Three-dimensional transient cooling simulations of a portable electronic device using
PCM in multi-fin heat sink
Yi-Hsien Wang , Yue-Tzu Yang
Department of Mechanical Engineering National Cheng Kung University, Tainan, 70101, Taiwan Corresponding author. Tel.: +886-6-2757575 ext. 62172; Fax: +886-6- 2352973. E-mail address: [email protected]
Abstract
Transient three-dimensional heat transfer numerical simulations were conducted to
investigate a hybrid PCM (phase change materials) based multi-fin heat sink.
Numerical computation was conducted with different amounts of fins (0 fin, 3 fins
and 6 fins), various heating power level (2W, 3W and 4W), different orientation tests
(vertical/horizontal/slanted), and charge and discharge modes. Calculating time step
(0.03s, 0.05s, and 0.07s) size was discussed for transient accuracy as well. The
theoretical model developed is validated by comparing numerical predictions with the
available experimental data in the literature. The results showed that the transient
surface temperatures are predicted with a maximum discrepancy within 10.2%. The
operation temperature can be controlled well by the attendance of phase change
material and the longer melting time can be conducted by using a multi-fin hybrid
heat sink respectively.
Keywords: Phase change material, Electronic cooling, Numerical simulation
1
Nomenclature
A, B, C Constant
b depth between heater to PCM (m)
pC specific heat (J/Kg K)
F melting volume fraction of PCM
g gravity ( ) 2m/sH height of heat sink (m)
specific enthalpy (J/kg)
h height of PCM (m) he length of heater (m)
a bL , L , L
c length of heat sink (m)
PCML length of PCM (m)
k thermal conductivity (W/m K)
P pressure (pa)
q wall heat flux (W/m2)
iS source term
T temperature (K)
t time step (s)
t time (s)
i ju ,u velocity (m/s)
W width (m) We width of heater (m) Wf width of fin (m) Wpb width of PCM in case B (m) Wpc width of PCM in case C (m)
,i jx x Coordinates (m)
Greek symbols
n volume fraction in nth phase thermal expansion coefficient liquid fraction dynamic viscosity (kg/m s) density ( ) 2kg/m
Constant
Subscripts ,i j component
L Liquid
2
3
m Melting
N nth phase
PCM phase change materials
S Solid
1. Introduction
Sensible and latent heat storages are the physical phenomena for thermal energy
storage which can be applied to different thermal applications, such as electronic
cooling and hot water insulation. A latent heat storage system requires lower weights
and fewer volume changes of material compared to conventional sensible heat energy
storage systems for a given amount of energy.
Reviewed the PCM experimental study, Setoh et al. [1] examined the cooling of
mobile phones using a phase change material (PCM). The result showed that using
suitable usage of PCM could keep the mobile phone under a stable temperature. Ho
and Viskanta [2] reported the heat transfer data during melting of n-octadecane from
an isothermal vertical wall of a rectangular cavity. The result showed that except in
the very early stages of melting, the rates of melting and of heat transfer were greatly
affected by the buoyancy driven convection in the liquid. LAfdi et al. [3] studied the
potential of using foam structures impregnated with phase change materials as heat
sinks for cooling of electronic devices. Humphries and Griggs [4] investigated the
heat transfer characteristics associated with the design and use of PCM thermal
capacitors. Hongbo et al. [5] investigated the cold storage with liquid/solid phase
change of water based on the cold energy recovery of Liquefied Natural Gas (LNG)
refrigerated vehicles. Water was adapted as the phase change material. Chan and Tan
[6] reported the findings of the solidification of an n-hexadecane inside a spherical
enclosure. The results showed that the solidification phase front progressed
concentrically inwards from the colder outer surface of the sphere.
The thermal management of battery modules with phase change materials (PCMs)
is investigated experimentally by Duan and Naterer [7]. PCM cylinder surrounding
the heater, and PCM jackets wrapping the heater are presented.
