DOI: 10.1002/er.5718
RESEARCH ARTICLE
Experimental and numerical investigation of the melting
process and heat transfer characteristics of multiple phase
change materials
Wei Li1,2,3
| Jun Wang1 |
Xu Zhang4 | Xueling Liu2
| Hongbiao Dong3
1
School of Energy and Safety Engineering,
Tianjin Chengjian University, Tianjin,
China
2
Key Laboratory of Efficient Utilization of
Low and Medium Grade Energy (Tianjin
University), Ministry of Education,
Tianjin, China
3
Department of Engineering, University of
Leicester, Leicester, UK
4
Tianjin Key Laboratory of Advanced
Mechatronics Equipment Technology,
Tiangong University, Tianjin, China
Summary
The melting and heat transfer characteristics of multiple phase change materials (PCMs) are investigated both experimentally and numerically. Multiple
PCMs, which consist of three PCMs with different melting points, are filled
into a rectangle-shaped cavity to serve as heat storage unit. One side of the cavity is set as heating wall. The melting rate of multiple PCMs was recorded
experimentally and compared with that of single PCM for different heating
temperatures. A two-dimensional mathematical model to describe the phase
change heat transfer was developed and verified experimentally. The properties of multiple PCMs, including the effect of the melting point difference
Correspondence
Wei Li, School of Energy and Safety
Engineering, Tianjin Chengjian
University, Tianjin 300384, China.
Email: anplw@126.com, anplw@tcu.
edu.cn
(combined type), thermal conductivity, and latent heat, on the heat transfer
Funding information
China Scholarship Council, Grant/Award
Numbers: 201908120031, 201908120062;
Excellent Scientific Special Commissioner
Foundation of Tianjin, Grant/Award
Number: 18JCTPJC60200; Ministry of
Education of China, Grant/Award
Number: 201604-505; Opening Funds
from the Key Laboratory of Efficient
Utilization of Low and Medium Grade
Energy; Project of Natural Science
Foundation of Tianjin, Grant/Award
Numbers: 17JCZDJC31400,
17JCYBJC20800, 15JCYBJC48600;
National Natural Science Foundation of
China, Grant/Award Number: 51805369
PCMs melt faster than the single PCM, and the multiple PCMs, with the melt-
performance of the PCM were analyzed numerically. The results show that,
the melting time decreases before it increases, with an increasing melting point
difference for the multiple PCMs. In addition, the melting point decreases with
increasing distance from the heating wall. Most of these types of multiple
ing point arranged as 322 K/313 K/304 K, has the shortest melting time in this
study. The melting rate of the multiple PCMs, 322 K/313 K/304 K, accelerates
faster than for the single PCM as the thermal conductivity, latent heat, and
heating wall temperature increase. Finally, generalized results are obtained
using a dimensionless analysis for both single and multiple PCMs.
KEYWORDS
dimensionless analysis, heat storage, melting rate, multiple phase change materials,
thermal conductivity
1 | INTRODUCTION
Abbreviations: HTF, heat transfer flow; PCM, phase change material;
Nu, Nusselt number; Ra, Rayleigh number; Ste, Stefan number; Fo,
Fourier number.
Int J Energy Res. 2020;1–14.
The trade-off between time and space limitations, which is
typically encountered with energy production, can be
improved by utilizing the latent heat of phase change materials (PCMs). These materials are very suited to both thermal
wileyonlinelibrary.com/journal/er
© 2020 John Wiley & Sons Ltd
1
2
energy storage and release, and to improve the reliability of
energy conversion systems.1 However, a low thermal storage
efficiency is associated with most PCMs, mainly by reason
of their low thermal conductivity.2 To alleviate the problem,
many studies were conducted, including adding conductive
nanoparticles,3-6 adopting fins,7-10 porous materials/
foams,11-14 microcapsules,15,16 and multiple PCMs.17
Farid18,19 suggested the use of a range of different
PCMs, with different melting points, to obtain phase
change heat storage units. Numerical simulation and experimental data indicate that both charge and discharge rates
can be improved by using thoroughly selected multiple
PCMs. This study has received extensive attention and is
considered to describe a new method to improve the performance of latent heat storage systems. When multiple PCMs
undergo a phase change, several different PCMs experience
the phase change simultaneously. When the theoretical
“homogeneous phase change process” occurs,20 the thermal
resistance keeps constant throughout the whole process.
This accelerates the heat transfer significantly.
There are two arrangement modes for multiple PCMs.
One is to arrange PCMs parallel to the direction heat
transfer flow (HTF). The other mode is to arrange PCMs
perpendicular to the direction of HTF.
