11affiliationtext: Eindhoven University of Technology, Mechanical Engineering
Department, The Netherlands22affiliationtext: Process and Energy department,
Delft University of Technology,Leeghwaterstraat 39,2628 CB Delft, the
Netherlands

#  The Impact of Thermosolutal Convection on Melting Dynamics of Nano-enhanced
Phase Change Materials (NePCM)

Yousef M. F. El Hasadi 111Email address for correspondence:
yme0001<EMAIL_ADDRESS>
###### Abstract

Nanoparticle-Enhanced Phase Change Materials (NePCM) have been a subject of
intensive research owing to their potential for enhanced thermo-physical
properties. However, their behavior during phase change processes, such as
melting or solidification, remains inadequately understood. This investigation
focuses on the melting process of NePCM in a square cavity, exploring distinct
cases of melting from both the top and bottom sides. The NePCM comprises
copper nanoparticles (2 nm in size) suspended in water. Our study involves
different combinations of constant temperature boundary conditions and
particle volume fractions. Utilizing a numerical model based on the one-fluid
mixture approach combined with the single-domain enthalpy-porosity model, we
account for the phase change process and particles’ interaction with the
solid-liquid interface. When melting NePCM from the top side, convection
effects are suppressed, resulting in a melting process primarily governed by
conduction. Both NePCM and pure water melt at the same rate under these
conditions. However, melting NePCM from the bottom side induces convection-
dominated melting. For pure water, thermal convection leads to the formation
of convection cells during melting. Contrastingly, melting NePCM triggers
thermosolutal convection due to temperature and particle concentration
gradients. The flow cells formed from thermosolutal convection in NePCM differ
from those in pure water driven by pure thermal convection. Our simulations
reveal that thermosolutal convection contributes to decelerating the solid-
liquid interface, thereby prolonging NePCM melting compared to pure water.
Surprisingly, the viscosity increase in NePCM plays a minimal role in the
deceleration process, contrary to prior literature attributing slow-downs of
the melting process of the NePCM primarily to increased viscosity.

Keywords: Phase change materials, NePCM, CFD, Melting,Colloidal Suspensions,
Thermosolutal Convection.

##  1 Introduction

In light of the substantial harm caused by climate change and the escalation
of global temperatures, numerous countries are actively pursuing the
decarbonization of their economies. Within modern economies, heat production
plays a pivotal role, where heating and cooling activities contribute to 50%
of global energy consumption and 45% of total worldwide carbon emissions [1].
Thermal energy storage (TES) can be essential in decarbonising the heat
production process. Thermal energy storage can be achieved mainly by two ways
the first is sensible heat storage, and the second is latent heat storage [2].
Latent heat storage is achieved using what we call phase change materials
(PCM), which they absorb, and realise a substantial amount of heat during
their phase change process [3]. PCMs could be organic, non-organic, or
eutectic mixtures [3]. Latent heat storage have substantial number of
applications such as energy storage in concentrated solar power, thermal
management, cold storage, thermal storage in buildings, district heating
systems, smart thermal grid, and thermal management [4, 5, 6, 7, 8, 9]. Phase
change materials suffer from low thermal conductivity, which hampers their
performance [10]. Several methods are used to enhance the thermal conductivity
of PCMs, such as incorporating metallic fillers, utilizing finned tubes, and
adding nanoparticles [11, 12, 13], enhancing the thermal conductivity of
liquids can be achieved also by self-propelled particles [14, 15] .

NePCM is a type of phase change material containing particles ranging in size
from nanometers to micrometers. In its liquid phase, NePCM can be regarded as
a colloidal suspension. Colloidal suspensions are well-studied materials, with
numerous theoretical and experimental investigations aimed at understanding
their behaviour [16]. Colloidal suspensions differ from atomistic solutions in
that their thermo-physical properties vary significantly with the volume
fraction of the particles [17]. The physics of colloidal suspensions’
solidification (freezing) is still not well understood. During the freezing
process, particles are significantly rejected from the frozen portion of the
suspension due to thermodynamic constraints [18]. This particle rejection
leads to an unstable shape of the solid-liquid interface, enabling the
formation of dendrites due to constitutional supercooling, a phenomenon well-
documented in many experiments [19, 20, 21, 22, 23]. Many investigators have
employed X-ray tomography and X-ray scattering to gain insights into the shape
of the solid-liquid interface and the redistribution of particles during the
freezing of colloidal suspensions. These techniques allow visualization of the
emergent solid-liquid interface and they map areas with significant particle
concentration [20, 24]. Thermosolutal convection is a phenomenon that has yet
to be thoroughly investigated for the cases that phase change occur.
Thermosolutal convection emerges when there are thermal and particle
concentration gradients within the system under considerations. This type of
convection profoundly influences the solidification process of colloidal
suspensions, as indicated by experiments conducted by [25, 26, 27]. The
experiments that are used for the investigation of the freezing process of the
NePCM are primitive compared to one used to investigate colloidal suspensions
in the material science community.

In modelling the freezing process of NePCM, the energy storage community has
predominantly employed single-phase models, incorporating the effects of
particles primarily into the thermophysical properties. For instance, Fan and
Khodadadi [28] utilized a well-known analytical solution for the one-
dimensional unidirectional freezing from the bottom, known as the Stefan
problem, to study NePCM with varying particle loadings. They compared their
theoretical predictions with experiments conducted using the same geometry.
Their theoretical predictions suggested that NePCM would freeze faster
compared to the pure solvent. However, their experimental results revealed a
non-linear dependence of the expedited freezing of NePCM on the volume
fraction of particles, which contradicted their developed theory. They
attributed this non-linear behaviour to the sedimentation of particles.
However, since the experiment is designed to operate in diffusive conduction
conditions Therefore, the layers formed by the highly conductive particles
sedimentation should aid in speeding up the freezing process of the NePCM,
rather than slowing it down. Similar observations to those of Fan and
Khodadadi [28] were made by Murugan et al. [29], who reported that NePCM with
a loading of 0.3% carbon nanotubes by volume solidified faster than NePCM with
a 0.9% loading by volume. The failure of Fan and Khodadadi’s one-dimensional
model [28] to capture and explain the non-linear dependence of expedited
freezing of NePCM on the volume fraction of particles in comparison to pure
PCM highlights that the mathematical frameworks of single-phase models cannot
fully capture the physics involved in the freezing process of NePCM. Do to the
incompetence of the single phase model to capture the physics that are
involved numerical simulations that are using this type of model are in their
majority erroneous [30, 31, 32, 33].