Wei et al. [8] reported a thermal energy storage system employing the phase change
material (PCM) FNP-0090 (product of NipponSeiro Co. Ltd.) for a rapid heat
discharge. The experimental study of PCM cooling application was studied by Fok et
al. [9]. The result indicated that PCM-based heat sinks with internal fins could be
viable for cooling hand-held electronic devices. A Phase Change Slurry (PCS) test is
conducted by Huang et al. [10]. The results indicated that the paraffin/water emulsion
containing a paraffin weight fraction of 30–50 wt.% is an attractive candidate. Mario
et al. [11] studied a new type of wall panel which is called phase change material
structural insulated panel (PCMSIP). Pal and Joshi [12] carried out an experimental
and computational study of melting in a tall enclosure by a constant heat flux source
and adiabatic boundaries which focus on the effect of natural convection. Mettawee
and Assassa [13] investigated the performance of a compact phase change material
(PCM) solar collector based on latent heat storage. The experimental results showed
that in the charging process, the average heat transfer coefficient increases sharply
with the increasing molten layer thickness, as the natural convection grows stronger.
Numerical simulation study have been discussed and predicted as following.
Shatikian et al. [14] carried out the transient three- and tow-dimensional simulations
using Fluent 6.0 software. Results showed that the transient phase-change process,
expressed in terms of the volume melt fraction of the PCM, depends on the thermal
and geometrical parameters of the system. The numerical study of melting natural
convection in a rectangular enclosure heated by three discreet protruding electronic
chips has been conducted by Faraji et al. [15]. The result presented that the working
time required by chips to reach the critical temperature depends closely on the
substrate thickness. Kandasamy et al. [16] studied various parameters with PCM.
Results showed that increased power input increased the melting rate. P. Lamberg and
Siren [17] presented a simplified analytical model to predict the solid–liquid interface
location and temperature distribution of the fin in the melting process with a constant
imposed end-wall temperature. The result was consistent with the numerical data.
Akhilesh et al.[18] presented a thermal design procedure for proper sizing of
composite heat sinks to get the maximizing energy storage and the longer melting
period in a given range of heat flux and height. Gong and Mujumdar [19] developed a
novel storage unit of multiple phase change materials. The finite element model has
been adapted to simulate the model involved as a result of alternating melting and
freezing processes. Huang et al. [20] found that 2D model prediction was compared
well with those of the 3D model under appropriate boundary conditions. Wang and
Mujumdar [21] studied the effect of orientation of heat sink on the thermal
performance. Brent et al. [22] used an enthalpy-porosity approach to model the
melting of pure gallium in a rectangular cavity with combined convection-diffusion
phase change. It showed that the method converges rapidly and is capable of
accurately predicting both the position and morphology of the melt front at various
time with relatively modest computational requirements.
Nayak et al. [23] studied the numerical model for heat sinks with phase change
materials and thermal conductivity enhancers. Kim et al. [24] conducted the
feasibility of using a novel cooling strategy that utilized the heat load averaging
capabilities of a phase change material (PCM). Vargas and Bejan [25] studied the
optimum thermodynamic match between two streams at different temperatures which
4
is determined by maximizing the power generation. Ravi et al. [26] studied the heat
transfer behavior of phase change material fluid (PCM) under laminar flow conditions
in circular tubes and internally longitudinal finned tubes.
Hao and Tao [27] studied the simulation of the laminar hydrodynamic and heat
transfer characteristics of suspension flow with micro-nano-size phase-change
material (PCM) particles in a microchannel. Saha and Dutta [28] studied the
characterization of melting process in a Phase Change Material (PCM)-based heat
sink with plate fin type thermal conductivity enhancers (TCEs). He et al. [29]
combined phase equilibrium considerations with DSC measurements, a reliable
design method to incorporate both the heat of phase change and the temperature
range.
The present study simulates the cooling technologies of portable hand-held
electronic devices with a constant and uniform volumetric heat generation using the
phase change material. Dimensions and conditions of the test model referred to the
experimental data reported by Fok et al. [9]. Correlations are developed for secured
working time and the corresponding surface temperature.
2. Mathematical formulation
2.1. Governing equations
The schematic diagram of the three-dimensional physical models is shown in
Fig.1. The system is designed based on the average dimensions of typical portable
hand-held electronic devices and all dimensions of the computational domain refer to
the experimental study by Fok et al. [9]. Dimensions of simulation for case A, B and
C are listed in Table.1. The PCM is based on the properties of N-eicosane as listed in
Table 2. The PCM base heat sink is composed by a solid aluminum block with a
cavity that contains with 44 mL phase change material inside each case, and the
height of air gap in each case is setting by 1 mm.
