (J kg

) or heat transfer coeffi-

cient (W/m

)

f fluid

H

height of the PCM enclosure (m) l liquid PCM

k thermal conductivity (W m

K

) m PCM, melt l

characteristic length (m), Eq. (19) max maximum value L

electronic component height (m) o initial

L

dimensionless electronic component height, L

L / l

ref reference value L

space between two successive sources (m) s substrate L

position of the bottom heat source (m) Greek symbols

L

width of the PCM enclosure (m) α thermal diffusivity (m² s

) Nu average heat source Nusselt number =

o m,l

h l k

β expansion factor (K

)

p pressure (Pa)

θ dimensionless temperature, ^{T T}

θ T

Δ

= − Pr Prandtl number, _Pr = ν

_/ α

τ dimensionless time,

o

l t τ = α

Q’ heat generation per unit length (W m

) δ

, δ Kronecker symbol, distance (m) Ra Rayleigh number,

3 o m,l m,l

g β l ΔT

Ra = υ α η distance perpendicular to the hot wall/liquid PCM interfaces

Ste Stefan number,

f

(c ) ΔT

Ste = ΔH μ dynamic viscosity (kg m

s

)

T temperature (K) υ kinematic viscosity (m² s

)

t time (s) ΔH

latent heat (J kg

)

X

X

electronic component, plate thickness (m)

ΔT characteristic temperature (K

), ³

m,l

T Q Δ = k X

dimensionless electronic component

thickness, X

X / l

ρ density (kg m

) 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. An experimental study of melting and natural convection heat transfer in an en- closure with three protruding heat sources, generating heat at a constant rate and attached on one of vertical walls, was conducted by Ju et al. [9]. Jianhua et al. [10]

studied the same configuration analyzed in [9], but heat sources are positioned on the bottom horizontal wall of the enclosure. It can be seen that no numerical study of melting and natural convection with protruding heat

sources was reported up to now. The present study con- siders the problem of the melting and 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 objective of the current study is to examine the ef-

fects of the plate thickness on the maximum temperature

of heat sources, the percentage contribution of substrate

heat conduction on the total removed heat, and tempera-

ture profiles in the substrate. Correlations were devel-

oped for the secured working time and the corresponding

realized melt fraction.

Mathematical model

Figure 1 presents the physical model, which consists of a rectangular enclosure containing a phase change material (PCM) with three identical protruding electronic components, denoted (1), (2) and (3), attached on the left wall. The height and thickness of each heat source are L

and X

, respectively. The distance between two consecu- tive heat sources is L

and the distance between the bot- tom enclosure wall and the bottom face of the lower heat source is L

. The height and width of the enclosure are H

and L

, respectively. The thickness of the plate (sub- strate) is X

.

Fig. 1 The physical model

The thermal conductivity, k

, specific heat at constant pressure, c

, and density, ρ

, of the heat sources are dif- ferent from those of the wall, k

, c

and ρ

. Each elec- tronic component generates a constant and uniform volumetric power. The particularity of the present cool- ing strategy is that there is no fan, and the device works till the end of the melting of PCM or if one of heat sources reaches the critical temperature T

. The flow is assumed to be two-dimensional, Newtonian, laminar and incompressible. The physical properties of the materials are constant at the temperature range under study. The density difference between 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

= T

. The reference tem- perature is equal to the melting temperature, T

. The density of the PCM is taken as a reference density. Based on the above mentioned assumptions, the governing equations for the PCM, heat sources and substrate are given by the following general form:

( )

( )

. u S

t

ρΦ ρ Φ Γ Φ

∂ + ∇ − ∇ =

∂ (1).

where Γ is the coefficient of diffusion and S is a source

term associated to the variable Φ (u, v or h). These quan- tities are summarized in Table 1.

Table 1 Terms of Eq. (1).

