10836


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 20, No 3, 2022, pp. 519 - 538 

https://doi.org/10.22190/FUME220603034G 

© 2022 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

CONVECTIVE FLOW AND HEAT TRANSFER OF  

NANO-ENCAPSULATED PHASE CHANGE MATERIAL (NEPCM) 

DISPERSIONS ALONG A VERTICAL SURFACE 

Mohammad Ghalambaz1,2, Haichuan Jin3, Amirhossein Bagheri4, 

Obai Younis 5,6, Dongsheng Wen3,7  

1Metamaterials for Mechanical, Biomechanical and Multiphysical Applications Research 

Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam 

2Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam 
3School of Aeronautic Science and Engineering, Beihang University, Beijing, P.R. China 

4Department of Mechanical Engineering, Shiraz University, Shiraz, Iran 
5Department of Mechanical Engineering, College of Engineering in Wadi Addwasir, Prince 

Sattam Bin Abdulaziz University, KSA 

6Department of Mechanical Engineering, Faculty of Engineering, University of Khartoum, 

Sudan 
7Lehrstuhl für Thermodynamik, Technische Universität München, Garching, Germany 

Abstract. Nano-encapsulated phase change suspension is a novel type of functional 

fluid in which the nanoparticles undergo phase change that contribute to heat transfer. 

Thus, the working fluid carries heat not only by sensible heat but also in the form of 

latent heat stored in the particles. The natural convection and heat transfer of Nano-

Encapsulated Phase Change Materials (NEPCMs) suspensions within a boundary layer 

along a heated flat surface are theoretically investigated in this work. The nanoparticles are 

core-shell structured with the core fabricated from PCMs covered by a solid shell. A 

similarity solution approach along with the finite element method is employed to 

address the phenomena. The outcomes indicate that a decisive factor in boosting the 

heat transfer is the temperature at which NEPCM particles undergo the phase transition. 

The heat transfer parameter can be enhanced by about 25% by just adding 5% of NEPCM 

particles, compared to the case with no NEPCM particles.  

Key words: Nano-encapsulated phase change materials; Phase change materials; 
Boundary layer heat transfer enhancement; similarity solution. 

 
Received: June 03, 2022 / Accepted July 30, 2022 

Corresponding author: Dognsheng Wen   

Technical University of Munich, Institute of Thermodynamics, Boltzmannstr. 15, 85747 Garching, Germany 
E-mail: d.wen@tum.de 



520 M. GHALAMBAZ, H. JIN, A. BAGHERI SARVESTSANI, O. YOUNIS, D. WEN 

 

1. INTRODUCTION 

Heat transfer is one of the important major demands of recent technological advancement 

in electronic packaging, aerospace and avionics systems, power engineering, and high-

power x-ray devices. These devices produce a tremendous amount of waste heat in confined 

space, requiring efficient cooling systems to extract the produced heat, and transfer it to the 

surrounding. The most challenging aspect of thermal control of surfaces with high heat-

flux-density is heat removal from the surface. Moreover, many of high-power devices are 

sensitive not only to the high temperatures but also to the temperature gradients across the 

surface of the device. The temperature gradient within a device would produce mechanical 

stress that may damage the sensitive structures of the device. Therefore, new working fluids 

or new techniques capable of enhancing heat transfer or inducing a uniform temperature 

distribution are of high interest. In this regard, phase change materials, nanofluids, and 

NEPCMs are introduced as promising passive ways to facilitate heat transfer and improve 

thermal management.  

A phase change material (PCM) benefits from a large latent heat thermal storage at a 

fusion temperature. This remarkable quality has drawn researchers' attention towards potential 

applications of PCMs in heat storage and heat sink technologies. Construction [1, 2], homes 

and offices air conditioning systems [3], space heating/cooling, and waste heat recovery in 

different industries are just a few areas that using PCM can exceptionally be beneficial. 

About two decades ago, nanofluid was introduced as a stable suspension of nanoparticles 

into a base fluid to enhance heat transfer [4]. Although nanoparticles can notably boost the 

heat transfer of a base fluid, they also have the drawback of reducing the working fluid's heat 

capacity, which is not good for many applications [5]. When the heat capacity of fluid is low, 

the temperature gradient along the cooling fluid increases. The Nano-Encapsulated PCM 

(NEPCM) are nanoparticles comprising a core and a shell. A phase change material such as 

paraffin, n-tetradecane, or octadecane is used in manufacturing the core, and the shell is 

usually made of a polymer [6, 7]. A stable suspension of NEPCM nanofluid can effectively 

remove a large amount of heat from a hot surface using the latent heat of NEPCM particles. 

The other important advantage of NEPCMs is the temperature control of a coolant at the 

fusion temperature. Indeed, NEPCMs tend to remain at their fusion temperature and go 

through a phase change.  

Due to important applications of PCMs and recent advancement in the improvement 

of these materials using nanotechnology, Sarkar et al. [8] and Sidik et al. [9] explored the 

thermal advantages of using nanofluids and phase change materials for energy storage 

applications. Sidik et al. [9] addressed the key factors that affect the enhancement of heat 

transfer of PCMs, including the shape and size of NEPCMs, and shell fraction of 

nanoparticles. Sarkar et al. [8] concluded employing encapsulated PCM in building 

components for cold storage applications could improve the energy saving. Shah [10] 

reviewed the nanotechnology aspect of PCMs by focusing on the synthesis technique and 

materials/morphology of PCMs and their influence on thermal-conductivity enhancement.  
The boundary layer theory for heat and fluid flow is an important and hot topic in 

applied mathematics and engineering applications. As a matter of fact, the formation of a 
boundary layer over a solid surface is a common phenomenon in a viscous flow. So, it 
has been the subject of various researches for decades. Considering heat transfer within 
the boundary layer of nanofluids, Manjunatha et al. [11], Sharma and Mishra [12], Roy et 
al. [13], Munir et al. [14], and Bilal et al. [15] addressed the boundary layer flow and heat 



 Convective Flow and Heat Transfer of Nano-Encapsulated Phase Change Material (NEPCM)... 521 

 

 

transfer flat surfaces. Yasin et al. [16] studied the stagnation point flow and boundary 
layer heat transfer over a stretching sheet. Rashed et al. [17] utilized a two-phase model to 
investigated the boundary layer heat transfer of nanofluids over a plate in motion. Madhu 
et al. [18] investigated the non-Newtonian effects of nanofluids over a stretching sheet. 
They noticed that the local Nusselt number varies inversely with the power-law index of 
the non-Newtonian fluid. 

Reddy et al. [19] studied the boundary layer heat transfer of nanofluids over an inclined 
vertical surface. They reported that the nanoscale effects of Brownian motion and 
thermophoresis tend to increase the plate's boundary layer thickness. Reddy and Chamkha [20] 
analyzed the natural convection boundary layer heat transfer of nanofluids over a cone in the 
presence of chemical reactions. The outcomes reveal that the presence of a chemical reaction 
decreases the amount of the temperature gradient of nanofluid at the surface. However, the 
boundary layer heat transfer of a suspension of NEPCMs has not been explored yet.  