The following assumptions are made to model the PCM based heat sink heat
transfer problem:
(1) three-dimensional,
(2) laminar flow,
(3) unsteady state,
(4) constant fluid properties,
(5) neglect radiation heat transfer.
The governing equations for the conservation of mass, momentum and energy for
this hybrid system can be written as
(1) continuity equation
5
0
n ni
i
ut x
(1)
(2) momentum equation 2
,
( )( ) n i j i
n i n n i n ij j j i
u u u pu
t x x x xg S
(2)
(3) Energy equation
n( ) ( ) (kn n ii i
Tu
t x x
)ix
(3)
wherenis the nth fluid’s volume fraction in the computational cells. For air
phase, the density depends on its temperature as shown in Table 1. For the aluminum
solid phase, constant thermo physical properties are specified. For the PCM phase,
considering computational continuity during phase change, the density can be
expressed as
LPCM
m(T-T )+1
(4)
Where L is the density of PCM at the melting temperature and mT is the thermal
expansion coefficient. is the dynamic viscosity of liquid PCM which is given by
B0.001 exp(A )
T (5)
Where A = and B = 1790 which is reported by Humphries and Griggs [4],
is the velocity component an is the specific enthalpy.
4.25
iu d is the liquid
fraction during the phase change which occurs over a range of temperatures lT T ,
defined by the following rel
T
ations:
. ,sT T 0, a . ,lT T 1, b . , ss l
l s
T TT T T
T T ,
c (6)
The source term in the momentum equation is given by iS
2
3
C(1 )iS
iu (7)
where 2
3
C(1 )
is the mimic ‘‘porosity function’’ defined by Brent et al. [22], so
that the source term will be active while the liquid fraction is larger than zero to
take the computational balance in momentum equation. The value of C is a mushy
zone constant reflecting the behavior of melting phase-change materials. The value of
6
C = has been used in the present study. The constant 510 = 0.001 is a constant to
avoid zero in the number.
2.2. Boundary conditions
Initial and boundary conditions follow the experimental setup by Fok et al. [9]. The
entire initial calculation domain is setting the temperature at 25°C besides the base
area of heater. The bottom wall of the heater ( 25 50 mm in x-z plane) in each case is
set at a constant heating power with 3 W. The boundary condition in each side of the
wall is setting adiabatic. The symmetry boundary condition in half domain is used to
reduce the calculation time as shown in Fig.2. The appliance of symmetry verification
is also discussed by Fok et al. [9]. The following initial and boundary conditions are
applied to solve the governing equations:
1. Initial condition: (given in case A, B and C)
25T °C, 0iu initial
2. Heat flux supplied from the bottom heater:
Case A ''
11~36 ,s0 ,
17.5~67.5
kx mmy mmz mm
T q
y
,
Case B ''
11~36 ,s0 ,
20.5~70.5
kx mmy mmz mm
T q
y
,
Case C ''
11~36 ,s0 ,
23.5~73.5
kx mmy mmz mm
T q
y
.
3. Symmetry boundary conditions at side:
Case A 36 ,0~18 ,0~85
0x mmy mmz mm
T
x
,
36 ,0~18 ,0~85
0i
x mmy mmz mm
u
x
Case B 36 ,0~18 ,0~91
0x mmy mmz mm
T
x
,
36 ,0~18 ,0~91
0i
x mmy mmz mm
u
x
Case C 36 ,0~18 ,0~97
0x mmy mmz mm
T
x
,
36 ,0~18 ,0~97
0i
x mmy mmz mm
u
x
4. Insulation boundary condition:
Case A 0
0x mm
T
x
,
0 ,18
0y mmy mm
T
y
, (not including heater),
0 ,85
0z mmz mm
T
z
7
Case B 0
0x mm
T
x
,
0 ,18
0y mmy mm
T
y
, (not including heater),
0 ,91
0z mmz mm
T
z
Case C 0
0x mm
T
x
,
0 ,18
0y mmy mm
T
y
, (not including heater),
0 ,97
0z mmz mm
T
z
3. Numerical computations
The numerical solution has been solved using the Fluent 12.0 software. The half
computational grid was built of 36 18 86 98
(55,728 cells) in case A,
(59,616 cells) in case B, (63,504 cells) in case C as shown in Fig.2. The
grid test was compared to the experimental data results by Fok et al. [9]. After
comparison to careful examination of grid refinement process, the grids are chosen to
simulate the following cases. Compared to different calculating time step size, the
transient temperature distribution of time step size
36 18 92 36 18
t =0.05s shows a good
alignment with the experimental data in Fig.3. The convergence criterion at each time
step was checked under for momentum equation and 410 610 for continuity and
energy equations.