Φ Γ S

1 0 0

u μ u

p S x

−∂ +

∂

v μ v

p S y

−∂ +

∂

h k/c ST

( )

m

T

p m

T

h = ∫ c dT+h T

(2)

2 3

( )

( )

u

S = C 1 f u f +b

− −

(3)

( )

( )

( )

v 3 ref m

S C 1 f v g T T

f ⁻ b ρ β

= − + −

+

(4)

1

(1

)

T f

c c

f Q'

S

δ

δ ρΔΗ

t

δ X .L

(5)

Boundary conditions

The interface conditions between two different mate- rials (1 and 2) (substrate, PCM or electronic component) are set as bellow:

1 2

int erface int erface

T T

k k

η η

∂ ∂

∂ = ∂ , T

=T

(6)

where η ∞ interfaces.

At the adiabatic walls:

= 0

w all

T η

∂

∂ (7) and

u=v=0 (8) Initial conditions

u=v=0, T=T

, f=0 (9) S

and S

are source terms used for the velocity suppres- sion in the solid regions (solid PCM, substrate and heat sources). One of the common models for the velocity suppression is to introduce a Darcy-like term [11] (C = 10

kg m

s

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 with taking a large value of the viscosity for solid regions.

The conductivity and the step function, δ , are set as

follows:

m s c

1

2

k for PCM k k for substrate

k for electronic components

1 for electronic components and PCM 0 for substrate

1 for electronic components 0 for PCM

δ δ

⎧ ⎪

= ⎨ ⎪

⎩

= ⎨ ⎧

⎩

= ⎨ ⎧

⎩

(10)

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

, thermal conductivity at interfaces, k

, and the specific heat, c

, are evaluated in terms of the liquid fraction as follows:

k

=f k

+(1−f) k

( )

s m p s

i

s p m s

k k k

k k

δ δ

δ δ

= +

+ (11)

( )

^c

^f ( )

^c

^{(1 f)} ( )

^c

where δ

is the distance from the interface to the first node in the fluid region and δ

is the distance between the interface and the first node inside the solid.

The discretized equations are obtained by integrating the governing equations in a staggered mesh, with M nodes in (x) direction and N nodes in (y) direction, using a finite volume method developed by Patankar [12]. 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 ve- locity equations. The discretized equations are derived over non uniform 60 × 120 grids. A fine grid size near solids was set to give more detailed hydrodynamic be- havior near interfaces, and the time step, 20 s, was found sufficient. Other grid sizes and small time steps has no effect on the numerical results. For every time step, the resulting algebraic equations are solved using the Tri-Diagonal Matrix Method (TDMA). The duration of a typical simulation exceeds 9 hours in a desk computer (CPU 2.6 GHz and RAM 1 Gb). The accuracy of the numerical results was examined by checking the energy and mass balances for every time step. Convergence is declared when the following criteria relating to mass and energy balances, ε

and ε

, defined as follows, are smaller than 10

and 10

, respectively.

( ) ( )

in out m

Max ϕ i,j − ϕ i,j ≤ ε (12) 1 ( − Q

+Q

+Q

+Q

) ≤ ε

(13) where ϕ

and ϕ

are the inlet and outlet mass flow per unit length:

( )

( )

( ) ( ) ( )

( ) ( ) ( )

1 1; 1 1

in out

i,j u i,j Δy v i,j Δx i,j u i 1,j Δy v i,j 1 Δx i M+ j N+

ϕ ρ

ϕ ρ

= + ⎫⎪

= + + + ⎬ ⎪⎭

≤ ≤ ≤ ≤

(14)

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

( ) /

3

s s sold substrate

sen, s

h h x y t

Q Q'

ρ − Δ Δ Δ

= ∑

(15)

( ) /

3

m mold liquid PCM

sen,l

h h x y t

Q Q'

ρ − Δ Δ Δ

= ∑

(16)

( ) /

3

1,2 ,3

old

c c c

heat source sen, c

h h x y t

Q Q'

ρ − Δ Δ Δ

= ∑

(17) Latent heat stored in the liquid PCM is defined as fol- lows:

3

f lat PCM

ΔH f x y Q t

Q' ρ ^∂ Δ Δ

= ∑ ∂

(18).

