Thermal Analysis of Melting Process with Conjugate Turbulent Flow Convection of Liquid

Metal Fluid in Mini-scaled Thermal Storage Channel

M Khamis Mansour1, 2, and M. Kamal El Youssef, 1

1,*
Department of Mechanical Engineering, Faculty of Engineering, Beirut Arab University
Email:m.mansour@bau.edu.lb
2
Department of Mechanical Engineering, Faculty of Engineering, Alexandria University-
1999 – until 2010

Abstract

This article presents thermal analysis of conjugate heat transfer problem of liquid metal fluid in mini-scaled
thermal storage. The heat transfer conjugate problem consists of turbulent convective flow of the heat transfer
fluid (HTF) and melting process in the phase change material (PCM). The Sodium-Potassium (NaK) liquid
metal fluid was selected as the HTF while the paraffin wax was used as the PCM. The local Nusselt (Nu)
number represents the heat characteristics of the HTF; alternatively, the melting thermal behavior was reported
by the average liquid fraction of the PCM. The present model was verified by comparison its results to those
obtained from previous work in the literature. There was a good agreement between both results with
maximum deviation of 10 %. A parametric study has been accomplished to show the effect of geometrical and
operational design parameters on the thermal behavior of the conjugate problem. The length to diameter ratio
(L/D) represents the geometrical parameter. However, Reynolds (Re) number, Stefan (Ste) number, and
Fourier (Fo) number represents the operational parameters. The parametric study has been carried out during
presence/absence of the natural convection of the PCM during the melting process. It was found from the
parametric study that the (L/D), Ste, and Fo have no impact on the local Nu number at fully developed region
except the Re number. On the other hand, sometimes the minichannel which is featured by the absence of
natural convection can be superior that during the presence of natural convection at variation of some
parameters. The result obtained from the parametric study showed that the fully-developed Nu number
approaches that of traditional standard correlation of constant wall temperature (CWT) problem.

Keywords

Melting process, conjugate problem, forced convection, natural convection, mini-channel thermal storage

List of Symbols L/D Length to diameter of the mini-channel

x, y, z Cartesian coordinates
do Outer mini-channel tube diameter, m T Temperature K
di Inner mini-channel tube diameter, m Y Incremental length of the unit cell, m
L Length of the mini-channel, m Y* Non-dimensionnel thermal axial length, Y/L
k Thermal conductivity, W/(m. oK) Tin Inlet fluid temperature, oK
K Dimensionless thermal conductivity (k/kp)
Nu Local Nusselt number THTF fluid temperature, oK
P Pressure, Pa (gauge value) TPCM Phase change material temperature, oK
R Dimensionless radial distance from the inner V Inlet fluid velocity, m/sec
pipe to the outer pipe r/ri Greek Symbols
Re Reynolds number
Ste Stefan number  Liquid Fraction
Fo Fourier number

1
 Subscripts Abbreviations
TC Traditional Channel
in inlet MC Mini-channel
t Time HTF Heat transfer fluid
f Heat transfer fluid PCM Phase change materials
p Phase change material WNC with natural convection
WONC without natural convection
CHF Constant heat flux
CWT Constant wall temperature

1. Introduction

Nowadays, carbon dioxide emissions and other harmful gases are spreading widely, which may be a challenge
in order to overcome this issue by showing and using another source then oil energies, for this reason renewable
energies are highly considered as good sources aiming to take the role of petroleum sources. Thermal energy
storage systems are used in many renewable energy applications in which these systems are prime important
nowadays presenting in several domains such as construction, manufacturing and electricity generation [1].
This makes the thermal energy storage system important in several engineering applications as using in heating
and cooling systems. The latent heat storage is widely used in industry to high value of stored energy per unit
volume [2]. On the other hand, recently there has been a fast development in the mini/micro scale heat
exchanger technology. This new technology motivates many researchers to investigate the thermal
characteristics of those heat exchangers. This study mainly focuses on mini-scaled thermal storage channel
design and its thermal behavior. The mini-channel (MC) consists of a PCM imbedded inside its outer tube
where a HTF flows inside the inner tube of this channel. The PCM absorbs the thermal energy released from
the HTF and stores it as a latent heat capacity and cooling the HTF, this leads to have a conjugate problem
associates the melting process of the PCM as well as the single-phase fluid flowing inside the MC (see Figure
1). The vertical position of the minichannel causes a presence of the natural convection currents during the
melting process. Therefore, all of these aspects make the heat transfer problem to be complicated to some
extent and gives some interests to be studied and investigated.