The melting three-dimensional Navier-Stokes and energy equations are solved
numerically by a finite-difference scheme, and an enthalpy-porosity approach
equation is adapted to simulate the phase-change (melting) boundary. The PCM-air
VOF model is implemented to solve the continuity equation and PCM-air gap
boundary due to the phase change volume expansion. In the VOF model, the
Geo-Reconstruct which is the volume fraction discretization scheme is used to
calculate the transient moving boundary. If the transient temperature of calculation
cell in the PCM domain is equal to or higher than the melting temperature, the
continuity and momentum equation will be calculated in the system. A uniform grid
system with a large concentration of nodes in regions of an even gradient is employed
due to highly unstable melting process. The numerical method used in the present
study is based on the SIMPLE algorithm of Patankar [30]. A comparison of theoretical
predictions with the experimental data in the literature was used to assess the grid
independence of the results.
Different sizes of meshes are employed to test the numerical model. The time step
duration of calculation is also discussed in the literature for further precise prediction
in Fig.4. The simulation result has been validated with the experimental data which
was reported in Fok et al. [9]. Certain discrepancies between calculations and the
available data by Fok et al. [9] may be caused by the round off and discretization or
measurement errors. In addition, the three dimensionality of the power level may
contribute to the discrepancy between model predictions and experimental data.
8
4. Results and discussion The numerical simulation of different amount of fins in case A, B, C have been
conducted with variable parameters including variations of power level (2W~4W),
different orientation test (vertical/horizontal/slanted) and charge and discharge modes.
The numerical algorithm and validation are evaluated in this study by comparing
numerical predictions with available experimental data provided by Fok et al. [9].
4.1. Numerical validation
To verify the present numerical model, the prediction domain under the conditions
power level of 3 W and the same geometry are compared with the available
experimental results by Fok et al. [9] as shown in Fig.3. The result of theoretical
predictions with the experimental data by Fok et al. [9] is used to assess the grid
refinement.
Different meshes ( ,30 16 80 36 18 86 and 42 22 88 ) in case A,
( , and30 16 86 36 18 92 42 20 96 104
92
) in case B and in case C
( , and ) are employed in testing the numerical
models. The results of the grid sensitivity study show that the simulations based on
the grid in case A,
30 16 92
36
36
18
18 98
86
42 20
36 18 in case B and 36 18 98 in case C
provide satisfactory numerical accuracy and are essentially grid independent. The
numerical results show that the transient surface temperatures are reasonably
predicted within a maximum discrepancy within 10.2% in case A, 8.7% in case B and
8.3% in case C. Besides, the time step refinement is discussed for further precise
calculation time step. Figure 4 shows that the time step size of 0.05 second in
different cases have good approaches to experimental data. For saving computer time
and avoid more round off errors, time step size of 0.05 second is chosen to simulate
the relatively discussed cases. Further decrease the time step size to 0.03 second does
not show any noticeable change in the instantaneous results for the transient
temperature distributions.
4.2. Power level test
The effect of different heating power level (2W~4W) on the performance of heat
sink with different fins (0~6fins) within PCM inside is examined in Fig.5. The results
show that the higher power level transferred into the system, the earlier melting
process will occur. The higher power source leads the earlier time from melting
temperature (36°C). This phenomenon indicates the end of PCM melting function and
no more temperature control ability after the melting temperature jumps. In each case,
the power level of 2W in present calculation is still an undergoing phase change
process after 150 min. The result of 2W predicts similar phase change process time
period with 3W power level of reference data [9]. The results of 3W in each case are
9
much more matched than other power level tests with the 4W test in reference data [9].