Validation

In order to check the physical validity of the proposed computational model, a comparison between predictions and experimental data obtained by Ju et al. [9] was con- ducted. The work conducted by Ju et al. [9] consists of an experimental study of the melting of PCM (n-octadecane, T

= 28 ℃ ) in an insulated rectangular enclosure heated with three protruding heat sources mounted on a vertical wall. The wall is made with Plexiglas. The heat flux den- sity, delivered by each heat source, was equal to 900 W/m² (Ste = 2.37, Ra = 3.74x10

). The geometric pa- rameters are summarized in Table 2.

Table 2 Geometric parameters of the experimental setup of Ju

et al. [9]

Xc(m) Lc(m) Le (m) Lh(m) Hm(m) Xs(m) Lm(m) 0.009 0.015 0.015 0.0075 0.09 0.02 0.06

The developed computer program was adjusted to re- produce experimental conditions, and the numerical pre- dictions have been, next, compared to experimental data.

Figure 2 displays the comparison between the nu- merical predicted and measured heat sources tempera- tures and temperatures obtained by Ju et al. [9]. Analysis of such a figure shows a satisfactory agreement between the measured and calculated temperatures. The maximum deviation is found less than 1 %. The slightly difference in the temperature history appears when heat sources reach the plateau region (quasi steady state). It is due, at prior, to the surrounding heat loss that occurred during experimental tests.

Results and discussion

In this numerical study, the volume of the PCM is kept

constant, and represented by the characteristic length, l

, given below :

3 0.06

o m m c c

l = H L − X L = m

(19) The power per unit length, Q’, delivered by each heat source is also kept constant and equal to 60 W/m [10].

For the baseline case, the modified Rayleigh number, Ra,

is

m,l m,l

g

Ra

= 5.01x10

. The values of the other pa- rameters are given in Table 3. They correspond to the values of geometric and thermo-physical parameters, frequently used in electronics [13]. The numerical inves- tigations are driven with varying the plate thickness, X

, from its reference value given in Table 3 (X

= 0.005 m).

Fig. 2 A comparison between numerical predicted surface

temperatures of discrete heat sources and measured temperatures obtained by Ju et al. [9].

Table 3

Parameters corresponding to the reference case.

Modules (ceramics)

Plate (alumina substrate)

PCM (n-eicosane) ρc= 3260 kg/m³

(cp)c = 740 J/kg kc = 170 W/m.K

Tcr = 75℃

Q’ = 60 W/m

ρs= 3900 kg/m³ (cp)s= 900 J/kg ks = 19.7 W/m.K

B=8.5

×

×

Tm= 36℃

(cp)m =2460 J/kg km= 0.1505 W/mK Xc(m) Lc(m) Le (m) Lh(m)

0.003 0.015 0.01 0.03

Hm(m) Xs(m) Lm(m)

0.121 0.005 0.035

ρm= 785 kg/m³ μm = 4.15x10⁻³kg/ ms

Pr = 67.83 Ste = 11.91

It should be noticed that the natural convection flow is laminar. In fact, based on the maximum temperature dif- ference, T

−T

= 39℃ (T

= T

), the Grashoff num- ber, Gr

, given by the following expression:

max m 3m

2m

g β ( T T ) H

Gr υ

= −

(20) is equal to 4.32×10

. This value is clearly lower than the transition value, 1.5×10

, as indicated in reference [14].

Since the plate and module thermal conductivity ratios, k

/k

and k

/k

are equal to 131 and 1130, respectively, conjugate heat conduction coupled to the fluid flow and phase change process is included in the mathematical model. A part of the heat generated by the heat sources is transmitted to the PCM by following two distinct paths.

Indeed, heat can reach the surface of the heat sources by conduction and convects away by the melted PCM stream. It also spreads in the printed circuit board (PC), which is the plate. Note that the particularity of the pre- sent cooling strategy is that the proposed PCM-based heat sink acts without fan, and the electronic chips work till the end of the melting of PCM or if one of the electronic components reaches the critical temperature T

.

