numerical analysis of latent thermal energy storage in a...
TRANSCRIPT
Bachelor of Science Thesis
KTH School of Industrial Engineering and Management
Energy Technology EGI-2017
SE-100 44 STOCKHOLM
Numerical Analysis of
Latent Thermal Energy Storage in a Cavity
Robert Olrog
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Bachelor of Science Thesis EGI-2017
Numerical Analysis of
Latent Thermal Energy Storage in a Cavity
Robert Olrog
Approved
2017-05-22
Examiner
Viktoria Martin
Supervisor
Amir Abdi
Commissioner
Contact person
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Abstract
Latent Thermal Energy Storage (LTES) has drawn attention because the technology is a
simple and cost-efficient method to store large amounts of energy. Latent energy is either
stored or released when the material inside the LTES undergoes phase-change. As LTES
operates at a constant temperature it can be utilized in several fields such as waste heat
management, building insulation, storage of solar energy and electronic cooling to name a
few. An obstacle to widespread use of LTES is its low energy recharge and discharge rate due
to the phase-change materials (PCM) thermophysical properties, namely thermal conductivity.
PCMs such as fatty acids, salt hydrates and paraffins are potential materials for domestic
application because of their melting temperature and are especially affected by low thermal
conductivity.
The objective is to numerically model a Latent Thermal Energy Storage and simulate the
melting and solidification process with different boundary conditions, and afterwards analyze
how it impacts natural convection, heat transfer rates, and the solid-liquid interface. Special
attention will be given to natural convection as a change in its strength can have a large
impact on heat transfer. Optimization and enhancing the rate of heat transfer is important as it
improves LTES effectiveness.
The geometry used in the numerical model is two-dimensional with 50 mm in width and
120 mm in height. The heat transfer surface area is the 120 mm wall. Four cases are
examined; two of which are melting and two of solidification. The geometry is identical in all
cases but placed in either a vertical or horizontal orientation.
Transient simulations are performed using ANSYS Fluent which is a computational fluid
dynamics software tool. The geometrical model used for ANSYS mimics the experimental
setup that Kamkari and Shokouhmand (2014) built to analyze melting in a rectangular
enclosure. This allows for a comparison between numerical data and experimental
observations in one of the melting cases.
The comparison between the numerical and experimental results show good agreement as the
solid-liquid interface is nearly identical and the amount of liquid in the enclosure differs by
less than 5 percent after two-hundred minutes. Natural convection is present in all cases to a
varying degree, and the amount of phase-change correlates to its strength and duration.
During melting convection is the main mode of heat transfer in both orientations, but in the
vertical case the strength tapers off as time progresses. The horizontal orientation produces a
natural convection for the entire duration of the simulation therefore leading to a higher
melting rate.
The solidification process entails conduction as the dominant mode of heat transfer. In the
horizontal orientation there is no detectable natural convection. The vertical position shows
convection in the early stages of solidification but disappears quickly. As a result there is a
higher amount of solid material in the vertical orientation by the end of the simulation.
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Sammanfattning
Latent värme energilagring(LTES) har fått ökad uppmärksamhet eftersom teknologin är en
simpel och kostnad-effektiv metod att lagra stora mängder energi. Latent värme lagras eller
frigörs när materialet inuti LTES byter fas. Eftersom LTES bibehåller en konstant temperatur
har den flera användningsområden inom isolering, solfångare och elektronisk kylning för att
nämna några. Ett hinder till utspridd användning av LTES är den långsamma laddnings- och
urladdningshastigheten på grund av fasbytesmaterialets(PCM) ämnesegenskaper, nämligen
termisk konduktivitet. Låg termisk konduktivitet drabbar PCM som fettsyror, salthydrater och
paraffin som är potentiella material för många LTES applikationer på grund av deras
smälttemperatur.
Målet är att numeriskt modellera en LTES och simulera smält och stelningsprocessen med
olika randvillkor, och därefter analysera hur dessa påverkar naturlig konvektion,
värmeöverföring och smältkonturen. Extra uppmärksamhet ges till naturlig konvektion
eftersom en ändring i dess styrka kan ha en stor påverkan på värmeöverföringen. Att försöka
optimera värmeöverföringen är viktig då det kommer öka LTES attraktivitet för termisk
energilagring.
Geometrin som används i den numeriska modellen är två-dimensionell med 50 mm i bredd
och 120 mm i höjd. Värmeöverföringsarean är väggen som är 120 mm. Fyra fall examineras:
två smältfall och två stelningsfall. Geometrin var identisk under alla fall men placeras i
antingen en vertikal eller horisontell orientering.
Transienta simuleringar utfördes i ANSYS Fluent som är en computational fluid dynamics
mjukvara. Modellen liknar Kamkari, Shokouhmand (2014) experimentella uppsättning som
byggdes för att analysera PCM smältning i en rektangulär behållare. Detta gjordes för att få
möjligheten till att jämföra numerisk data till experimentella observationer i ett av fallen.
Jämförelsen mellan simuleringens och experimentets resultat visar god likhet eftersom både
smältkonturen och mängden vätska i behållaren är snarlika, samt skiljer sig mindre än 5%
efter två-hundra minuter. Naturlig konvektion närvarar i alla fall, och mängden fasbyte
korrelerar till dess styrka och varaktighet. Under smältning är konvektion den huvudsakliga
drivaren av värmeöverföring i båda orienteringar, men i det vertikala fallet minskar styrkan
under simuleringen. Det horisontala fallet producerar konvektion under hela simuleringen
vilket leder till en högre smälthastighet jämfört med den vertikala.
I stelningsprocessen är konduktion den huvudsakliga drivaren av värme. Det horisontella
fallet visar ingen konvektion. I den vertikala positionen finns tecken på konvektion i det tidiga
skedet, men minskar snabbt. Därför finns det mer fast materiel i den vertikala positionen vid
slutet av simuleringen på grund av konvektion vid starten.