Figure 1. 3D and 2D view of the presented MC model

2
A melting process during the presence of natural convection has been investigated by many researchers, all of
those researchers analyzed the effect of natural convection on the melting process itself either theoretically
(numerically or analytically) or experimentally or both [3]. They concluded that the natural convection currents
improve the melting process by shorting the time required for the melting process. However, most of the
previous research works focused only on the melting process during the presence of the natural convection
without attention to the heat characteristics of the HTF, or to solving the conjugate heat transfer coupling
between the HTF and PCM [3].

According to the available literature, a few studies treated the conjugate heat transfer problem during the
melting process. Some of the research works [4-11] treated the HTF flow inside the channel as steady fully
developed and employed standard correlations to calculate the convective heat transfer coefficient of the HTF
for CWT and constant heat flux (CHF) boundary conditions. Other researchers [12-15] analyzed the energy
and momentum equations for the HTF while ignoring the natural convection occurred in the melted PCM
liquid.

Zhang and Faghri [16] studied a hollow cylinder containing PCM with HTF pumped in the interior of the
cylinder by semi-analytical method forming a conjugate problem. Nu number was studied along the cylinder
with different lengths. The results showed that the convection inside the tube never reached the fully developed
state either with maximum tube length (L/D > 100), this leads to realize that the laminar forced convective
heat transfer for HTF of moderate Prandtl number (water) must be solved as a conjugate problem with the
PCM. The results were achieved during absence of natural convection mode. Cao and Faghri [17] have
numerically studied a thermal storage system. They concluded that the phase change for PCM and the transient
forced convection heat transfer for HTF with low Prandtl number must be treated as conjugate melting
problem. The study was carried out during absence of natural convection. Also, they carried out an
optimization process for different L/D ratios. Recently, M Khamis Mansour [3] studied numerical and
experimental simulation of 3-D conjugate heat transfer problem in MC. The paraffin wax was used as a PCM
while the water was used as a HTF. Experimental work has been done to validate the numerical results. The
effect of inlet temperature and velocity of the HTF were taken into account. Results showed that natural
convection during the PCM melting shortens the fusion time and makes the wax at the outlet face of the MC
to melt faster than that at the inlet face. The results also showed that the local Nu number trends in the conjugate
heat transfer problem during melting at the presence of natural convection are different compared to the
traditional trends at CHF and CWT as boundary conditions.

It can be concluded from pervious paragraphs that the study of the conjugate problem in the melting process
is very scare, particularly at with natural convection (WNC) currents. In addition, the effect of the geometrical
and the operational parameters at the conjugate problem thermal characteristics are not extensively addressed
yet according to the author’s knowledge. The main objective of this study is to conduct a parametric study for
the effect of the key design parameters on the conjugate heat transfer problem during the melting process as
well as to determine the Nu number correlation of the convective heat transfer of the HTF.

2. Mathematical model

Again, the conjugate heat transfer problem consists of co-axial tubes where the HTF flows inside the inner
tube of diameter (di) and the PCM bounds the outer tube of diameter (do). A single cell is chosen as a
benchmark in order to simulate the heat transfer process of the MC as well as to solve the problem in 2D
forms. This is attributed to the symmetrical design of the many cells forming the MC. The model geometry is
presented in Figure 1.
3
2.1 Model assumptions

The 2D flow is studied using commercial software ANSYS FLUENT 14.5 under transient condition with the
following assumptions:
 Turbulent and incompressible
 Constant thermo-physical properties
 The liquid phase’s density varies linearly with the temperature as in the Boussinesq approximation;
 Viscous heat dissipation is neglected
 Natural convection is incompressible, Newtonian, and turbulent flow
 Density difference between the liquid phase and solid phase of the PCM is negligible
 Radiation heat is neglected
 No heat generation

2.2 Governing equations

HTF domain

Continuity equation
The continuity equation for 2D model is given by:
 V V y
 ( x  )0 (1)
t x y
Momentum equation
The momentum conservation for 2D model equation is given by:
Vx V V P   2V  2V 
(  Vx x  Vy x )      2x  2x   g x (2)
t x y x  x y 

V y V y P   V y  V y 
V y 2 2

(  Vx Vy )      2  2   g y (3)
t x y y  x y 
Energy equation
The energy equation for 2D model is given by:
 T f T f T f    2T f  2T f


 f Cp  Vx  Vy   
t x y   K f  x 2  y 2
 (4)
   
Where ρ is the density, V is the velocity component, p is the pressure,  is the dynamic viscosity, P is the
pressure, g is the gravity, Cp is the specific heat, Kf is fluid thermal conductivity, Tf is the fluid temperature
and x and y are the 2-D Cartesian coordinates.