Current studies focus on the first mode. For example,
Adebiyi, Gong et al21-24 studied, numerically, a heat storage
system with five different types of PCMs. Their results show
that a suitable phase change temperature distribution accelerates the melting rate significantly. When the latent heat
of the PCM decreases, as the melting point increases, the
required time could be reduced by about 13% to 26% compared to single PCMs. A heat-sink model, which had several kinds of PCMs with different melting points, was
investigated by Hou and Cui.25,26 It was found that the
mass of the heat sink could be reduce by using multiple
PCMs than single PCM, and the operation parameters had
a significant impact on the heat sink's performance.
Aldoss27 proved, experimentally, that the charge/discharge
rate of multiple PCMs was larger than for a single PCM,
and the heat storage performance improved as the number
of multiple PCMs increased. Seeniraj and Narasimhan28
studied, numerically, the impact of multiple PCMs on the
performance of a solar energy latent heat storage device.
The results show that the melting rate of a single PCM was
significantly lower than that of multiple PCMs. Moreover,
the HTF outlet temperature was almost constant. This
study used the effective thermal conductivity, and a specific
relationship with the Rayleigh number, to replace the
effect, which the natural convection of the liquid PCM has,
on the melting properties. Adine and Qarnia29 introduced
the concept of thermal energy storage efficiency to analyze
the thermal behavior of two PCMs. These were arranged in
the direction of the HTF to form a shell-and-tube phase
LI ET AL.
change thermal energy storage unit. The group studied,
quantitatively, the effect of the HTF inlet temperature and
flow rate on the heat storage system performance. Kanzawa
and Farid30 studied the performance of a shell-and-tube
heat storage unit by investigating the available energy.
Their results show that the heat transfer within the unit
was more uniform, when the melting points of the PCM
decreased in the direction of the HTF. He and coworkers31
used the method to establish a mathematical model for the
PCMs heat storage units, with different melting temperatures, which is based on the enthalpy method. They32 also
found a mathematical model, based on exergy efficiency, to
study the charge/discharge processes of three PCMs.
Studies that focus on arranging the PCM perpendicular to the heat transfer direction, however, are relatively
rare. Gong and Mujumdar33 studied a phase change heat
storage system with one-dimensional laminated composite PCMs. Instead of arranging PCMs in the direction of
HTF, the PCMs were arranged perpendicular to the flow
direction of the HTF. The results show that this method
could also increase the charge/discharge rate of the heat
storage system. Influenced by this study, Wang et al20,34
proposed the idea of a “homogeneous phase change
process.” It was found that the phase change temperature
should be distributed according to a quadratic parabola
in the vertical HTF flow direction to enable homogeneous phase change in PCMs. The group35 also used
experimental methods to analyze the heat storage efficiency of cylindrical heat storage units. It was found that
the efficiency of a heat storage unit, which was filled with
multiple PCMs, was significantly higher than for single
PCMs. Shaikh and Lafdi36 studied the phase change heat
transfer processes of different types of multiple PCMs.
Their results show that the total heat storage rate, for
absorption using a single PCM, was significantly lower
than the heat storage rate, for absorption using multiple
PCMs. Fang and Chen37 quantitatively analyzed the mass
of multiple PCMs shell-and-tube thermal energy storage
units to study their thermal storage capabilities. The group
found that the mass fraction for each PCM, which was
used in a multiple PCMs heat storage system, affects both
the charge and discharge rate of the heat storage system
significantly. Also, the melting temperature of the PCM
alters the heat storage unit's performance. Taghilou et al38
explored the phase change process in a double tube multiple PCMs heat exchanger, which had two separate sections
and contained two PCMs with different melting points.
The literature, as described above, indicates that,
our understanding, in particular, of the second arrangement mode for multiple PCMs is still insufficient. Few
researchers studied the effects of the thermophysical properties (melting point, thermal conductivity) and operating
parameters (such as the HTF temperature) on the phase
LI ET AL.
3
change and heat transfer of multiple PCMs, which are
arranged perpendicular to the direction of HTF. Therefore,
to better understand the phase change and heat transfer
characteristics of multiple PCMs, the melting process of
multiple PCMs, which are situated inside a rectangular cavity, where one side of the rectangular cavity is at a constant
temperature, was simulated in this study. In addition, a
corresponding experiment was performed to verify the
model. Compared with a single PCM, the effects of PCM
combination, thermal conductivity, and latent heat of
multiple PCMs, and the heating surface temperature on
the melting rate and heat transfer of the PCM were analyzed
both numerically and experimentally. Some generalized
results could be obtained using dimensionless analysis.
1. Each PCM was not mutually melted and isotropic.
2. The Boussinesq approximation was used to simulate
natural convection due to density difference in the liquid PCM.