As mentioned, the suspension of a NEPCMs shows promising properties as a heat transfer 
working fluid. In this regard, several studies addressed the effect of nano/microencapsulation 
of PCMs in micro channel heat transfer applications. Seyf et al. [21] numerically studied the 
flow field and thermal performance of a slurry of water-NEPCM over an isothermal unconfined 
square-cylinder in a channel. The core of the NEPCM was produced from n-octadecane, and 
the average size of capsules was 100 nm. In another research, Seyf et al. [22] investigated the 
thermal responses of octadecane NEPCM suspended in polyalphaolefin (PAO) in a microtube 
heat sink. The results reveal that using NEPCMs considerably enhances the heat transfer rate, 
while it tremendously increases the pressure drop. They also reported that when the mass-
concentration of the nanoparticles increases, the heat transfers more evenly, and it accelerates 
temperature uniformity. In another numerical investigation conducted by Rehman et al. [23] the 
thermal enhancement and hydrodynamics characteristics of a confined slot jet impingement 
for a mixture of NEPCM as a coolant were examined. The water-base coolant was composed 
100 nm n-octadecane NEPCM particles. They found that dispersing NEPCM particles in the 
base fluid resulted in notable enhancement of the thermal performance. However, the presence 
of NEPCMs also increased the viscosity of the slurry, consequently increasing the pressure 
drop. In a recent study, Edalatifar et al. [24] employed an artificial intelligence approach to 
estimate the free convection heat transfer of NEPCMs in an enclosure.   

Ho et al. [25] conducted an experiment to inspect the impact of using encapsulated 
PCMs on the heat transfer enhancement of micro channels. Their report shows up to 52% 
improvement in heat transfer in some cases. However, in some cases, the possibility of a 
reduction in heat transfer due to utilizing encapsulated particles is detected. Wang et al. 
[26] analyzed the thermal performance of encapsulated PCM particles flowing through 
microchannels. They suggested that using 2% of encapsulated nanoparticles can boost the 
rate of heat transfer up to 1.36 times higher than that of distilled water when the particles 
underwent the phase transition. Ho et al. [27, 28] synthesized samples of nanofluids 
(water-Al2O3 nanofluids) and samples of a slurry of encapsulated PCMs. They examined 
the thermal performance of the samples in a tube [27] and a mini channel heat sink [28]. 
The results reveal that the performance of each type of nanofluid or slurry depends on 
variables such as the flow rate, heat transfer, and the downstream or upstream position.  

Although there are noteworthy studies addressing the flow and thermal behavior of 

NEPCMs in channels, there is no study so far, which explores the motion and thermal 

behavior of NEPCMs in boundary layer flows. The main goal of the present investigation 

is to theoretically study the free convection boundary layer flow and heat transfer of 

NEPCMs over a hot flat plate for the first time.  



522 M. GHALAMBAZ, H. JIN, A. BAGHERI SARVESTSANI, O. YOUNIS, D. WEN 

 

2. MATHEMATICAL MODEL 

Consider a stabilized dilute mixture of NEPMC particles suspended in a pure liquid, 
which flows over a flat plate forming a boundary layer. These particles are capsules of 
PCMs comprising a shell layer and a PCM core. The plate is placed in a quiescent 
suspension of a base fluid and NEPCMs. The plate is at the constant temperature of Tw, 
and the surrounding fluid is at the cold temperature of T∞. Fig.1 illustrates the details of 
the coordinate system and the physical model. The gravity acts in a downward direction. 
The thermal buoyancy force gives rise to natural convection, which flows across the plate 
in the upward direction. The transient temperature of NEPCM particles (Tf) is lower than 
the temperature of the plate, even though it remains above T∞ the cold ambient temperature. 
The NEPCMs move with the natural convection flow of the base fluid. When NEPCM 
particles reach hot regions (T>Tf), a part of the thermal energy is absorbed as the latent heat 
through the phase transition of the nanoparticles from solid to liquid. Accordingly, when they 
reach cold places (T<Tf), they cool down and undergo solidification by releasing their energy.  

Due to the significance of heat of transformation, a considerable advance in the convection 
can be expected by using a suspension of NEPCM nanoparticles. However, the presence of 
NEPCM particles changes other thermophysical properties of the suspension, including the 
density, the thermal conductivity, the dynamic viscosity, and the thermal volume expansion of 
the suspension. Consequently, these thermophysical properties can alter the convection of the 
suspension. Thus, modeling and systematic inspection of convective heat transfer of NEPCM 
suspensions are essential to evaluate the thermal behavior of NEPCMs. 

 

Fig. 1 Schematic view of the physical model and the coordinate system 

When it comes to modeling the boundary layer and heat transfer of NEPCMS, considering 
some presumptions and simplifications is inevitable. In the present study, the deviation of the 
wall's temperature from ambient is reckoned limited. Hence, aside from density which is 
modeled using the Boussinesq model and prompts the buoyancy forces, the temperature does 
not affect the other thermophysical properties. The temperature of the nanoparticles and the 
fluid around them is the same due to very tiny size of nanoparticle. The volume fraction of 



 Convective Flow and Heat Transfer of Nano-Encapsulated Phase Change Material (NEPCM)... 523 

 

 

NEPCM particles is considerably small (ϕ), and hence, the suspension of NEPCMs 
nanoparticles is a dilute suspension, obeying the Newtonian fluid model. Finally, the particles 
are considered uniformly dispersed within the base fluid, and the suspension remains 
homogeneous and stable. 

By employing the buoyancy effects, the governing equations for the NEPCM's suspension 

within a base fluid can be introduced as [23, 29]: 

 0,   
u v

x y

 
+ =

 
 (1) 

  
2 2

b b b b b2 2

u u P u u
u v g T T

x y x x y
    



       
+ = − + + + −    

         
 (2a) 

 
2 2

b b b 2 2

v v P v v
u v

x y y x y
  

       
+ = − + +    

         
 (2b)

 

 ( ) ( )
2 2

b 2 2b b

T T T T
Cp u Cp v k

x y x y
 

      
+ = +    

        
 (3) 

where u and v are the velocity components in the x and y directions, respectively. T represents 

the temperature and P the pressure. In the aforementioned equations, ρ and μ are presenting 

the density and the dynamic viscosity, respectively. The specific heat capacity (Cp), and the 

thermal conductivity (k) are also thermal characteristics of the suspension. The gravitational 

constant is denoted by g, and the coefficient of thermal volume expansion is represented by β. 

The subscript of b indicates the bulk properties attributed to a suspension of NEPCM 

fluid. By employing the usual boundary layer approximations, the control equations (1)-(3) 

are simplified as: 

 0,   
u v

x y

  
+ = 

    

(4)

 

 

 
2

b b b b b2

u u P u
u v g T T

x y x y
    



     
+ = − + + −  

        

(5a) 

 

0
P

y


=


 

(5b) 

 

( ) ( )
2

b 2b b

T T T
Cp u Cp v k

x y y
 

    
+ =  

     
 (6) 

Following the representation of the physical model, the appropriate boundary conditions 

are introduced as: 

 w
0 : 0,     y u v T T= = = =

 
(7a)

 

 
: , 0,  .y u v T T


→  → →

 
(7b)

 



524 M. GHALAMBAZ, H. JIN, A. BAGHERI SARVESTSANI, O. YOUNIS, D. WEN 

 

3. BULK PROPERTIES OF THE SUSPENSION 

The thermophysical properties in the governing equations of Eqs. (4-6) with the subscript 

of b represent the suspension (a mixture of NEPCMs and the pure liquid) thermophysical 

properties. Thus, some thermophysical models are required to evaluate the thermophysical 

features of the mixture. Following [30], it is possible to obtain the density of NEPCMs' 

mixture and the base fluid using: 

 
( )b f p1   = − +  

(8) 

Here, ρ indicates the density, and ϕ represents the nanoparticles concentration. The 

subscripts of f and p represent the pure liquid and the nanoparticles, respectively.  