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Contents
Abstract ...................................................................................................................................... 3
Figures ........................................................................................................................................ 7
Tables ......................................................................................................................................... 8
Nomenclature ............................................................................................................................. 9
1 Introduction ....................................................................................................................... 11
1.1 Literature Survey ....................................................................................................... 13
1.1.1 Phase-change Material ....................................................................................... 13
1.1.2 Dimensionless numbers ...................................................................................... 14
1.1.3 Numerical formulations ...................................................................................... 14
1.1.4 Solidification and melting characteristics .......................................................... 15
1.2 Objectives .................................................................................................................. 17
2 Model ................................................................................................................................ 18
3 Method .............................................................................................................................. 19
3.1 Meshing ..................................................................................................................... 19
3.2 Numerical Modelling ................................................................................................. 19
3.3 Enthalpy-Porosity Method ......................................................................................... 19
3.4 Governing equations .................................................................................................. 20
3.5 Numerical Procedure ................................................................................................. 21
3.6 Limitations ................................................................................................................. 22
3.7 ANSYS Setup ............................................................................................................ 23
3.8 Parametrical Study ..................................................................................................... 23
4 Comparison between numerical and experimental results ................................................ 24
4.1 Melt Fronts ................................................................................................................ 24
4.2 Temperature Distributions ......................................................................................... 25
4.3 Energy ........................................................................................................................ 26
5 Results and Discussion ..................................................................................................... 27
5.1 Case 1: ....................................................................................................................... 27
5.1.1 Melt Front Evolution .......................................................................................... 27
5.1.2 Nusselt ................................................................................................................ 27
5.1.3 Streamlines ......................................................................................................... 28
5.2 Case 2: ....................................................................................................................... 29
5.2.1 Melt Front&Melt Fraction .................................................................................. 29
5.2.2 Temperature & Point Temperature..................................................................... 30
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5.2.3 Nusselt Number .................................................................................................. 31
5.2.4 Energy ................................................................................................................ 32
5.2.5 Streamlines ......................................................................................................... 32
5.3 Case 3: ....................................................................................................................... 33
5.3.1 Melt Front & Melt Fraction ................................................................................ 33
5.3.2 Temperature Distributions .................................................................................. 34
5.3.3 Energy ................................................................................................................ 35
5.3.4 Streamlines ......................................................................................................... 36
5.4 Case 4 ........................................................................................................................ 36
5.4.1 Melt Front & Melt Fraction ................................................................................ 37
5.4.2 Temperature Distribution ................................................................................... 37
5.4.3 Energy ................................................................................................................ 38
5.4.4 Streamlines ......................................................................................................... 38
6 Conclusion ........................................................................................................................ 40
6.1 Future work ................................................................................................................ 41
References ................................................................................................................................ 42
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Figures
Figure 1: Temperature distributions at the vertical mid-plane of the enclosure for hot wall temperature of 60°C
(Kamkari and Shokouhmand, 2014) ..................................................................................................................... 16
Figure 2:Model Geometry ..................................................................................................................................... 18
Figure 3: ANSYS Fluent software usage flow ...................................................................................................... 22
Figure 4: Melt Front (left side simulation and right side experiment) (Kamkari and Shokouhmand, 2014) ........ 24
Figure 5: Melt Fractions as function of time: Simulation VS Experiments (Kamkari and Shokouhmand, 2014) 25
Figure 6: Temperature Distributions (right side simulation and left side experiment) (Kamkari and
Shokouhmand, 2014) ............................................................................................................................................ 25
Figure 7: Amount of absorbed Energy: Simulation VS Experiments (Kamkari and Shokouhmand, 2014) ......... 26
Figure 8: Case 1, Nusselt number as a function of time – vertical melting ........................................................... 28
Figure 9: Case 1, Streamlines (Left is t=50, Middle t=90, Right t=170)- vertical melting .................................. 29
Figure 10: Case 2, Melt Front and Melt Fraction as a function of time - horizontal melting ................................ 30
Figure 11: Case 2; Temperature Distribution and Point Temperature as a function of time – horizontal melting 31
Figure 12: Case 2, Location of Points in horizontal orientation ............................................................................ 31
Figure 13: Case 2, Nusselt as a function of time ................................................................................................... 32
Figure 14: Case 2, The amount of absorbed energy- horizontal melting .............................................................. 32
Figure 15:Case 2, Streamlines (Left is t=50, Middle t=90, Right t=170) ............................................................. 33
Figure 16: Case: 3 Melt Front and Melt Fraction as a function of time ................................................................ 34
Figure 17: Case 3, Temperature Distribution and Point Temperature – vertical solidification ............................. 35
Figure 18:Case 3, The amount of energy discharged from the enclosure – vertical solidification ....................... 36
Figure 19: Case 3, Streamlines (Left is t=50, Middle t=90, Right t=170) ........................................................... 36
Figure 20: Case 4, Temperature Distribution Figure 21: Case 4, Melt Fraction ........................ 37
Figure 22: Case 4 Temperature Distributions and Point Temperature – horizontal solidification ........................ 38
Figure 23: Case 4, The amount of released energy- horizontal solidification ....................................................... 38
Figure 24: Case 4 Streamlines (Left is t=50, Middle t=90, Right t=170 ). No streamlines are drawn ................. 39
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Tables
Table 1: Comparative advantages and disadvantages with organics versus inorganics (Zalba et al., 2003) ......... 13
Table 2: Review of the different scenarios ............................................................................................................ 17
Table 3: Thermophysical properties of lauric acid (Kamkari and Shokouhmand, 2014) ...................................... 18
Table 4: Limitations .............................................................................................................................................. 22
Table 5: ANSYS Fluent Setup .............................................................................................................................. 23
Table 6: ANSYS Case Setup and Initial Conditions ............................................................................................. 23
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Nomenclature
Symbol Designation Unit
cp Specific heat [kJ/kg-K]
𝑘 Thermal Conductivity [W/m°C]
h Enthalpy (kJ/kg)
L Heat of Fusion (kJ/kg)
T Temperature [°C]
𝑚 Mass [kg]
l Characteristic Length [m]
ℎ́ Heat transfer coefficient, [W/𝑚2°C]
𝑁𝑢 Nusselt number
𝛾 Liquid fraction
𝑝 Density [kg/𝑚3]
𝛼 Thermal diffusivity [𝑚2
𝑠]
𝜇 Dynamic viscosity [𝑘𝑔
𝑠∗𝑚]
𝑇𝑠𝑜𝑙𝑖𝑑𝑢𝑠 Solidification temperature [℃]
𝑇𝑙𝑖𝑞𝑖𝑑𝑖𝑢𝑠 Melting temperature [℃]
Q Energy absorbed or released [𝐽]
Pr Prandtl number
L Latent heat of fusion [J/kg]
H Enthalpy [J]
𝛽 Expansion coefficient [1/K]
Acronyms
PCM Phase-change material
TES Thermal Energy Storage
LTES Latent Thermal Energy Storage
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Acknowledgments
The report would not have been possible without the guidance, support, and insightfulness
from my supervisor Amir Abdi and I would therefore like to express my gratitude towards
him.
Robert Olrog
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1 Introduction
Within the recent years renewable energy technology has been solidified as a viable
alternative to fossil fuel production. The threat of climate change, possible depletion of fossil
resources, increased political will and improvement of technology has contributed to the fact
that renewables(including hydropower) now account for 30% of the total global installed
power generating capacity(World Energy Perspectives, 2016). The renewable trend shows no
sign of slowing down, and will likely accelerate if the prediction that renewables will
eventually stand for 60% of the global total capacity holds true(World Energy Outlook, 2016).
Despite the capacity currently being at 30%, renewables only account for 23% of the global
total energy production. This discrepancy arises from the fact renewable production is largely
dependent on intermittent environmental factors and will therefore not run at installed
capacity most of the time. In addition, even when environmental conditions allow for
maximal production the demand might be too low. If the current trend in investing in
renewable technology continues its utilization will require technologies to match the
intermittent production with variation in demand.
One potential solution to reduce the mismatch between supply and demand is Thermal Energy
Storage (TES), which is gaining attention because the technology is a simple and cost-
efficient method to store large amounts of energy. TES stores thermal energy in the form of
either sensible or latent energy. Sensible energy is stored or released in TES when there is a
change in temperature without any phase-change. Latent energy occurs when the energy is
used to melt or solidify the material inside the TES which is referred to as phase-change
material(PCM). Latent Thermal Energy Storage (LTES) is in most cases a more attractive
option than sensible storage. One reason is that the energy required to change phase is in
almost all cases significantly larger compared to changing the temperature of a PCM, which
translates to a larger storage capacity for LTES devices. In addition, latent heat storage
operates at a constant temperature which allows for a larger degree of control when using it
for thermal storage purposes. These properties allows PCM to be utilized in several fields
such as waste heat management, building insulations, storage of solar energy and climate
control of automobiles to name a few (Vijayakumar, Prabhu, 2014). LTES unique selling
point is the ability to store large amounts of thermal energy in a cost efficient and
technological simple way.
An example where a LTES system can contribute to a more sustainable future is to capture
waste heat from industrial processes. Surplus heat accounts for 20-50% of the industrial
energy input, and to re-introduce it into the energy system will increase overall efficiency
(Chiu, Martin 2015). A potential method is allowing TES devices to store the energy and to
release it in a district heating when demand is high. The solution proposed by Chiu and
Martin (2015) would use several mobile-TES units for transportation of surplus heat from the
production site to end district heating plant. In their paper they show that the economic
viability of the solution is dependent on transportation costs and energy storage opportunities.
The authors decide to pick a LTES based mobile unit as technological solution to transport
thermal energy. A potential barrier to implementation of LTES unit is due to slow recharge
and discharge rate. If significant development in LTES heat transfer rate would occur it would
increase the economic and technical viability of this solution, as perhaps they can reduce the
amount of mobile units. Efficient technology for re-use and transportation of surplus heat is
one of the building blocks to reduce unnecessary costs and resource consumption.
