Alexandria Engineering Journal (2015) xxx, xxx-xxx

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Simulation of melting of a nano-enhanced phase change material (NePCM) in a square cavity with two heat source-sink pairs

Aziz Ebrahimi, Abdolrahman Dadvand *

Department of Mechanical Engineering, Urmia University of Technology, Urmia, Iran Received 8 May 2015; revised 24 August 2015; accepted 13 September 2015

KEYWORDS

Phase change materials; Melting; Nanoparticles; Thermal energy storage; Heat source-sink pairs

Abstract Melting of a NePCM in a square cavity with different arrangements of two heat source-sink pairs flush-mounted on the vertical sidewalls is investigated numerically. The governing equations were solved on a non-uniform mesh using a pressure-based finite volume method with an enthalpy porosity technique to trace the solid-liquid interface. Four different cases are studied: Case I where the sources and sinks are separately placed on two vertical sidewalls; Case II where the sources and sinks are alternately placed on two vertical sidewalls; Case III where the sources are placed below the sinks on the vertical sidewalls; and Case IV where the sources are placed above the sinks on the vertical sidewalls. It was found that, Case II has the highest liquid fraction and Case IV possesses the lowest liquid fraction at the final stages of the melting process. In addition, the impacts of the nanoparticle loading are analyzed. In all the cases studied, the volumetric concentration of nanoparticles of 2% would result in the highest melting rate.

© 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

After the oil crisis in 1973 and the rise in energy prices, researches started replacing the conventional energy resources with the much more efficient ones. Thermal energy can be stored in the form of sensible heat, latent heat and thermo-chemical reactions. Among them, the latent heat, i.e. the heat

* Corresponding author. Tel.: +98 441 3980264; fax: +98 441 3554184.

E-mail address: a.dadvand@mee.uut.ac.ir (A. Dadvand).

Peer review under responsibility of Faculty of Engineering, Alexandria

University.

that is stored during phase change process, plays a key role in melting process. The solid to liquid phase change during the melting process can store a large amount of energy provided an appropriate material is chosen. Latent heat storage materials, also called phase change materials (PCMs), due to saving-releasing large quantities of energy within a small volume and ability to keep their temperature constant during the melting-solidification process have found diverse engineering applications such as energy conservation in buildings, solar energy equipment, transport and storage containers, human body and greenhouses (see the elaborate reviews by Khudhair et al. [1], Zhou et al. [2], Zalba et al. [3] and Sharma et al. [4]). Recently, a new type of PCM room was introduced and its thermal performance was examined experimentally and

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1110-0168 © 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Nomenclature

cp specific heat at constant pressure (J/kg K) Greek symbols

g gravitational acceleration (m/s2) DH latent heat (J/kg)

H enthalpy of the NePCM (W/m K) K Boltzmann constant (J/K)

h sensible enthalpy (J/kg) a liquid fraction

k thermal conductivity of NePCM (W/m K) l dynamic viscosity (N s/m2)

L latent heat (J/kg) P density (kg/m3)

Tliquidus liquidus temperature (K) s stress tensor

T 1 solidus solidus temperature (K) u volumetric fraction of nanoparticle

p static pressure (N/m2) 4 correction factor

T temperature of NePCM (K) Subscripts np nanoparticle

Tref t reference temperature time (s)

V velocity (m/s) pcm npcm pure PCM nano-PCM

numerically by Meng et al. [5] in order to control the indoor air temperature.

There have been many numerical and experimental studies on the melting of PCMs in different enclosures. Zhang et al. [6] studied experimentally the melting of n-octadecane in an enclosure discretely heated at a constant rate from one side and thermally insulated from the other sides. Their results showed that natural convection had a significant effect on

the shape of the solid-liquid interface. The effect became more pronounced when the Stefan number was increased. Faraji and El Qarnia [7] simulated the melting of a PCM in a vertical rectangular enclosure where three volumetric protruding heat sources mounted on one of the vertical walls. They demonstrated that the bottom heat source contributed the highest heat transfer rates. Kousksou et al. [8] studied numerically the melting of a PCM in a rectangular enclosure, where the

(C) (D)

Figure 1 Four different arrangements (cases) of two heat source-sink pairs: A: Case I, B: Case II, C: Case III and D: Case IV.

vertical walls and the top wall of the cavity were insulated and the bottom wavy wall was maintained at constant temperature. It was found that by increasing the amplitude value of the bottom wavy surface, the rate of melting increased.