The NEPCM nanoparticles are consist of a shell and a core PCM [30, 31]. Therefore, 

the density of NEPCM composed of a shell and a core PCM can be evaluated as [30, 32]: 

 

( ) c s
p

s c

1   


 

+
=

+
 

(9) 

where c and s as subscripts indicate the core and the shell, respectively, and ι signifies the 

weight ratio of the core-shell. In Eq. (9), the density of the core PCM is considered as the 

mean of its solid and liquid densities. The subscript p also denotes the nanoparticles.  

As discussed by Khanafer and Vafai [5], the assumption of thermal equilibrium between 

the particles and the pure liquid makes it possible to drive the effective heat capacity of both a 

suspension of nanoparticles and the pure liquid from the energy equations as follow: 

 

( ) f f p p
b

b

1 Cp Cp
Cp

  



− +
=

 

(10) 

However, many of the previous researchers have preferred to use a more simplified 

relation as [30, 32]: 

 
( )b m f m p1Cp C Cp C Cp= − +  

(11) 

where Cpp is the specific heat of encapsulated particles. Cm represents the mass fraction 

of NEPCM particles, which is related to the particles concentration by ϕ = Cm×ρb/ρp [32]. 

Here, we use the definition of Eq. (8) for the evaluation of the bulk specific heat capacity 

of the suspension. When there is no phase transition, Cpp is calculated using the following 

relation [32]:  

 

( )
( )

c,l s c s

p

s c p

Cp Cp
Cp

  

  

+
=

+
 

(12) 

where the core PCM's specific heat capacity (Cpc,l) can be taken through the mean of its 

solid and liquid specific heat capacities. Providing that the phase transition is taken into 

account in evaluating the heat transfer and the latent heat of the confined particle is at 

hand, it is possible to employ an approximate profile with an integral value equal to the 

latent heat of phase change. For example, a rectangular, a triangle, or a sinusoidal profile 

is feasible for modeling the temperature dependent specific heat of encapsulated particles 

[22, 30, 33, 34]: 



 Convective Flow and Heat Transfer of Nano-Encapsulated Phase Change Material (NEPCM)... 525 

 

 

 

sf
c c,l

Mr

h
Cp Cp

T
= +

 

(13a) 

 

1
, ,

sin
2

sf

c c l c l

Mr Mr

h T T
Cp Cp Cp

T T




    −
= + −     

     

(13b) 

 

( )
,

, 1 2
2

fs c l

c c l

MrMr

h Cp
Cp Cp T T

TT

 
= + − − 

   

(13c) 

where TMr is the phase-change temperature range (TMr = T2 - T1). Indeed, the phase 

change takes place in the range of T1 to T2 instead of the exact fusion temperature of Tf. 

Taking the fusion temperature as the middle of the transition temperature range, T1 and T2 

can be introduced as T1 = Tf - TMr/2 and T2 = Tf + TMr/2. In the case of temperatures out of 

the melting range, the magnitude of the specific heat of the nanoparticle is matching Cpc, 

and regarding temperatures within the range, the specific heat of the particle is estimated 

to solve the aforementioned equation. Here, following the literature studies [22, 30, 33, 

34], it is presumed that the specific heat of the molten and the solid PCM is similar. As 

stated by Alisetti and Roy [34], the divergence between applying different profiles for 

evaluation of the specific heat of PCM in a circular duct is under 4%. Thus, in the present 

study, the simple rectangle profile of Eq. (11a) is employed to be a representative of the 

latent heat of NEPCM as a function of temperature. As a result, by considering the phase 

change latent heat capacity and the sensible heat capacity, the total heat capacity of the 

suspension of NEPCM capsule can be evaluated as: 

 

( )

( )
c s c s

p

s c p

Cp Cp
Cp

  

  

+
=

+
 

(14a) 

where

 

1

sf
c c,l 1 2

Mr

2

0

0

T T

h
Cp Cp T T T

T

T T

 



= +  

   

(14b) 

The coefficient of the thermal volume expansion, βp, is evaluated through the upcoming 

equation, which was discussed by Khanafer and Vafai [5] for a base fluid containing 

nanoparticles: 

 
( )b f p1   = − +  

(15) 

Now, closure models for the estimation of the dynamic viscosity and the thermal 

conductivity of the suspension are required. Many of the previous studies have evaluated the 

thermal conductivity of Micro/Nano-EPCM as two parts of static and dynamic thermal 

conductivity. In most of the literature-works, the mixture's static thermal conductivity was 

estimated by using the Maxwell model [5, 35], and then the dynamic thermal conductivity of 

the mixture was modeled using the fluid shear rates [22, 29, 30, 36]. However, Buongiorno 

[37], using a scale analysis approach, demonstrated that the shear rate effects are not of 

importance in the nanoscales. Hence, it can be concluded that the thermal conductivity 



526 M. GHALAMBAZ, H. JIN, A. BAGHERI SARVESTSANI, O. YOUNIS, D. WEN 

 

relations for the microencapsulated PCMs are not applicable to the case of nano-encapsulated 

suspensions. In most of the literature studies, the dynamic viscosity of the suspension of 

Micro-Encapsulated PCMs (MEPCMs) was approximated using the Vand [38] model as 

[22, 30]: 

 

( )
2.5

2b

f

1 A


 


−

= − −

 

(16) 

In Eq. (16), the parameter of A depends on the size of particles and can be evaluated 

using a curve-fitting on the experimental data. For example, the value of A has been 

reported as A=1.16 for 0.13mm diameter glass spheres [38], A=3.4 (10–30 μm diameter 

MEPCM particles) [39], A=3.7 (6.3μm average diameter) [40], and A=4.45 (about 10 μm 

diameter MEPCM particles) [41]. It should be noted that the model of Vand [35] has been 

introduced for the MEPCM particles and not for Nano-EPCM particles. 

Regarding nanoparticles, Buongiorno [42] and Venerus et al. [43] addressed the effects 

that the presence of nanoparticles can lay upon the dynamic viscosity and the thermal 

conductivity of dilute suspensions of nanofluids in two comprehensive benchmark studies. 

More than thirty authors around the world have studied the aforementioned properties of 

nanofluids in different experimental setups and through various types of measurement 

devices. The outcomes demonstrate the possibility of expressing the dynamic viscosity and 

the thermal conductivity of fluids containing nanometer-sized particles by linear relations. 