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An obstacle to widespread use of PCM, as discussed above, is its low energy discharge and
recharge rate. This is mainly due to the materials thermophysical properties, namely thermal
conductivity. Low thermal conductivity decreases the rate of heat transfer as Fourier’s Law of
Heat Conduction states that they are directly proportional to each other.
𝑄 = −𝑘𝐴𝑑𝑇(𝑥,𝑡)
𝑑𝑥 1
One way to increase the rate of heat transfer is increasing the surface area through the use of
fins or partitions. Thermal conductivity may also be increased by dispersing high conductivity
particles into the PCM, though the technique has not been implemented because of
sedimentation issues.
Conduction will not be the only mechanism to drive heat transfer in the phase-change process.
Natural convection will also occur as the density differences in the liquid will initiate a
motion. The driving force behind this mechanism is buoyancy as the motion is not generated
by a pump or a fan, but only by the density difference due to temperature gradients. As the
hotter fluid rises, cold fluid will replace it. When the cooler fluid heats up it will also start to
rise which is the beginning of natural convection as a continuous process starts. Studying the
natural convection currents will often expose how long it takes to release or recharge a latent
heat storage enclosure. Therefore a lot of research has been undertaken to understand how it
behaves in different geometries and boundary conditions.
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1.1 Literature Survey
1.1.1 Phase-change Material
Two important factors that enable LTES to compete with other forms of energy storage is the
phase-change materials ability to capture large amounts of energy and the fact that they
operate at a constant temperature (Dutil et al., 2011). Low thermal conductivity is considered
a major drawback of PCMs as it inhibits heat transfer and reduces phase-change (Sharif et al.,
2015). To quantify the total amount of latent energy available in a PCM the equation below
can be used:
𝑄 = 𝑚 ∗ 𝐿 2
𝑄 = 𝑇ℎ𝑒 𝑒𝑛𝑒𝑟𝑔𝑦 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑜𝑟 𝑟𝑒𝑙𝑒𝑎𝑠𝑒𝑑 𝑖𝑛 𝑎 𝑝ℎ𝑎𝑠𝑒 𝑐ℎ𝑎𝑛𝑔𝑒
m = 𝑡ℎ𝑒 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑃𝐶𝑀
𝐿: 𝐻𝑒𝑎𝑡 𝑜𝑓 𝐹𝑢𝑠𝑖𝑜𝑛
PCM can in most cases be categorized into two distinct groups which are organics and
inorganics (Giriswamy et al., 2014). Organics are often preferred as they are non-corrosive
and have a low volume change. Inorganics are in most cases corrosive, though they often
exhibit a higher latent heat of fusion. Table 1 shows the comparative advantages and
disadvantages with organics versus inorganics and can originally be found in Zalba et al.
(2003) article.
Table 1: Comparative advantages and disadvantages with organics versus inorganics (Zalba et al., 2003)
Organics Inorganics Advantages Advantages
No corrosives Greater phase change enthalpy Low or none undercooling
Chemical and thermal stability
Disadvantages Disadvantages Lower phase change enthalpy Undercooling
Low thermal conductivity Corrosion Inflammability Phase separation
Phase segregation, lack of thermal stability
The most common materials used for PCM are paraffin waxes, fatty acids and hydrated salts
(Farid et al., 2004). The cost of paraffin waxes is generally been low and they usually have
energy densities of roughly 200kJ/kg. Fatty acids(capric, lauric, palmitic and stearic acids) is
attractive for domestic heat storage purposes as the melting range is usually between 30 to 65
degrees. Their latent heat of fusion is around 180 kJ/kg. Hydrated salts have the highest
energy density of the three (250kJ/kg) but their application is limited due to sub-cooling and
phase segregation.
Giriswamy et al. (2014) argued that a material needs to meet several criteria before being
classified as a PCM which are:
High latent heat of fusion
Clearly determined phase-change temperature
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Avoids super-cooling
Remains chemically stable after several phase-change cycles
Be non-hazardous/corrosive
Be economical
1.1.2 Dimensionless numbers
The use of different dimensionless parameters has been important in studying PCM and LTES
as it enables researches to compare results between experiments (Agyenim et al., 2009). One
of the most important dimensionless parameters is the Nusselt number which represents how
much heat transfer is increased by convection. (Cengal, Ghajar (2011))
𝑁𝑢 =ℎ́ ∗ 𝑙
𝑘=
𝑞𝑐𝑜𝑛𝑣
𝑞𝑐𝑜𝑛𝑑 3
The Prandtl number represents the ratio between viscose diffusion rate and thermal diffusion
rate. Numbers around one are usually gases where both momentum and heat dissipates with
the same strength. Low Prandtl number materials often have high thermal conductivity and
are in most cases metals. Hills et al. (1975) reasoned that low Prandtl numbers indicate that
conduction was the dominant mode of heat transfer and therefore convection could be
ignored. Their results contradict other research described in the literature review. PCMs are in
most cases classified as high Prandtl number
𝑃𝑟 =𝑣
𝛼 4
1.1.3 Numerical formulations
There are several numerical methods to solve solidification and melting problems. Dutil et al.
(2011) wrote that the solutions often can be categorized to fixed grid and adaptive grid. They
argued that fixed grid is better suited method to handle phase-change problems when the
melting interface is not on a macroscopic level, which is not a reasonable assumption when
natural convection currents occur. Using the enthalpy-porosity method proposed by Voller et
al. (1987) the governing equations would be the same for both the solid and liquid domain and
takes into consideration the effects of natural convection. Dutil et al. (2011) showed that
enthalpy-porosity method has successfully solved a range of solidification and melting
problems before. In addition, the technique would avoid sharp discontinuities at the solid-
liquid interface by introducing a mushy zone, which conceptually can be viewed as a porous
material. To account for the porosity Voller et al. (1987) considered adding a source term to
the governing equations. The source term would represent to what degree the liquid´s velocity
is decreased when it flowed through the mushy zone. Al-bidid et al. (2013) recent review on
computational fluid dynamics application on latent thermal heat storage, find that a majority
of researchers use the enthalpy-porosity method as;
The governing equations are equal in both phases
No extra conditions needs to be satisfied at the solid-liquid interface
Enthalpy-porosity model can extrapolate conditions within the mushy zone
Phase-change is easier to model
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1.1.4 Solidification and melting characteristics
Hills et al. (1975) studied the solidification process of a liquid metal. The metal was enclosed
in a rectangular cavity with one side cooled below 𝑇𝑠𝑜𝑙𝑖𝑑𝑢𝑠 and the rest were insulated. Some
of the limits to the numerical model presented in the paper were that the heat transfer was
solely one-dimensional and to disregard any movement of the liquid which in extent limited
all effects of natural convection. The authors argued that any variation from experimental data
was due to the small levels of thermal energy dissipating through the insulation and side
walls. If the heat dissipation was taken into account the one-dimensional assumptions would
hold true.
Gau and Viskanta (1985) constructed an experimental setup to analyze the effect of natural
convection on melting and solidification with a low Prandtl number material. In this case
gallium was used and the geometry was a rectangular enclosure. The top part of the enclosure
was used as a heat sink and the bottom was a heat source. Therefore the solidification
occurred from above and melting from below. The experiment showed that in early stages of
solidification the main mode of heat transfer was conduction. But in contrast with Hills et al.
(1975) the heat conduction was highly anisotropic and therefore could not be considered one-
dimensional. Gau and Viskanta (1985) determined that natural convection currents initiated
shortly after cooling circulation began and did not play a crucial role during solidification.
The experiment shows that during melting from below thermal conductivity was a less crucial
component, compared to the solidification process, to understand the rate of melting. When
the bottom surface heats up, the adjacent fluid started to rise due to the density difference. As
the solid seceded, the fluid started exhibiting turbulent and intense circulation. This was
deduced from acceleration of the melting that took place and irregular solid-liquid interface
that formed. Despite gallium with a high conductivity and low Prandtl number the melting
process in this experiment showed that convection forces was the dominant mode of heat
transfer in melting and played a key role shaping the interface.