Figure 3 shows the transient maximum temperatures of the heat sources and total liquid fraction for substrate thickness ratios, X

/X

, ranging from 0.4 to 2. By ana- lyzing this figure, it is seen that the maximum tempera- ture goes through three stages. At early times (first stage), the maximum temperature increases, reaches a maximum

Fig. 3 Transient maximum temperatures of the electronic

components (solid lines) and liquid fraction (dashed

lines) for various substrate thicknesses, X

/X

.

and decreases slightly. The second stage begins when the maximum temperature reaches a plateau (quasi steady state), during which the temperatures of chips remain constant. After this period, the maximum temperature quickly rises to the critical temperature, T

= 75℃ (third stage). Note that higher maximum temperatures and minimum working time (time required to reach the criti- cal temperature, T

= 75℃) are obtained for thinning substrates, due to their relative lower thermal inertia. For example, the quasi steady state temperature reached by the heat sources are 56.9 ℃ and 54.9 ℃ for X

/X

= 0.4 and 2, respectively. Analysis of this figure, also, shows that the duration of the second stage is nearly independ- ent on the substrate thickness ratio. In fact, whatever the ratio X

/X

is, the duration of this stage is about 1500 s.

The stability of the maximum temperature during this stage is explained by the fact that all the heat generated by the heat sources is convected away by the molten PCM currents, when natural convection is developed.

Note that the second stage is achieved when about 57 % of the solid PCM is melted, for all cases. The duration of this stage decreases with decreasing substrate thickness.

At the end of the melting process, the heat source tem- perature goes inwards to the threshold value.

Figure 4 shows the effect of the substrate thickness ra- tio, X

/X

, on the substrate temperature distribution at different times, t = 320 s, 2300 s and 4300 s, for x = Xs/2.

These times are selected in order to represent all stages of the phase change process. As it can be seen from this figure, at t = 320 s, the substrate thickness ratio has no effect on the temperature distribution in the substrate, and the maximum temperature is located near the middle heat source. As time progresses (t = 2300 s), the molten PCM region expands and a lower portion of substrate becomes well cooled in comparison to the other substrate portions as a result of natural convection currents. With an increase in substrate thickness ratio, the most upper and lower portions of substrate become slightly heated.

However, the picture is reversed for the upper portion at t = 4300 s. Also, it can be seen that, with an increase in

substrate thickness ratio, the overall temperature differ- ence in the substrate drops. For example, at t = 2300 s, this difference is 14℃ and 8℃, for X

/X

= 0.4 and 2, respectively. So the use of isothermal or isoflux boundary conditions at the plate does not properly represent the heat transfer characteristics in the present situation. Note that Figure 4 also shows that the maximum temperature location is changed from the middle heat source to the upper one, and quickly approaches to the critical tem- perature, when time progresses.

Table 4 summarizes the percentage of heat generated in modules and evacuated from the left solid/liquid inter- face during the second stage (quasi steady state). By analyzing this table, we see that the heat removal in- creases from 14 % to 22.4 % for the bottom plate portion and from 7 % to 8.73 % for the upper plate portion, as the substrate thickness ratio increases. Indeed, following the usual fin approximate analysis [15], the thermal re- sistance of the downstream (upstream) substrate may be approximated as:

0.5

1

( )

s

s c s m

R ≈ k h X H

(22)

Fig. 4 Substrate temperature distributions for various sub-

strate thickness ratios X

/X

at x = X

/2.

Table 4 Plate and modules heat transfer contribution for various substrate thickness ratios.