The melting of PCMs is normally accompanied by natural convection heat transfer. Researchers have found that the discrete heat source and magnetic field have remarkable influences on natural convection heat transfer [9-11]. In addition, research in heat transfer using suspensions of nanometer-sized solid particles in base liquids has indicated that the suspended nanoparticles significantly change the transport properties and heat transfer characteristics of the suspension [12-18]. Low thermal conductivity is a key feature of PCMs. Therefore, many researches took the advantage of adding nanoparticles to these materials to enhance their thermal conductivity [19-24]. As an example, Wu et al. [21] employed the Hot Disk thermal constants analyzer and infrared monitoring methods for investigating the effects of Cu nanoparticles on the thermal conductivity and the phase change heat transfer

Table 1 Thermo-physical properties of paraffin wax [37] and Al2Ö3 [38].

Thermo-physical property

Al2Ö3

nanoparticles

PCM (paraffin wax)

Thermal conductivity (W/mK)

Solidus temperature (K)

Liquidus temperature (K)

Specific heat (J/kg K) Latent heat of fusion (J/kg)

Dynamic viscosity (N s/m2) Density (kg/m3) Diameter of nanoparticle (m)

3600 59 x 10"

0.21 if T < Tsc

0.12 if T > 1

liquidus

2890 173,400

0.001 exp (-4.25 +

0.001(t-319.15)+1

1790-j

Temperature (K)

300 320 340 360

Temperature (K)

Figure 2 Variations with temperature of dynamic viscosity (A) and thermal conductivity (B) of PCM dispersed with different volume fractions of Al2O3 nanoparticles.

Figure 3 Typical computational non-uniform grid.

0.7 0.6 0.5

ä 0.4 -d

'•3 0.3 J

2000 Time (s)

Figure 4 Effect of the grid size on the variations of the liquid fraction with time associated with Case I for pure PCM.

of PCMs. They concluded that by adding 2 wt.% of Cu nanoparticles to paraffin, the thermal conductivity enhancement of 14.2% and 18.1% occurred in solid and liquid states, respectively.

Several numerical and experimental studies have been carried out on the solidification and melting of nano-enhanced PCMs (NePCMs). Kashani et al. [25] performed numerical modeling of solidification of Cu-water nanofluid as a PCM in an enclosure with vertical wavy walls at different Grashof (Gr) numbers and considered the effects of surface waviness and nanoparticle dispersion on the solidification rate. The results showed that dispersion of nanoparticle in the PCM caused a decrease in solidification time and that the surface waviness could be considered as a controlling factor for solidification time. In addition, in all the Gr numbers tested, by

increasing the waviness the total solidification time increased. Hosseinizadeh et al. [26] simulated numerically the unconstrained melting of NePCM inside a spherical container using RT27 and copper particles as base material and nano-particle, respectively. They conducted the simulations for three different Stefan numbers and volume fractions of nano-particles. The results revealed that an increase in thermal conductivity in conjunction with a decrease in latent heat would result in higher melting rate of NePCM. Arasu and Mujumdar [27] carried out a numerical study on the melting of paraffin wax dispersed with different volume fractions of Al2O3 in a square enclosure heated from two different sides, i.e., from below and from one vertical side. The results indicated that due to dominated natural convection in the case of vertical wall heating, the melting rate was increased. Also they found that increasing

Present work Arasu and Mujumdar [ 7]

t = 500 s

t = 1000 s

t = 3000 s

Figure 5 Comparision of present results in terms of velocity vectors and isotherms with those of [27] for paraffin wax with 2 wt.% of Al2O3 nanoparticles.