Following this outcome, Ghalambaz et al. [44] utilized the upcoming linear equations to 

study the heat and mass transfer of nanofluids: 

 

 b
f

1
k

Nc
k


 

= + 
   

(17a) 

 

 1b
f

Nv





 
= + 

 
   

(17b) 

Here Nc is the thermal-conductivity number, and Nv indicates the dynamic-viscosity 
number. The magnitudes of Nc and Nv depend on several parameters, including the shape 
of the particles, the nanoparticles material, the pure liquid, and the operating temperature 
[44]. It is obvious that the higher the value of Nc or Nv, the greater the amplification of 
the thermal conductivity or the dynamic viscosity when the flow contains nanoparticles.  

Nc and Nv are attainable by applying linear curve fitting over gathered experimental data 
for the suspensions thermophysical properties. A list of values of Nc and Nv is evaluated and 
reported in [44]. It is noteworthy to mention that these relations are valid for a diluted 
suspension of nanoparticles, and more advance relations or actual experimental data shall be 
used for high volume fractions of nanoparticles. Here, Eqs. (17a) and (17b) are adopted to 
determine the thermal conductivity of the suspension and the dynamic viscosity within the 
flow field. 

4. SIMILARITY SOLUTION 

For the purpose of transforming the equations governing the boundary layer flow field 
along with the attributed boundary conditions into a selection of ordinary differential 
equations, some similarity variables are required to be proposed. First, in order to satisfy 



 Convective Flow and Heat Transfer of Nano-Encapsulated Phase Change Material (NEPCM)... 527 

 

 

the continuity equation, the stream function is introduced into it as u=∂ψ/∂y and v=-
∂ψ/∂x. The pressure between equations of (2a) and (2b), is excluded between momentum 
equations in x-direction and y-direction by taking the cross. For obtaining a similarity 
solution through which the equations can be rearranged as a scaled expression, the 
similarity variables are proposed as: 

 

1

4η Ra
x

y

x
=

 
(18a) 

  

(18b) 

where η is the similarity variable, S and θ are the scaled flow and temperature. Here, the 
local Rayleigh number (Rax) is introduced as: 

 

( ) 3f f

f f

Ra
x

g T T x 

 


−

=

 

(19) 

By positioning the similarity variables of Eq. (18) and using the thermal conductivity 
and dynamic viscosity relations of Eq. (17), the partial differential equations of Eqs. (5) 
and (6) along with the boundary conditions of Eq. (7) are explained into a series of non-
linear ordinary differential equations as:  

 

( ) ( ) bf f
b b f

1
2 3 1

4 Pr
S S S S Nv S

 
 

  
   − = + +

 

(20) 

 

( )
( )

( )
b

f

3
1 0

4

c
Nc S

c


  


 + + =

 

(21) 

where Pr=μf /(ρf αf). The density ratio of ρb/ρf can be simply evaluated using Eq. (7) as  
(1-ϕ)+ϕρp/ρf. The thermal expansion ratio of βb/βf can be rewritten as ϕβp/βf +(1-ϕ) by 
using Eq. (15). The other important thermophysical property would be the heat capacity 
ratio (ρCp)b/(ρCp)f which is evaluated using Eqs. (8), (10) and (14) as follow: 

 

( )

( )
( )b

f

1
1

Cp
f

Cp Ste


  

 
= − + +

 

(22) 

where θf represents the phase-change temperature, δ is the parameter of fusion range, 
which should be a very small number. When δ approaches zero, the phase change temperature 
approaches the scaled phase change temperature. λ is the scaled heat capacity introduced 
as the proportion of the sensible heat capacity of NEPCM nanoparticles to the sensible thermal 
capacity of the base fluid, where Ste is the Stefan number. These scaled parameters and 
numbers are introduced as: 

    
( )

f
f

w

T T

T T
 



−
=

− ( )
Mr

w

T

T T




=
−

,
( )
( ) ( )

c,l s c s

s cf

Cp Cp

Cp

  


  

+
=

+
, 

 

( ) ( )( )w s cf
sf c s

Cp T T
Ste

h

  

 


− +

=  (23) 

1

4

,  
w

x

T T
S

T T
Ra










−
= =

−



528 M. GHALAMBAZ, H. JIN, A. BAGHERI SARVESTSANI, O. YOUNIS, D. WEN 

 

and

 
f f

2
othe 0r

2
wi1 sef if f

 
  
 

= −   + = 
   

(24) 

By substituting the final scaled form of the thermo-physical ratio parameters, the 

governing equations of the flow and heat transfer are rewritten as: 

 

( ) ( ) ( ) ( )
p p

f f

1
1 2 3 1 1

4 Pr
S S S S Nv S

 
     

 

   
   − + − = + + − +   

     

(25) 

 
( ) ( )

3 1
1 1 0

4
Nc f S

Ste
     



 
 + + − + + = 

   

(26) 

In the above equations, it should be noticed that f is not a constant, and it a function of 

scaled temperature as defined by Eq. (24). The transformed form of the boundary conditions 

of Eq. (7) are: 

 
( ) ( ) ( )0 : 0 0, 0 0, 0 1,at S S = = = =

 
(27a) 

 
( ) ( ): 0, 0.as S   → → →

 
(27b) 

The important characteristics of heat transfer of the NEPCMs suspension are the heat 

transfer at the surface. The balance in the heat transfer at the surface of the plate is given 

using the convective heat transfer coefficient (h) as: 

 

( )b w
T

k h T T
y




− = −


 

(28) 

The local Nusselt number is achieved through using the above equation and implementing 

the similarity variables brought in Eq. (18), and equals to:  

 

( )
1

b 4

f

0 R
x x

k
Nu a

k
= −

 

(29) 

where, using Eq. (17a), the local Nusselt number can be rewritten as: 

 
( ) ( )

1

4Nu Ra 1 0
x x

Nc 
−

= − +
 

(30) 

It is worth noticing that the term NuxRax
-1/4 is known as the reduced Nusselt number (Nur).  

5. NUMERICAL METHOD AND CODE VALIDATION 

Eqs. (25) and (26) are a set coupled, high order, and non-linear ordinary differential 

equations. It is very hard to obtain an analytical solution for non-linear high-order 

differential equations. Hence, a numerical approach is required to integrate the equations. 

Here, bvp4c finite element code with automatic grid adaptation and error control is 

employed for the purpose of finding a solution for the governing equations along with the 

boundary conditions. In order to implement the ODE equations in bvp4c code, they need 

to be written in the form of a set of first-order coupled equations as follow:  



 Convective Flow and Heat Transfer of Nano-Encapsulated Phase Change Material (NEPCM)... 529 

 

 

 

( ) ( ) ( )

( )

( )

( )

p p2

2 1 3 4

f f

1 2 2 3 3

1 5

4 5 5

1
1 2 3 1

4 Pr
, , ,

1

1
1

3
,

4 1

y y y y

y y y y y
Nv

f y y
Ste

y y y
Nc

 
   

 



  




   
− + − − − +   

   
  = = =

+

 
− + + 

 
 = = −

+
 

(31) 

 
f 5 f

1 otherwise 0
2 2

f if y f
 

 
 

= −   + = 
   

(32) 

Accordingly, the corresponding boundary conditions using the new notation are: 

 
( ) ( ) ( )1 2 4η 0: 0 0, 0 0, 0 1,y y y= = = =  

(33a) 

 
( ) ( )2 4η : η 0, η 0.y y→ → →  

(33b) 

Two of the boundary conditions are approaching infinity. Therefore, the governing 

equations have to be solved in a large domain to reach an asymptotic solution. In the 

present study, the value of η∞ is increased starting from η∞=10 to higher values, and the 

reduced Nusselt number is monitored to ensure that the asymptotic solution is reached 

and it is independent of the selected value of η∞. The absolute error was set as 10-8, and 

the absolute relative error was set as 10-6. More details about the utilized code can be 

found in [45]. 