Following the miss match between the theory, and experiments El Hasadi and
Khodadadi[34] developed a version of the Stefan problem that includes mass
transfer known alternatively as Rubinstein problem suitable to model the
solidification process of colloidal suspensions. The model predicted the non-
linear dependence of the expedited freezing process similar to observations
made by Fan and Khodadadi [28] in their experiments, also the its numerical
predictions of the experimental data of Fan and Khodadadi[28] are better
compared to the theoretical model of Fan and Khodadadi[28]. The enhancement of
the prediction accuracy of the El Hasadi and Khodadadi [34] model indicates
the importance of adding the mass transfer of particles in the numerical
framework that models the phase change process of the NePCM. Also, El Hasadi
and Khodadadi[34] showed that the solidification process of the NePCM is
governed by the thermal transport, and particle transport due to particle
rejection from the solid-liquid interface, and the parameter that controls the
process is the transition segregation coefficient $k_{t}$. $k_{t}$ is given
from the following relation:

$k_{t}=\frac{\phi_{w}m_{l}}{T_{c}-T_{m}}$ (1)

Where $\phi_{w}$, $m_{l}$, $T_{c}$, and $T_{m}$ represent the mass fraction of
the particles, the liquidus slope, the cold-side temperature of the system,
and the melting temperature of the solvent, respectively. The physical
interpretation of the transition segregation coefficient is that the
solidification process of colloidal suspensions or that of the NePCM is
governed by the removal of heat only if the segregation coefficient $k_{0}$,
defined as the ratio between the mass fraction of the particles in the solid
and liquid sides of the interface, is greater than the transition segregation
coefficient. However, if the segregation coefficient $k_{0}$ is lower than the
transition coefficient, the solidification process is controlled by the mass
transport of the particles. Consequently, any enhancements in thermal
conductivity will not expedite the freezing process if $k_{o}\leq k_{t}$,
explaining the deceleration observed in various experiments on the freezing of
NePCM[28]. El Hasadi and Khodadadi[35, 36] extended the one-dimensional model
they developed for the freezing process of colloidal suspensions to ‘two
dimensional space. Additionally, they incorporated the effect of convection
into their model. Their findings highlight the essential role played by the
size and concentration of the particles in the development of the solid-liquid
interface. El Hasadi and Khodadadi[35, 36] were the first to predict the
dendritic, unstable shape of the solid-liquid interface observed during
freezing experiments with colloidal suspensions[20]. Furthermore, El
Hasadi[35, 36] were also, the first to predict the existence of thermosolutal
convection, which was later confirmed by sophisticated experiments conducted
by Scotti et al.[25]. The prediction of thermosolutal convection by the model
of El Hasadi and Khodadadi[35, 36] underscores the necessity of including mass
transfer of particles in the modelling of the freezing process of colloidal
suspensions and NePCM. Only a few investigations in the literature regarding
the freezing process of colloidal suspensions and NePCM include the transport
of particles[37, 38, 39].

For the case of melting the solid form of colloidal suspensions NePCM, in
contrast to the case of freezing colloidal suspensions, where we had a
significant number of investigations coming from the physical sciences, we
will rely on the research being conducted by the energy storage community.

There have been numerous experimental investigations exploring the melting
process of NePCM using different types of solvents and nanoparticles [40, 41,
42, 43]. Chintakrinda et al. [40] performed tests on a NePCM composed of
organic PCM and graphene nanofibers. They found that suspending 11% of
graphene suppressed the convection currents and did not led to any enhancement
in the melting process, despite the addition of thermal conductivity enhancers
such as graphene nanofibers. Ho and Gao [41] experimentally investigated the
performance of n-octadecane and alumina nanoparticles, confirming similar
results to Chintakrinda et al. [40]. They observed that increasing the
nanoparticle loading significantly degraded the melting performance of the
NePCM. Fan et al. [42] and Li et al. [43] investigated the melting process of
nano-enhanced phase change materials (NePCM) within a rectangular cavity,
heated from the bottom, and vertical sides respectively. They used cavities
with different geometries and varying aspect ratios, examining NePCM samples
prepared by incorporating graphene nanoplatelets (GNP) into 1-tetradecanol. To
stabilize the NePCM colloidal suspensions, they employed surfactants. The
addition of these surfactants usually increase the suspension’s viscosity by
forming polymeric brushes around the particle surfaces, thereby impeding fluid
movement. For instance, Fan et al. [42] noted a tenfold increase in NePCM
viscosity upon adding 3% GNP particles compared to that of 1-tetradecanol.
This increase in viscosity significantly surpassed what the dilute colloidal
hydrodynamic theory predicts, indicating that the addition of the surfactant
was the primary reason for the viscosity increase. The two previous
experimetal investigations’ results indicated that an increase in GNP loading
led to a decline in both heat storage and heat transfer rates during melting.
This effect was attributed to a significant rise in viscosity, which
negatively impacted natural convective heat transfer. These findings suggest
that utilizing NePCM in a rectangular cavity geometry may not necessarily
enhance heat storage rates due to the adverse impact of increased viscosity,
which outweighs the thermal conductivity enhancement. Moreover, the
experimental investigations imply that numerical predictions, as we will
demonstrate shortly, regarding melting accelerations and heat transfer
enhancements reported in the literature by adding nano-paeritles to phase
change materilas, may have been overestimated, primarily due to an
underestimation of the effects of mass transport of the particles in their
models. Additionally, a deceleration of the solid-liquid interface during the
NePCM melting process was observed in the experiments by [44, 45].

Numerical simulations of the melting process of NePCM often rely on the
single-phase model, which neglects the transport of particles, including them
solely in the transport properties. Xiong et al.’s extensive review [46] on
numerical investigations related to the modeling of NePCM’s phase change
process and Dhaidan and Khodadadi’s review [47] on experimental investigations
into the NePCM melting process both highlight a significant discrepancy
between numerical and experimental predictions. They partially attribute this
discrepancy to the absence of particle transport in numerical simulation
models. Feng et al. [48] utilized the Lattice Boltzmann numerical method to
study the melting process of NePCM heated from the bottom side of a
rectangular cavity. They reported the classical finding that increasing the
volume fraction led to a significant enhancement in the NePCM melting process
compared to the pure solvent. However, this enhancement contradicts the
findings of many experimental investigations, such as that of Fan et al. [42].
The Lattice Boltzmann method, in its current form, does not offer new physical
insights into the NePCM melting process due to its inability to account for
the mass transport of particles or their interaction with the solid-liquid
interface. Practically speaking, both the Lattice Boltzmann method and the
single-phase model based on finite volume yield similar physical insights into
the NePCM melting process, offering no advantage in using the former. Numerous
numerical investigations employ the single-phase model to simulate the NePCM
melting process [49, 50, 51, 52]. However, most of their results contradict
experimental observations.

El Hasadi [53] was the first to investigate the mass transport of particles
during the melting process of colloidal based NePCM using a similar model as
that of of El Hasadi and Khodadadi [35, 36] which is based on the one-fluid
mixture model. He used a rectangular cavity as a geometry of cohice, the NePCM
is melted from the the vertical left side. The major findings were that the
solid-liquid interface takes unstable shape, and that the size of the
particles plays an essential role in the thermal transport, and into the
redistribution of the particles as well. Also, another outocme of the
investigation is the report about the existence of the thermo-solutal
convection currents which thier intensity depends on the mass fraction of the
particles. In another investigation, El Hasadi [54] further investigated the
melting process of NePCM within a rectangular cavity heated from two sides,
cooled from one side, and insulated on the last side. He observed that an
increase in the volume fraction of particles resulted in the deceleration of
the NePCM melting process as particle loading increased. For instance, at a
particle volume fraction of 10%, the NePCM’s melting process slowed down by
approximately 2.2% compared to the pure solvent, which in this case was water,
aligning closely with observations from experiments.

In this study, we delve into the influence of thermosolutal convection on the
melting process of NePCM within a square cavity geometry, subjected to melting
from both the top and bottom sides. We employ a numerical model akin to those
employed in [35] [53], which incorporates the transport of nanoparticles.
Distinct values for particle concentration, imposed temperature boundary
conditions, and segregation coefficients are implemented to examine the impact
of these factors on the melting process.