3. All the PCMs' thermophysical properties kept constant except the melting point.
We then created a mathematical model based on
the assumptions, which can describe the continuity
equations, momentum and energy equations during the
charge/discharge process as follows:
For the PCM:
The continuity equation is:
∂u ∂v
+ =0
∂x ∂y
2 | PHYSICAL AND
MATHEMATICAL MODEL
2.1 | Physical model
To highlight the key points and simplify the study, a
short section of heat accumulator was selected as the
research object. The ratio of tube length to diameter was
not large, the inlet and outlet temperature of the HTF
were considered identical, that is, the temperature of
the wall remained constant. This way, the study of the
heat storage device could be transformed from a threedimensional problem to a two-dimensional problem. This
reduced the difficulty of the analysis significantly.
The physical model is a cavity that is divided into
three equal sections by two plates, as shown in Figure 1.
The thickness of the plate is δ. One side of the cavity represents the tube wall, where the temperature of heating
wall was constant. The other walls were adiabatic. In the
three chambers of the cavity, PCMs were placed that
have different melting points.
The momentum equation is:
in x-direction
∂u
∂u
∂u
1 ∂p u ∂ 2 u ∂ 2 u
+ Sx
+u +v = −
+
+
∂t
∂x
∂y
ρ ∂x ρ ∂x 2 ∂y2
ð2aÞ
in y-direction
∂v
∂v
∂u
1 ∂p u ∂ 2 v ∂ 2 v
+u +v = −
+
+
+ Sy
∂t
∂x
∂y
ρ ∂x ρ ∂x 2 ∂y2
ð2bÞ
The energy equation is:
2
∂H
∂v
∂H
∂ T ∂2 T
+u +v
=α
+ 2
∂t
∂x
∂y
∂x 2
∂y
ð3Þ
In these equations, ρ is the density of paraffin, p is the
pressure, u is the velocity component in the x direction,
v is the velocity component in the y direction, and H is
the enthalpy and as defined by the following equation.
H = h + ΔH 0
2.2 | Mathematical model
For the convenience of the study, some assumptions were
made for the mathematical model:
ð1Þ
ðT
h = href +
Cp dT
ð4Þ
ð5Þ
T ref
ΔH 0 = f ΔH
ð6Þ
In Equation (5), the reference enthalpy is represented
by href, and ΔH represents the specific melting enthalpy.
Furthermore, S represents the momentum source term
that can be obtained using the following equations.
FIGURE 1
Schematic of the physical model
Sx = −Að f Þu
ð7Þ
Sy = −Að f Þv
ð8Þ
4
LI ET AL.
Að f Þ =
C ð1 − f Þ2
f3 +ε
ð9Þ
where ε is set to 0.001 to make sure it is not divided by
0. C is a constant and set to 106. Furthermore, f is the liquid fraction and defined as:
8
>
>
>
<
f=
0
T < Ts
T −T s
Ts < T < Tl
>
T
l −T s
>
>
:
1
T > Tl
ð10Þ
For the plate:
2
∂T plate
∂ T plate ∂ 2 T plate
λ
=
+
∂t
∂y2
∂x 2
ρc plate
ð11Þ
The solver used unsteady, implicit, and 2D methods.
Furthermore, the effect of gravity was considered. The
Boussinesq model was used to account for the buoyancy
drive in the phase change region. Second-order Upwind
scheme was applied for the discrete equation. Furthermore,
pressure-velocity coupling scheme was the SIMPLE. The
underrelaxation factors for momentum, pressure, energy,
correction, and liquid fraction were set to 0.7, 0.3, 1.0, and
0.9, respectively. The convergence criterion was 10−5 for
the continuity and momentum equations, and it was 10−7
for the energy equation. The time step was set to 1 second.
When the total number of grids was 28 800, the accuracy
requirements of the modeling results could be satisfied.
Because the copper inside the device was thin, the copper
plate was divided into smaller parts with a spacing of 0.2.
3 | EXPERIMENTAL
With the boundary conditions:
3.1 | Experimental setup
x = 0,
u = v = 0,
x = X 1 −δ,
u = v = 0,
−λPCM1
x = X 2 −δ,
u = v = 0,
−λPCM2
x = X 3 −δ,
u = v = 0,
−λPCM3
x = X 1,
u = v = 0,
−λplate
x = X 2,
u = v = 0,
−λplate
x = X3
y=Y
u = v = 0,
u = v = 0,
T = Tw
ð12aÞ
∂T plate
∂T PCM1
= −λplate
∂x
∂x
ð12bÞ
∂T plate
∂T PCM2
= −λplate
∂x
∂x
ð12cÞ
∂T plate
∂T PCM3
= −λplate
∂x
∂x
ð12dÞ
∂T plate
∂T PCM2
= −λPCM2
∂x
∂x
ð12eÞ
∂T plate
∂T PCM3
= −λPCM3
∂x
∂x
ð12fÞ
∂T plate
=0
∂x
∂T PCM
= 0,
∂y
∂T plate
=0
∂x
The experimental system includes a constant temperature
water bath, a multiple PCMs heat storage unit, some thermocouples, a data acquisition instrument, and a computer,
see Figure 2. The multiple PCM heat storage unit was a
rectangular cavity with the dimensions 120 × 60 × 60 mm.