The reliability of the present code is evaluated by comparing the obtained results with 

the literature. In Fig. 2 the temperature profiles brought by Bejan [46] are compared with 

those reported in the present research.  

 

Fig. 2 A comparison between the current data and those of Bejan [46] 



530 M. GHALAMBAZ, H. JIN, A. BAGHERI SARVESTSANI, O. YOUNIS, D. WEN 

 

Table 1 shows a comparison between the outcomes of the present study and the results 

that can be found in the literature for the reduced Nusselt number (Nur) when ϕ=0.  

Table 1 Comparison Nur calculated in the present investigation and the results available 

in literature when ϕ=0 

Pr Present study Bejan [46] 

0.72 0.3874 0.387 

1 0.4010 0.401 

2 0.4260 0.426 

10 0.4650 0.465 

100 0.4900 0.490 

6. RESULTS AND DISCUSSION 

Here, for estimating the range of the scaled parameters, n-octadecane [47] for the core 

PCM, poly (methyl methacrylate) PMMA for the shell [48], and water for the base fluid 

[47] are adopted. The core-shell weight proportion is selected as ι=0.7. In this case, λ and 

Ste (Stefan number) can be approximated as 0.4 and 0.313, respectively. The volume 

fraction of the NEPCM nanoparticles within the water is adopted to be fixed as ϕ=5%. 

The fusion temperature is picked in the range of 0<θf<1 where the scaled ambient 

temperature and the scaled hot wall temperature equal to zero and unity, respectively. To 

illustrate the results, the scaled parameters are adopted in the following range: 0<Nc<6 

and 0<Nv<6 for a mixture where nanoparticles are distributed throughout a base fluid. 

Since the PCM core is less dense than water [49, 50], the density ratio is implemented as 

ρp/ρf=0.9, and the buoyancy ratio of ϕ×βp/βf is assumed negligible because of the 

inconsiderable volume fraction of nanoparticles (0.05) and very low thermal expansion of 

solid particles compared to a liquid. The Prandtl number (Pr) is fixed 6.2, which is about 

the Prandtl number of water. For a fixed value of Pr=6.2, the asymptotic value of η∞ is 

about η∞=20. As a summary, the preset values of the scaled parameters would be assigned 

as Pr=6.2, θf=0.2, ι=0.7, λ=0.4, Ste = 0.313, Nc=Nv=3.0, (ρp/ρf)=0.9, and (ϕ×βp/βf)=0. In 

this study, the results are presented as the preliminary values of the scaled parameters, 

and in other circumstances, the value of the parameter will be stated.  

For the first part of the numerical calculations, it is examined how the average Nusselt 

number may vary by the changes in the value of δ. The Delta parameter (δ) is the scaled 

fusion range, and it should be a small parameter. The calculations for various values of δ 

were carried out. The outcomes indicate that the results are independent of this parameter 

for δ<0.025. It is important to keep in mind that very small values of δ lead to a very low 

fusion temperature range. Since the latent heat of phase change varies according to a 

temperature within the fusion process, the quality of the utilized grid is of utmost 

importance for capturing the phase change within acceptable accuracy. The utilized bvp4c 

code in the present study enjoys the advantage of automatic error control and automatic grid 

adaptation. Although the automatic error control and automatic grid adaptation can 

significantly reduce the computational cost of the calculations to deal with a small value of δ, 

using a small value of δ results in a dramatical increase of computational costs. Hence, for the 

sake a balance between the accuracy and computational cost, the appropriate value of 

δ=0.025 is adopted for the calculations. 



 Convective Flow and Heat Transfer of Nano-Encapsulated Phase Change Material (NEPCM)... 531 

 

 

Since all of the presented results are derived from scaled governing equations, the 

scaled suffix is left out for brevity. Figures 3 and 4 depict the effect of the fusion 

temperature (θf) on the velocity and temperature profiles. The computational domain of 

solution is about η∞=20; however, here, the results are plotted in a smaller domain for the 

sake of graphical clarity. As mentioned, the velocity profiles equal to zero at the surface 

and tend to zero at the infinity, which satisfies the hydrodynamic boundary conditions. 

There is a maximum velocity peak next to the heated surface. Any rise in the phase-

change temperature elevates the maximum value of the maximum-velocity peak. It also 

shifts the maximum-velocity peak to the right. Attention to the temperature profiles of 

Fig. 4 illustrates that when the fusion temperature rises, the temperature gradient 

dwindles. Moreover, in the case of θf=0.1, a sharp change in the path of the temperature 

profile can be observed at the location of about η=2.4. The similar trends of the path 

changes but smoother can be seen for other fusion temperature profiles. The surge of the 

fusion temperature shifts the mentioned path changes towards the surface. Indeed, the 

location of this path change is the place that the change phase of the NEPCM particles 

occurs. In the case of θf=0.1, the path change occurs at η=2.4 corresponding to the 

temperature of θ=0.1. Indeed, when the fusion temperature elevates, the magnitude of the 

temperature profile extends. Consequently, the increase in temperature corresponds to the 

increase of the buoyancy forces. Hence, as it was observed in Fig. 3, the elevation of the 

fusion temperature promotes the fluid velocities and the maximum-velocity peaks. The 

reason behind the shift of the maximum-velocity peaks toward the surface (by the rise in 

the fusion temperature) is that the velocity profiles and the buoyancy forces tend to 

follow the trend of the change path in the temperature profiles due to the growth of the 

transient temperature. As mentioned, a surge of the fusion temperature shifts the temperature 

profiles toward the hot plate. 

 

Fig. 3 The velocity profiles for various values of the scaled fusion temperature (θf) when 

Nc=Nv=3.0 



532 M. GHALAMBAZ, H. JIN, A. BAGHERI SARVESTSANI, O. YOUNIS, D. WEN 

 

 

Fig. 4 The temperature profiles for various values of the scaled fusion temperature (θf) 

when Nc=Nv=3.0 

Figures 5 and 6 show the effect of fusion temperature (θf) and the thermal conductivity 

number (Nc) and the dynamic viscosity number (Nv) on the magnitude of the temperature 

gradient and the reduced Nusselt number at the wall surface. For each set of Nc and Nv 

numbers, Fig. 5 shows that there is a maximum-peak for the temperature gradient about 

θf=0.2. Starting from a low value of θf ~ 0.025, the temperature gradient and Nusselt number 

raise by the increase of θf until about θf=0.2, which corresponds to a peak for the maximum 

temperature gradient. After that, whenever the fusion temperature (θf) ascends, the Nusselt 

number and the temperature gradient follow a descending trend. A smooth local-minimum can 

be seen about θf = 0.975. In fact, the growth of the fusion temperature physically means that 

the phase transition at the core of nanoparticles will occur in a temperature closer to the wall 

temperature. When the fusion temperature and the wall temperature are close to each other, 

the NEPCM particle undergoes phase change just next to the wall, and hence, it would keep 

the temperature of the liquid at a high level next to the wall. However, it should be heeded that 

the phase transition occurs, a substantial amount of energy would be absorbed because of the 

latent heat of the phase change, which results in the increase of the heat transfer rate. 

