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A numerical analysis of solid–liquid phase change heat transfer
around a horizontal cylinder
M. Mahdaoui
a
, T. Kousksou
a,
⇑
, S. Blancher
a
, A. Ait Msaad
b
, T. El Rhafiki
b
, M. Mouqallid
b
a
Laboratoire des Sciences de l’Ingénieur Appliquées à la Mécanique et au Génie Electrique (SIAME), Université de Pau et des Pays de l’Adour – IFR – A. Jules
Ferry, 64000 Pau, France
b
Ecole Nationale Supérieure des Arts et Métiers, ENSAM Marjane II, BP 4024 Meknès Ismailia, Morocco
article info
Article history:
Received 22 May 2012
Received in revised form 19 May 2013
Accepted 5 August 2013
Available online 11 August 2013
Keywords:
Melting
PCM
Natural convection
Horizontal cylinder
abstract
A numerical study is conducted to analyze the melting process around a horizontal circular
cylinder in the presence of the natural convection in the melt phase. Two boundary condi-
tions are investigated one of constant wall temperature over the surface of the cylinder and
the other of constant heat flux. A numerical code is developed using an unstructured finite-
volume method and an enthalpy porosity technique to solve for natural convection cou-
pled to solid–liquid phase change. The validity of the numerical code used is ascertained
by comparing our results with previously published results.
Ó2013 Elsevier Inc. All rights reserved.
1. Introduction
Latent heat storage is required to insure the continuity of a thermal process in energy systems where a temporal differ-
ence exists between the supply of energy and its utilization [1,2]. A good understanding of the heat transfer during melting
process is essential for predicting the storage system performance with accuracy and avoiding costly system overdesign. The
purpose of this paper is to study numerically the solid–liquid interface motion during phase change heat transfer around
heated horizontal cylinder. The presence of natural convection in the melt region during melting may lead to increasing
the heat transfer rate and causes the predictions based upon pure conduction to be non-realistic. The interest for this prob-
lem may be attributed to its relevance to shell-and-tube latent heat storage exchangers with the phase change material
(PCM) on the shell side. This problem has received increasing attention over the last decade. Bathelt and co-workers [3]
and Ramsey et al. [4] appear to be among the first authors to report on experimental studies of melting around circular cyl-
inders. Bathelt et al. [3] first conducted experiments on the melting of a paraffin around a single horizontal heated cylinder.
Sasaguchi and Viskanta [5] investigated experimentally the phase change heat transfer during melting and solidification of
melt around two cylindrical heat exchangers spaced vertically. All these experiments provided conclusive evidence of the
important role played by natural convection on the time-wise variation of the melt shape, the surface temperature, and
the instantaneous local as well as circumferentially averaged heat transfer coefficients around the embedded heat sources.
The problem of melting around heated horizontal circular cylinders has also been analyzed by means of numerical meth-
ods. Several researchers [6–8] tackled this problem by formulating the conservation equations in terms of stream function,
vorticity and temperature. Body-fitted coordinates were employed to track the time-wise changing physical domain (moving
solid–liquid interface). A finite difference method was then used for the solution of the resulting conservation equations.
0307-904X/$ - see front matter Ó2013 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.apm.2013.08.002
⇑
Corresponding author. Tel./fax: +33 629668430.
E-mail address: tarik.kousksou@univ-pau.fr (T. Kousksou).
Applied Mathematical Modelling 38 (2014) 1101–1110
Contents lists available at ScienceDirect
Applied Mathematical Modelling
journal homepage: www.elsevier.com/locate/apm
There are two methods for formulating the problem of heat transfer with phase change. The first one uses the temper-
ature as a dependent variable while the second uses the enthalpy as a dependent variable in the energy equation. In the
models based upon the temperature, the energy equation is written separately for each phase and the coupling between
the two equations is done through the energy balance at solid–liquid interface. In this type of formulation it is necessary
to know explicitly the position of the interface in order to determine the temperature. This moving interface is not known
before hand which complicates the problem and its solution. One of the methods to solve the problem is to immobilize the
interface as done by Landau [9], Duda et al. [10], Caldwell and Savovic [11], Mitchell and Vynnycky [12] and Ismail et al. [13].
These methods are not suitable for problems in which phase change takes place over a temperature interval although they
are accurate for isothermal phase change problems. Also, for problems involving complex geometry and/or large geometry
change in the molten zone, the implementation of these methods is extremely difficult, sometimes is impossible, e.g. for
multi-dimensional problems with multi-interface in the phase change processes. A second method is to use the enthalpy
as a dependent variable and hence write a single energy equation for the whole domain, liquid and solid [14]. This method
has the advantage of not requiring to determine the interface position in order to solve the energy equation.