The k-epsilon turbulence model was used to solve the turbulent flow of the HTF.

 (  k )  (  ku i )   t  k 
     2t E ij E ij  ò (5)
t xi x j   k  x j 

4
 ( ò) ( òu i )   t ò  ò ò2
     C 1ò 2t E ij E ij  C 2ò  (6)
t x i x j   ò x j  k k

Where K is the turbulence kinetic energy while Ɛ represents the turbulence dissipation u i represents velocity
component in y direction, E ij represents component of rate of deformation, t represents eddy viscosity,
k2
t  C  ,also the equations consist of some constant terms: C   0.09 ,  k  1 ,  ò  1.3 , C 1ò  1.44 ,
ò
C 2ò  1.92 .

PCM domain

In the PCM domain, enthalpy-porosity technique was used to stimulate the model in ANSYS FLUENT during
solidification/melting process. The melt interface is not tracked explicitly by this technique. Instead, a quantity
called the liquid fraction, which indicates the fraction of the cell volume that is in liquid form, is associated
with each cell in the domain. The liquid fraction is computed during each iteration, based on an enthalpy
balance. The mushy zone is a region in which the liquid fraction lies between 0 and 1. The mushy zone is
modeled as a porous medium in which the porosity decreases from 1 to 0 as the material solidifies. When the
material has fully solidified in a cell, the porosity becomes zero and hence the velocities also drop to zero. The
equations are the following, given in Cartesian co-ordinates as follows:

Continuity equation
The continuity equation for 2D model is given by:

Vi   0
xi ( 7)
Momentum equation
The momentum conservation for 2D model equation is given by:

V j    ViV j     V2j  P  g j  Si

2

t xi x j (8)
xi
Energy equation
The energy equation for 2D model is given by:
  
H    Vi H     k T  (9)
t xi xi  xi 
Where ρ is the density, k is the thermal conductivity, μ is the dynamic viscosity, P is the pressure, Si is the
momentum source term, V is the velocity component, x is the Cartesian coordinate and H is the enthalpy of
the material is computed as the sum of the sensible enthalpy, h, and the latent heat, ΔH:
H  h  H ( 10 )

T
Where, h  href   C p dT and H  L f
Tref

href is the reference enthalpy at the reference temperature Tref , C p is the specific heat, L f is the latent heat, and
 is the liquid fraction during the phase change which occurs over a range of temperatures Ts < T < Tl defined
by the following relations:
5
 H ( 11 )
  0 if Ts  T [Solid]
Lf

H T  Ts ( 12 )
   if Ts  T  Tl [Mushy]
Lf Tl  Ts

H ( 13 )
  1 if T  Tl [Liquid]
Lf

The source term S i in the momentum equation is given by:

C 1   
2
S i   A( )Vi  Vi ( 14 )
 3 

Where A( ) is defined as the "porosity function" which reduces the velocities gradually from a finite value
as 1 in fully liquid to 0 in fully solid state within the computational cells involving phase change.  is a small
number (0.001) to prevent division by zero and C is the mushy zone constant, it measures the amplitude of the
damping; the higher this value, the steeper the transition of the velocity of the material to zero as it solidifies.

2.3 Boundary Conditions

Inlet section

At the inlet section for the HTF domain, the velocity and temperature are considered as uniform while for the
PCM the velocity is zero and the walls are adiabatic.
d
At y = 0, 0  x  i , at t ≥ 0 Vx  0,V y  Vin , T f  Tin
2 ( 15)
Outlet section

If in ANSYS FLUENT, the details of the flow velocity and pressure are not known prior to the solution of the
flow problems, the outlet boundary conditions are used to model flow exits. At the outflow boundary
conditions, it is first assumed a zero normal gradient for all the variables except pressure. Then, ANSYS
FLUENT updates those gradients in the model calculations.