##  2 Mathematical Model

One of our foundational assumptions hinges on treating the NePCM as a binary
mixture, a simplification that significantly streamlines the complexity of our
mathematical model. By adopting this assumption, we notably disregard direct
interaction forces between particles and between particles and the fluid. The
particle interactions will be preserved in the values of thermo-physical
properties. The current investigation employs a model built upon the author’s
prior work and it was undergone rigorous testing and validation across various
scenarios of NePCM solidification and melting [35, 34, 36, 53, 55, 56, 57].
The basic governing equations are the following:
Continuity:

$\nabla\cdot\vec{U}=0,$ (2)

Momentum

$\rho\frac{\partial\vec{U}}{\partial
t}+\rho(\vec{U}\cdot\nabla)\vec{U}=-\nabla
p+\mu\nabla^{2}\vec{U}-\rho\frac{C_{m}(1-\lambda)^{2}}{\lambda^{3}}\vec{U}+(\rho\beta)g\left(T-T_{\text{ref
}}\right)\vec{e}_{y},$ (3)

Energy:

$\rho c_{p}\frac{\partial T}{\partial t}+\rho c_{p}\vec{U}\cdot\nabla
T=\nabla\cdot(k\nabla T)-\rho L\frac{\partial\lambda}{\partial t},$ (4)

Species:

$\frac{\partial\phi_{w}}{\partial
t}+\vec{U}\cdot\nabla\phi_{w}=\nabla\cdot\left(D_{B}\nabla\phi_{w}+D_{T}\frac{\nabla
T}{T}\right)$ (5)

Where $\vec{U}$ represents the velocity vector, $T$ denotes temperature, and
$\phi_{w}$ stands for the mass fraction of the particles. The Darcy term
within the momentum equations assumes a pivotal role in modulating velocity
values during the melting process. This term functions as a damping mechanism,
diminishing flow rates in scenarios characterized by high porosity or
permeability. Consequently, it prevents instabilities, ensuring a stable and
controlled melting process. Within the mushy zone, the local mass fraction is
computed using a straightforward mixture formula as follows:

$\phi_{w}=\lambda\phi_{wf}+(1-\lambda)\phi_{ws},$ (6)

Where $\lambda$ represents the liquid fraction, and the subscripts $f$ and $s$
denote the liquid and solid states, respectively. The momentum equations
incorporate Darcy law damping terms, wherein we assign a porosity constant
$C_{m}$ with a value of $10^{5}kg/m^{3}s$. Additionally, we disregard particle
diffusion within the heat flux vector due to the nano-particles’ low diffusion
coefficient [58].

###  2.1 Initial and boundary conditions

In our current investigation, we’ll focus on a square cavity geometry with
thermally insulated left and right sides. For our simulations, we’ll explore
two variations of this geometry. In the first configuration, the melting
process of the NePCM solid composite will commence from the top side of the
cavity. While, in the second configuration, the NePCM solid composite will
initiate melting from the bottom side of the cavity. Both the bottom and top
sides of the cavity will have a constant temperature boundary condition, and
the verical sides of the cavity will be isolated thermally. The NePCM solid
composite will start melting by elevating the temperature above the melting
temperature for the given particle concentration. We’ll use various nano-
particle concentrations, with a particle size of $d_{p}=2nm$. The geometry
under-consideration is shown in Figure 1.

###  2.2 Effective thermo-physical properties

The NePCM mixture that we will use is suspension of water, and copper nano-
particles. We’ll make the assumption that the suspension in its liquid form
comprises hard-sphere particles, thereby preventing particle clustering.
Additionally, we’ll consider the properties of the pure water in solid and
liquid phases as constant for simplification purposes. However, these
properties vary with the concentration of the particles, a characteristic
often observed in colloidal particles.

To derive the density, heat capacity, and a portion of the Boussinesq term,
we’ll utilize the following equations.

$\displaystyle\rho=(1-\phi)\rho_{f}+\phi\rho_{p}$ (7) $\displaystyle\rho
c_{p}=(1-\phi)\left(\rho c_{p}\right)_{f}+\phi\left(\rho c_{p}\right)_{p},$
$\displaystyle\rho\beta=(1-\phi)(\rho\beta)_{f}+\phi(\rho\beta)_{p}.$

Where $\phi$, $\rho$, and $\beta$ represent the volume fraction of the
particles, the density, and the thermal expansion coefficient, respectively.
The subscripts $f$ and $p$ denote the liquid (solid) and particle phases,
respectively. The relationship between the volume fraction of the particles
and their mass fraction is as follows:

$\phi=\frac{\rho_{f}\phi_{w}}{\rho_{f}\phi_{w}+\rho_{p}\left(1-\phi_{w}\right)}$
(8)

The thermal conductivity of the NePCM is calculated as the following:

$k=k_{1}+k_{2}$ (9)

Where $k_{1}$ is the Maxwell contribution to the thermal conductivity:

$k_{1}=k_{f}\frac{k_{p}+2k_{f}-2\phi\left(k_{f}-k_{p}\right)}{k_{p}+2k_{f}+\phi\left(k_{f}-k_{p}\right)}$
(10)

and $k_{2}$ is the contribution due to the dispersion:

$k_{2}=C_{k}\left(\rho c_{p}\right)|\vec{U}|\phi d_{p}$ (11)

The redistribution of particles will notably impact the mass diffusivity of
the particles. To account for this effect, we will define a compressibility
factor as a function of the volume fraction, following Peppin et al. [19]. The
constant $C_{k}$ is obtained from Wako and Kaguei [59]. The compressibility
equation is expressed as follows:

$z(\phi)=\frac{1+\left(4-\frac{1}{0.64}\right)\phi+\left(10-\frac{4}{0.64}\right)\phi^{2}+\left(18-\frac{10}{0.64}\right)\phi^{3}}{1-\frac{\phi}{0.64}}$
(12)

The mass diffusivity of the particles can be calculated from the following
equation:

$D_{B}=D_{o}(1-\phi)^{6}\left(\frac{d(\phi z)}{d\phi}\right)$ (13)

Where $D_{0}$ is the Einstein diffusivity:

$D_{0}=\frac{Tk_{B}}{3\pi d_{p}\mu}$ (14)

The theromophoretic diffusivity is evaluated from the following relation:

$D_{T}=\beta_{k}\left(\frac{\mu}{\rho}\right)\phi$ (15)

Where $\beta_{k}$ is a non-dimensional parameter that is a function of the
thermal conductivities described by [58], the variation of the latent heat of
fusion of the NePCM is given as the following:

$\rho L=(1-\phi)(\rho L)_{f}$ (16)

To incorporate the change of the liquidus and solidus temperatures of the
NePCM with the volume fraction of the particles, we’ll employ the following
straightforward linear phase diagram:

$\displaystyle T_{\text{Liq }}=T_{m}-m_{l}\phi_{w}$ (17) $\displaystyle
T_{\text{Sol }}=T_{m}-\frac{m_{l}}{k_{0}}\phi_{w}$

Where $T_{\text{Liq}}$ and $T_{\text{Sol}}$ represent the solidus and liquidus
temperatures of the NePCM suspension, respectively. $k_{0}$ denotes the
segregation coefficient, governing the ratio of the mass fraction of the
particles on the solid and liquid sides of the interface. For our simulations,
we’ll set the segregation coefficient to a value of 0.1. We’ll utilize the
following equation to calculate the melting temperature:

$m_{l}=\frac{k_{B}T_{m}^{2}}{v_{p}\rho L_{f}}$ (18)

The relation that relates the liquid fraction $\lambda$ with the
$T_{\text{Liq}}$, and $T_{\text{Sol}}$ temperatures is the following:

$\displaystyle\lambda=0$ for $\displaystyle\ T<T_{\text{Sol}}$ (19)
$\displaystyle\lambda=\frac{T-T_{\text{Sol}}}{T_{\text{Liq}}-T_{\text{Sol}}}$
for $\displaystyle\ T_{\text{Sol}}<T<T_{\text{Liq}}$ $\displaystyle\lambda=1$
for $\displaystyle\ T>T_{\text{Liq}}$

The thermophysical properties of water and copper particles are listed in
Table 1. Additionally, Table 2.2 presents the properties of the NePCM for a
mass fraction of 10%. From Table 2.2, it is evident that even at the maximum
particle mass fraction that we will use, both the thermal conductivity and
viscosity do not exhibit significant enhancements compared to pure water.

| Water | Copper nanoparticles
---|---|---
Density | $997.1\mathrm{~{}kg}/\mathrm{m}^{3}$ | $8954\mathrm{~{}kg}/\mathrm{m}^{3}$
Viscosity | $8.9\times 10^{-4}\mathrm{~{}Pa}\mathrm{~{}s}$ | $-$
Specific heat | $4179\mathrm{~{}J}/\mathrm{kg}\mathrm{K}$ | $383\mathrm{~{}J}/\mathrm{kg}\mathrm{K}$
Thermal conductivity | $0.6\mathrm{~{}W}/\mathrm{m}\mathrm{K}$ | $400\mathrm{~{}W}/\mathrm{m}\mathrm{K}$
Thermal expansion coefficient | $2.1\times 10^{-4}\mathrm{~{}K}^{-1}$ | $1.67\times 10^{-5}\mathrm{~{}K}^{-1}$
Heat of fusion | $3.35\times 10^{5}\mathrm{~{}J}/\mathrm{kg}$ | $-$
Table 1: Thermophysical and transport properties for the solvent and the
nanoparticles

o 1.0 — X[c] — X[c] — Transport Property |  $\phi_{w}$ = 10% ($\phi$=* $\frac{\rho_{f}\phi_{w}*100}{\rho_{f}\phi_{w}+\rho_{p}(1-\phi_{w})}$ [ρ_f = 997.1,ϕ_w = 0.1,ρ_p = 8954][x2]%)
---|---
Thermal Conductivity |  6.23$\times 10^{-1}$ $\mathrm{~{}W}/\mathrm{m}\mathrm{K}$
Density |  1.10$\times 10^{3}$ $\mathrm{~{}kg}/\mathrm{m}^{3}$
Dynamic viscosity |  9.18 $\times 10^{-4}$ $\mathrm{~{}Pa}\mathrm{~{}s}$
Specific heat |  3.8$\times 10^{3}$ $\mathrm{~{}J}/\mathrm{kg}\mathrm{K}$
Thermal expansion coefficient |  *$10^{4}\frac{(1-\phi)(\rho_{f}\beta_{f})+\phi(\rho_{p}\beta_{p})}{\rho_{m}}$[ϕ=0.0122,ρ_f = 997.1,β_f = 0.00021,ρ_p = 8954, β_p = 0.0000167,ρ_m = 1100][x2] $\times 10^{-4}$ $\mathrm{~{}K}^{-1}$
Table 2: The values of the transport properties of the NePCM at different
nano-particle volume fractions.

##  3 Results

The results will include cases of melting the NePCM from the top, and bottom
sides receptively.

###  3.1 Melting from the top side

In this subsection will consider a single case with $Ra$ = *$10^{-4}\frac{\rho
g\beta(Th-Tc)H^{3}}{\mu\frac{k}{\rho cp}}$[ρ= 1.1×10^3, g = 9.8,β= 1.9×10^-4,
Th = 290,Tc = 270, H = 0.005, μ= 9.18×10^-4, k = 6.23×10^-1, cp =
3.8×10^3][x2] $\times 10^{4}$, and specifically will impose the hot side on
the top side of the cavity, with $T_{h}$ = 290 K, while the bottom side is
kept at $T_{c}$ = 270 K, the initial mass concentration $\phi_{w}$ is equal to
10%, and $k_{o}$ = 0.1. This particular configuration suppresses thermal
convection because the temperature field is stably stratified, and thermal
conduction will transport the heat. The solid-liquid interface develops a
stable shape, which is a particular property of melting controlled by
conduction. At later times of the melting process, the mushy zone is extended,
and we can observe small pockets of an unstable interface, this is a result of
the particle rejection from the solid phase to the liquid phase as shown in
Figure 2. The melted part of the NePCM contains a higher concentration of
particles, as shown in Figure 3, consistent with experimental observations
[60], which report particle sedimentation during the melting process of the
NePCM. The high concentration nano-particle pockets are concentrated in the
solid interface area, and they remain attached to the interface as the melting
process proceeds. Since there is no convection mechanism to redistribute the
particles away from the solid-liquid interface. The temperature contours
reflect the domination of conduction as the main heat transfer mechanism, in
which the heat is diffused from the high-temperature top side to the cold
bottom side, as shown in Figure 4.

We will compare the thermal performance of the NePCM with $\phi_{w}$ = 10%
with that of pure water, by comparing the temperature fields, and the average
liquid fraction. We will evaluate the temperature profile along a vertical
line that crosses the bottom, and top sides of the cavity at the center
$\dfrac{x}{H}$, where $H$ is the length of the cavity, and it is equal in our
case to 0.005 m. The results of the compression is shown in Figure 5, the two
temperature profiles of the NePCM, and that of pure water are following each
other closely, which shows that adding the particles did not enhance
sufficiently the thermal properties especially that of thermal conductivity,
that is the main thermal transport property that controls the transport during
the conduction process. If we compare the values of the thermal conductivities
of the pure water, and the NePCM with $\phi_{w}$ = 10% from the Table 1, and
Table 2.2 we will find that the thermal conductivity only increased about
4.0%, which is consistent with the effective medium theory, and the findings
of the world wide benchmark exercise for measuring the thermal conductivity of
nanofluids [61].

We used low concentration of particles for our simulations, for the two
following reasons. The first reason is that it will reduce the computational
time needed to perform the simulations.While the second reason is that we
wanted to follow the recommendations made by different experimental
investigations that recommend the use of suspensions that contain particle
loadings of about 1% to 5% by volume, in order to for the suspensions to stay
in stable condition [61]. The volume concentration of particles that we used
is 1.22 % . The comparison of the average liquid fraction for different time
instances is shown in Figure 6, for the NePCM, and pure water, the liquid
fraction evolution for the two cases is identical indicating of no enhancement
in the melting rate. Investigations in the literature show enhancement in the
melting rate, only when the they use concentrations of particles that are
higher than 5%, for example they used concentration of particles $\phi$ equal
to 5%, and 10% in their simulations to show enhancement of the performance of
the NePCM compared to pure water [48], suspensions with such high
concentration of particles, it will be very difficult to kept in stable
condition, due to the exsistance of many types of interparticle, fluid -
particles interactions that they will intiate the formation of clusters [62].