Pal and Joshi (2001) investigated numerically and experimentally the effects of different
boundary conditions on melting of PCM. They utilized the enthalpy-porosity technique to
solve the melting problem numerically which treats the liquid and solid as the same domain.
A tall rectangular enclosure was used for Pal and Joshi’s (2001) experiment. The side wall
exhibited a constant heat flux and the rest was insulated. The experimental data showed that
in the early stages of melting conduction was the dominant mode of heat transport. Later in
the experiment natural convection would increase the rate of melting and determine solid-
liquid interface formation. The numerical model was two-dimensional as the authors argued
all three-dimensional effects can be neglected as the code was previously validated. The
comparison between numerical and experimental data showed the shape of the interface was
in agreement, but the rate of melting was higher in the numerical method. A potential reason
for this was the perfect adiabatic conditions.
Kamkari and Shokouhmand (2014) performed an experimental investigation on melting of
Lauric Acid, a high Prandtl number PCM. The thermophysical properties of the PCM were
determined for the experiment through calorimetric studies of the material. The rectangular
container had an inside dimension of 50 mm in width, 120 mm in height and 50 mm in depth.
Nusselt number was calculated during the experiment and the results depict the different
modes of heat transfer during the melting process which were “conduction, transition, strong
convection and vanishing convection” in that order. Transition to convection began when the
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adjacent liquid reached a temperature distinctly higher than the solid. As the liquid region
developed and its temperature became less uniform, the convection currents grew stronger
which resulted in a higher Nusselt number and consequently a higher melting rate. When the
warm fluid reached the top it eroded the solid PCM and afterward slowly flowed down the
interface exchanging heat with the solid. The authors reasoned that downward flow was the
cause of the top part of the PCM exhibiting a higher melting rate than the bottom and
eventually resulting in an interface shape of a concave curve, which is observed in Figure 1.
When the liquid touched the opposite wall of the enclosure the experiment showed that the
top level started to develop a uniform temperature whilst the liquid PCM below the peak
height of the solid region exhibited a temperature gradient. The authors observed that a
stratified liquid PCM layer developed with two distinct temperatures and convection currents.
They said that the “imaginary” stratification layer can be assumed to be at the same level the
peak height of the solid PCM. Above the stratification layer the convection currents were
weaker compared to the ones below. They reasoned that in the lower level the convection was
driven by the interaction between the ascending hot fluid and the descending cold fluid which
generated stronger currents. In the upper level there was no cold source to enhance the
convection rates. As the PCM seceded laterally the stratification layer moved downwards.
Figure 1 shows the temperature distribution of the enclosure in the experiment. The yellow
area can be regarded as the solid-liquid interface, the red as liquid, and blue as solid. The heat
source is the right wall. Stratification starts to occur at roughly 130 minutes into the
experiment when the liquid reached the other wall, and followed the peak height of the solid.
Below the peak height the temperature is not uniform.
Figure 1: Temperature distributions at the vertical mid-plane of the enclosure for hot wall temperature of 60°C
(Kamkari and Shokouhmand, 2014)
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1.2 Objectives
Studying melting or solidification processes can either be done experimentally or through a
numerical model. Experimental setups are expensive and time consuming which is
disadvantageous if the goal is to optimize natural convection in an enclosure. Using numerical
analysis instead allows for faster modelling of the enclosure, more parameters to tweak and a
reduction in cost.
The study aims to numerically model the melting and solidification process in a rectangular
enclosure and numerically explore what mechanism drives heat transfer in four different
scenarios.
Case 1 attempts to re-produce the results of Kamkari and Shokouhmand (2014) experiment to
validate the model and if any anomalies appear try to explain them. Once the model is
producing results similar to the experimental observations, further review of the numerical
data will be made. Case 2 will rotate the enclosure 90° so heat transfer is from below and test
the effect it has on melting. It has previously been shown that natural convection is directly
affected by the inclination (Ozoe, Sayama (1975)). In Case 3 the enclosure resumes its
original vertical orientation but boundary and initial conditions are changed to induce
solidification. Case 4 will examine solidification again when the enclosure is in a horizontal
position.
A side-objective is to compare convection, the rate of heat transfer and phase-change rate
between orientations. Special attention will be given to natural convection as a change in its
strength can have a large impact on heat transfer rates in both low and high Prandtl number
materials, as stated in the literature survey. A high heat transfer rate implicates that the Latent
Thermal Energy Storage can quickly charge or discharge its contents and therefore is often
the focus of optimization attempts. The reason why orientation is chosen as an optimization
parameter is because changing it infers no additional costs to the LTES, but might have a
large impact on performance.
Table 2: Review of the different scenarios
Phase-Change Orientation
Case 1 Melting Vertical
Case 2 Melting Horizontal
Case 3 Solidification Vertical
Case 4 Solidification Horizontal
-18-
2 Model
The numerical model is made to mimic the geometry used in the experimental setup by
Kamkari and Shokouhmand (2014). The inside dimensions of the rectangular container are
50 mm in width, 120 mm in height and 50 mm in depth. Right wall of the enclosure is
covered with an aluminum isothermal heat source that will provide constant temperature to
the enclosure. In the experiment by Kamkari and Shokouhmand, insulation was provided to
the other walls to reduce heat loss. Therefore in the numerical model the outside layer is
represented by an adiabatic boundary condition.
𝑘𝑑𝑇(𝑥𝑤𝑎𝑙𝑙,𝑡)
𝑑𝑥= 0 5
Commercially available Lauric Acid is used as the PCM material. The materials
thermophysical properties as phase-change temperatures, thermal conductivity, density,
viscosity and sensible heat are shown in Table 3 and are gathered from Kamkari and
Shokouhmand (2014) article. Transient simulations are performed using ANSYS Fluent
which is a computational fluid dynamics software tool. The model is shown in Figure 2 and is
simplified to two-dimensions because of the long simulation time required to handle three
dimensions.
Figure 2:Model Geometry
Table 3: Thermophysical properties of lauric acid (Kamkari and Shokouhmand, 2014)
Specific heat capacity solid/liquid (kJ/kg K) 2.18/2.39
Melting temperature range (°C) 43.5–48.2
Latent heat of fusion (kJ/kg) 187.21
Thermal conductivity solid/liquid (W/m K) 0.16/0.14
Density solid/liquid (kg/m3) 940/885
Dynamic Viscosity(kg/m∙s) (44/75°C) 0.008/0.004
-19-
3 Method
Simulation programs have become very capable of handling complex models. Many have the
ability to integrate several kind of physical phenomenon such as fluid dynamics, transient heat
and structural mechanics. In most cases the physics are often described by systems of non-
linear differential equations. A common method to solve the non-linear systems is to
discretize the model through either the finite difference method, finite element method or
finite volume method. This is accomplished through a process called meshing where the
model is divided into discrete cells. Each cell will have a certain number of nodes surrounding
them. ANSYS Fluent utilizes the finite volume method which creates small control volumes
surrounding each node in the mesh. The finite volume method ensures that in a typical
scenario the governing equations are solved for each discrete volume cell and that mass,
energy and momentum are conserved.
3.1 Meshing
To be able to analyze the underlying system of non-linear equations it is often necessary to
discretize the equations and solve them for each cell. The collection of cells is called a mesh.
The model´s mesh is consists of 3717 quadrilateral cells within the PCM domain. The grid
size was based on previous numerical studies on the enthalpy-porosity method. Groulx and
Kheirabadi (2015) analyzed the performance of three different grid resolutions on a similar
two-dimensional and rectangular enclosure. They concluded that the grid size with the least
amount of cells produced results similar to higher density meshes, which required upwards to
60-70 hours of simulation. Considering computing time and insignificant variation in results
they deemed a grid size with 6000 cells satisfactory for the numerical experiment. The model
that is used in the current simulation has a less complex geometry and therefore the amounts
of cells can be reduced without compromising the results.