Consequently, when the phase changes next to the wall, the wall temperature gradient 

decreases, which eventually drops the magnitude of heat transfer enhancement. Although 

this transition next to the wall declines the wall temperature gradient, still the other 

thermophysical properties, including the enhancement of conduction, can result in an 

overall improvement of the heat transfer in the presence of NEPCM particles. Hence, as 

seen in Fig. 6, generally, the heat transfer rate is higher within the NEPCM suspension 

than the pure fluid. The Nusselt number of a pure base fluid with no NEPCM particle is 

depicted as a gray narrow dashed line in Fig. 6. 



 Convective Flow and Heat Transfer of Nano-Encapsulated Phase Change Material (NEPCM)... 533 

 

 

 

Fig. 5 The surface temperature gradient vs the scaled fusion temperature (θf) and for 

different Nc and Nv 

 

Fig. 6 The reduced Nusselt vs the scaled fusion temperature (θf) and for different Nc and Nv 

At very low fusion temperatures, the phase of NEPCM particles changes at the boundary 

layer's edge, which is quite far from the hot surface. A small increase in the fusion 

temperature shifts the phase transition of the nanoparticles inside the boundary layer, which 

leads to an increase in the heat capacity of the cold fluid at a cool temperature. As it was 

observed in Fig. 4, as an example for θf=0.1, the phase transition occurs at η=2.4. It was 

also observed that any rise in the fusion temperature (θf) would shift the phase change of 

NEPCM particles towards the wall. The phase change tends to uniform the temperature 

profiles about the fusion temperature. So, when it occurs with a sufficient distance from the 



534 M. GHALAMBAZ, H. JIN, A. BAGHERI SARVESTSANI, O. YOUNIS, D. WEN 

 

wall, indeed it decreases the actual the thermal boundary layer's thickness, and the hot wall 

sees a uniform cold region in nearby. This is where the higher fusion temperature improves 

the reduced Nusselt number.  

Attention to the case of Nc=Nv=0  in Figs. 5 and 6 show that the temperature gradient 

and Nusselt number are higher than the pure fluid (the gray line) in most cases. The case 

of Nc=Nv=0 indeed neglects the major effect of the increment of thermal conductivity 

and dynamic viscosity of the NEPCM suspension because of existing particles. In this 

case, only the thermal capacity ratio, the density ratio, and the buoyancy ratio parameters 

remain effective. In these figures, it is obvious that the heat transfer of NEPCM suspension is 

higher than the pure fluid for the case of θf<0.8. However, for the fusion temperature 

higher than 0.8, the heat transfer would be less than that of the pure fluid since the phase 

transition affects the surface temperature gradient. 

The change of Nc and Nv only shifts the observed trend of behavior for temperature 

gradient and Nusselt number. The growth of the thermal conductivity (Nc) number declines 

the magnitude of the temperature gradient but boosts the Nusselt number. Indeed, when Nc 

increases, the thermal conductivity of the suspension surges in the presence of NEPCM 

particles. The increase of conduction tends to uniform the temperatures next to the wall, 

which causes a drop of the temperature gradient. However, the heat transfer can be 

calculated as the thermal conductivity multiple by the temperature gradient. Hence, 

although the magnitude of the temperature gradient of the surface drops, a better thermal 

conductivity would boost the overall heat transfer rate.  

Following any rise in the dynamic viscosity (Nv) number, the suspension's viscosity in 

the presence of the particles grows, and the rate of heat the flow accordingly plummets. 

The increase of viscosity would diminish the fluid velocity, and as a consequence, the 

overall rate of heat transfer. Based on the results of Fig. 6, improvement in the heat 

transfer is expected in most cases of regular suspensions of NEPCM particles. Fig. 7 

depicts the reduced Nusselt number as a function of Stefan number. As seen, the increase 

of Stefan number reduces the thermal performance of NEPCMs. It should be pointed out 

that the phase transition's latent heat is reversely related to Stefan number. This means 

that a large value of Stefan number represents a very low latent heat of phase change, and 

in contrast, a small value of Stefan number indicates a large amount of latent heat of 

phase change. Fig. 7 reveals that NEPCM particles with low Stefan number are very effective 

on the increase of heat transfer. It also confirms that not only the Stefan number of NEPCM 

particles but also the fusion temperature of NEPCM particles are important. At fusion 

temperatures close to the hot wall temperature, the presence of NEPCM particles only shows a 

minor enhancement in heat transfer rate regardless of the Stefan number.  

Table 2 lists a brief summary of the impact of the scaled parameters on the amount of 

the temperature and the velocity gradients at the wall. Adopting the first row of the table 

as a reference and comparing the results of the other rows with this row, Table 2 shows 

that the magnitude of the wall temperature gradient (-θ'(0)) is a decreasing function of λ, 

θf, and Ste. The effect of the variation of the density ratio (ρp/ρf) is negligible. The surface 

velocity gradient is a degreasing function of λ and (ρp/ρf) and an increasing function of θf 

and Ste.  



 Convective Flow and Heat Transfer of Nano-Encapsulated Phase Change Material (NEPCM)... 535 

 

 

 

Fig. 7 The reduced Nusselt number vs Stefan number for various scaled fusion temperatures 

(θf) 

Table 2 The magnitude of temperature gradient (-θ'(0)) and the velocity gradient (S''(0)) for 

various value of scaled parameters when ϕ=0.05, (ϕ×βp/βf)~0, and Nv=Nc=3.0 

Ste λ θf (ρp/ρf) -θ'(0) S''(0) 

0.3 0.4 0.2 0.9 0.4469 0.8823 

0.3 0.4 0.2 0.8 0.4469 0.8825 

0.3 0.3 0.2 0.9 0.4463 0.8830 

0.3 0.4 0.3 0.9 0.4456 0.8931 

0.3 0.4 0.1 0.9 0.4464 0.8696 

0.2 0.4 0.2 0.9 0.4606 0.8640 

7. CONCLUSIONS 

The flow and heat transfer behavior of NEPCM particles dispersed in a pure fluid 

along  a heated vertical plate was studied. The boundary layer governing equations were 

introduced and rearranged as a group of non-linear ordinary differential equations and 

solved numerically. The influence of some important scaled parameters, including Prandtl 

number (Pr), Stefan number (Ste), dynamic viscosity number (Nv), thermal conductivity 

number (Nc), and fusion temperature (θf), as well as density ratio of the NEPCM (ρp/ρf), 

were investigated, as summarized below: 

▪ Generally, the presence of NEPCM nanoparticles enhances the natural convection.  
▪ The decrease in the Stefan number enhances the heat transfer rate.  
▪ At the optimum scaled fusion temperature, θf ~0.2, the heat transfer rate reaches its 

maximum value. At θf = 0.2, the reduced Nusselt number is increased from 

Nur=0.445 for the case of the base fluid to 0.56 at 5% Vol. of NEPCM particles, 

showing 25% enhancement in heat transfer. 