di Vx V y T
At y = L, 0  x  , at t ≥ 0  , 0 ( 16)
2 y y y
Walls

Walls surrounded the PCM are assumed adiabatic except the interface wall between the PCM and HTF.
di d  d i  y = 0, at t ≥ 0 Ts
x o ,  0,Vx  V y  0
2 2 y ( 17 )

di d  di Ts
x o , y = L, at t ≥ 0  0,Vx  V y  0
2 2 y ( 18 )

do Ts ( 19 )
x , 0 ≤ y ≤ L, at t ≥ 0  0,Vx  V y  0
2 x

6
 V y T ( 20 )
at x = 0, 0  y  L , at t ≥ 0 Vx   0 (symmetry)
x x
di
At 0  y  L, x  , at t≥ 0
2
T T
Vx  V y  0, ( K PCM ) PCM  ( K HTF ) HTF , and TPCM  THTF
y y
At PCM – HTF interface, no-slipping condition is assumed and no heat generation or thermal contact
resistance does not exist.

Initial conditions

At the beginning of the charging process, the temperatures of the HTF is Tin and PCM is 300 ◦K.

3. Model validation

In this section, the numerical results obtained from the present model are tested through the comparison with
obtained from Zhang and Faghri [16] with low Prandtl number fluid of 0.004 was used as the HTF. Figure 2
is prepared to show the comparison between the results of the two models at specific conditions listed in this
Figure. There is a good agreement between both results with maximum deviation of to 1.33% occurs at X =7.5
as shown in Figure 2

30
L/D = 50
Ste = .5
Re = 5*105
25
Fo = 4
Pr = 0.004
20 Kf = 0.1
Ro = 2.5

present model
Nu

15
reference model

10

0
0 5 10 15 20 25 30 35 40 45 50
X=Y/D
Figure 2. Nu number along the MC length at Fo=4; calculated and from Zhang and Faghri [5]

7
4. Results and Discussion

This section is divided into three parts; the first part present a comparison between the two studied cases during
the presence and the absence of natural convection at the same initial condition. The second part discusses the
effect of the geometrical and operational key parameters on the conjugate problem’s thermal characteristics.
The length-to-diameter ratio (L/D), Reynolds (Re) number, Stefan (Ste) number, and Fourier (Fo) number are
employed as the key design parameters. The local Nu number for the HTF and average liquid fraction (γ) for
the PCM represent the thermal characteristics of the conjugate heat transfer problem. The third part of this
section focuses on comparison between the outcome result of the average Nu number from the present model
with the traditional correlations in the references (constant wall temperature (CWT) and constant heat flux
(CHF)). Also, a comparison was done between minichannel (MC) and traditional channel (TC) of large
diameter in case of utilizing Turbulent liquid metal as HTF.

4.1 Comparison Between WNC and WONC Cases

As noticed in Figure 3, both trends of local Nu number are coincident except at the end of the channel. In
WNC, there is a drop in the wall heat flux at the end of the channel as a result of an accommodation of natural
convection currents (see Figure 4), therefore, the trends of the local nu number for the WNC channel tend to
decline at this zone.

55
Re = 10000
50 Fo = 10.9
Ste = 0.44
45 L/D = 30
Pe = 210
40 Kf = 138.6
Kw = 1094
35 Rw = 1.1
Nu

Ro = 21
30 WNC

25
WONC

20

15

10

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Y*

Figure 3. Local Nu number along the MC length WNC and WONC channels

8
 (a) (b)
Figure 4. Liquid fraction profile for (a) WNC channel and (b) WONC channel

Although most of the local nu number trends for both cases (WNC and WONC) are coincident except at the
end of the channel, the liquid fraction profiles are different for both cases (WNC and WONC). At the beginning
of the melting process, it was observed that the presence of natural convection enhances the melting process.
However, after a certain time, the melting process during the absence of natural convection melts completely
before that at WNC and as shown in Figure 5.

This observation can be explained by Figures 4 and 6, at the beginning of melting process the natural
convection takes place a significant role in enhancement the melting process until there is a zone settled at the
channel bottom is formed as shown in Figures 4. This zone takes longer time in the melting process and the
heat transfer to the PCM is converted to sensible heat transfer for the PCM instead of latent heat.

Therefore, by looking to Figure 6, it was found the ratio Qs/Qt for the melting process at WNC is lower than
that at WONC case at the beginning. Then, this ratio becomes higher for WNC channel than that at WONC
case after a certain time in which the so-called unmelted residual zone (at the channel bottom) start to be
formed.