###  3.2 Melting form the bottom side

Now will consider the case of melting from the bottom side, the thermal
Rayleigh number equal to *$10^{-4}\frac{\rho g\beta(Th-
Tc)H^{3}}{\mu\frac{k}{\rho cp}}$[ρ= 1.1×10^3, g = 9.8,β= 2.1×10^-4, Th =
290,Tc = 270, H = 0.005, μ=8.9×10^-4, k = 6.0×10^-1, cp = 4.179×10^3][x2]
$\times 10^{4}$, for the case of pure water, and $Ra$ = *$10^{-4}\frac{\rho
g\beta(Th-Tc)H^{3}}{\mu\frac{k}{\rho cp}}$[ρ= 1.1×10^3, g = 9.8,β= 1.9×10^-4,
Th = 290,Tc = 270, H = 0.005, μ= 9.18×10^-4, k = 6.23×10^-1, cp =
3.8×10^3][x2] $\times 10^{4}$ for the NePCM with $\phi_{w}$ = 10% . The value
of the $Ra$ for the case of the pure water is higher than that of the NePCM,
this is due to the slight increase in the viscosity with volume fraction of
the particles, and combined with the very marginal increase in the thermal
conductivity of NePCM. The hot side is the bottom side of the cavity with
$T_{h}$ = 290 K, while the cold side is the top wall of the cavity, and its
temperature is set to ($T_{c}$) equal to 270 K. In this specific
configuration, thermal convection will form due to the temperature and density
differences. The evolution of the melting process for NePCM with a mass
fraction of particles ($\phi_{w}$) = 10% is shown in Figure 7. At times $t$ =
8 and 52 seconds, heat is transported through conduction. Since the morphology
of the solid-liquid interface is flat (planar) in shape, there is no
indication that convection has started to form. As we proceed further in time,
at $t$ = 127 and 143 seconds, convection begins to form, and the shape of the
solid-liquid interface changes from flat to curved due to the effects of
natural convection.

The accompanying concentration profiles are shown in Figure 8. During the
early stages of the melting process, layers with different concentrations of
particles are formed, with a zone of high particle concentration forming above
the hot bottom wall. The concentration of particles is higher in the melting
part of the NePCM due to the rejection of the particles from the solid phase
during the phase change process. At $t$ = 125 and 143 seconds, we observe a
significant redistribution of the particles due to the formation of convection
cells. Convection cells facilitate the dissipation of high particle
concentration layers that form due to particle diffusion during the initial
stages of melting. With the help of convection cells, the particles are more
evenly distributed throughout the melting part of the NePCM, and only small
high-concentration pockets exist on the left and vertical at the vicinity of
the solid-liquid interface.

At times $t$ = 8 and 52 seconds, the temperature is diffusing from the hot
bottom side. At later times, however, the hot temperature at the bottom wall’s
centre rises due to convection, bouncing the convection current to warm layers
of water away from the bottom wall. The temperature profiles for the case of
bottom NePCM melting are shown in Figure 9.

We are comparing the flow fields (streamlines) superimposed on the liquid
fraction contours for the cases of pure water and NePCM with a mass
concentration of particles of 10%, Ra =$\eval*{$10^{-4}\frac{\rho g\beta(Th-
Tc)H^{3}}{\mu\frac{k}{\rho cp}}$}[\rho=1.1\times 10^{3},g=9.8,\beta=1.9\times
10^{-4},Th=290,Tc=270,H=0.005,\mu=9.18\times 10^{-4},k=6.23\times
10^{-1},cp=3.8\times 10^{3}][x2]\times 10^{4}$, as shown in Figure 10. The
comparison reveals that both cases’ flow fields generated by convection are
nearly similar. However, the flow intensity is higher in the case of pure
water compared to NePCM, with a mass concentration of particles of 10%. The
solid-liquid interface advances further in the case of pure water compared to
NePCM with a mass concentration of particles of 10% due to the high intensity
of the convection for the case of pure water. This deceleration of the melting
front is also observed in the experiments conducted by Zeng et al. [63] for
the bottom melting for the NePCM.

To have a better understanding of the melting performance of the NePCM in the
case of bottom melting, we will compare the time evolution of the average
liquid fraction with that of water at different time instances, as shown in
Figure 11. In the melting process’s early stage, pure water and the NePCM melt
at the same rate. However, in the late melting stage, pure water melts faster
than the NePCM, which can be observed from Figure 10, where convection becomes
the dominant mechanism controlling the melting process. The water melts faster
by about *$(\frac{(0.59727-0.584)}{0.59727})\times 100$[x2] %.

To further clarify our observations, we plot the velocity and temperature
along a vertical line extending from the bottom to the top sides of the
cavity, located at<EMAIL_ADDRESS>Figure 12 shows the velocity
distribution. Interestingly, near the bottom wall, the velocity of pure water
and the NePCM are equal. However, as we move further from the wall, the
velocity of the water becomes higher than that of the NePCM due to stronger
convection currents. After reaching a maximum value, the velocity decreases
due to a reduction in convection intensity . From the peak point onwards, the
velocity of pure water is higher than that of the NePCM.

The temperature profiles are depicted in Figure 13. The temperature of the
NePCM is slightly higher than that of pure water near the bottom wall, which
can be attributed to the higher thermal conductivity of the NePCM. However, as
we move away from the bottom wall, the temperature of pure water becomes
higher than that of the NePCM due to better mixing facilitated by convection.

####  3.2.1 effect of the Rayleigh number

We will increase the Rayleigh number by raising the temperature of the bottom
wall to further investigate the natural convection effect on the melting
process of the NePCM. We will increase the bottom temperature to $T_{h}$ = 300
K. The corresponding value of the thermal Rayleigh number is to
*$10^{-4}\frac{\rho g\beta(Th-Tc)H^{3}}{\mu\frac{k}{\rho cp}}$[ρ= 1.1×10^3, g
= 9.8,β= 2.1×10^-4, Th = 300,Tc = 270, H = 0.005, μ=8.9×10^-4, k = 6.0×10^-1,
cp = 4.179×10^3][x2] $\times 10^{4}$ for the case of the pure water, and for
the case of NePCM with $\phi_{w}$ = 10%, $Ra$ = $\eval*{$10^{-4}\frac{\rho
g\beta(Th-Tc)H^{3}}{\mu\frac{k}{\rho cp}}$}[\rho=1.1\times
10^{3},g=9.8,\beta=1.9\times 10^{-4},Th=300,Tc=270,H=0.005,\mu=9.18\times
10^{-4},k=6.23\times 10^{-1},cp=3.8\times 10^{3}][x2]\times 10^{4}$ .

The superimposed contours of streamlines and the solid-liquid interface for
the case of pure water ($\phi_{w}$ = 0) are shown in Figure 14. At t = 8 sec,
the flow is predominantly governed by conduction, with the fluid velocity
approximately equal to zero, and the solid-liquid interface being flat.
However, at t = 52 sec, the effects of convection start to become apparent,
leading to the formation of two symmetric counter-rotating vortices of nearly
equal strength. As time progresses, these two counter-rotating cells occupy a
significant portion of the melted water zone, while the solid-liquid interface
takes on a curved and symmetric shape, in contrast to the initial flat shape
during the early stages of the melting process. This change in shape of the
solid-liquid interface is a result of the substantial increase in convection
strength within the cells. Both cases of water, with melting occurring from
the bottom with $Ra$ values of $3.74\times 10^{4}$ and $7.31\times 10^{4}$
respectively, have nearly identical, flow fields and solid-liquid interface
evolution. The only difference is that the latter case experiences a faster
melt rate than the former.