The discrepancy may be caused by the round off error and discretization or
measurement error. Considering these factors, the overall comparison with numerical
test data is satisfactory with experimental one. The results of the grid sensitivity study
and power level test showed that the simulations based on the grid provide
satisfactory predictions.
4.3. Transient performance of charge and discharge modes
Fig.6 shows the comparisons of the transient surface temperature distributions of
charge and discharge modes with the present numerical study and experimental result
of Fok et al. [9]. The heating period is 4W on for 30 minutes, 0W off for 10 minutes,
4 W on for 30 minutes and 0W off for last 100 minutes. Total test time is 170 minutes
to compare with available data Fok et al. [9]. Under the charging stage, the numerical
result of peak temperature using PCM base heat sink model are 41.5°C in case A,
39.3°C in case B and 37.4 °C in case C. Due to the discrepancy of different fin
numbers, the case C has a better temperature control than other cases. Under high
power heating and high flux energy diffusion, 6 fin in case C has a lower temperature
jump and smooth operation temperature distribution. In the same trying stage, the
multi-fin heat sink with PCM inside presented powerful temperature control and
maintained the system under a stable temperature gradient. Such temperature control
will provide better system performance and prolong products life time.
4.4. Effect of orientation on PCM-based heat sink performance
The effect of multi-orientation is an important test for the PCM based heat sink
cooling technique to validate the thermal performance. Fig.7 indicates three different
orientations which are vertical, horizontal and slanted (at 45 ). In the front stage
before melting process, the result shows no difference in three different tests. After the
melting process, the temperature difference of orientation effect in the present study is
less than 2°C. This test shows that orientation has a limited effect on the transient
thermal performance in the hybrid cooling system. The results are similar to the
references which are reported by Fok et al. [9], and Wang et al. [21].
4.5. Flow characteristics and heat transfer performance
Fig.8 to 10 illustrates the different stages in the melting process with the melting
fraction of 0.1 to 1 and transient time of 3001.2s to 10480s. The morphology of the
melting front is parallel to the vertical fin in fraction of 0.1 in each case. Similar
melting front is discussed by Shatikian et al. [14]. Due to gravity effect and lower
density, the air will keep on the top edge and the volume will change with the melting
10
fraction of PCM. The distribution of the each melting front is relatively matched with
the melting temperature distribution. From Fig.8, the more influence of gravity effect
is accompanied with the growth of melting area in fraction F = 0.3 and F = 0.6 in each
case. At the same melting fraction stage, the fraction F = 0.1 in case A is 3001.2, 2401
and 3202 in case B and C. During the very early stage, heat transfer is dominated by
conduction and the latent heat absorption of n-eicosane phase change will be
accompanied. Therefore the temperature of each case are almost the same under
melting temperature. In the melting fraction of F = 0.6 in each cases, case A has a
higher temperature distribution ( = 308.6K) than case B ( =308.2K) although
the melting time of case A is longer than case B. In case C, F = 0.9 has a smooth heat
distribution and much longer melting time than others. After these results it can be
realized that the overall temperature will be increased by the decreasing of melting
fraction and PCM shows the powerful dominant of temperature control using a hybrid
heat sink with PCM inside which is consistent with Kandasamy et al. [16].
minT minT
4.6. PCM exploitation
Based on Fok et al. [9] experimental study and present numerical study, the usage
of PCM with 6 fins prolonged the melting stage and provided stable and lower
temperature in the exam module. Compared to case A, B and case C in Fig.8 to 10,
the melting stage of case A was finished at 8201.2 seconds, case B was finished at
8187.0 seconds and case C was finished at 10480.0 seconds. Case B provided more
stable temperature distribution than case A as shown in Fig.8 and Fig.9. From case C
in Fig.10, the suitable fins arrangements can provide the system heat dissipation status
which help to explore new hybrid efficiently cooling system under same operation
temperature. Therefore present results could provide future exploitation based on
same geometric and heat source.