the nanoparticle content to more than 2wt.% would weaken the natural convection due to increase in viscosity. Sebti et al. [28] conducted a comprehensive numerical study to investigate heat transfer enhancement during the melting process of a PCM in a 2D square cavity through dispersion of Cu nanoparticles. They found that the nanofluid heat transfer rate increased and the melting time decreased as the volume fraction of nanoparticles increased. Ho and Gao [29] carried out the melting experiments of n-octadecane dispersed with Al2O3 nanoparticles in a vertical square enclosure. The vertical side walls were differentially heated isothermally while the remaining side walls were thermally insulated. The results indicated that natural convection heat transfer into the melted region of the enclosure degraded significantly with increasing the mass fraction of nanoparticles dispersed in the NePCM as compared to that of the base PCM. Zeng et al. [30]

accomplished experimental investigation on melting of 1-dodecanol dispersed with various loadings of multi-walled carbon nanotubes (CNTs) in a bottom-heated vertical cylindrical cavity. Their investigations showed that in the presence of the CNTs, the melting rate was reduced as a result of increased viscosity, leading to significant weakness of natural convection during melting. They concluded that the competing effect between the enhanced heat conduction and weakened natural convection determines the melting rate of NePCMs. El Hasadi and Khodadadi [31] simulated numerically the effect of nanoparticles size on the solidification process of NePCM. Their results showed that by decreasing the nanoparticle size from 5 nm to 2 nm, solid-liquid interface was changed from a stable planar shape to an unstable dendritic structure. Dhai-dan et al. [32-34] investigated experimentally and numerically melting of paraffin (n-octadecane) with CuO nanoparticle sus-

t = 1000 s t = 3000 s

Present results

t = 1000 s t = 3000 s

Arasu and Mujumdar [27]

[27] - 1000s Present - 1000s [27] - 3000s Present - 3000s

Figure 6 Qualitative and quantitative comparison of solid-liquid interface between present work and that of Arasu and Mujumdar [27] at t = 1000 s and t = 3000 s for paraffin wax with 2 wt.% of Al2O3 nanoparticles.

pensions in geometrically different enclosures including a square cavity, a horizontal cylindrical capsule and an annulus subjected to a constant heat flux. Their results clarified that the heat transfer augmented with increasing the loading of nanoparticles due mainly to an increase in thermal conductivity. Jourabian et al. [35] investigated numerically the effects of nanoparticles volume fraction and the position of the hot cylinder in the melting process of Cu/water nanofluid PCMs in an annulus using enthalpy-based LBM.

The objective of this study was to investigate numerically the effect of different arrangement of two heat source-sink pairs located on the vertical sidewalls of a square cavity on the melting rate of paraffin wax as PCM dispersed with Al2O3 as nanoparticle in the cavity. The rest parts of the cavity walls are adiabatic. Three different volume fractions of nanoparticles (i.e., 0 wt.%, 2 wt.% and 5 wt.%) are examined. The dimension of the cavity is H x H (25 mm x 25 mm) and the heat sources and sinks are considered to be H/4 in size located separately on the vertical sidewalls.

2. Physical model

The problem under consideration is the melting of a NePCM in a 2-D square cavity of side length 25 mm x 25 mm with different arrangements of two heat source-sink pairs flush-mounted on the vertical sidewalls. Horizontal walls and other parts of the vertical walls are thermally insulated (see Fig. 1). The heat sources are maintained at a constant temperature of 330 K above the melting point. The sinks' temperature is maintained at the constant temperature of 300 K. The enclosure is completely filled with paraffin wax containing 0 wt.%, 2 wt.% and 5 wt.% of Al2O3 as nanoparticles for enhancement of thermal conductivity. The initial temperature of PCM/ Al2O3 is considered 300 K. In addition, the following assumptions are made in present work:

(1) Paraffin wax as PCM is pure and in its liquid phase is incompressible and Newtonian.

t = 100 s

Figure 7 Liquid-solid interface (left), isotherms (middle) and velocity vectors (right) associated with Case I for paraffin wax with 0 wt.% of Al2O3 nanoparticles.