A heating panel was set to act as one wall of the cavity.
The constant temperature water from the constant water
bath flows inside the heating panel to keep its surface temperature constant. The other walls were made of plexiglass,
which enables monitoring the melting of the PCM in the
cavity. The cavity was evenly divided into three parts by
two copper plates, with a thickness of 2 mm, and three
kinds of PCMs, with different melting points were, respectively, filled into the three parts. The outer side of the
rectangular cavity was covered with thermal insulation
material to reduce heat dissipation.
To keep the heating wall at a constant temperature,
the water velocity was set to very high, and thermocouples were placed at different positions on the heating
panel to monitor its surface temperature.
ð12gÞ
3.2 | Material properties
and experimental procedures
ð12hÞ
In this paper, the PCMs are three paraffins with different
melting points. A differential scanning calorimeter was
used to detect the melting point, latent heat, and specific
heat of the paraffins. In the experiments, there were
slight differences between the physical properties of the
three PCMs. However, in order to fully study the effects
of the experiment on the PCM, these differences were
2.3 | Numerical method
The physical model with the square-shaped cavity has a
simple structure and suitable for a quadrilateral mesh.
LI ET AL.
5
ignored in the numerical simulation, and the physical
properties shown in Table 1 were used.
After the cavity had been heated for a period of time,
a part of the PCM was melted into a liquid, and the color
of liquid PCM was different from that of the solid PCM.
The melt fraction of the PCM in the cavity was calculated
by using the software ImageJ. The calculation principle
F I G U R E 2 The experimental setup [Colour figure can be
viewed at wileyonlinelibrary.com]
TABLE 1
mainly used the software to pick up the liquid PCM part
and calculate the area of the liquid PCM as A. The total
area of the PCM in the cavity was B, which means that
melt fraction can be expressed as f = A/B.
Experimental procedure:
1. The three PCMs of equal mass were melted and filled
into three chambers of the cavity. This was done in a
way that the melting point decreased gradually, away
from the heating panel.
2. The rectangular cavity filled with multiple PCMs, was
kept at ambient temperature for more than 12 hours
to ensure the PCM temperature was uniform.
3. We performed a constant temperature bath to reach
the set temperature, circulated the constant temperature water into the heating panel, and started the
experiment.
4. The melting process was recorded graphically by a
camera until the PCMs melted completely.
5. We put the liquid PCMs at ambient temperature, for
enough time to let the PCM solidify and ensure a uniform temperature distribution. The water bath was set
to a new temperature (348, 353, and 358 K) and the
procedures 3 and 4 were repeated.
6. The PCM melt fraction was calculated using the
ImageJ software, and the effect of the heating wall
temperature on the melting time was studied.
Thermophysical properties of the used paraffins and
3.3 | Uncertainty analysis
copper
Property
PCM1
PCM2
PCM3
Copper
ρ (g/cm )
850
850
850
8978
Cp (J/kgK)
2000
2000
2000
381
3
λ (W/mK)
0.2
0.2
0.2
387.6
Tm ( C)
35
40
50
—
β (1/K)
7.6 × 10−4
7.6 × 10−4
7.6 × 10−4
—
μ (Pas)
0.004
0.004
0.004
—
H (J/g)
≈210
≈210
≈210
—
Abbreviation: PCM, phase change material.
TABLE 2
To determine the uncertainty of ImageJ software calculation for the PCM melt fraction during the experiment,
polygons with different side lengths were drawn and colored, and printed as pictures with the 1:1 ratio. The area
of these polygons could be easily obtained by analytical
method. The printed pictures were scanned into a computer, and the area of these images was calculated using
the ImageJ software. A comparison of analytical area and
calculated area was performed to get the uncertainty of
software processing method, see Table 2.
Comparison between the software calculated area and the theoretical area
Graph
Length or
radius
Theoretical
area
Software calculated
area
Uncertainty
(%)
Circle
1
3.1415
3.16144
0.63
Regular triangle
1.73
1.2975
1.31536
1.35
Square
1.41
1.994
2.01936
1.25
Regular pentagon
1.18
2.3866
2.41616
1.22
Hexagon
1
2.598
2.62688
1.09
Total average uncertainty
1.06
6
LI ET AL.
It can be seen from Table 2, the total average uncertainty of the method of ImageJ software was 1.06% after
weighting the uncertainty relative to the theoretical area
of five different shapes.