5. Conclusions
Various transient three-dimensional heat transfer investigations of hybrid PCM
based different fin amount heat sink numerical cooling technique are carried out in
this study. Markedly flow field and heat transfer characteristics are found with
variations of different power level (2W-4W), different orientation tests
(vertical/horizontal/slanted), and charge or discharge modes. The theoretical model
developed using laminar Navier-Stokes equations of motion, energy equation,
porosity like source term and VOF model, is capable of predicting the flow and heat
transfer characteristics correctly for a PCM based heat sink system. The numerical
results show that the transient surface temperatures are reasonably predicted with a
maximum discrepancy within 10.2%. Through this study, the finding indicated that
11
the use of PCM in the aluminum heat sink would give electronic packages a more
stable operation temperature. The heat transfer performance of case C with 6 fins
provides more stable temperature control, better cooling ability and lower maximum
operation temperature than case B and case C. The orientation tests for various fins
heat sink show the limited effect on the phase change performance of the system in
each case. The results show that the operation temperature can be controlled well by
the attendance of phase change material and the longer melting time can be conducted
by using a multi-fin hybrid heat sink respectively.
Acknowledgements
The authors would like to sincerely thank Professor F. L. Tan, School of
Mechanical Engineering and Aerospace, Nanyang Technological University for his
providing the detailed dimensions and valuable experimental data for us to complete
the numerical validations.
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Table Captions
Table 1
Geometric parameters in the present numerical study.
Table 2
Properties of PCM, aluminum, and air used in the numerical calculation.
15
Table 1
Geometric parameters in the present numerical study.
Case A
W(mm) La (mm) H(mm) b(mm) h(mm) he(mm) We(mm) Lpcm(mm) Wpcm(mm)
72 85 18 3 14 50 50 63 50
Case B
W(mm) Lb (mm) H(mm) b(mm) h(mm) he(mm) We(mm) Wb(mm) Wf(mm) Wpb(mm)
72 91 18 3 14 50 50 50 2 15.75
Case C
W(mm) Lc (mm) H(mm) b(mm) h(mm) he(mm) We(mm) Wpcm(mm) Wf(mm) Wpc(mm)
72 97 18 3 14 50 50 50 2 9
16
Table 2
Properties of PCM, aluminum and air used in the numerical calculation.
Material k(W/m K) ( ) 3kg/m pC ( J/k ) g KmT ( 0 ) C (g m ) /s ( ) J/kg
N-eicosane 0.15 785
0.001(T-308)+1
2460 35-37 1790exp( 4.25 )
T
247300
Aluminum 202.4 2719 871 660.4 __ __
Air 0.0242 5 21.2 10 0.01134 3.498 T T 1006.4 __ __ __
17
Figure Captions
Fig. 1. The physical model.
Fig. 2. Computational grid distributions of half domain.
(a) case A (without fin) (b) case B (3fins) (c) case C (6fins)
Fig. 3. Effect of grid refinement on temperature distributions in the present study.
Fig. 4. Effect of time step refinement on the temperature distributions in the
present study.
Fig. 5. Comparison of three different powers in the present study.
(a) case A (without fin) (b) case B (3fins) (c) case C (6fins)
Fig. 6. Effect of charge and discharge on the temperature distributions.
Fig. 7. Effect of orientation on the temperature distributions at 3W.
Fig. 8. Evolution of melting process and temperature contour of the case A:
(a) F= 0.1, t = 3001.2 s, (b) F= 0.6, t = 5601.2 s
(c) F= 0.9, t = 6801.2 s, (d) F= 1, t = 8201.2 s
Fig. 9. Evolution of melting process and temperature contour of the case B:
(a) F= 0.1, t = 3001.2 s, (b) F= 0.6, t = 5601.2 s
(c) F= 0.9, t = 6801.2 s, (d) F= 1, t = 8201.2 s
Fig. 10. Evolution of melting process and temperature contour of the case C:
(a) F= 0.1, t = 3001.2 s, (b) F= 0.6, t = 5601.2 s
(c) F= 0.9, t = 6801.2 s, (d) F= 1, t = 8201.2 s
18
Lb
HeaterPCMHeat sink
w
Wf
Wpb
he
bh
Wf
Wpc
he
bh
Lc
w
HeaterPCMHeat sink
(A)
(C)
(E)
(B)
(D)
(F)
b
Lpcm
hw
HeaterPCMHeat sink
he
Air gap
H
x
z
Yz
Y
gLa
gwpcm
wpcm
wpcm
H
H
Fig.1. The physical model.