536 M. GHALAMBAZ, H. JIN, A. BAGHERI SARVESTSANI, O. YOUNIS, D. WEN 

 

▪ As the fusion temperature elevates, the velocities of the NEPCM within the 
boundary layer rises as well, which, however, does not always lead to the 

enhancement of heat transfer. For instance, the velocity corresponding to the case 

of θf=0.2 is lower than that of θf=0.5, but corresponding to a large heat transfer 

enhancement. This indicates the significance of how the latent heat of NEPCM 

particles can affect the heat transfer behavior of suspension.  

It shall be noted that one assumption made in this work is that the suspension of 

NEPCM is diluted (i.e., ϕ<5%). Hence, the interactions among the NEPCM particles and 

the non-Newtonian effects were neglected. It was also assumed that NEPCM particles 

remain homogeneous in the fluid. However, in practice, some slip mechanisms such as 

the Brownian motion and thermophoresis effects could change the local concentration of 

NEPCM particles, and a suspension with a high volume-fraction of NEPCM particles 

may behave like slurries. The slip mechanisms of NEPCM particles and non-Newtonian 

effects should be the subject of future studies.  

Acknowledgments: This project is supported by China 111 project (B18002). 

REFERENCES 

1. Fokaides, P.A., Kylili, A., Kalogirou, S.A., 2015, Phase change materials (PCMs) integrated into 
transparent building elements: a review, Materials for renewable and sustainable energy, 4(2), 6. 

2. Huang, X., Alva, G., Jia, Y., Fang, G., 2017, Morphological characterization and applications of phase 
change materials in thermal energy storage: A review, Renewable and Sustainable Energy Reviews, 72, 

pp. 128-145. 

3. Moreno, P., Solé, C., Castell, A., Cabeza, L.F., 2014, The use of phase change materials in domestic heat 
pump and air-conditioning systems for short term storage: A review, Renewable and Sustainable Energy 

Reviews, 39, pp. 1-13. 

4. Angayarkanni, S., Philip, J., 2015, Review on thermal properties of nanofluids: Recent developments, 
Advances in colloid and interface science, 225, pp. 146-176. 

5. Khanafer, K., Vafai, K., 2011, A critical synthesis of thermophysical characteristics of nanofluids, 
International Journal of Heat and Mass Transfer, 54(19-20), pp. 4410-4428.  

6. Liu, C., Rao, Z., Zhao, J., Huo, Y., Li, Y., 2015, Review on nanoencapsulated phase change materials: 
preparation, characterization and heat transfer enhancement, Nano Energy, 13, pp. 814-826. 

7. Ghasemi, K., Tasnim, S., Mahmud, S., 2022, PCM, nano/microencapsulation and slurries: A review of 
fundamentals, categories, fabrication, numerical models and applications, Sustainable Energy 

Technologies and Assessments, 52, 102084. 

8. Sarkar, S., Mestry, S., Mhaske, S., 2022, Developments in phase change material (PCM) doped energy 
efficient polyurethane (PU) foam for perishable food cold-storage applications: A review, Journal of 

Energy Storage, 50, 104620. 

9. Sidik, N.A.C., Kean, T.H., Chow, H.K., Rajaandra, A., Rahman, S., Kaur, J., 2018, Performance 
enhancement of cold thermal energy storage system using nanofluid phase change materials: a review , 

International Communications in Heat and Mass Transfer, 94, pp. 85-95. 

10. Shah, K.W., 2018, A review on enhancement of phase change materials-A nanomaterials perspective, 
Energy and buildings, 175, pp. 57-68. 

11. Manjunatha, S., Puneeth, V., Gireesha, B.,Chamkha, A., 2021, Theoretical Study of Convective Heat 
Transfer in Ternary Nanofluid Flowing past a Stretching Sheet, Journal of Applied and Computational 
Mechanics, 8(4), pp. 1279-1286. 

12. Sharma, R., Mishra, S., 2020, Metal and Metallic Oxide Nanofluid over a Shrinking Surface with 
Thermal Radiation and Heat Generation/Absorption, Journal of Applied and Computational Mechanics, 
8(2), pp. 557-565. 

13. Roy, N.C., Hossain, M.A., Pop, I., 2021, Analysis of Dual Solutions of Unsteady Micropolar Hybrid 
Nanofluid Flow over a Stretching/Shrinking Sheet, Journal of Applied and Computational Mechanics, 

7(1), pp. 19-33. 



 Convective Flow and Heat Transfer of Nano-Encapsulated Phase Change Material (NEPCM)... 537 

 

 

14. Munir, S., Maqsood, A., Farooq, U., Hussain, M., Siddiqui, M.I., Muhammad, T., 2022, Numerical 
analysis of entropy generation in the stagnation point flow of Oldroyd -B nanofluid, Waves in Random 

and Complex Media, doi:10.1080/17455030.2021.2023782. 

15. Bilal, M., Ayed, H., Saeed, A., Brahmia, A., Gul, T., Kumam, P., 2022, The parametric computation of 
nonlinear convection magnetohydrodynamic nanofluid flow with internal heating across a fixed and 

spinning disk, Waves in Random and Complex Media, doi: 10.1080/17455030.2022.2042621. 

16. Yasin, M.H.M., Ishak, A., Pop, I., 2015, MHD stagnation-point flow and heat transfer with effects of 
viscous dissipation, Joule heating and partial velocity slip, Scientific reports, 5(1), 17848. 

17. Rashed, A.S., Mahmoud, T.A., Kassem, M.M., 2021, Behavior of Nanofluid with Variable Brownian and 
Thermal Diffusion Coefficients Adjacent to a Moving Vertical Plate, Journal of Applied and 
Computational Mechanics, 7(3), pp. 1466-1479. 

18. Madhu, M., Kishan, N., Chamkha, A., 2016, Boundary layer flow and heat transfer of a non-Newtonian 
nanofluid over a non-linearly stretching sheet, International Journal of Numerical Methods for Heat & 
Fluid Flow, 26(7), pp. 2198-2217. 

19. Reddy, P.S., Chamkha, A.J., Al-Mudhaf, A., 2017, MHD heat and mass transfer flow of a nanofluid over 
an inclined vertical porous plate with radiation and heat generation/absorption, Advanced Powder 
Technology, 28(3), pp. 1008-1017. 

20. Reddy, P.S., Chamkha, A., 2017, Heat and mass transfer analysis in natural convection flow of nanofluid 
over a vertical cone with chemical reaction, International Journal of Numerical Methods for Heat & 
Fluid Flow, 27(1), pp. 2-22. 

21. Seyf, H.R., Wilson, M.R., Zhang, Y., Ma, H., 2014, Flow and heat transfer of nanoencapsulated phase 
change material slurry past a unconfined square cylinder, Journal of heat transfer, 136(5), 051902. 