3.4 | Validation of the numerical model
Figure 3 shows a comparison of numerical predictions
for the contour of the liquid-solid interfaces based on the
experimental results, when the surface temperature of
heating panel was 343 K, and the initial temperature was
283 K. It can be seen, from Figure 3, that the numerical
predictions match the experimental results well. However, the numerical simulation data were higher than the
experimental data during the later period of the melting
process. This is likely because some of the heat storage
units in the experiment lost heat and could not be reliably insulated, while the numerical simulation assumed
complete insulation. Therefore, for the same melting
time, at the later stage of PCM melting, the overall heat
storage in the numerical simulation was higher than
using experimental data.
4 | R ES U L T S A N D D I S C U S S I O N
The melting process of multiple PCMs was
numerically. The simulation was done until
melted completely. The results are shown as a
tion, heat flux of heating surface, location of
simulated
the PCM
melt fracthe solid-
F I G U R E 3 Comparison between experimental and numerical
melt fraction vs time [Colour figure can be viewed at
wileyonlinelibrary.com]
liquid interface, and the Nusselt number. The simulation
was started with an initial temperature (Tini = 283 K)
which was below the PCM melting temperature.
4.1 | Effect of the melting-point
difference of the PCM on melting process
The amount of heat flux through the heating wall determines how much heat is transferred during heat transfer.
The effective heat transfer of the PCM during thermal
energy storage can be obtained using the following formula:
Qtotal = Qsen + Qlat
Qsen =
N
X
ð13Þ
mi Cp,l,i f i T l,i − T m,i − f i,ini T l,i,ini −T m,i
i=1
+ C p,s,i ð1− f i ÞT s,i − 1 −f i,ini T s,i,ini
+ C p,s,i T m,i f i −f i,ini
Qlat =
N
X
mi hsfi f i
ð14Þ
ð15Þ
i=1
hðtÞ =
Qtotal
Surw ðT w −T m ÞΔt
q = hðtÞ ðT w −T m Þ
ð16Þ
ð17Þ
where f is the melt fraction, N represents the number
of phase change heat storage unit cavities (in this paper
N = 3), and the subscript i represents the ith phase
change thermal storage unit. The sum of the latent heat
and sensible heat in the phase change heat storage unit
equals its total heat. The formula to calculate the sensible
heat storage capacity is Equation (14), and the calculation
formula for the latent heat storage capacity is Equation (15). The mass of the PCM is expressed by m, while
T and T represent the temperature and the average temperature, Tm represents the phase change temperature.
The temperature of wall is represented by Tw, the subscript ini is the initial state during heating, and the average heat transfer coefficient is represented by h.
As mentioned above, the thermophysical properties
of all PCMs in this experiment were the same, with
the exception of the melting points. Therefore, a multiple
PCMs system can be characterized by its melting point,
such as multiple PCMs 313 K/313 K/313 K (single PCM),
multiple PCMs 319 K/313 K/307 K, multiple PCMs 328 K/
313 K/298 K, multiple PCMs 307 K/313 K/319 K, multiple
PCMs 298 K/313 K/328 K. It can be seen that all the multiple PCMs have the same average melting points which
LI ET AL.
are equal to the single PCM's. This provides a uniform
standard to analyze the effect of melting point differences
on the melting processes of multiple PCMs. Figure 4 shows
the melt fraction and heat flux vs time for different multiple PCMs at the same heating wall temperature. Figure 4A
shows that, for the multiple PCMs, where the melting point
decreases gradually, away from the heating wall, which is
referred to as “forward multiple PCMs” in this paper, the
complete melting time decreases before it increases with
increasing of melting point difference. For example, the
melting time of multiple PCMs 322 K/313 K/304 K, which
was the fastest melting material of all multiple PCMs,
was reduced by 9.6% compared with the single PCM.
On the other hand, the melting times of multiple PCMs
319 K/313 K/307 K and multiple PCMs 325 K/313 K/304 K
were reduced by 8.5% and 7.8%, respectively. However, for
the multiple PCMs where the melting point increases away
F I G U R E 4 Melt fraction and heating wall heat flux for
different multiple phase change materials (PCMs): A, melt fraction
and B, heat flux [Colour figure can be viewed at
wileyonlinelibrary.com]
7
from the heating wall, which is called “backward multiple
PCMs” in this paper, the complete melting time increases
with increasing melting point difference. For example,
the melting time of multiple PCMs 310 K/313 K/316 K,
multiple PCMs 307 K/313 K/319 K and multiple PCMs
304 K/313 K/322 K were 7.4%, 17.9%, and 30.2% higher
than for the single PCM. Therefore, to speed up the melting
process and improve the heat storage performance, forward
multiple PCMs with suitable melting point differences
should be selected.