The contribution of the present study to the actual knowledge lies in the fact that the problem of the melting process
around a circular cylinder is formulated using the enthalpy based method. A numerical code is developed using an unstruc-
tured finite-volume method and to solve for natural convection coupled to solid–liquid phase change. The model is first
tested against experimental data and then used to study the thermal behavior of the PCM during the melting process for
two boundary conditions: constant wall temperature over the surface of the cylinder and constant heat flux.
2. Physical model and basic equations
The physical problem is shown in Fig. 1. The horizontal cylinder of diameter D= 16 mm is immersed in a solid PCM. At the
instant t= 0 the PCM is at the phase change temperature T
m
. When the cylinder surface temperature is at T
w
>T
m
, the fusion
starts. Two fusion cases will be treated here. The first, when the surface temperature of the cylinder T
w
>T
m
whilst the second
specifies the heat flux q. The diameter Dof the cylinder serves as the characteristic length scale on which the Rayleigh num-
ber is based.
Unsteady two-dimensional melting of PCM is governed by the basic laws represented by the continuity, momentum and
energy equations and by the following assumptions:
– The thermophysical properties of the PCM are constant but may be different for the liquid and solid phases.
– The Boussinesq approximation is valid, i.e., liquid density variations arise only in the buoyancy source term, but are other-
wise neglected.
– The liquid is Newtonian.
– Viscous dissipation is neglected.
– Fluid motion in the melt is laminar and two-dimensional.
Since the present formulation deals with solutions on unstructured grids, it is essential to represent the conservation laws
in their respective integral forms.
Fig. 1. Layout of the physical model and correspondant mesh.
1102 M. Mahdaoui et al. / Applied Mathematical Modelling 38 (2014) 1101–1110
Z
S
~
u~
ndS ¼0;ð1Þ
d
dt Z
V
q
~
udV þZ
S
q
~
u~
u~
ndS ¼Z
V
~
r
pdV þZ
S
s
~
ndS þZ
V
~
A
U
dV;ð2Þ
d
dt Z
V
q
c
p
TdV þZ
S
q
c
p
T~
u~
ndS ¼Z
S
k~
r
T~
ndS þZ
V
q
L
F
@f
@t;ð3Þ
where ~
uis the velocity vector, pthe pressure and Tthe temperature.
s
is the viscous stress tensor for a Newtonian fluid:
s
¼
l
r
uþ
r
u
T
:ð4Þ
The integration occurs over a control volume Vsurrounded by a surface S, which is oriented by an outward unit normal
vector ~
n. The source term in Eq. (2) contains two parts:
~
A
U
¼
q
bTT
m
ðÞ
~
gþA~
u;ð5Þ
where bis the coefficient of volumetric thermal expansion and ~
gthe acceleration of gravity vector. The first part of the
source term represents the buoyancy forces due to the thermal dilatation. T
m
is the melting temperature of the PCM. The last
term is added to account for the velocity switch-off required in the solid region. During the solution process of the momen-
tum field, the velocity at the computational cell located in the solid phase should be suppressed while the velocities in the
liquid phase remain unaffected. Also, as the solid region melts the mass in the corresponding computational cell should be-
gin to move. The present study adopts a Darcy-like momentum source term to simulate the velocity switch-off [15]
A¼Cð1fÞ
2
f
3
þ
e
:ð6Þ
The constant Chas a large value to ensure a smooth transition between solid and liquid region and eis small number used
to prevent the division by zero when a cell is fully located in the solid region, namely f= 0. In this work, C=110
15
kg/m
3
s
and e= 0.001 are used.
3. Numerical procedure
The conservation Eqs. (1)-(3) are solved by implementing them in an in house code. This code has been successfully val-
idated in several situations involving flow and heat transfer as in [1,2]. The present code has a two dimensional unstructured
finite-volume framework that is applied to hybrid meshes. The variables values are stored in cell centers in a collocated
arrangement. All the conservation equations have the same general form. By taking into account the shape of control vol-
umes, the representative conservation equation to be discretized may be written as
d
dt Z
V
q
/dV þX
i
Z
S
j
q
u
i
/
C
/
@/
@x
i
dS
i
¼Z
V
S
/
dV:ð7Þ
Solidification and melting are generally transient phenomena, where the explicit schemes are too restrictive owing to sta-
bility limitations. Hence implicit schemes are often preferred and the simplest choice is the first order Euler scheme. The cell
face values, appearing in the convective fluxes, were obtained by blending the upwind differencing scheme (UDS) and the
central difference scheme (CDS) using the differed correction approach [16,17]. The coupling of the dependent variables
was obtained on the basis of the iterative SIMPLE algorithm developed by Patankar and Spalding [18,19].