0.9

0.8

0.7

0.6
Re = 10000
Fo = 10.9 WNC
0.5 Ste = 0.44
ɣ

0.4 L/D = 30 WONC

Pe = 210
0.3
Kf = 138.6
0.2 Kw = 1094
0.1
Rw = 1.1
Ro = 21
0
0 50 100 Fo 150 200 250

Figure 5. Liquid fraction profile along with Fo number for PCM (turbulent flow)

9
 1
Re = 10000
0.9 Ste = 0.44
0.8 L/D = 30
Pe = 210
0.7
Kf = 138.6
0.6 Kw = 1094
Qs/Qt

0.5 Rw = 1.1 WNC

0.4
Ro = 21 WONC

0.3
0.2
0.1
0
0 50 100 Fo 150 200 250

Figure 6. Comparison between WNC and WONC in terms of Qs/Qt ratio along with Fo number

4.2 Parametric study

4.2.1 Effect of L/D ratio

As observed in Figures 7 and 8 that the length to diameter ratio has no effect on determination the Nu
number neither for WNC nor for WONC case particularly at the fully-development zone.

50
Re = 10000
L/d = 10
Ste = 0.44
Fo = 109
40 L/d = 20
Pe = 210
Kf = 138.6 L/d = 30
Kw = 1094
30 Rw = 1.1 L/d = 40
Ro = 21
Nu

L/d = 60
20
L/d = 80

10

0
Y*
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 7. Effect of L/D ratio on the local Nu number along MC length for WNC channel

10
 60
Re = 10000
Ste = 0.44 L/d = 10
50 Fo = 109 L/d = 20
Pe = 210
L/d = 30
40
Kf = 138.6
Kw = 1094 L/d = 40
Rw = 1.1
30
L/d = 60
Ro = 21
Nu

L/d = 80
20

10

0
Y*
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 8. Effect of L/D ratio on the local Nu number along MC length for WONC channel

On the other hand, the increasing in the L/D ratio in case of the presence of natural convection leads to increase
the melting time of the process as shown in Figure 9. As the characteristic length of the MC increases the
melting time for PCM increases. The explanation is due to the relationship between the heat transfer coefficient
for the wax and the characteristic length of the channel (i.e. its length).
It is interesting to note that the channels, which have L/D ≥ 40 have almost the same melting time length for
WNC, channel. This refers to that it is no need to use larger channels (i.e. larger than 40D) during turbulent
flow of the HTF at WNC channel.

Furthermore, Figure 10 indicate that the L/D ratio has no effect on the melting progress in case of the absence
of natural convection. This means, that different values of L/D ratio will expose to the same melting time
during turbulent flow of HTF (liquid metal fluid) at WONC case.

0.9

0.8
Re = 10000 L/d = 10
0.7
Ste = 0.44 L/d = 20
0.6 Pe = 210
Kf = 138.6 L/d = 30
0.5 Kw = 1094 L/d = 40
ɣ

Rw = 1.1
0.4
Ro = 21 L/d = 60
0.3
L/d = 80
0.2

0.1

0
0 50 100 150 Fo 200 250 300 350

Figure 9. Effect of L/D on liquid fractional profile along with Fo number for WNC channel

11
 1 Re = 10000
Ste = 0.44
0.9
Pe = 210
0.8 Kf = 138.6
Kw = 1094
0.7 Rw = 1.1
Ro = 21 L/d = 10
0.6
L/d = 20
0.5
ɣ

L/d =30
0.4
L/d = 40
0.3
L/d = 60
0.2
L/d = 80
0.1

0
Fo
0 25 50 75 100 125 150

Figure 10. Effect of L/D on liquid fractional profile along with Fo number for WONC channel

4.2.2 Effect of Reynolds (Re) number

The Re number has a direct slight influence on the determination of the local Nu number and this is clearly
shown in Figures 10 and 11. These Figures show that as the Re number increases the local Nu number increases
in both cases (WNC and WONC). This confirms to the relationship between the Nu number and Re number
for the traditional standard correlation at constant wall temperature (CWT) and constant wall heat flux. (CHF).

70 L/D = 30
Re = 10000 Ste = .44
60 Re = 15000 Fo = 54
Kf = 138.6
Re = 20000 Kw = 1094
50
Re = 25000 Rw = 1.1
40
Ro = 21
Nu

30

20

10

0
0 0.1 0.2 0.3 0.4 Y* 0.5 0.6 0.7 0.8 0.9 1

Figure 10. Effect of Re number on the local Nu number along MC length for WNC channel

12
 70
L/D = 30
60
Ste = .44
Fo = 54
Kf = 138.6
50
Kw = 1094
Re = 10000 Rw = 1.1
40
Ro = 21
Nu

Re = 15000
30
Re = 20000

20 Re = 25000

10

0
Y*
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 11. Effect of Re number on the local Nu number along MC length for WONC channel

The effect of Re number increasing on the melting process behavior at WNC and WONC cases is presented
in Figures 12 and 13.