For the case of the NePCM with $\phi_{w}$ = 10% and $Ra$ = $5.61\times
10^{4}$, the superimposed contours of streamlines and the solid-liquid
interface are depicted in Figure 18. At t = 8 sec, the solid-liquid interface
is flat, and the melting process is primarily governed by conduction, similar
to the pure water case. However, as time progresses, the flow field evolves to
consist of three vortexes. The middle vortex rotates counter-clockwise, while
the other two surrounding vortexes rotate clockwise. The left vortex exhibits
a higher strength compared to the remaining two. As time further elapses to t
= 127 sec and 143 sec, the symmetry of the solid-liquid interface is
disrupted, and the three vortexes grow in size. The melting near the right
vertical wall occurs at a faster rate, whereas the growth of the solid-liquid
interface near the left vertical wall is considerably slower. The breaking of
symmetry in the solid-liquid interface can be attributed to the generation of
thermosolutal convection resulting from the redistribution of the particles
during the melting process. This physical mechanism plays a crucial role in
the observed phenomena, and thus using single phase model to model the melting
process of the NePCM is an oversimplification that leads to significant error
in the predictions. Also, the difference in the flow fields between the pure
water and the NePCM is responsible for the clear deceleration of the melting
process of the NePCM compared to the water, not the increase of the viscosity
as several investigations show in the literature show references.

To further investigate the non-symmetric shape of the solid-liquid interface
we observed in Figure 14, we plotted the particle redistribution as shown in
Figure 17 . For $t=8$ sec and $t=52$ sec, we observe that the particles in the
melting zone are distributed in alternating high and low-concentration layers
of particles . When the thermosolutal convection starts, the particles are
redistributed along the melted portion of the NePCM. Even though the particles
are about *$\frac{\rho_{p}}{\rho_{f}}$ [ρ_p = 8954,ρ_f = 1000] [x2] times
heavier than the water. The particles are not distributed evenly, and a large
portion of them accumulates at the left wall of the cavity, the accumulation
of particles at the left wall explains the slowing down of the solid-liquid
interface in this particular location of the melting zone . Overall, the non-
uniform distribution of the particles leads to the non-symmetric solid-liquid
interface observed .

The temperature distribution for the case of pure water with $Ra=7.3\times
10^{4}$ is depicted in Figure 15. At the beginning of the melting process, the
temperature profile indicates conduction as the dominant heat transfer
mechanism. However, thermal convection becomes the dominant factor influencing
the temperature profile as time progresses. The warmest temperature zone is
situated along the vertical line crossing the bottom wall of the system. On
the other hand, for the NePCM case with $Ra=5.61\times 10^{4}$, the
temperature profiles are illustrated in Figure 18. Initially, during the early
stages of melting, the temperature profiles resemble those observed for pure
water. However, as thermosolutal convection becomes more prominent, the
temperature profiles become asymmetric. In the case of the NePCM, two warm
zones are observed, one near the left bottom corner of the duct and the other
to the right of the duct’s center. These warm zones are closely related to the
locations where the particles accumulate the most. The flow, temperature, and
concentration fields are strongly coupled. Neglecting the transport of
particles would lead to significant overestimations that do not reflect
reality.

We will compare the liquid fraction time evolution for the case of pure water
with that of the NePCM with mass fraction of particles ($\phi_{w}$ ) = 10% for
case of bottom wall temperature ($T_{h}$) = 300 K, and the top wall
temperature ($T_{c}$) = 270 K, as shown in Figure 19. At the early stage of
melting the pure water and the NePCM are melting with the same rate, however
when the convection starts, the water melts faster than the NePCM. The water
melts faster by about *$(\frac{(0.9376-0.883)}{0.9376})\times 100$[x2] % . If
we check the thermal Rayleigh numbers for the case of pure water, and the
NePCM we will find that they differ by *$\frac{(7.3-5.61)}{7.3}\times 100$ [1]
%, the difference in the Rayleigh number is much higher than the difference in
the melting rate slowdown, which suggest that the water most melted by higher
rate than 5.82% which leads us to conclude that the thermal Rayleigh number,
is not the only parameter that is contributing to the deceleration of the
solid-liquid interface, but the combined thermosolutal convection is the
leading parameter that controls the melting process of the NePCM, as will show
with extra evidence later . If the thermal convection is not the leading
parameter of controlling the melting process of the NePCM, thus increasing the
viscosity of the NePCM due to the suspension of the particles, is not the only
reason for the deceleration of melting as it was thought before [42, 47].
Another observation is that if we increase the temperature of bottom wall
decseleration of the melting rate of NePCM will increase compared to water.

We will further analyse the cases of the pure water, and NePCM for the case of
melting from the bottom side for the thermal boundary conditions of $T_{h}$ =
300 K, and $T_{c}$ = 270 K, by plotting different quantities along a vertical
line that it is extended from the middle of the bottom wall to the middle of
the top wall.

Figure 20 the velocity profiles for the pure water, and the NePCM, for the
case of the pure water the velocity profile is nearly symmetric, with a local
maximum. While, for the case of the NePCM the velocity profile is distorted
due to the thermosolutal convection effect, and the velocity profile consists
of two local maxima points, and one local minimum around the middle the hight
of the duct. Beyond the distortion of the velocity profile the thermosolutal
convection effects also the magnitude of the velocity significantly, for
example the maximum velocity for the case of pure water is about 5 times
greater than the maximum velocity of the NePCM . The concentration profile for
the case of the NePCM is shown in Figure 22 . There is a high concentration of
particles near the bottom wall for distance of about $\dfrac{y}{H}$ $\simeq$
0.1, then the particles have a uniform concentration throughout the melting
zone, with concentration higher than the initial concentration of 10% . Near
the solid-liquid interface, we observe a local maximum value, then a sharp
decrease in the concentration of the particles in the mushy zone. In the non-
melted part of the NePCM, the concentration of the particles retains its
initial value. Figure 21 shows the variation of the temperature. For the case
of water, in which the melting process is controlled purely by thermal
convection, the mixing process is better compared to NePCM, which is reflected
in the temperature profile, in which the water temperature is higher than that
of the NePCM along the vertical line. Even in the region near the wall where
there is an accumulation of nanoparticles, which creates a zone of higher
thermal conductivity, we do not observe an increase in the values of the
temperature of the NePCM compared to that of the water.

Figure 23 shows the variation of the thermal conductivity ratio of the NePCM
to that of the pure water, in the melted region of the NePCM the thermal
conductivity is about 3.7% higher than that of the water, this type of
enhancement is consistent with the findings of global thermal conductivity
benchmark exercises for nanofluids [61]. In Figure 24 we show the ratio of the
viscosity of the NePCM to that of water, the viscosity increase is about 3.3%
which is less than the increase of the thermal conductivity, the increase in
the viscosity is consistent with the international banshmark exercises of the
viscosity of stable nanofluids [64]. The viscosity of the NePCM did not
significantly increased due to the redistribution of the particles, and thus
is not the main reason for the significant deceleration of the melting rate of
the NePCM compared to pure water that is observed. We plot also the ratio
between the melting temperature of the NePCM, and that of pure water, as shown
in Figure 25, the two melting temperatures are nearly equal, which shows that
the redistribution of the particles did not play a significant role on
alerting the Liquids temperature, and thus the liquids temperature variation
did not play a significant role in the deceleration of the solid-liquid
interface observed for the case of the NePCM compared to that of water.