Figure 4A also shows that the forward multiple PCMs
melt only slowly in the early melting stage but faster later.
By contrast, the backward multiple PCMs melt fast in
the early stage but slowly later. This is because the melting
point of the forward multiple PCMs, near the heating
wall, is high. This means that the temperature difference
between the heating surface and the adjacent PCM is
small, which results in lower heat transfer and lower melting speed during the early melting stage. With time passing, the temperature at the interface between the front
PCM and the adjacent PCM was higher than the melting
point of the latter. The latter also began to melt, which produced more than one melting interfaces, see Figure 3. As
a result, natural convection could be triggered simultaneously in every PCM, which increased both the heat
transfer coefficient and heat flux. This is also shown in
Figure 4B. The heat flux decreased rapidly from its initial
high value before it increased, which was due to the direct
contact between the PCM and the heating wall initially.
This occurred because the thermal resistance between
PCM and the heating wall increased sharply with the melting of PCM, and the heat flux decreased correspondingly.
When the melted PCM increased by a certain amount, natural convection began to have an important effect on the
increase of the heat flux.
The heat flux curves of some multiple PCMs, especially the multiple PCMs 322 K/313 K/304 K with the
shortest melting time increased slightly a second time.
This is due to the natural convection of the melted
PCMs with medium and low melting points, which
increased the heat transfer coefficient later. Other multiple PCMs did not show a second increase. The reason
for this was that the heat flux increment, which was caused by natural convection cannot offset the decrement
caused by the increase in thermal resistance due to more
melted PCM.
The main purpose of this paper is to attempt to
improve the performance of a heat storage device with
multiple PCMs. The multiple PCMs 322 K/313 K/304 K
can be considered the most suitable multiple PCMs in
this study. To highlight the key points, only multiple
PCMs 322 K/313 K/304 K and the single PCM are compared in the following sections.
8
4.2 | Effect of the thermal conductivity
of PCM on the melting process
Figure 5 shows the time evolution of both melt fraction
and heat flux for the single PCM and multiple PCMs with
different thermal conductivities (0.2 W/mK, 0.5 W/mK,
1.0 W/mK). Figure 5A indicates that, all PCMs melt fast
with increasing thermal conductivity, and multiple PCMs
shows a slightly faster melting rate than the single PCM.
For example, as the thermal conductivity raised from
0.2 W/mK to 0.5 W/mK and 1.0 W/mK, the melting
time of single PCM was shortened by 48.6% and 70%,
respectively. On the other hand, the melting times of
multiple PCMs decreased by 49% and 73%, respectively.
It can also be drawn from Figure 5A that, when the
thermal conductivity was 0.2 W/mK, the melting time of
multiple PCMs was shortened by 9.6% than that of the
single PCM. Furthermore, when the thermal conductivity
F I G U R E 5 Melt fraction and heat flux at the heating wall for
the single phase change material (PCM) and multiple PCMs with
different thermal conductivities: A, melt fraction vs time and B,
heat flux vs time [Colour figure can be viewed at
wileyonlinelibrary.com]
LI ET AL.
increased to 0.5 and 1.0 W/mK, the melting time was
shortened by 10.46% and 19.6%, respectively.
Figure 5B shows that, the higher the PCM's thermal
conductivity is, the higher the heat flux is. The heat flux
decreases with time, and the higher thermal conductivity
of the PCM accelerates the heat flux decrease. With the
same thermal conductivity, the multiple PCMs heat flux
is less than for the single PCM during the initial melting
stage. However, it is higher than a single PCM during the
later melting stage. In general, a multiple PCM can make
heat transfer more uniform.
4.3 | Effect of the latent heat of the PCM
on the melting process
Figure 6 presents the time evolution of the melt fraction
and heat flux for the single PCM and multiple PCMs, for
F I G U R E 6 Melt fraction and heat flux at the heating wall for
single phase change material (PCM) and multiple PCMs with
different latent heat: A, melt fraction vs time and B, heat flux
vs time [Colour figure can be viewed at wileyonlinelibrary.com]
LI ET AL.
different latent heats (ΔH = 180, 210, 240, 270 kJ/kg).
According to Figure 6A, the multiple PCMs have less melting time than the single PCM for the same latent heat.
When the latent heat was 180, 210, 240, and 270 kJ/kg, the
melting time for the multiple PCMs was reduced by 9.3%,
9.6%, 9.65%, and 10.5% compared to the single PCM.
It can be seen from Figure 6B that, the heat flux
increases slightly with the latent heat, for both single PCM
and multiple PCMs. The heat flux for single PCMs is always
higher than for multiple PCMs with the same latent heat
during the early stage, and conversely during the late stage.