1
0.9
0.8 L/D = 30
0.7
Ste = .44
Pe = 2.1
0.6
Kf = 138.6
0.5 Re = Kw = 1094
ɣ

0.4 10000 Rw = 1.1

Re = Ro = 21
0.3 15000
0.2 Re =
20000
0.1
0
0 50 100 Fo 150 200 250

Figure 12. Effect of Re number on liquid fraction profile along with Fo number for WNC channel
1
Re =
0.9
10000
0.8
Re =
0.7 15000 L/D = 30
0.6 Re = Ste = .44
0.5 20000 Pe = 10.5
ɣ

0.4 Kf =
0.3 138.6
0.2
Kw =
1094
0.1
Rw = 1.1
0
Ro = 21
0 25 50 Fo 75 100 125 150

Figure 12. Effect of Re number on liquid fraction profile along with Fo number for WONC channel

13
As observed from Figure 12, at Re > 15,000 there is no effect of increasing the velocity of the HTF on the
melting on the process behavior at WNC channel. However, at any value of Re number, the
increasing/decreasing of the inlet velocity for the HTF has no influence on liquid fraction profile for WONC
case as shown in Figure 12.

At Re = 10,000, the channel of WONC case has faster melting time than that for WNC case. However, after
Re > 10,000, the wax melts faster at WNC than that at WONC. This can be emphasized by looking to Figure
13, the ratio Qs/Qt for the PCM during the presence of natural convection is larger at WONC case.
The reason is attributed to the effect of unmelted zone at the channel bottom. However, at high turbulent flow
(Re > 10,000), the heat transfer coefficient is high enough to accelerate the melting process at this unmelted
zone at the bottom of the channel.

1
L/D = 30
0.9 Re = 10000
Ste = .44
0.8 Pe = 10.5
Kf = 138.6
0.7 Kw = 1094
0.6
Rw = 1.1
Ro = 21
Qs/Qt

WNC
0.5
WONC
0.4

0.3

0.2

0.1

0
0 50 100 Fo 150 200 250

Figure 13. Comparison between WNC and WONC in terms of Qs/Qt ratio along with Fo number

4.2.3Effect of Stefan (Ste) number

In Figures 14 and 15, the Ste number has no impact on the determination of the local Nu number in both
cases (WNC and WONC). This is due to the physical properties for the PCM and HTF in the present model
are considered independent of the temperature

14
 45 Re = 10000
Fo = 10.9
40 L/D = 30
Pe = 210
35 Kf = 138.6
Kw = 1094
30 Rw = 1.1
Ro = 21 Ste = 0.44
25
Nu

Ste = 2.64
20

Ste = 3.74
15

Ste = 4.84
10

0
0 0.1 0.2 0.3 0.4 Y* 0.5 0.6 0.7 0.8 0.9 1

Figure 14. Effect of Ste number on the local Nu number along MC length for WNC channel

45
Re = 10000
40 Fo = 10.9
35 L/D = 30
Pe = 210 Ste = 0.44
30 Kf = 138.6
25 Kw = 1094 Ste = 2.64
Nu

Rw = 1.1
20 Ro = 21 Ste = 3.74
15
Ste = 4.84
10

0
0 0.1 0.2 0.3 0.4 Y* 0.5 0.6 0.7 0.8 0.9 1

Figure 15. Effect of Ste number on the local Nu number along MC length for WONC channel

As it is noticed from Figures 16 and 17, the Ste number produces significant effect on the trends of the liquid
fraction performance either at the presence or at the absence of the natural convection.
It may be interesting to note that the minichannel with WNC case at low inlet temperature of 100 oC (Ste =
0.44) has lower liquid fraction performance, compared to that of WONC case at the same temperature (100
o
C, Ste = 0.44). In contrast, the increasing in inlet HTF temperature poses shorting in the melting time length
of the PCM.

Here, it may be concluded that at the working of low inlet HTF temperature (less than or equal to 100 oC
related to Ste = 0.44) in minichannel the absence of natural convection situation is favorable for the thermal

15
performance i.e. the channel should be placed in horizontal orientation. Conversely, the presence of natural
convection is recommended with higher HTF inlet temperature of 200 oC (Ste = 1.54).