###  3.3 effect of the segregation coefficient

To gain more substantial confidence in our conclusion regarding the dominant
role of thermosolutal convection in retarding the development of the NePCM
melting process compared to pure water, we will set the segregation
coefficient to a value of one ($k_{0}$ = 1.0). This choice signifies that
particle concentration will remain uniform within the cavity, indicating that
particles will not interact with the solid-liquid interface, or with fluid it
self. This scenario, represented by $k_{0}$ = 1.0, aligns with the single-
phase models commonly referenced in existing literature. As demonstrated in
Figure 26, the case where particle redistribution is absent ($k_{0}$ = 1.0)
shows that the NePCM melts at a rate equivalent to that of pure water,
consistent with the predictions of many models found in the literature [48].
However, when particle redistribution is present, as in the case of the NePCM
with $k_{0}$ = 0.1, the melting process of the NePCM is notably slower than
that of pure water, which is in line with the majority of experimental
findings [42]. The observation depicted in Figure 26 strongly supports the
notion that particle redistribution by thermosolutal convection serves as the
primary factor responsible for the deceleration of the NePCM melting process
rather than an increase in viscosity. Particle redistribution significantly
influences both the flow structure and the shape of the solid-liquid
interface, as illustrated in Figure 27. In the NePCM scenario with
$k_{o}$=1.0, the flow structure closely resembles that of pure water,
featuring two counter-rotating flow cells, distinct from the three-cell flow
structure observed with $k_{o}$ =0.1. This highlights the essential role of
particle redistribution in NePCM’s melting process and its overall
performance. Also, both the NePCM with segregation coefficient of 1.0, and
0.1, have the same thermal Rayleigh number = $5.61\times 10^{4}$ however,
their flow fields, and melting behaviour is different. Which shows that
thermal Rayleigh number is not the controlling non-dimensional number for the
melting process of the NePCM.

We also plot the viscosity ratio of NePcM to that of pure water for a vertical
line extending from the bottom to the top walls located at $x/H$ = 0.5 for
$k_{o}$ = 1.0 and 0.1, as depicted in Figure 28. A slight increase in the
viscosity of the liquid NePCM is observed with $k_{o}$ = 0.1 compared to NePCM
melted with $k_{o}$ = 1.0. This increase in viscosity is due to the rejection
of particles during the melting process for the case of NePCM with $k_{o}$ =
0.1. The viscosity increase amounts to approximately
*$\frac{1.03124-1.03122}{1.03124}\times 10^{2}$%, which is negligible and
highlights that the reduction in melting speed observed for NePCM with $k_{o}$
= 0.1 is attributable not to viscosity increase, but rather to thermosolutal
convection. Our simulations have conclusively demonstrated the significant
role of thermosolutal convection in shaping the NePCM melting process and the
corresponding flow fields. Furthermore, it is essential to note that
thermosolutal convection represents a novel physical mechanism that must be
considered to explain the degradation of NePCM observed in experiments.

####  3.3.1 Effect of mass fraction of particles

We will investigate the effect of the mass fraction of the particles. We
tested various mass loadings of particles while keeping the segregation
coefficient at 0.1. Consequently, in all the considered NePCM samples, the
ratio of the particles’ mass fraction at the interface’s solid and liquid
sides remains equal. The liquid fraction distribution and the flow field
during the late melting stage of the NePCM are depicted in Figure 29.

When $\phi_{w}=0.5\%$, the flow structure and the solid-liquid interface shape
resemble waters. The melting process is predominantly controlled by thermal
convection alone, given the reduced intensity of thermosolutal convection with
lower particle mass fractions. With an increase in mass fraction to
$\phi_{w}=5\%$, the flow transitions to a non-symmetric structure featuring
three cells, accompanied by a non-symmetric solid-liquid interface shape. The
flow structure and interface for $\phi_{w}=5\%$ closely resemble those of
$\phi_{w}=10\%,$, underscoring the significance of particle mass fraction in
shaping the flow structure and the liquid-solid interface shape during the
NePCM melting process. This phenomenon, a novel contribution, is reported for
the first time in the literature.

##  4 Conclusions

Our investigation employed numerical simulations to comprehensively analyze
the melting dynamics of NePCM (Nano-enhanced Phase Change Materials),
incorporating intricate considerations of particle mass transport. The current
investigation encompassed two distinct melting configurations: top-side and
bottom-side melting scenarios. Through this comprehensive study, our findings
yield the following key conclusions:

  * •

Melting the NePCM from the top cavity side ensures the suppression of
convection effects, presenting the sole configuration where thermal
conductivity enhancement can be optimally utilized.

  * •

In scenarios where melting occurs from the bottom cavity side with pure water,
the process is primarily controlled by thermal convection. However, in the
case of NePCM melting, it is dominantly governed by thermosolutal convection.

  * •

The presence of thermosolutal convection results in a slower solid-liquid
interface compared to pure water, consistent with numerous experimental
observations. Notably, for a mass fraction of particles ($\phi_{w}$) of 10%,
the NePCM demonstrates approximately a 6% deceleration in comparison to pure
water.

  * •

For our investigations the enhancement of thermal conductivity surpasses that
of viscosity in all cases considered. Consequently, we exclude enhanced
viscosity as the primary physical mechanism influencing the deceleration of
the NePCM melting process.

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$T_{c}$ $T_{h}$ LiquidSolid(a)$T_{c}$$T_{h}$SolidLiquid(b) Figure 1:  The
geometry under-consideration (a) melting from the top side (b) melting from
the bottom side

(a) t= 8 sec

(b) t = 50 sec

(c) t = 140 sec

(d) t = 200 sec

Figure 2: The evolution of liquid fraction profiles for the case of melting
from the top for $T_{h}$ = 290 K, $T_{c}$ = 270 K, and $\phi_{w}$ = 10%.

(a) t= 8 sec

(b) t = 50 sec

(c) t = 140 sec

(d) t = 200 sec

Figure 3: The evolution of concentration profiles of nano-particles for the
case of melting from the top for $T_{h}$ = 290 K, $T_{c}$ = 270 K, and
$\phi_{w}$ = 10%.

(a) t= 8 sec

(b) t = 50 sec

(c) t = 140 sec

(d) t = 200 sec

Figure 4: The evolution of temperature profiles for the case of melting from
the top for $T_{h}$ = 290 K, $T_{c}$ = 270 K, and $\phi_{w}$ = 10%. Figure 5:
The variation of the temperature along a vertical line that crosses the
bottom, and top sides of the cavity at $\dfrac{x}{H}$ = 0.5, for the case of
for the case of melting from the top for $T_{h}$ = 290 K, $T_{c}$ = 270 K, and
$\phi_{w}$ = 10%.  Figure 6: Comparison between the variation of the average
liquid fraction for different time intenseness, for the case of pure water,
and NePCM $\phi_{w}$ = 10%, for melting from the top with $T_{h}$ = 290 K,
$T_{c}$ = 270 K.

(a) t= 8 sec

(b) t = 52 sec

(c) t = 127 sec

(d) t = 143 sec

Figure 7: The evolution of liquid fraction profiles for the case of melting
from the top for $T_{h}$ = 290 K, $T_{c}$ = 270 K, and $\phi_{w}$ = 10%.