22. Seyf, H.R., Zhou, Z., Ma, H., Zhang, Y., 2013, Three dimensional numerical study of heat-transfer 
enhancement by nano-encapsulated phase change material slurry in microtube heat sinks with tangential 

impingement, International journal of heat and mass transfer, 56(1-2), pp. 561-573. 
23. Mohib Ur Rehman, M., Qu, Z.G., Fu, R.P., 2016, Three-dimensional numerical study of laminar confined 

slot jet impingement cooling using slurry of nano-encapsulated phase change material, Journal of 

Thermal Science, 25(5), pp. 431-439. 
24. Edalatifar, M., Tavakoli, M.B., Setoudeh, F., 2021, A Deep Learning Approach to Predict the Flow Field 

and Thermal Patterns of Nonencapsulated Phase Change Materials Suspensions in an Enclosure, Journal 

of Applied and Computational Mechanics, 8(4), pp. 1270-1278. 
25. Ho, C.J., Chen, W.C.,Yan, W.M., 2013, Experimental study on cooling performance of minichannel heat 

sink using water-based MEPCM particles, International communications in heat and mass transfer, 48, 

pp. 67-72. 
26. Wang, Y., Chen, Z., Ling, X., 2016, An experimental study of the latent functionally thermal fluid with 

micro-encapsulated phase change material particles flowing in microchannels, Applied Thermal 

Engineering, 105, pp. 209-216. 
27. Ho, C.J., Huang, J., Tsai, P., Yang, Y.M., 2011, Water-based suspensions of Al2O3 nanoparticles and 

MEPCM particles on convection effectiveness in a circular tube, International Journal of Thermal 
Sciences, 50(5), pp. 736-748. 

28. Ho, C.J., Chen, W.C.,Yan, W.M., 2014, Correlations of heat transfer effectiveness in a minichannel heat 
sink with water-based suspensions of Al2O3 nanoparticles and/or MEPCM particles, International 
Journal of Heat and Mass Transfer, 69, pp. 293-299. 

29. Kuravi, S., Kota, K.M., Du, J., Chow, L.C., 2009, Numerical investigation of flow and heat transfer 
performance of nano-encapsulated phase change material slurry in microchannels, Journal of heat 
transfer, 131(6), 062901. 

30. Chai, L., Shaukat, R., Wang, L., Wang, H.S., 2018, A review on heat transfer and hydrodynamic 
characteristics of nano/microencapsulated phase change slurry (N/MPCS) in mini/microchannel heat 
sinks, Applied Thermal Engineering, 135, pp. 334-349. 

31. Du, X., Fang, Y., Cheng, X., Du, Z., Zhou, M.,Wang, H., 2018, Fabrication and characterization of 
flame-retardant nanoencapsulated n-octadecane with melamine–formaldehyde shell for thermal energy 
storage, ACS Sustainable Chemistry & Engineering, 6(11), pp. 15541-15549. 

32. Chen, B., Wang, X., Zeng, R., Zhang, Y., Wang, X., Niu, J., Li, Y., Di, H., 2008, An experimental study 
of convective heat transfer with microencapsulated phase change material suspension: laminar flow in a 
circular tube under constant heat flux, Experimental Thermal and Fluid Science, 32(8), pp. 1638-1646. 

33. Rajabifar, B., 2015, Enhancement of the performance of a double layered microchannel heatsink using 
PCM slurry and nanofluid coolants, International Journal of Heat and Mass Transfer, 88, pp. 627-635. 

34. Alisetti, E.L., Roy, S.K., 2000, Forced convection heat transfer to phase change material slurries in 
circular ducts, Journal of thermophysics and heat transfer, 14(1), pp. 115-118. 



538 M. GHALAMBAZ, H. JIN, A. BAGHERI SARVESTSANI, O. YOUNIS, D. WEN 

 

35. Dinarvand, S., Yousefi, M., Chamkha, A.J., 2022, Numerical Simulation of Unsteady Flow toward a 
Stretching/Shrinking Sheet in Porous Medium Filled with a Hybrid Nanofluid , Journal of Applied and 

Computational Mechanics, 8(1), pp. 11-20. 

36. Kuravi, S., Du, J.,Chow, L.C., 2010, Encapsulated phase change material slurry flow in manifold 
microchannels, Journal of thermophysics and heat transfer, 24(2), pp. 364-373. 

37. Buongiorno, J., 2006, Convective transport in nanofluids, Journal of heat transfer, 128(3), pp. 240-250. 
38. Vand, V., 1945, Theory of viscosity of concentrated suspensions, Nature, 155(3934), pp. 364-365. 
39. Mulligan, J., Colvin, D., Bryant, Y., 1996, Microencapsulated phase-change material suspensions for 

heat transfer in spacecraft thermal systems, Journal of spacecraft and rockets, 33(2), pp. 278-284. 

40. Yamagishi, Y., Takeuchi, H., Pyatenko, A.T., Kayukawa, N., 1999, Characteristics of microencapsulated 
PCM slurry as a heat‐transfer fluid, AIChE Journal, 45(4), pp. 696-707. 

41. Wang, X., Niu, J., Li, Y., Wang, X., Chen, B., Zeng, R., Song, Q., Zhang, Y., 2007, Flow and heat 
transfer behaviors of phase change material slurries in a horizontal circular tube, International journal of 
heat and mass transfer, 50(13-14), pp. 2480-2491. 

42. Buongiorno, J., Venerus, D.C., Prabhat, N., Mckrell, T., Townsend, J., Christianson, R., Tolmach ev, 
Y.V., Keblinski, P., Hu, L.W., Alvarado, J.L., 2009, A benchmark study on the thermal conductivity of 
nanofluids, Journal of Applied Physics, 106(9), 094312. 

43. Venerus, D.C., Buongiorno, J., Christianson, R., Townsend, J., Bang, I.C., Chen, G., Chung, S. J., Chyu, 
M., Chen, H.,Ding, Y., 2010, Viscosity measurements on colloidal dispersions (nanofluids) for heat 
transfer applications, Applied rheology, 20(4), 44582. 

44. Ghalambaz, M., Doostani, A., Izadpanahi, E., Chamkha, A., 2017, Phase-change heat transfer in a cavity 
heated from below: the effect of utilizing single or hybrid nanoparticles as additives , Journal of the 
Taiwan Institute of Chemical Engineers, 72, pp. 104-115. 

45. Shampine, L.F., Kierzenka, J., Reichelt, M.W., 2000, Solving boundary value problems for ordinary 
differential equations in MATLAB with bvp4c, Tutorial notes, pp. 1-27. 

46. Bejan, A., 2013, pp. 186-189, Convection heat transfer, John Wiley & Sons, New Jersey, United States. 
47. Rao, Y., Dammel, F., Stephan, P., Lin, G., 2007, Convective heat transfer characteristics of microencapsulated 

phase change material suspensions in minichannels, Heat and Mass transfer, 44(2), pp. 175-186. 
48. Hasan, M.I., 2017, Improving the cooling performance of electrical distribution transformer using 

transformer oil–Based MEPCM suspension, Engineering Science and Technology, an International 

Journal, 20(2), pp. 502-510. 
49. Li, W., Wan, H., Jing, T., Li, Y., Liu, P., He, G., Qin, F., 2019, Microencapsulated phase change 

material (MEPCM) saturated in metal foam as an efficient hybrid PCM for passive thermal management: 

A numerical and experimental study, Applied Thermal Engineering, 146, pp. 413-421. 
50. Siao, Y.H., Yan, W.M., Lai, C.M., 2015, Transient characteristics of thermal energy storage in an 

enclosure packed with MEPCM particles, Applied Thermal Engineering, 88, pp. 47-53.