0.9

0.8 Re = 10000
L/D = 30
0.7
Pe = 210
0.6 Kf = 138.6
Kw = 1094
0.5
Rw = 1.1
ɣ

0.4 Ste = 0.44 Ro = 21

0.3 Ste = 2.64

0.2 Ste = 3.74

0.1 Ste = 4.84

0
0 50 100 150 200 250
FO

Figure 16. Effect of Ste number on liquid fraction profile along with Fo number for WNC channel

0.9

0.8

0.7

0.6 Re = 10000
L/D = 30
0.5 Pe = 210
ɣ

Kf = 138.6
0.4
Kw = 1094
Ste = 0.44
0.3 Rw = 1.1
Ste = 2.64 Ro = 21
0.2
Ste = 3.74
0.1
Ste = 4.84
0
0 25 50 FO 75 100 125 150

Figure 17. Effect of Ste number on liquid fraction profile along with Fo number for WONC channel

16
 1 1
Ste 0.44
Re = 10000 Re = 10000
Ste = 0.44 0.9
0.9 Ste 1.54 Ste = 0.44
L/D = 30 0.8 L/D = 30
0.8 Ste 2.64
Pe = 210 Pe = 210
0.7 Kf = 138.6 0.7 Kf = 138.6
Kw = 1094 0.6 Kw = 1094
0.6
Rw = 1.1 Rw = 1.1

Qs/Qt
0.5 Ro = 21 0.5 Ro = 21
Qs/Qt

Ste 0.44
0.4 0.4
Ste 1.54
0.3 0.3
Ste 2.64
0.2 0.2

0.1 0.1

0 0
0 50 100 Fo 150 200 250 0 50 Fo 100 150

(a) (b)
Figure 18. Effect of Ste number on melting process for (a) WNC (b) WONC case in terms of Qs/Qt ratio
with Fo number

At inlet temperature of 500 oC (Ste = 4.84), the melting time is 275 secs (i.e. Fo = 30) in case of WNC,
however; the complete melting process need more than 470 secs (i.e. Fo = 51.3) in case of WONC case at the
same Ste number. Moreover, Figures 18 a and b show the variation of Qs/Qt ratio with different inlet
temperature (Ste number) at WNC and WONC, respectively. These Figures show that the increase in the inlet
HTF temperature can enhance the melting process significantly and especially in case of WNC since as the
Ste number increase from 0.44 to 1.54 the melting time is reduced by 88 %, while in case of WONC it is
reduced by 65 %.
Therefore, it can be concluded that the presence of natural convection is recommended in case of large Ste
number (Ste > 0.44 i.e. Tfin – Tm> 40K) for turbulent liquid metal flow. This conforms perfectly to the effect
of Re number. The effect of the presence of natural convection is recommended at higher Re number (Re>
10,000). Accordingly, at the presence of natural convection the design of the minichannel storage system
should be accomplished at high temperature if the inlet velocity is small and the vice versa is correct.

4.2.4 Effect of Fourier (Fo) number

Figures 19 and 20 reveal that the turbulent liquid metal flow is not affected by the variations of the boundary
layer of the melting process. Therefore, the local Nu number remains at constant value either at WNC or
WONC cases for turbulent flow during the progress of the melting process.

17
 45
Re = 10000
40 Ste = .44
L/D = 30
35 Pe = 210
Kf = 138.6
30 Kw = 1094
Rw = 1.1 Fo = 10.9
25 Ro = 21 Fo = 54
Nu

20 Fo = 109
Fo = 164
15

10

0
0 0.1 0.2 0.3 0.4 Y* 0.5 0.6 0.7 0.8 0.9 1

Figure 19. Effect of Fo number on the local Nu number along MC length for WNC channel
45

Re = 10000
40 Ste = .44
L/D = 30
35 Pe = 210
Kf = 138.6
30 Kw = 1094
Rw = 1.1 Fo = 10.9
25 Ro = 21
Fo = 54
Nu

20 Fo = 109

15 Fo = 131

10

0
0 0.1 0.2 0.3 0.4 Y* 0.5 0.6 0.7 0.8 0.9 1

Figure 20. Effect of Fo number on the local Nu number along MC length for WONC channel

4.3 Comparison Between Minichannel (MC) and Traditional (CHF and CWT) Correlations

In this study, the correlations of Chen and Chiou [18] for constant wall heat flux (CHF)
Nu = 5.6 + 0.0165 Pe0.85 Pr 0.01 while for constant wall temperature (CWT) Nu=4.5 + 0.0156 Pe 0.85 Pr 0.01 are
used.