3.2 Numerical Modelling
As Al-bidid et al. (2013) stated in the literature survey, the most common method to
numerically solve solidification and melting problems is with a fixed grid domain method
employing the enthalpy-porosity formulation of the governing equations. This numerical
analysis can be performed without the need for explicit information about the position of the
solid-liquid interface. In addition, it can handle melting that is either isothermal or over a
temperature range.
3.3 Enthalpy-Porosity Method
Dutil et al. (2011) argue enthalpy-porosity method is widely used by researchers as it allows
for the governing equations to be the same for both the liquid and solid domain and that one is
not required to track the interface. The enthalpy-porosity method handles these issues by
introducing a mushy zone and calculating the liquid fraction of each cell for every iteration.
The liquid fraction 𝛾 is the ratio between liquid and solid and can conceptually be regarded as
-20-
the porosity of the material. If 𝛾 is 1 then the cells control volume only contains liquid and
vice versa. If 0 < 𝛾 < 1 then both phases are present.
Voller and Prakash(1987) introduced a simple way to formulate enthalpy-porosity method by
defining the liquid fraction and enthalpy in each cell. The enthalpy is the sum of both the
sensible heat and latent heat of the PCM. The authors defined it as:
𝐻 = ℎ + 𝛥𝐻 6
where
ℎ: 𝑠𝑒𝑛𝑠𝑖𝑏𝑙𝑒 ℎ𝑒𝑎𝑡
ℎ = 𝑐𝑝 ∗ 𝑇 7
and
𝛥𝐻: 𝐿𝑎𝑡𝑒𝑛𝑡 ℎ𝑒𝑎𝑡
𝐿: 𝐻𝑒𝑎𝑡 𝑜𝑓 𝐹𝑢𝑠𝑖𝑜𝑛
∆𝐻 = 𝛾 ∗ 𝐿 8
In order to define each cell in the mushy zone the authors regard the liquid fraction as a
function of temperature:
𝛾 = 0 𝑖𝑓 𝑇 ≤ 𝑇𝑠𝑜𝑙𝑖𝑑𝑢𝑠
𝛾 = 1 𝑖𝑓 𝑇 ≥ 𝑇𝑙𝑖𝑞𝑢𝑖𝑑𝑢𝑠
𝛾 = 𝑇−𝑇𝑙𝑖𝑞𝑢𝑖𝑑𝑢𝑠
𝑇𝑠𝑜𝑙𝑖𝑑𝑢𝑠−𝑇𝑙𝑖𝑞𝑢𝑖𝑑𝑢𝑠 𝑖𝑓 𝑇𝑠𝑜𝑙𝑖𝑑𝑢𝑠 ≥ 𝑇 ≥ 𝑇𝑙𝑖𝑞𝑢𝑖𝑑𝑢𝑠 9
To decide the dampening effect the porous region has on the motion of the fluid, Voller and
Pakash use a linear function dependent on the liquid fraction. 𝑣𝑙 is the free velocity where the
porosity is 1 which can be regarded as being in liquid region of the model.
𝑣 = { 0 𝛾 ≤ 0𝛾 ∗ 𝑣𝑙 0 < 𝛾 < 1𝑣𝑙 1 = 𝛾
10
3.4 Governing equations
The numerical software that is utilized is called ANSYS Fluent. To solve
solidification/melting problems it employs the enthalpy-porosity method proposed by Voller
and Prakash(1987). The information was primarily gathered from the ANSYS Fluent Theory
(2013) guide and was supplemented by the original research paper.
-21-
The governing equations are formulated as:
The continuity equation:
0yx
vv
x y
11
The momentum equation:
2
( ) ( v ) ii x y i
vv v S
t i x y i
𝑖 = 𝑥, 𝑦 12
The 𝑆𝑖 is a source term which accounts for the momentum sink or friction upon the fluid
caused by the porosity in the mushy zone. In ANSYS porosity in each cell can conceptually
be regarded as equal to the liquid fraction. As the interface solidifies the porosity will
decrease from 1 until it reaches 0. The migration from high porosity to low porosity will cause
dampening impact on the velocity of the fluid. Near the solid region the porosity function
reaches an extremely large value, but when it fully solidifies the velocity is set to 0 and
therefore the source term as well. In the liquid region the porosity function becomes 0 so the
fluid velocity or momentum equation is not affected by the source term. In ANSYS Fluent the
porosity function 𝐴(𝛾) is based on the Carman-Kozeny relation for porous media.
𝐴(𝛾) = = (1−𝛾)2
(𝛾3−𝜀)∗ 𝐴𝑚𝑢𝑠ℎ (porosity function) 13
𝑆𝑥 = 𝐴(𝛾)𝑣𝑥 14
𝑆𝑦 = 𝐴(𝛾)𝑣𝑦 + 𝜌𝑔𝛽(𝑇 − 𝑇𝑚) 15
In the porosity function 𝜀 is a very small number to avoid division by zero. 𝐴𝑚𝑢𝑠ℎ is the
mushy zone constant and is a user defined value and it represents how quickly the source
terms increases in value when the liquid fraction changes. 𝛽 is the thermal expansion
coefficient.
The energy equation:
( H) ( ) (k T) ShvHt
16
In this case 𝑆ℎ represents the enthalpy change due to the release or absorption of latent heat
and is defined by:
h
p HS
t
17
3.5 Numerical Procedure
The creation of a model in ANSYS can often be divided into three categories called pre-
processing, solution and post-processing. The typical steps in each category are shown in
Figure 3.
-22-
Figure 3: ANSYS Fluent software usage flow
3.6 Limitations
As melting and solidification problems rarely have an analytical solutions one is required to
rely on numerical models. It is important to remember that these models are an approximation
and can sometimes produce unrealistic results if the setup is not correct. The setup includes
picking the right scheme and solver so the simulation accounts for the right physical
phenomenon like if it is turbulent or laminar flow in the enclosure.
Choosing the wrong mesh size might lead to the simulation not accurately displaying the
physics. Using two-dimensions reduces the models validity and might ignore important
physical phenomenon as well. The accuracy of a numerical result is dependent on the
convergence of the solution, and even if the residuals are low the solution will always have an
error. Because of the very long processing time the simulation only ran to two-hundred
minutes and it is possible that data beyond that time would lead to other conclusions.
In addition, ANSYS Fluent has some built in limitations that influence the results. Finally the
comparison between the simulation and experimental results of Kamkari and Shokouhmand’s
(2014) has a lot of uncertainty as it is impossible to be sure that the thermophysical properties
they give are correct, that their heated wall is isothermal, and the total amount of energy
losses through their insulation.
The following Table 4 shows what limits apply to the numerical model and what has been
done to mitigate the effect of them.
Table 4: Limitations
Limitations Action
The model cannot be used on compressible
flows
The model does not allow one to specify
material properties based on phase
Created user-defined functions that allows
thermophysical properties to be
approximated by a piece-wise linear function
dependent on temperature. Thermophysical
properties are based on Table 3.
Large discretization reduce the validity of
results
The model uses a grid size that was deemed
appropriate by Groulx and Kheirabadi (2015)
Low accuracy in the solutions Set convergence criteria to 10−6 or limit to
100 iterations per time step.
-23-
3.7 ANSYS Setup
The tailored ANSYS setup is presented in Table 5. The parameters not specified below
remains the standard setting that ANSYS chooses for melting and solidification problems, and
are not changed due to their proven track record.
Table 5: ANSYS Fluent Setup
Thermophysical properties: Lauric
Acid
Temperature dependent thermophysical properties,
defined by Table 3.
𝐴𝑚𝑢𝑠ℎ = 1 ∗ 106
Value was deemed appropriate for a similar model
by Groulx and Kheirabadi (2015).