4.4 | The effect of heating temperature
on the melting process
Figure 7 shows the melt fraction and heat flux of the
single PCM and multiple PCMs for different heating
9
temperatures over time. As expected, a higher heating
temperature reduces the melting time, considerably see
Figure 7A. For example, when the heating surface temperature was 348 K, it took 316 minutes for the single
PCM to complete melting. However, it took only 273 or
241 minutes, respectively, when the heating temperature
was 353 or 358 K. This represents a reduction by 13.6% or
23.7%. The multiple PCMs needed 284 minutes, when the
heating temperature was 348 K, and 244 or 214 minutes,
respectively, when the heating temperature was 353 or
358 K. This represents a reduction by 10.9% or 21.9%. It
can also be seen that, the multiple PCMs melts faster
than the single PCM with increasing heating temperature. The melting time of the multiple PCM was 10.1%
shorter than that of single PCM when the heating temperature was 348 K. The former was 10.9% or 11.2%
shorter than the latter when the heating temperatures
were 353 or 358 K.
As shown in Figure 7B, for the same heating temperature, the heat flux of the multiple PCMs was lower than
for the single PCM in the early stage but, higher in the
later stage. This means that multiple PCMs make the heat
transfer more uniform and decrease the melting time.
Figure 8 shows the contour images of the melting
process for single PCM and multiple PCMs at different
heating temperatures. Compared to the single PCM, for
the same heating temperature, the multiple PCMs melted
slower during the initial period, but faster during the
later period. Multiple PCMs melt faster in general.
Figure 9 indicates that, the trend of change for the
liquid-solid interface is inclined from lower left to the
upper right, and the trend of change is clearly visible for
large melt fractions. This is because of the growing influence of natural convection as the liquid region increases.
The upward flow at the heating wall and the downward
flow at the solid PCM interface generate, overall a clockwise flow, which improves the heat transfer. The internal
curve slope of multiple PCMs was greater than that of the
single PCM. At the same time, multiple PCMs show more
vortexes, which also increase the natural convection.
4.5 | Dimensionless analysis of the
melting process
F I G U R E 7 Melt fraction and heat flux at the heating wall for
the single phase change material (PCM) and multiple PCMs with
different heating temperatures: A, melt fraction vs time and B, heat
flux vs time [Colour figure can be viewed at
wileyonlinelibrary.com]
It can be seen from Figures 4-7 that the single PCM and
the multiple PCMs have similar melting characteristics
under different conditions. In other words, it should be
helpful and possible to obtain generalized results via
dimensionless analysis.
For the generalized result, two dependent dimensionless parameters are used: the melt fraction of PCM (f )
and the Nusselt number (Nu), which are defined as:
10
LI ET AL.
F I G U R E 8 Solid-liquid interface image for the single phase change material (PCM) and multiple PCMs during the melting process for
a heating temperature of 348 K/353 K/358 K at different times: A, single PCM and B, multiple PCMs [Colour figure can be viewed at
wileyonlinelibrary.com]
F I G U R E 9 Streamline diagram for the single phase change material (PCM) and multiple PCMs during the melting process for a
heating temperature of 348 K: A, single PCM and B, multiple PCMs [Colour figure can be viewed at wileyonlinelibrary.com]
LI ET AL.
11
f=
V melt fraction of PCM
V total mass of PCM
Nu =
qw
L
Tw − Tm λ
ð18Þ
ð19Þ
In Equation (19), the heat flux at the heating wall
is represented by qw, the characteristic length is L,
and the PCM thermal conductivity coefficient is λ. The
heat, which is stored in the PCM at a certain time, can be
expressed using the melt fraction. In other words, the
heat, which is transferred from the heating wall to the
PCM, can be expressed using the Nusselt number.
In addition, several dimensionless parameters are
needed,
Fo =
λ t
ρC p L2
ð20Þ
F I G U R E 1 0 Melt fractions as a function of Fo, for single
phase change material (PCM) and multiple PCMs: A, single PCM
and B, multiple PCMs [Colour figure can be viewed at
wileyonlinelibrary.com]
Cp ðT w −T m Þ
4H
ð21Þ
ρCp gβL3 ðT w − T m Þ
λ
ν
ð22Þ
Ste =
Ra =
where the Fourier number (Fo) represents the dimensionless time of unsteady heat transfer process. The
Stefan number (Ste) reflects the temperature difference
between heating wall and the PCM. Ste varies from 0.33
(ΔT = 35 K) to 0.42(ΔT = 45 K) in this study. The intensity of natural convection in liquid PCM is expressed
using the Rayleigh number (Ra).
Figure 10 shows the curves for the melt fraction vs
Fo for multiple PCMs and single PCM with different
Ste. The figure indicates that no generalization was
achieved.