Summation of the fluxes through all the faces of a given CV results in an algebraic equation which links the value of the
dependent variable at the CV centre with the neighbouring values. The equation may also be written in a conventional man-
ner as
A
P
/
P
¼X
nb
A
nb
/
nb
þb
/
:ð8Þ
The coefficients A
nb
contain contributions of the neighboring (nb) CVs, arising out of convection and diffusion fluxes as
defined by Eqs. (1)-(3). The central coefficient A
P
on the other hand, includes the contributions from all the neighbours
and the transient term. In some of the cases, where sources term linearization was applied, it also contained part of the
source terms. b
/
contains all the terms those are treated as known (source terms, differed corrections and part of the unstea-
dy term).
The evaluation of the source term in the energy equation has been made using the new source algorithm proposed by
Voller [20] where the new value of fat iteration nin cell Pis calculated as follows:
M. Mahdaoui et al. / Applied Mathematical Modelling 38 (2014) 1101–1110 1103
f
nþ1
P
¼f
n
P
þA
n
P
D
t
q
L
F
VT
P
T
m
ðÞ:ð9Þ
This update is followed by an overshoot/undershoot correction:
f
nþ1
P
¼0iff
nþ1
P
<0
1iff
nþ1
P
>1
(ð10Þ
The momentum, pressure correction and temperature are solved sequentially using an ILU-preconditioned GMRES pro-
cedure implemented in the IML++ library [21]. All of the computational meshes were generated using the open-source soft-
ware Gmsh [22].
4. Model validation
Before conducting such numerical investigations, the physical model was first used for validation against experimental
data obtained by Gau and Viskanta’s [23], for the two-dimensional problem of melting of PCM with the presence of natural
convection in a rectangular enclosure. The dimension of the cavity used in the experiments are L= 6.35 cm and W= 8.89 cm.
The left hot wall and the right cold wall are maintained at temperatures, T
H
= 38.35 °C and T
C
= 28.3 °C, respectively, and it
was filled by solid PCM initially at temperature T
C
. The horizontal walls are insulated. The physical properties for pure gal-
lium used in this investigation are given in Table 1. The physical property values were chosen at 32 °C, a temperature that is
representative of the temperature range of the experiment. The Prandtl number for liquid gallium at this temperature is
0.0216 and the situation shown in the rectangular cavity corresponds to a Stefan number of 0.039 and a Raleigh number
of 6 10
5
.
Numerical investigations were conducted using 16,000 cells and the time step of 10
3
s was found to be sufficient to give
accurate results. Once the calculation is started, the solid gallium melts. Fig. 2 shows the shape and location of the solid–li-
quid interface at several times during the melting process. The black and red lines indicate the experimental [23] and cal-
Table 1
Physical properties of pure Gallium.
Density (liquid) 6093 kg m
3
Reference density 6095 kg m
3
Reference temperature 29.78 °C
Volumetric thermal expansion coefficient of liquid 1.2 10
4
Thermal conductivity 32 W m
1
K
1
Melting temperature 29.78 °C
Latent heat of fusion 80160 J kg
1
Specific heat capacity 381.5 J kg
1
Dynamic viscosity 1.81 10
3
kg m
1
s
1
Prandtl number 2.16 10
2
Fig. 2. Numerical and experimental melting front position.
1104 M. Mahdaoui et al. / Applied Mathematical Modelling 38 (2014) 1101–1110
t = 60 s
t = 200 s
t = 800 s
Fig. 3. Stream lines, temperatures contours and interfaces liquid-solid at times 60 s, 200 s and 800 s (constant wall temperature T
w
= 311 K).
t = 60 s
t = 200 s
t = 800 s
Fig. 4. Stream lines, temperatures contours and interfaces liquid-solid at times 60 s, 200 s and 800 s (constant wall heat flux q= 60 kW/m
2
).
M. Mahdaoui et al. / Applied Mathematical Modelling 38 (2014) 1101–1110 1105
culated data respectively. Before a time of 2 min, the shape of the interface is nearly flat because convection is still weak and
melting is driven by conduction. After 2 min, the interface becomes wavy due to the circular flow inside the molten region.
The position of the melt front near the top surface in the calculation before a time of 17 min is over-estimated compared to
experiment, and after 10 min, it is underestimated. However, the overall trend shows good agreement with experiment and
we can safely say that our model and code are fairly validated to track the melting boundary satisfactorily. The discordance
observed may be explained by the difficulty encountered by Gau and Viskanta [23] to ensure the stability in vertical wall’s
temperatures during the experiment. The same remarks were made by other authors when validating their numerical code.
t=40s t=200s
t=260 s
t =
600 s
Fig. 5. Temperatures contours at times 40 s, 200 s, 260 s and 600 s (constant wall temperature T
w
= 315 K).
Fig. 6. Nusselt number versus time (constant wall temperature T
w
= 315 K).