(a) t= 8 sec

(b) t = 52 sec

(c) t = 127 sec

(d) t = 143 sec

Figure 8: The evolution of concentration profiles of nano-particles for the
case of melting from the bottom for $T_{h}$ = 290 K, $T_{c}$ = 270 K, and
$\phi_{w}$ = 10%.

(a) t= 8 sec

(b) t = 52 sec

(c) t = 127 sec

(d) t = 143 sec

Figure 9: The evolution of temperature profiles for the case of melting from
the bottom for $T_{h}$ = 290 K, $T_{c}$ = 270 K, and $\phi_{w}$ = 10%.

(a) t= 127 sec

(b) t = 143 sec

(c) t = 127 sec

(d) t = 143 sec

Figure 10: Comparison between evolution of the flow field (streamlines)
superimposed on the contours of the liquid fractions for pure water ($\phi$=0)
(a), (b), and for the case of the NePCM ($\phi_{w}$ = 10%), for $T_{h}$ = 290
K, $T_{c}$ = 270 K. Figure 11: Comparison between the variation of the average
liquid fraction for different time intenseness, for the case of pure water,
and NePCM $\phi_{w}$ = 10%, for melting from the bottom with $T_{h}$ = 290 K,
$T_{c}$ = 270 K. Figure 12: The variation of the velocity along a vertical
line that crosses the bottom, and top sides of the cavity at $\dfrac{x}{H}$ =
0.5, for the case of for the case of melting from the bottom for $T_{h}$ = 290
K, $T_{c}$ = 270 K, and $\phi_{w}$ = 10%.  Figure 13: The variation of the
temperature along a vertical line that crosses the bottom, and top sides of
the cavity at $\dfrac{x}{H}$ = 0.5, for the case of for the case of melting
from the bottom for $T_{h}$ = 290 K, $T_{c}$ = 270 K, and $\phi_{w}$ = 10%.

(a) t= 8 sec

(b) t = 52 sec

(c) t = 127 sec

(d) t = 143 sec

Figure 14: Development of the flow field (shown by the stream lines)
superimposed on the contour of the liquid fraction for $T_{h}$ =300 K, $T_{c}$
= 270 K, for case of pure water.

(a) t= 8 sec

(b) t = 52 sec

(c) t = 127 sec

(d) t = 143 sec

Figure 15: The evolution of temperature profiles for the case of melting from
the bottom for $T_{h}$ = 300K, $T_{c}$ = 270 K, for the case of pure water.

(a) t= 8 sec

(b) t = 52 sec

(c) t = 127 sec

(d) t = 143 sec

Figure 16: Development of the flow field (shown by the stream lines)
superimposed on the contour of the liquid fraction for $T_{h}$ =300 K, $T_{c}$
= 270 K, for case of NePCM with $\phi_{w}$ = 10%.

(a) t= 8 sec

(b) t = 52 sec

(c) t = 127 sec

(d) t = 143 sec

Figure 17: The evolution of concentration profiles of nano-particles for the
case of melting from the bottom for $T_{h}$ = 300 K, $T_{c}$ = 270 K, and
$\phi_{w}$ = 10%.

(a) t= 8 sec

(b) t = 52 sec

(c) t = 127 sec

(d) t = 143 sec

Figure 18: The evolution of temperature profiles for the case of melting from
the bottom for $T_{h}$ = 300 K, $T_{c}$ = 270 K, and $\phi_{w}$ = 10%. Figure
19: Comparison between the variation of the average liquid fraction for
different time intenseness, for the case of pure water, and NePCM $\phi_{w}$ =
10%, for melting from the top with $T_{h}$ = 300 K, $T_{c}$ = 270 K.  Figure
20:  Comparison between the variation of the velocity along a vertical line
that crosses the bottom, and top sides of the cavity at $\dfrac{x}{H}$ = 0.5,
for the case of for the case of melting from the bottom for $T_{h}$ = 300 K,
$T_{c}$ = 270 K, for pure water, and NePCM with $\phi_{w}$ = 10%.  Figure 21:
Comparison between the variation of the temperature along a vertical line that
crosses the bottom, and top sides of the cavity at $\dfrac{x}{H}$ = 0.5, for
the case of for the case of melting from the bottom for $T_{h}$ = 300 K,
$T_{c}$ = 270 K, for pure water and NePCM with $\phi_{w}$ = 10%.  Figure 22:
The variation of the mass fraction of the particles along a vertical line that
crosses the bottom, and top sides of the cavity at $\dfrac{x}{H}$ = 0.5, for
the case of for the case of melting from the bottom for $T_{h}$ = 300 K,
$T_{c}$ = 270 K, for NePCM with $\phi_{w}$ = 10%.  Figure 23: The variation of
the ratio between the thermal conductivity of the NePCM and that of pure water
along a vertical line that crosses the bottom, and top sides of the cavity at
$\dfrac{x}{H}$ = 0.5, for the case of for the case of melting from the bottom
for $T_{h}$ = 300 K, $T_{c}$ = 270 K, for NePCM with $\phi_{w}$ = 10%.  Figure
24: The variation of the ratio between the viscosity of the NePCM and that of
pure water along a vertical line that crosses the bottom, and top sides of the
cavity at $\dfrac{x}{H}$ = 0.5, for the case of for the case of melting from
the bottom for $T_{h}$ = 300 K, $T_{c}$ = 270 K, for NePCM with $\phi_{w}$ =
10%.  Figure 25: The variation of the ratio between the liquid phase melting
temperature of the NePCM and that of pure water along a vertical line that
crosses the bottom, and top sides of the cavity at $\dfrac{x}{H}$ = 0.5, for
the case of for the case of melting from the bottom for $T_{h}$ = 300 K,
$T_{c}$ = 270 K, for NePCM with $\phi_{w}$ = 10%.  Figure 26: Comparison
between the variation of the average liquid fraction for different time
intenseness, for the case of pure water, and NePCM $\phi_{w}$ = 10%, for
melting from the top with $T_{h}$ = 300 K, $T_{c}$ = 270 K with different
segregation coefficients.

(a) Pure Water

(b) NePCM $k_{o}$ = 1.0

(c) NePCM $k_{o}$ = 0.1

Figure 27: Development of the flow field (shown by the stream lines)
superimposed on the contour of the liquid fraction for $T_{h}$ =300 K, $T_{c}$
= 270 K, for case of pure water, and NePCM with $\phi_{w}$ = 10%, $t$ = 142
sec, and different segregation coefficients . Figure 28: The variation of the
ratio between the viscosity of the NePCM and that of pure water along a
vertical line that crosses the bottom, and top sides of the cavity at
$\dfrac{x}{H}$ = 0.5, for the cases of melting from the bottom for $T_{h}$ =
300 K, $T_{c}$ = 270 K, for NePCM with $\phi_{w}$ = 10%, and different
segregation coefficients.

(a) $\phi_{w}$ = 0.5%

(b) $\phi_{w}$ = 5%

(c) $\phi_{w}$ = 10%

Figure 29: Development of the flow field (shown by the stream lines)
superimposed on the contour of the liquid fraction for $T_{h}$ =300 K, $T_{c}$
= 270 K, for case NePCM with different volume fractions, $t$ = 142 sec, and
$k_{o}$ = 0.1.