Scheme: Presto! and Laminar flow
Shmueli et al. (2010) Letan showed that Presto!
scheme produced more accurate results compared to
others in melting/solidification problems.
Mesh
Grid Size: 3717
Grid size was deemed appropriate for a similar
model by Groulx and Kheirabadi (2015)
3.8 Parametrical Study
As the objective of the study is to examine convection, heat transfer, and phase-change rates
during different scenarios, the parametrical study is encompassed in the results and not in a
separate chapter. The parameters that will be tested are those coupled to deciding if the
process is melting or solidification, which are boundary and initial conditions. In addition, the
orientation will be tested to understand to what degree convection is affected by a vertical or
horizontal geometry. These parameters are crucial to analyzing why convection and melting
rate differs in different scenarios. Therefore to discuss the sensitivity of the model four cases
are being examined; which two are of melting and two of solidification.
Table 6: ANSYS Case Setup and Initial Conditions
Phase-Change Orientation Wall temperature Initial temperature
Case 1 Melting Vertical 60℃ 26℃
Case 2 Melting Horizontal 60℃ 26℃
Case 3 Solidification Vertical 28℃ 62℃
Case 4 Solidification Horizontal 28℃ 62℃
-24-
4 Comparison between numerical and experimental
results
A comparison is made between the results of the numerical model and Kamkari and
Shokouhmand’s (2014) experiment.
4.1 Melt Fronts
Figure 4 is composed of two sets of pictures that are of the melt front in the ANSYS
simulation and from Kamkari and Shokouhmand’s (2014) experiment. The sequential photos
are taken at an interval of twenty minutes beginning at ten minutes into the experiment.
Comparing the solid-liquid interfaces in Figure 4 Picture (c) one is able to observe that both
the simulation and experiment will experience an increased melting in the upper region of the
enclosure. This is likely caused by convection currents and buoyancy which will be described
in detail in chapter 5.1.1. Both melt fronts shortest distance from the left wall are roughly the
same, but the simulation’s erodation of the upper region begins further down. A probable
cause of this discrepancy is that when defining the properties of the liquid in ANSYS one is
required to define density as a function of the temperature. As that specific information is
unavailable a trial and error approach was conducted until the results were sufficiently
close. A possible consequence of an imperfect density function is an increase or decrease of
buoyancy compared to the experimental data, which is a possible reason for the different melt
fronts. Figure 5 is a graph of the melt fraction which is the percentage of liquid in the
enclosure. The numerical model and experimental results are in good agreement.
Figure 4: Melt Front (left side simulation and right side experiment) (Kamkari and Shokouhmand, 2014)
-25-
Figure 5: Melt Fractions as function of time: Simulation VS Experiments (Kamkari and Shokouhmand, 2014)
4.2 Temperature Distributions
The numerical and experimental temperature distributions in Figure 6 are in good agreement.
Three observations indicate this:
1. In both sets of temperature distributions the liquid adjacent to the wall appears to be
heated to 60 degrees independent of time. Therefore one can assume that heat transfer
and the conduction near the wall in the simulation reflects the experiment.
2. Kamkari and Shokouhmand (2014) hypothesized that above the peak height of the
PCM an “imaginary” stratification layer will develop with a little to none natural
convection. This would cause a uniform temperature in the upper region. Both sets of
temperature distributions exhibit this behavior after Picture (g).
3. Below the stratification layer in both sets the liquid has a non-uniform temperature
that relates to strong motion in the lower region.
Figure 6: Temperature Distributions (left side simulation and right side experiment) (Kamkari and Shokouhmand, 2014)
0
0,2
0,4
0,6
0,8
1
0 50 100 150 200 250
Time (min)
Experimental
Simulation
-26-
4.3 Energy
Figure 7 exhibits the total amount of energy stored in either sensible or latent heat in the
system. The left graph is taken from the simulation and the right is Kamkari and
Shokouhmand (2014) experiment. As the numerical model is two-dimensional the mass of
PCM is not an accurate representation of reality. Therefore the amount of energy inserted in
the system cannot be described and compared with meaning. Though it is possible to analyze
the relationship between sensible and latent energy, which seems to be very similar in the two
figures. It is not surprising that the latent energy matches between the systems because the
melt fractions shown in Figure 5 are nearly identical, but also the sensible heat seems to be in
good agreement which indicates that the convection currents distribute the energy similarly in
the liquid. The graph shows one of the key reasons why Latent Thermal Energy Storage is
preferable as roughly three times more energy is stored in latent energy compared to only
sensible.
Figure 7: Amount of absorbed Energy: Simulation VS Experiments (Kamkari and Shokouhmand, 2014)
0
200
400
600
800
1000
1200
1400
0 100 200 300
Ene
rgy
(J)
Time (min)
TotalEnergyLatentEnergy
-27-
5 Results and Discussion
5.1 Case 1:
The simulation mimics the experiment performed by Kamkari and Shokouhmand (2014). The
right wall is set at a constant temperature at 60℃ and the others have adiabatic boundary
conditions. Initial temperature is 26℃ as water was circulated beforehand to give the PCM a
uniform temperature. As the initial temperature is below 𝑇𝑠𝑜𝑙𝑖𝑑𝑢𝑠 the PCM enclosure will
experience melting.
5.1.1 Melt Front Evolution
Review of the time dependent melt front details what mode of heat transfer is currently
dominating. In the beginning of the simulation the PCM is uniformly heated parallel to the hot
wall indicating conduction as the mode of heat transfer. Shortly after initiation buoyant forces
overcome viscous forces and the hotter fluid rises to the top creating a pocket in the upper
region, which can be seen thirty minutes into the simulation in Figure 4 Picture (b). As the
liquid region expands, the convection currents gather strength which eventually increases the
size of the pocket. Inside the pocket a natural circulating current called a vortex is generated
increasing the curvature of the melt front.
5.1.2 Nusselt
Figure 8 plots the right walls Nusselt number versus time. The value correlates to different
modes of heat transfer. Initially the Nusselt number is zero which indicates that no heat is
transferred which is most likely a result of the first iteration of the simulation where the heat
transfer has not yet been determined. After a time step a sharp increase in the Nusselt number
represents a small thermal resistance caused by the thin liquid region and a large temperature
difference. As a small liquid region develops against the wall the Nusselt number drops to a
local minimum of 14 and quickly increases to 15. The increase might indicate the generation
of convection currents. After one-hundred minutes the Nusselt number monotonically
decreases until the end of the simulation. Theoretically this can be explained by weakening of
convection currents that occurs by an increase of bulk temperature or decrease of convection
above the stratification layer. Reviewing Figure 4 Picture(g) shows that after one-hundred ten
minutes the solid domain of the PCM no longer touches the top part of the enclosure, which
will initiate stratification. It is roughly the same time the Nusselt number starts to decrease.
Above the stratification layer little to none natural convection takes place which means the
Nusselt number decreases as well. As the stratification layer moves downwards a larger part
of the surface will be facing a region with no convection, which explains the monotonically
decreasing value.
-28-
Figure 8: Case 1, Nusselt number as a function of time – vertical melting
5.1.3 Streamlines
The streamlines presented below show the path taken by the particles during a certain moment
in time. The goal with this chapter is to analyze the streamlines and compare it with predicted
motion of the fluid and convection. Unfortunately the enclosure boundary cannot be shown
using the available post-processing tools and it is difficult analyzing the speed of the motion.
Streamlines will also appear even if conduction is the mode of heat transfer. Therefore these
results are only used to clarify and give insight to the motion of the liquid region.
The left streamline in Figure 9 shows that a particle starting adjacent to the wall flows up to
the top wall and deflects towards the melt front. Once it reaches the front the particle cools
down and flows downwards heating up the solid along the way. As it reaches the bottom it
regains heat and continues its circulatory path. The middle Picture, which is streamlines at
ninety minutes, has a larger liquid region and therefore a more developed natural convection.