(2) Viscous dissipation, thermal radiation, three-dimensional convection and volumetric expansion are negligible.

(3) The paraffin wax and Al2O3 nanoparticles are considered as continuous media, they are assumed to be in thermodynamic equilibrium and no-slip boundary condition is applied between them.

(4) The thermo-physical properties of the PCM and NePCM are assumed temperature dependent.

(5) The melting is an unsteady process and the flow is laminar and two-dimensional.

3. Mathematical formulation and numerical implementation

3.1. Governing equations

An enthalpy-porosity technique is used to simulate the melting of a NePCM in a square cavity with different arrangements of two heat source-sink pairs. Instead of tracking the solid-liquid interface explicitly, the enthalpy porosity technique computes a quantity called the liquid fraction in each cell in the

computational domain at any iteration, based on enthalpy balance. In the melting process, the liquid fraction changes between 0 and 1. It possesses the values of 0 and 1 respectively for solid state and liquid state and lies between 0 and 1 for the so-called mushy state.

The continuity, momentum, and energy equations for the 2D transient laminar flow including buoyancy-driven convection can be expressed as follows [36]: Continuity equation:

@p+r-(Pv) = o (1)

In this study @p is zero due to incompressibility of the fluid flow after melting occurs. Momentum equation:

d - -- -— (p V) + V-(p VV) = -VP + P~ +V-T + V (2)

where P denotes the static pressure, s is the stress tensor, p~ and F are the gravitational and external body forces, respectively.

0.06 0.19 O.jl 0.44 0.56 0.69 0.S1 0.94 302 306 309 313 317 321 324 328

t = 4000 s

Figure 8 Liquid-solid interface (left), isotherms (middle) and velocity vectors (right) associated with Case I for paraffin wax with 2 wt.% of Al2O3 nanoparticles.

Energy equation:

д(ри)

V ■ (pVH) = V ■ (kVT) + S

where H denotes the enthalpy of the NePCM, q is density of the NePCM, k stands for the thermal conductivity of NePCM, ~ is the velocity and Tis the temperature. Here S is volumetric heat source term, which is set to zero in the present study.

The total enthalpy H is computed as the sum of the sensible enthalpy, h and the latent heat, AH:

H = h + AH (4)

h = href +

' Tref

Here, href and Tref denote the reference enthalpy and reference temperature, respectively and CP represents the specific heat at constant pressure.

The latent heat in terms of the latent heat of the PCM, L is written as follows:

DH = aL

where a is the liquid fraction and is defined as

0; if T < Tsolidus

a = ^ T ^ _t s, ; if Tsolidus < T < Tliquidus

Tliquidus Tsolidus ' 1;

if T > Tt

liquidus

The energy Eq. (3) and the liquid fraction Eq. (6) are coupled. This would necessitate an iteration between them to solve for temperature. In the enthalpy-porosity technique the mushy region (partially solidified region) is considered as a porous medium. The porosity in each cell is set equal to the liquid fraction in that cell. The porosity in fully solidified regions is set equal to zero, which extinguishes the velocities in these regions.

3.2. Boundary and initial conditions

The heat sources and sinks are considered as constant-temperature boundary conditions and the rest portions of the cavity walls are made adiabatic. The initial condition is set equal to the temperature of heat sinks.

0.06 0.19 0.31 0.44 0.56 0.69 Ü.S1 0.94 302 306 309 313 317 321 324 328

t - 4000 s

Figure 9 Liquid-solid interface (left), isotherms (middle) and velocity vectors (right) associated with Case I for paraffin wax with 5 wt.% of Al2O3 nanoparticles.

0.6 0.5 0.4 0.3 0.2 0.1 0

Time (s)

Figure 10 Variations of the liquid fraction for three different loadings of Al2O3 nanoparticles for Case I.

3.3. Thermo-physical properties

The thermo-physical properties of the pure paraffin wax and the solid Al2O3 nanoparticles, which were used in the present work are found in [37,38] and are also given in Table 1.