The combination of Fo and Ste was sufficient to melt,
using only conduction. However, the heat transfer in the
PCM used in the present study is also affected by natural
convection, which means it is necessary to include Ra in
the analysis. Therefore, the dimensionless combination
parameter FoSteRa0.12 was used to obtain the generalized
results, see Figure 11.
Figure 12 shows the change for the Nu vs the
combination parameter FoSteRa0.12, for single PCM and
multiple PCMs. This produces a uniform result. The
following conditions are true in this study: 0.33 < Ste <
0.42, 4431 < Ra < 5697, 0 < f < 1, single PCM 0 <
FoSteRa0.12 < 5.14, multiple PCMs 0 < FoSteRa0.12 < 4.58.
The melt fraction as a function of FoSteRa0.12,
for multiple phase change materials (PCMs) and single PCM
[Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 11
12
LI ET AL.
along the heating wall (away from the constant wall
temperature), which can enhance the PCM's heat
transfer and shorten its melting time.
2. Compared with the single PCM, the multiple PCMs
has a faster melting rate with increasing thermal conductivity, a reduced melting time for the same latent
heat, and melts faster with increasing heating temperature. The multiple PCMs make the heat transfer
more uniform and shorten the melting time, which
improves the performance of the heat storage unit.
3. A dimensionless analysis was carried out and presented using the melt fraction and Nusselt number as
function of, a combination parameter, which consists
of the Fourier, Stefan, and Rayleigh number. This way
it was possible to obtain (within the scope described
in this paper) generalized results for both single PCM
and multiple PCMs.
Nu as a function of FoSteRa0.12, for multiple
phase change materials (PCMs) and single PCM: A, single PCM
and B, multiple PCMs [Colour figure can be viewed at
wileyonlinelibrary.com]
FIGURE 12
5 | C ON C L U S I ON S
Experimental and numerical studies were carried out to
investigate melting and heat transfer in multiple PCMs,
in a rectangular cavity heat storage unit, with a constant
wall temperature. The properties of PCMs, including
the melting point difference, thermal conductivity, latent
heat, heating temperature, together with the melting
and heat transfer of the multiple PCMs, were analyzed
and compared with the single PCM. The main conclusions are:
1. The order for the PCM has a great impact on the melting time of the multiple PCMs. Both the melting process and heat transfer can be improved using forward
multiple PCMs. Then, for a reasonable melting point
difference, the melting point difference decreases
A C KN O WL ED G EME N T S
This research was funded by National Natural Science
Foundation of China, grant number 51805369; the
Project of Natural Science Foundation of Tianjin,
grant numbers 15JCYBJC48600, 17JCYBJC20800, and
17JCZDJC31400; the Opening Funds from the Key Laboratory of Efficient Utilization of Low and Medium Grade
Energy (Tianjin University); Ministry of Education of
China, grant number 201604-505; and Excellent Scientific
Special Commissioner Foundation of Tianjin, grant number 18JCTPJC60200. W. L. (201908120062) and X. Z.
(201908120031) gratefully acknowledge financial support
from China Scholarship Council.
NOMENCLATURE
Symbols
cp
H
h
ΔH
m
f
C
S
T
ΔT
T
A
L
p
ε
g
t
Q
specific heat at constant pressure (J/kgK)
total enthalpy (J/kg)
sensible enthalpy (J/kg)
latent heat (J/kg)
mass of PCM (kg)
melt fraction of PCM
mushy zone constant
source term
temperature ( C)
temperature difference ( C)
average temperature ( C)
mushy zone constant
characteristic size
pressure (Pa)
small number
gravity (m/s2)
time (s)
heat (W)
LI ET AL.
q
h
Sur
N
u, v
X1 ,
X2, X3
13
heat flux (W/m2)
average heat transfer coefficient (w/m2K)
surface area (m2)
number of phase change heat storage unit
cavities
velocity component in the x, y coordinates (m/s)
distance from origin (mm)
Greek letters
λ thermal conductivity (W/mK)
α thermal diffusivity (m2/s)
μ dynamic viscosity (kg/ms)
ν kinematic viscosity (m2/s)
β thermal expansion coefficient (1/K)
ρ density (kg/m3)
δ copper plate thickness (mm)
Subscripts
s
solid
l
liquid
w
wall
m
melt
ref
reference
ini
initial
sen
sensible
lat
latent
i
ith phase change thermal storage unit
x, y
coordinate in x, y direction
plate plate
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How to cite this article: Li W, Wang J, Zhang X,
Liu X, Dong H. Experimental and numerical
investigation of the melting process and heat
transfer characteristics of multiple phase change
materials. Int J Energy Res. 2020;1–14. https://doi.
org/10.1002/er.5718