Because of the size of the pocket in the upper region a small vortex has formed further
increasing convection and melting in its vicinity. In the right figure the current seems to
deflect at the stratification layer proposed by Kamkari and Shakaoumound.
-29-
Figure 9: Case 1, Streamlines (Left is t=50, Middle t=90, Right t=170)- vertical melting
5.2 Case 2:
Case 2 tests the same geometry as Case 1 except it is rotated 90°. Initial conditions and
thermophysical properties are identical otherwise.
5.2.1 Melt Front&Melt Fraction
Melt fronts in Figure 10 initially show a straight solid-liquid interface parallel to the heated
wall. When the liquid region grows the solid-liquid interface begins to exhibit a wavy
curvature. This feature is likely a result of many separate convection currents forming. As
time progresses these currents will merge and become quasi-stable, which means that they
continue to have roughly the same circulatory path which is known as Bernard convection. A
consequence of this merger is seen in Figure 10 Picture (e) where larger and deeper grooves
form.
The melt fraction rate seems to hold steady during the entire simulation as seen in Figure 10.
Compared to Case 1 the melt fraction increased by roughly 5-10% at a given time which
indicates that a horizontal geometry absorbs more energy than the vertical. Though given
indefinite time both cases can absorb the same amount of energy.
-30-
Figure 10: Case 2, Melt Front and Melt Fraction as a function of time - horizontal melting
5.2.2 Temperature & Point Temperature
The first three sets of pictures in Figure 11 show a different temperature scale. There is no
apparent reason why this occurs, but the effect is negligible as one is still able to accurately
compare the temperatures relative to each other. In Figure 11 Picture (c) small pockets of heat
arise which can be ascribed to convection. One explanation is that a small circular current
exist between the heat pockets which will stir up the heated liquid. In Figure 11 Picture (i) the
pockets have disappeared in favor of a more uniform temperatures in the liquid region. A
potential cause of this behavior could be the formation of Bernard convection which will be
described in chapter 5.2.5.
Point temperature graphs represent the temperature value at certain locations in the model.
Graphical representation of the points can be viewed in Figure 12. As Point 1 is located
beneath Point 2 it will exhibit melting first. After sixty minutes Point 1 enters the liquid
region which can be inferred from the curve when it stops being smooth. Roughly one-
hundred minutes into the simulation Point 1 temperature will reach a ceiling and remain
qausi-stable afterwards. This can be indicative of a strong motion in the liquid which creates a
uniform temperature as it is constantly mixed. When Point 2 enters the liquid region the
temperature jumps to the same value as Point 1 instead of a slightly lower value. Because this
happens in the end of the simulation it is safe to assume the strong convection is present
during the entire time
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 50 100 150 200 250
Time(min)
Melt Fraction
-31-
Figure 11: Case 2; Temperature Distribution and Point Temperature as a function of time – horizontal melting
Figure 12: Case 2, Location of Points in horizontal orientation
5.2.3 Nusselt Number
The Nusselt number in Figure 13 resembles the values in Case 1. Both have an early drop
with a local minimum of around 14. Afterwards convections kicks in increasing the Nusselt
number. At around fifty minutes the behavior between charts starts to differ as Case 2´s
Nusselt number diminishes until seventy minutes when it starts plateauing. The initial loss in
strength can be attributed to the increased bulk temperature. There might several reasons why
the Nusselt number becomes quasi-stable instead of continuing to decrease as in Case 1, but a
strong contender is the fact the entire heat transfer surface is experiencing convection as the
heated wall is in a horizontal position. Compared to Case 1 only the surface below the
stratification layer(which grew as time progressed) experienced any convection, which
reduced its surface Nusselt number.
0
10
20
30
40
50
60
0 50 100 150 200 250
Tem
pe
ratu
re (
C)
Time (min)
Point 1
Point 2
-32-
Figure 13: Case 2, Nusselt as a function of time
5.2.4 Energy
Initially energy is both stored in sensible and latent heat equally. As time progresses the
amount of latent heat stored increases faster than the amount of sensible. Both Case 1 and 2
exhibits similar relationships between the energies.
Figure 14: Case 2, The amount of absorbed energy- horizontal melting
5.2.5 Streamlines
The streamlines in Figure 15 reveal the formation of Bernard Cells, which is a type of natural
convection where the fluid develops a regular pattern. This indicates a constant convection
current during the entire simulation, and is confirmed by the constant presence of streamlines.
As postulated the currents merge as time progresses which leave deeper grooves in the
interface.
0
10
20
30
40
50
60
0 50 100 150 200 250
Nu
sselt
Time(min)
Nusselt number
0
200
400
600
800
1000
1200
1400
0 50 100 150 200 250
Ene
rgy
(J)
Time (min)
Total Energy
Latent Energy
Sensible Energy
-33-
Figure 15:Case 2, Streamlines (Left is t=50, Middle t=90, Right t=170)
5.3 Case 3:
For Case 3 solidification process in a vertical position will be examined. As Kamkari and
Shokouhmand (2014) never performed a solidification experiment there are no boundary
conditions or initial conditions that are given. It is shown by Shatikian, (2004) who performed
both melting and solidification simulations on PCM materials that different wall temperatures
produced vastly different results. Therefore both the initial conditions and temperature is
chosen to reflect the values for Case 1&2 but for a solidification problem. If the same
differences in initial temperature, boundary temperature and melting temperature are used for
solidification purposes the initial temperature of the PCM would be 62℃ and boundary
condition temperature 28℃.
5.3.1 Melt Front & Melt Fraction
Early stages of the melt front development in Figure 16 show a thin layer of solid PCM
forming. Traveling downward the y-axes the solid layer grows thicker and near the bottom
extends further into the liquid region, which can be explained by the buoyancy. When the
liquid cools the density decreases and it starts to flow downwards, gathering in the lower
region. After a while a cold liquid blanket forms over lower region which induces a large
mushy-zone that can be observed Figure(16) Picture (f) and forward.
Figure 16 charts the melt fraction during solidification. Initially the PCM is completely liquid
which means that the melt fraction is 100 %. A sharp decrease in melt fraction is experienced
the first twenty minutes and can be explained by the thin solid layer developing and inducing
conduction, which can be seen in in Figure 16 Picture (a). After fifty minutes when a thicker
liquid domain appears no change in trend is observed. Compared to the melting in Case 1 with
the same geometry one can conclude that solidification is a slower process. At two-hundred
minutes solidification has 35% phase change rate whilst melting is 80%. The lower rate is due
to the weak convection and is described in the next chapter.
-34-
Figure 16: Case: 3 Melt Front and Melt Fraction as a function of time
5.3.2 Temperature Distributions
Figure 17 Picture(a) shows a relatively uniform temperature distribution parallel to the hot
wall. As time progresses the cooled fluid begins to sink to the bottom of the storage which can
be inferred from Figure 17 Picture(b). Later on the simulation shows that convection currents
have formed as the cold liquid starts to stir up in Figure 17 Picture(c). Eventually convection
will mix the liquid so the temperature becomes uniform. It is difficult to tell if natural
convection is still present in latter stages when there is a low temperature gradient in the
liquid.
Figure 17 maps the temperature at Point 1 and Point 2 in the geometry. Between ten minutes
to sixty minutes into the simulation, convection is the dominant mode of heat transfer as the
temperature drops rapidly and erratically. This might also explain the high melt rate in the
first fifty minutes of the simulation as seen in Figure 16. Eventually bulk temperature
decreases and the temperature distribution becomes uniform as seen in Figure 17 & Picture
(f). Point 1 shows that after one-hundred ten minutes the bulk temperature is hovering above
𝑇𝑠𝑜𝑙𝑖𝑑𝑢𝑠 and remains relatively constant in the liquid region.
-35-
Figure 17: Case 3, Temperature Distribution and Point Temperature – vertical solidification
5.3.3 Energy
Figure 18 charts the amount of energy removed from the enclosure due to solidification.