1106 M. Mahdaoui et al. / Applied Mathematical Modelling 38 (2014) 1101–1110
5. Results and discussion
Using the above-described model simulations were carried out for melting of a PCM (Gallium) around a horizontal cyl-
inder. Numerical investigations were conducted using 38,000 cells and the time step of 10
3
s was found to be sufficient to
give accurate results. The effect of the melting process on the streamlines, temperature contours and the solid–liquid inter-
face for several times is gauged through the result illustrated in Figs. 3 and 4. Two cases are investigated: one of constant
wall temperature T
w
over the surface of the cylinder and the other of constant heat flux q. For two cases, during the early
phase the liquid is confined between the rigid heated cylinder and a concentric moving solid–liquid interface. Inspection
of the figures reveals also that at early times the melt regions are similar in shape and that when heat transfer to the gallium
is predominantly by conduction the melt region is symmetrical about the axis of the cylinder. After some time natural con-
vection develops and intensifies, influencing the melt shape in general and the melt region above the cylinder in particular.
The strong upward thrust of the melting zone is caused by natural convection. At early stages of natural convection a plume
rises from the top of the heated cylinder and at later times circulation conveys the hot liquid to the upper part of the melt
zone and in this manner continues to support the upward movement of the solid–liquid interface. The thermal plume which
originates near the top produces nonsymmetrical melting about a vertical plane through the axis of the cylinder. Comparison
of the melt contours reveals a more slender shape for constant heat flux than for the constant wall temperature boundary
condition. The increase in the wall temperature can provoke oscillation of the plume above the cylinder (see Fig. 5). As the
wall temperature of the cylinder increases the solid above the cylinder melts faster.
Fig. 7. Nusselt number versus time (constant wall heat flux q= 60 kW/m
2
).
Fig. 8. Nusselt number around the cylinder (constant wall temperature T
w
= 315 K).
M. Mahdaoui et al. / Applied Mathematical Modelling 38 (2014) 1101–1110 1107
The local Nusselt number at the hot wall is a good indicator of how convection affects overall conduction around the cyl-
inder. Figs. 6 and 7 show the evolution of the local Nusselt number at the top (90°) and the bottom (90°) parts of the cyl-
inder for several times and for the two boundary conditions. The influence of natural convection on the melting process is
indicated by the non-symmetric melting patterns in the top (90°) and the bottom (90°) parts of the cylinder caused the
convective currents. At early times when heat transfer by conduction predominates the local Nusselt number decreases
monotonically with time for both boundary conditions and is practically independent of the angular position around the cyl-
inder (see Figs. 8 and 9). As melting continues and natural convection develops, the Nusselt number becomes non uniform.
The Nusselt number is then almost independent of angular position at the lower part of the cylinder and is most non-uniform
at the upper half. The Nusselt number reaches a constant quasi-steady value at large time even though the solid liquid inter-
face continues to move as melting progresses. This suggests that the processes which occur in the neighbourhood of the
interface do not contribute significantly to the overall thermal resistance to heat transfer. The variation of the local Nusselt
at the top part of the cylinder can also be used to detect the oscillation of the plume in the top (90°) part of the cylinder
which becomes important between 200 s and 300 s.
From an engineering point of view, the rate of melting is one of the most important parameters of the problem. The time
evolution of the total liquid fraction in the cavity (ratio of volume of melt to volume cavity) is a factor that has been widely
used as a monitoring parameter in earlier publications. From the liquid fraction versus time plot, one can get both the rate
melting (slope of the tangent line at a given time) and the average melting rate (ratio of current liquid fraction and time).
Figs. 10 and 11 display the time evolution of the total fraction of the liquid in the cavity for the two boundary conditions.
Fig. 9. Nusselt number around the cylinder (constant wall heat flux q= 60 kW/m
2
).
Fig. 10. Liquid fraction versus time for different T
w
.
1108 M. Mahdaoui et al. / Applied Mathematical Modelling 38 (2014) 1101–1110
We show that as the wall temperature of the cylinder increases the PCM around the cylinder melts faster. We can also note
that for the imposed temperature, once the convection is established, the variation of the liquid fraction with time seems to
be linear. In the case of the imposed heat flux, the liquid fraction increases linearly with time. This occurs because for the
value of wall temperature T
w
used, liquid superheating (sensible heat gain) is unimportant. Almost all of the incoming ther-
mal energy is being used to melt the solid PCM.
6. Conclusion
A physical model using an unstructured finite-volume method and an enthalpy porosity technique is developed to study
the effect of natural convection on the melting process around horizontal circular cylinder. Two boundary conditions were
investigated one of constant wall temperature over the surface of the cylinder and the other of constant heat flux. It is found
that, independent of the boundary condition, melting of PCM in the bottom part is very inefficient because the energy
charged to the system is mainly transferred to the upper part of the cylinder by the convective flow in the melt.
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