The thermo-physical properties of the NePCM are calculated from the following relations where the subscripts np and pcm stand for nanoparticles and PCM, respectively. Density [22]:

Pnpcm = UPnp + (1 - U)Ppcm (8)

Specific heat capacity [22]:

9(pCp)np + (1 - u)(PCp)p

p,npcm

Latent heat [22]:

L _(1 - U)(pL)pcm

Dynamic viscosity [39]:

I = 0 983e(12-959u)i

r'npcm v-sv^c. r-pcm

The effective thermal conductivity is calculated from the following correlation proposed by Vajjha et al. [40], which is a combination of Maxwell's theory (first term on the right hand side) and Brownian motion (second term on the right hand side):

knp ^ 2kpcm 2(kpcm knp)u т

7 \ kpcm

knp + 2k,

+ 5 X 104MUPpcm cp,pcm

',pcm Л

V pnpdnp

-AT u)

pnpdnp

where the factor pk for Al2O3 is given by

bk = 8.4407(100u)-1'07304

^ is the correction factor in the Brownian motion term defined as

T Tsolidus

^ ^ Tliquidus Tsolidus 1

if T < Tsolidus

if Tsolidus < T < Tliquidus

if T > Tliquidus

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Time (s)

Case II

2000 Time (s)

Case III

0 1000 2000 3000 4000

Time (s)

Case IV

Figure 11 Variations of the liquid fraction for three different loadings of Al2O3 nanoparticles for Cases II, III and IV.

The Boltzmann constant k takes the value of 1.381 x 10-23 (J/K), and f(T, u) is obtained from the experi-

mental data as [23] follows:

A(T, u) = (2.8217 x 10-2u + 3.917 x 10-3)-

-3.0669 x 10-2u - 3.91123 x 10-3)

where T0 is set equal to 273 K.

Fig. 2 depicts the variations of the effective dynamic viscosity and thermal conductivity of paraffin wax dispersed with 0%, 2% and 5% of Al2O3 nanoparticles as a function of temperature obtained respectively from Eqs. (11) and (12). The variations of thermal conductivity and dynamic viscosity of NePCM with volume fraction and temperature are reasonably in good agreement with the experimental results reported with Ho and Gao [24]. It is seen from Fig. 2A that the dynamic viscosity increases with volumetric concentration of Al2O3 nanoparticles and decreases with temperature. However, the increase in the dynamic viscosity due to the volumetric concentration of nanoparticles is more appreciable at (relatively) low temperatures. The enhancement of the dynamic viscosity can affect the melting process of the NePCM especially when the natural convection is the dominant heat transfer mechanism. Fig. 2B shows that the thermal conductivity of the NePCM increases with the volumetric concentration of nanoparticles and decreases with the temperature.

3.4. Solution procedure

The governing equations subject to the boundary and initial conditions are solved by the collocation finite volume method. A non-uniform grid is used (see Fig. 3). The PISO algorithm is used for pressure-velocity coupling whereas the PRESTO scheme is adopted for the pressure correction equation. The first order upwind differencing scheme is used to discretize the momentum and energy equations. The thermo-physical properties of PCM are considered temperature-dependent. The time step for integrating the temporal derivatives was considered to be 0.001 s for the first several iterations and changed to 0.1 s afterward. The number of iterations for every time step is fixed at 10 to satisfy the convergence criteria of 10~3 for the continuity and momentum equations, and 10~6 for the energy equation. The under-relaxation factors for x- and y-components of the momentum equations, pressure correction equation, energy equation, and liquid fraction are set equal to 0.5, 0.3, 1 and 1, respectively.

0.06 0.19 0.31 0.44 0.56 0.69 0.81 0.94 302 306 309 313 317 321 324 328

t - 4000 s

Figure 12 Liquid-solid interface (left), isotherms (middle) and velocity vectors (right) associated with Case II for paraffin wax with 2 wt. % of Al2O3.