In contrast with Case 1 the graph shows that most of the energy is removed by lowering
sensible energy. Ninety minutes into the simulation the trend shifts and the amount of latent
energy accelerates. This can be attributed to convection dominating heat transfer in the
beginning as it removes sensible energy from the entire liquid region until the bulk
temperature reaches a little above 𝑇𝑠𝑜𝑙𝑖𝑑𝑢𝑠. Before a further decrease in temperature the
material needs to phase-change therefore increasing the amount of latent energy.
0
10
20
30
40
50
60
70
0 50 100 150 200 250
Tem
pe
ratu
re (C
)
Time (min)
Point 1
Point 2
Melt Temp
-36-
Figure 18:Case 3, The amount of energy discharged from the enclosure – vertical solidification
5.3.4 Streamlines
The streamlines exhibit similar behavior in all pictures and show a circulation occurring
throughout the liquid area. A weak convection therefore exists throughout the simulation and
seems probable considering the geometry. The particle velocity is most likely very slow in
latter stages of the simulation as both temperature, and therefore density, have a very small
gradient in the liquid as seen in the Point temperatures in Figure 17. The middle picture
appears to have a particle flows through the solid domain. Such a path is impossible in reality
and is probable the consequence of a numerical error or that the velocity inside the solid
domain is set to zero.
Figure 19: Case 3, Streamlines (Left is t=50, Middle t=90, Right t=170)
5.4 Case 4
Case 4 explores solidification when the lower wall is set at a constant temperature of 28℃ and
the initial temperature is 62℃. The geometry is in a horizontal position.
0
100
200
300
400
500
600
700
800
0 50 100 150 200 250
Ene
rgy
(J)
Time (min)
Total Energy
Latent Energy
Sensible Energy
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5.4.1 Melt Front & Melt Fraction
Initially a very thin layer of solid liquid developed parallel to the horizontal heated wall in
Figure 20. As time progresses growth of the solid layer continues laterally without any
variations in the solid-liquid interface. The lack of a wavy melt front indicates that convection
does not exist or is so weak it has no impact on solidification.
The phase-change rate starts steep but slows down at twenty minutes. The evolution seems to
be monotonically decreasing with time. Compared to the Melt Fraction of Case 3 both the
percentage of liquid and the rate of change are lagging behind. A potential cause of the
decreased heat transfer is the lack of convection currents.
Figure 20: Case 4, Temperature Distribution Figure 21: Case 4, Melt Fraction
5.4.2 Temperature Distribution
Throughout the simulation temperature will remain isothermal in the horizontal plane and a
function of time and distance in the vertical plane. As the liquid cools down and density
increases it will settle on the bottom. The cold fluid in the lower region has no opportunity to
increase its temperature which means there is no density difference to make the fluid rise.
This inhibits a continuous natural convection process from taking place. In Figure 22 the
green region (which represents the temperature above 𝑇𝑠𝑜𝑙𝑖𝑑𝑢𝑠) grows as more cold fluid
settles in the bottom.
The point temperature in Figure 22 reveals the impact of buoyancy and stratification as the
colder liquid with a higher density settles on the bottom. Without any motion in the liquid
region during the simulation both Points will experience different temperatures as the liquid
has stratified. A probable cause of this phenomenon is the total lack of convection that
circulates liquid and disperses heat.
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Figure 22: Case 4 Temperature Distributions and Point Temperature – horizontal solidification
5.4.3 Energy
The evolution between sensible and latent energy shows a similar development and seems to
increase at the same rate. Towards the end of the simulation the sensible energy rate of change
seems to decrease. A probable cause of this is the lack of convection which reduces
temperature change in the liquid region. Compared to Case 3 the energy stored at two hundred
minutes is about 16% less.
Figure 23: Case 4, The amount of released energy- horizontal solidification
5.4.4 Streamlines
The streamlines pictures in Figure 24 lack the appearance of particle path indicating very little
to no motion in the liquid. Even the streamlines in Case 3 detect some motion even during
weak convection in solidification. Therefore the absence of any streamlines argues the total
lack of convection in the simulation.
0
100
200
300
400
500
600
700
0 50 100 150 200 250
Ene
rgy
(J)
Time (min)
Latent Energy
Total Energy
SensibleEnergy
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Figure 24: Case 4 Streamlines (Left is t=50, Middle t=90, Right t=170 ). No streamlines are drawn
in the post processing.
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6 Conclusion
A study of melting and solidification processes was conducted in four different scenarios. The
objective was to compare natural convection, rate of heat transfer and phase-change rates
between the cases. To give an accurate depiction of phase-change the simulation accounted
for conduction, convection, density change and porosity in the solid-liquid interface.
Four cases were examined; which two were of melting and two solidification. The geometry
was identical in all cases but placed in either a vertical or horizontal position. Detailed solid-
liquid interface and temperature fields were analyzed to understand what drives heat transfer.
In addition, numerical results of the sensible and latent energy, Nusselt number, melt fraction
and streamlines were discussed.
Case 1 tries to re-produce Kamkari and Shokouhmand’s (2014) experimental results in an
attempt to validate the numerical model. Both the solid-liquid interface shapes and melt
fractions are in good agreement with each other. In addition, the simulation and experiment
both exhibit a similar initially strong convection which vanishes as time progresses, and can
be attributed to the development of a stratification layer.
Case 2 shows the results of melting in a horizontal position with isothermal temperature from
below. The melting rate is faster compared to Case 1 and the Nusselt number does not
monotonically decrease which indicates a longer lasting strong convection. This is likely due
to the geometry and horizontal heated wall. Also the values of the Point temperatures imply
the presence of strong convection as both the lower and top Point exhibit the same
temperature when they enter the liquid region, which occurs because motion mixes the liquid.
A strong natural convection is therefore present during the entire simulation.
Case 3´s enclosure resumes the vertical position and solidification is taking place. In the
beginning there is slight motion in the liquid region. This can be concluded by the steep and
erratic drop in point temperature until it nearly reaches 𝑇𝑠𝑜𝑙𝑖𝑑𝑢𝑠, which indicates that a natural
convection disperses the heat throughout the liquid region. The relationship between sensible
and latent energy indicates an initial convection and is described in the result section.
Therefore an early stage of the energy storage is driven by convection and later by
conduction.
Case 4 simulates solidification in a horizontal enclosure. The results indicate almost
exclusively conduction as the mode of heat transfer during the entire process. This is likely
caused by the colder liquid settling on the bottom, enabling temperature stratification. As a
consequence it inhibits natural convection currents from forming.
The results suggest that melting in a horizontal enclosure is more efficient as a strong
convection is present during the entire process, increasing heat transfer and melting rates. In
the vertical case the convection tappers of as time progresses. At two-hundred minutes
horizontal orientation had 10% more liquid in the enclosure.
For solidification the vertical enclosure triggers an initial convection, which causes a higher
solidification rate during the beginning of the simulation. The horizontal solidification process
is driven by conduction which decreases the phase-change rate. At two-hundred minutes the
vertical orientation had 16% more solid in the enclosure.
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6.1 Future work
The study included detailed reviews of four cases. In the two melting cases convection plays
an important role in transferring heat. It is shown in the literature review that convection is
influenced by the inclination of the enclosure. Therefore it would be suitable for future studies
to map in detail the optimum inclination angle. In addition, it would be relevant to get a better
understanding of how increasing the surface area by including fins or partitions affects
convection and what the trade-offs are.
For solidification convection plays a minor role on determining the heat transfer. This fact
together with the low thermal conductivity leads to lower solidification rates. Therefore it will
be interesting to study techniques to increase convection. Simulating a top wall boundary
condition might increase convection during solidification because the cold fluid at the top
would sink and mix with the hotter liquid, which could be relevant to study. Conduction
might also be improved by inserting nano-particles with high thermal conductivity, though it
might include drawbacks such as sedimentation.
Future studies should therefore include parametrical studies to try to optimize the
melting/solidification rate with relevant parameters such as inclination angle, enclosure
geometry and boundary condition.
-42-
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