19
(a) case A (without fin) (b) case B (3fins) (c) case C (6fins)
Fig.2. Computational grid distributions of half domain.
20
0 40 80 120 16020 60 100 14020
40
60
80
30
50
70
Tem
per
atu
re(0 C
)
Grid test Exp. of Fok et al. [9] of case A36x18x86 of case AExp. of Fok et al. [9] of case B36x18x92 of case BExp. of Fok et al. [9] of case C36x18x98 of case C
Time (min)
Fig.3. Effect of the grid refinement on temperature distributions.
21
0 40 80 120 16020 60 100 14020
40
60
80
30
50
70
Tem
per
atu
re(0 C
)
Time step test Exp. of Fok et al. [9] of case At=0.05s of case AExp. of Fok et al. [9] of case Bt=0.05s of case BExp. of Fok et al. [9] of case Ct=0.05s of case C
Time (min)
Fig.4. Effect of time step refinement on the temperature distributions.
22
(a) case A (without fin) 0 40 80 120 16020 60 100 140
0
20
40
60
80
10
30
50
70
Tem
per
atu
re(0 C
)
Power test of case AExp. of 4W of Fok et al. [9] Simulation of 2WSimulation of 3WSimulation of 4W
Time (min)
0
20
40
-10
10
30
50
err
of
3W
pre
dic
tio
n t
o e
xp. (
%)
(b) case B (3fins) 0 40 80 120 1620 60 100 140 0
0
20
40
60
80
10
30
50
70
Tem
per
atu
re(0 C
)
Power test of case BExp. of 4W of Fok et al. [9] Simulation of 2WSimulation of 3WSimulation of 4W
Time (min)
0
20
40
-10
10
30
50
err
of
3W p
red
icti
on
to
exp
. (%
)
(c) case C (6fins) 0 40 80 120 1620 60 100 140 0
0
20
40
60
10
30
50
Tem
per
atu
re(0 C
)
Power test of case CExp. of 4W of Fok et al. [9] Simulation of 2WSimulation of 3WSimulation of 4W
Time (min)
0
20
40
-10
10
30
50
err
of
3W p
red
icti
on
to
exp
. (%
)
Fig.5. Comparison of three different powers.
23
0 40 80 120 16020 60 100 140 180
Time (min)
10
20
30
40
50
60
15
25
35
45
55
Tem
pe
ratu
re(0 c
)
Effect of charge and discharge usage
Numerical prediction of case ANumerical prediction of case BNumerical prediction of case CExp. of Fok et al. [9] for case A
Fig.6. Effect of charge and discharge on the temperature distributions.
24
0 40 80 120 16020 60 100 14020
40
60
80
30
50
70
Tem
per
atu
re(0 C
)
Orientation effect Present result (vertical) of case APresent result (horizontal) of case A
Present result (slanted at 450) of case APresent result (vertical) of case BPresent result (horizontal) of case B
Present result (slanted at 450) of case BPresent result (vertical) of case CPresent result (horizontal) of case C
Present result (slanted at 450) of case C
Time (min)
Fig.7. Effect of orientation on the temperature distributions at 3W.
25
(a) F = 0.1, t = 3001.2 s (b) F = 0.6, t = 5601.2 s
(c) F = 0.9, t = 6801.2 s (d) F = 1, t = 8201.2 s.
Fig.8. Evolution of melting process and temperature contour of the PCM
(Case A).
26
(a) F = 0.1, t = 2401.0 s (b) F = 0.6, t = 4302.0 s
(c) F = 0.9, t = 5298.0 s (d) F = 1, t = 8187.0 s.
Fig.9. Evolution of melting process and temperature contour of the PCM
(Case B).
27
(a) F = 0.1, t= 3202.0 s (b) F = 0.6, t= 7191.0 s
(c) F = 0.9, t= 9183.0 s (d) F = 1, t= 10480.0 s.
Fig.10. Evolution of melting process and temperature contour of the PCM
(Case C).
28