3.5. Grid independence test

In order to check the grid independence of the obtained results, a mesh refinement was carried out, as shown in Fig. 4 for liquid fraction pertaining to Case I (see Fig. 1) with 0 wt.% of Al2O3. Three different grid sizes of 80 x 80, 100 x 100 and 110 x 110 are utilized. As there is negligibly small difference between the results associated with the last two grid sizes, the grid with 100 x 100 cells is mainly used for all the simulations carried out in the present work.

4. Validation of the model

In order to substantiate the accuracy of the current numerical results, first we simulate the melting of paraffin wax as PCM dispersed with 2 wt.% of Al2O3 nanoparticles in a square enclosure and compare the results at three different times with those reported by Arasu and Mujumdar [27]. The vertical side hot wall is at a constant temperature of 330 K while the cold wall, facing the hot wall, is at 300 K and the bottom and top

walls are kept adiabatic. The results are depicted in terms of streamlines and isotherms (see Fig. 5) and liquid-solid interface (see Fig. 6). A reasonably good agreement is obtained.

5. Results and discussion

As mentioned above, in the present work, the melting of paraffin wax as PCM dispersed with different volumetric concentration of Al2O3 nanoparticles in a square cavity with discrete heating is investigated. As shown in Fig. 1, four different cases are studied: Case I where the sources and sinks are separately placed on two vertical sidewalls (Fig. 1A); Case II where the sources and sinks are alternately placed on two vertical sidewalls (Fig. 1B); Case III where the sources are placed below the sinks on the vertical sidewalls (Fig. 1C); and Case IV where the sources are placed above the sinks on the vertical sidewalls (Fig. 1D).

5.1. Case I

The results associated with Case I for 0wt.%, 2 wt.% and 5 wt.% of Al2O3 nanoparticles are shown in Figs. 7-10,

0.06 0.19 0.31 0.44 0.56 0.69 0.81 0.94 302 306 309 313 317 321 324 328

Kit--,'.

/' liilU

t = 4000 s

Figure 13 Liquid-solid interface (left), isotherms (middle) and velocity vectors (right) associated with Case III for paraffin wax with 2 wt.% of Al2O3.

respectively and are given in terms of liquid-solid interface, isotherms, velocity vectors and liquid fraction.

5.1.1. Liquid-solid interface

For all the volumetric concentrations, the liquid-solid interface, which is fairly flat at the initial stages of the melting process, becomes more and more distorted as the time passes. During the melting process, natural convection of the liquid phase is developed, which causes the hot liquid PCM near the sources to ascend and the cold liquid PCM to descend. As a result, the temperature in the upper region of the liquid becomes higher than that in the lower region, hence accelerating the melting process in the upper part. Therefore, the liquid-solid interface is more advanced near the upper region of the cavity.

5.1.2. Isotherms

For all the volumetric concentrations, it can be seen that at the initial stages of melting process (see the images at t = 100 s) the isothermal lines are parallel to the heat sources. This would imply that the conduction is the dominated heat transfer

mechanism. However, even the very small contribution of the natural convection can cause the heat to transfer toward the top part of the sources and more amount of PCM be melted there. As the time advances, the natural convection becomes the dominant mode of heat transfer, which can also be deduced from the distorted isotherm lines. This causes the heat to traverse from the sources toward the top region of the cavity, hence accelerating the melting process in this region.

5.1.3. Velocity vectors

At the onset of melting process, the solid NePCM starts changing to liquid in close proximity to the heat sources. At the early stages of melting process, two semi-identical clockwise circulations are generated. Finally, the two circulations combine together to form one greater circulation.

5.1.4. Liquid fraction (a measure of melting rate)

Although the liquid-solid interface images seem identical for all the three loadings of Al2O3 nanoparticles, however there are differences between the amounts of melted NePCM for the three loadings as shown in Fig. 10. It can be seen from this

302 306 309 313 317 321 324 328

t = 100 s

t = 2000 s

t = 4000 s

Figure 14 Liquid-solid interface (left), isotherms (middle) and velocity vectors (right) associated with Case IV for paraffin wax with 2 wt.% of Al2O3.

figure that the rate of melting increases with the increase in volumetric concentration of nanoparticles by 2wt.%. Also, it is seen that the melting rate of 5% Al2O3 is almost the same as that of the pure paraffin wax. This may be explained as follows: when the nanoparticles are added to the PCM, its conductivity as well as viscosity is increased. The conductivity increment has positive effect while the viscosity enhancement has negative effect (weakens the buoyancy effect) on the heat transfer and hence on the melting process. At high volumetric concentration of nanoparticles, the negative effect of viscosity enhancement may become equivalent to the positive effect of conductivity increment of NePCM.