Xs/Xs,ref % heat generated in modules

0.4 14.00 19.00 7.70 21.00 7.20 18.50 7.00 0.6 16.05 18.60 7.60 19.87 7.00 17.80 7.70 1 18.74 18.63 7.24 19.85 6.78 17.30 8.50 1.6 20.60 18.10 7.10 18.30 6.40 15.90 8.70

2 22.40 18.10 6.90 17.70 6.00 15.40 8.73

EC1 EC2 EC3

plate

y(m) y(m) y(m)

So as the thermal resistance of substrate decreases with an increase of the substrate thickness, X

, the ther- mal spreading in substrate intensifies. Consequently, the heat transfer rate from the most bottom and upper por- tions of the substrate increases with the increase in the substrate thickness ratio, X

/X

. The total heat trans- ferred from the faces of the heat sources and substrate is enhanced from 94.4 % to 95.23 % by increasing X

/X

from 0.4 to 2, which explains the decrease of the maxi- mum temperature, during the second stage (quasi steady state), as it was shown in Figure 3.

The average Nusselt number for heat source “j”, Nu

, based on the maximum temperature, is defined as follow:

2 1,2,3

0

1 2

c c

L X

i

c c max i

Nu k θ ds

η (L X ) θ

+

∂

= − + ∫ ∂

(21) Figure 5 displays the average heat sources Nusselt numbers and maximum temperature dependency on the substrate thickness ratio X

/X

during the second stage (quasi steady state). As it was shown in Table 4, the heat removal from the faces of heat sources decreases from 58.5 % to 51.2 %, which explains the slight drop in the mean heat sources Nusselt numbers, Nu, as shown in Figure 5.

Fig. 5 Average heat sources Nusselt numbers and maximum

temperature dependency on the substrate thickness ratio, X

/X

.

The reliability of the electronic devices depends on the maximum temperature. The chip temperature increases transiently from 0 to T

. When the chip temperature approaches the threshold value (T ~T

= 75℃), the de- vice has to stop working. The proposed concept based on the PCM heat storage reservoir has to work intermittently.

The heat stored in the PCM is naturally rejected to the ambient, and the melted PCM re-solidifies during the

‘switch off’ periods. As the melted PCM re-solidifies, it can be used in the next cycles. It was found that, from

Figure 6, the maximum dimensionless working time, τ

, and the total liquid fraction, f

,

, achieved at the end of the charging process (T ~T

) vary, approximately, like

0.5 s s,ref

X X

⎛ ⎞

⎜ ⎟

⎜ ⎟

⎝ ⎠

, within the range 0.4 to 2. The following numerical correlations were derived:

0.5

0.059628 + 0.041925

max

s,ref

= X

τ ^⎛ ⎜ ⎜ ⎝ X ^⎞ ⎟ ⎟ ⎠ (23)

0.5

0.69558 + 0.167647

t,max

s,ref

f X

X

⎛ ⎞

= ⎜ ⎜ ⎝ ⎟ ⎟ ⎠ (24).

The parity plot, Figure 7, shows a good agreement of the above correlations with numerical solutions. Maxi- mum deviation was found less than 3 % for the working time, τ

, and less than 1 % for the liquid fraction, f

.

Fig. 6

Maximum electronic component dimensionless work- ing time, τ

, and liquid fraction, f

, as a function of X

/X

.

Fig. 7

Parity plot.

Conclusion

In the current work, melting of a phase change mate- rial (PCM) in a rectangular cavity heated with three pro- truding heat sources mounted on a vertical conducting plate has been numerically studied. It has been shown that PCMs can be used to absorb power dissipation by electronics. The usage can also reduce the size of the cooling system (saving money, space, or other system resources). A mathematical model was developed and numerical investigations were conducted to study the effects of the substrate thickness ratio on the thermal performance of the cooling PCM based heat sink. It has been concluded that, within the same geometry, the se- cured working time and the melt fraction achieved de- pend on the substrate thickness. Higher working time and lower maximum temperature are obtained for higher sub- strate thickness. Correlations, giving maximum working time and melt fraction were developed. The results clearly showed that the working time, τ

, required by chips to reach the critical temperature (T

) depends closely on the substrate thickness, so neglecting the effect of the plate in electronics thermal analysis is not suitable.

The maximum duration of the plateau region (second stage), corresponding to the stable values of chips tem- peratures, depends also on the substrate thickness. The approach developed herein can be used in the design of PCM-based cooling systems.

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