Referring to Figure 2, the profile of the local Nu in case of utilizing MC approaches closely with the profile
of the CWT at fully developed region.

18
 80
L/D = 30
70 Ste = .44
Re = 20000
Fo = 109
60
Pe = 420
Kf = 138.6
50
Kw = 1094
Rw = 1.1 CHF
Nu

40
Ro = 21 CWT
30
Present
Model
20

10

0
0 0.2 0.4 0.6 0.8 1
Y*

Figure 21. Comparison between the local Nu number along MC length with traditional correlations

In turn, the minichannel thermal storage for turbulent liquid metal flow behaves such as isothermal channel
and the isothermal standard correlations for CWT conditions can be used in the designing of those systems.

4.4 Comparison between Mini Channel (MC) and Traditional Channel (TC)

30

L/D = 50
Ste = .5 CHF
25
Re = 5*105
CTW
Fo = 4
20 Pr = 0.004 MC
Ro TC = 2.5
TC
Ro MC = 21
Nu

15

10

0
0 10 20 30 40 50
X=Y/D
Figure 22. Comparison the local Nu number between MC and TC and the traditional correlations

19
A comparison between the thermal performance of the minichannel (MC) and traditional channel (TC) has
large diameter are represented by the Figure 22. As depicted in this Figure, the trend of Nu number for the
minichannel approaches closely to CWT case while the trend of the traditional channel approaches closely to
CHF condition. This can be explained by the effect of using of minichannel causes an amplification to the heat
transfer coefficient dictates the channel to behaves as isothermal wall conditions

5. Conclusion
In this study, a 2D conjugate melting heat transfer problem of turbulent forced convection of liquid metal
inside a minichannel has been numerically investigated. The numerical model has been solved during
presence/absence of natural convection inside the melting process. The numerical results have been verified
against those obtained analytically from the previous work in the literature. The effect of the key parameters
(L/D, Re, Ste, and Fo numbers) on the convection heat transfer and the melting process has been investigated.
The major findings from this study can be summarized as follows:
a) The trends of the Nu number at the presence and at the absence of the natural convection current are
coincident except at the end of channel they deviate.
b) In case of the presence of natural convection, the L/D has direct effect on the melting process but this effect
has limitation until L/D = 40, after this value, the melting time is approximately the same. On the other hand,
the WONC channel has the same melting time at all different L/D ratios. However, the L/D ratio has no impact
on the local Nu number in both cases. Therefore, the designer has no need to have longer channel than 40D
except large capacity of stored energy is required.
C) In both cases (WNC and WONC), the Re number has a direct influence on the determination of the local
Nu number as expected in traditional turbulence correlations. However, for melting behavior, it was found that
the variation in the Re number has no impact on the melting thermal performance for both cases except only
at Re > 10,000 for WNC case only. In addition, the melting process at WONC channel has shorter time length
compared with that at WNC channel for the smaller value of Re = 10,000 only. Therefore, the net outcome
can be summarized here as follows; there is no need for the designer to work with high inlet velocity for the
HTF either on WNC or WONC for turbulent flow. Furthermore, at low inlet HTF velocity, the absence of
natural for the PCM is recommended to achieve a fast melting time.
d) The Ste number has a noticeable impact on the melting process in the two cases (WNC and WONC).
However, at low HTF inlet temperature the melting time in WONC case is superior than that of WNC case,
but at high inlet temperature, the vice versa is occurred. As a result, it is recommended to accompany the
presence of natural convection with high inlet HTF temperature particularly at low inlet HTF velocity inlet in
MC.
e) The variation of the melting boundary layer (represented by Fo number) and the inlet fluid temperature
(represented by Ste number) has no impact on determination the trends of the Nu number for the HTF.
f) It was found that the value of Nu number at turbulent flow fully-develop coincides with that of the traditional
values of Nu number at CWT standard correlation with maximum deviation of 10 %.

20
Therefore, it can be concluded that the conjugate melting heat transfer problem of turbulent liquid metal flow
can be treated as CWT case for traditional heat transfer problem. However, by comparing the performance of
traditional large channel (ro/ri < 2.5), it was found that the traditional large channel behaves as CHF standard
correlation as reported in the previous work [5].

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