5.2. Cases II-IV

It was observed in Case I that u = 2% gave rise to the higher melting rate. This is also the case for the other three cases studied in the present work (see Fig. 11). In Cases II and III, it is seen that the melting rate for u = 5% is even lower than that for the pure PCM. As observed from Fig. 2, both the dynamic viscosity and the thermal conductivity of the PCM increase with volumetric concentration of Al2O3 nanoparticles and decrease with temperature. However, the increase in the dynamic viscosity with the volumetric concentration of nanoparticles is more appreciable at (relatively) low temperatures. The enhancement in the dynamic viscosity can affect the melting process of the NePCM especially when the natural convection is the dominant heat transfer mechanism. At low temperatures, the dynamics viscosity of NePCM has negative effect on the natural convection so that it can deteriorate the positive effect of increasing the conductivity. This is the reason why for u = 5% the melting rate is lower than its counterpart for u = 0% and u = 2%.

Since u = 2% gives the higher melting rate, for cases II-IV, the Liquid-solid interface, isotherms and velocity vectors are given only for u = 2%. In Case II (see Fig. 12), two circulations having opposite directions (i.e., one clockwise and the other counterclockwise) are formed, but unlike Case I, they never combine together to constitute a unicellular circulation. However, the overall effect of these two circulations can enhance the melting performance of the system. In cases III and IV (see Figs. 13 and 14), like Case II, the two circulations have opposite directions and they would not combine to form a unicellular circulation. However, the arrangements of the heat sources and sinks are such that, appropriate heat transfer would not happen between them and hence the melting rate is low compared to cases I and II.

5.3. Effect of arrangement of heat source-sink pairs on the melting rate

Comparison between different cases studied in the present work shows that for all the volumetric concentration of Al2O3 nanoparticles, Case II has the highest liquid fraction and Case IV possesses the lowest liquid fraction at t = 4000 s (see Fig. 15). It may be noted that at initial stages of the melting process, cases I, III and IV involve higher liquid fractions than Case II. This can be interpreted by the consideration of the effects of viscosity, conductivity and difference in circulations due to difference in arrangement of heat source-sink pairs.

0.7 0.6 0.5

•3 0.3

0.2 0.1 0

case I case II case III case IV

2000 Time (s)

0.7 0.6

0.2 0.1 0

case I case II case III case IV

/ V - -

2000 Time (s)

0.7 0.6 0.5

case I case II case III case IV

0 1000 2000 3000 4000

Time (s)

Figure 15 Comparision of liquid fraction between different cases for different loadings of Al2O3 nanoparticle; (A) u = 0%, (B) u = 2% (C) and u = 5%.

6. Conclusions

In the present work, melting of a NePCM in a square cavity with different arrangements of two heat source-sink pairs

flush-mounted on the vertical sidewalls is investigated numerically. The governing equations were solved on a non-uniform mesh using a pressure-based finite volume method with an enthalpy porosity technique to trace the solid-liquid interface. Four different cases are studied: Case I where the sources and sinks are separately placed on two vertical sidewalls; Case II where the sources and sinks are alternately placed on two vertical sidewalls; Case III where the sources are placed below the sinks on the vertical sidewalls; and Case IV where the sources are placed above the sinks on the vertical sidewalls. It was found that, Case II has the highest liquid fraction and Case IV possesses the lowest liquid fraction at the final stages of the melting process. In addition, the impacts of the nanoparti-cle loading are analyzed. In all the cases studied, the volumetric concentration of nanoparticles of 2% would result in the highest melting rate.

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