Accepted Manuscript

Numerical analysis of charging and discharging performance of a thermal energystorage system with encapsulated phase change material

Selvan Bellan, Jose Gonzalez-Aguilar, Manuel Romero, Muhammad M. Rahman, D.Yogi Goswami, Elias K. Stefanakos, David Couling

PII: S1359-4311(14)00557-2

DOI: 10.1016/j.applthermaleng.2014.07.009

Reference: ATE 5784

To appear in: Applied Thermal Engineering

Received Date: 4 February 2014

Revised Date: 3 July 2014

Accepted Date: 5 July 2014

Please cite this article as: S. Bellan, J. Gonzalez-Aguilar, M. Romero, M.M. Rahman, D.Y. Goswami,E.K. Stefanakos, D. Couling, Numerical analysis of charging and discharging performance of a thermalenergy storage system with encapsulated phase change material, Applied Thermal Engineering (2014),doi: 10.1016/j.applthermaleng.2014.07.009.

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Numerical analysis of charging and discharging performance of a thermal energy

storage system with encapsulated phase change material

Selvan Bellana, Jose Gonzalez-Aguilara*, Manuel Romeroa, Muhammad M.

Rahmanb,c, D. Yogi Goswamib,d, Elias K.Stefanakosb,e, David Coulingf

aIMDEA Energy Institute, Ramon de la Sagra 3, 28935 Móstoles, Spain bClean Energy Research Center, University of South Florida, Tampa, Florida, USA cDepartment of Mechanical Engineering, University of South Florida, Tampa, Florida,

USA dDepartment of Chemical & Biomedical Engineering, University of South Florida,

Tampa, Florida, USA eDepartment of Electrical Engineering, University of South Florida, Tampa, Florida,

USA f E.ON New Build & Technology Limited, Nottingham, NG11 0EE, UK

*Corresponding author: jose.gonzalez@imdea.org

Abstract

The objective of this paper is to develop a two dimensional two-phase model to

study the dynamic behavior of a packed bed thermal energy storage system, which is

composed of spherical capsules of encapsulated phase change material (PCM- sodium

nitrate) and high temperature synthetic oil (Therminol 66) as heat transfer fluid. The

heat transfer coefficient is calculated based on the phase change process inside the

capsule by enthalpy formulation model and the flow inside the system is predicted by

solving the extended Brinkman equation. After model validation, the developed model

is used to investigate the influence of capsule size, fluid temperature (Stefan number),

tank size (length and diameter), fluid flow rate and the insulation layer thickness of tank

wall on the performance of the system. The dynamic behavior of the system, subjected

to partial charging and discharging cycles, is also analyzed. It is found that increasing

the capsule size, fluid flow rate, or decreasing the Stefan number, results in an increase

in the thermocline region which finally decreases the effective discharge time and the

total utilization.

Keywords: Latent thermal energy storage, Thermocline system, Encapsulated

phase change material, Molten salt, Concentrating solar power

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1. Introduction

Concentrating solar power (CSP) technologies have been projected as one of the

most promising candidates for substituting conventional power generation technologies

[1]. Although it is variable as most of the renewable energy systems, like solar

photovoltaic and wind, due to the sunlight availability, clouds, aerosol, etc., it can be

coupled with the thermal energy storage system (TES), which stores the solar thermal

energy for later use and increase the energy source availability beyond normal daylight

hours. Hence, it can significantly increase the hours of electricity generation and

improve the dispatchability of CSP plants. Various TES systems have been proposed

and implemented in the past few decades; thermal energy can be stored as either

sensible or latent heat [2].

Several works on the sensible heat storage in packed beds are found in the

literature [3-4, 5]. As a pioneering work, Schumann [3] presented the first numerical

study on modeling of the packed bed, which has been widely adopted in subsequent

studies. The temporal variation of heat transfer fluid (HTF) and filler bed temperatures

at the axial symmetry of the tank is predicted by this model. The performance of

thermal storage system filled with quartzite rocks, for parabolic trough CSP plants, was

investigated by Yang and Garimella [6] using a CFD model, and then the cyclic

behavior of sensible thermocline storage system was investigated and found that the

cycle efficiency was intimately influenced by the filler particle diameter and radius of

the tank for a given mass flow rate [4]. Researchers in the National Renewable Energy

Laboratory (NREL) numerically modeled the packed-bed molten salt thermocline

system in which the solid fillers were in the form of hexagonal rods or a honeycomb-

like structure [7]. Using the adopted Schumann model [3], the performance of

thermocline energy storage system, filled with rocks as filler material, was studied by

Van Lew et al. [8] and the effects of particle diameter, bed dimensions, fluid flow rate

and the solid filler material on the dynamic performance of thermocline storage system

were studied by Hänchen et al. [9].

Storing the thermal energy in the form of latent heat of fusion of a phase change

material (PCM) significantly increases the energy density, thus potentially reduces the

storage size and cost compared to the sensible heat storage system. Hence, several

authors have been focused on the use of phase change materials for thermal energy

storage. Experimental and numerical studies have been conducted to characterize the

PCMs [10-11]. Various studies on the latent heat storage in packed beds have been

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found in the literature [12-14]. Theoretical and experimental investigations on the

transient thermal characteristics of a phase-change thermal energy storage system

packed with spherical capsules were conducted by Saitoh and Hirose[12], the influence

of capsule diameter and the fluid flow rate on the overall thermal response of the TES

was studied. Brief reviews of the work performed on thermocline storage system with

PCM capsules were presented [13-14]. The dynamic discharging characteristics of TES

system with coil pipes were studied by [15]. The n-tetradecane was taken as PCM and

the aqueous ethylene glycol solution with 25% volumetric concentration was used as

heat transfer fluid and the influence of the inlet temperature of HTF, flow rate and the

diameter of coil pipes on the outlet temperature of the heat transfer fluid was analyzed.

An experimental study was carried out to evaluate the thermal behavior of different TES

units coupling with a micro-CHP system [16]; a cylindrical TES tank was used to

compare the performance of two phase change materials with different melting

temperature and encapsulation method. The mathematical models reported in the

literature can be subdivided into three major groups: Single phase model [17],

continuous solid phase model [18] and the concentric dispersion model [19]. Comparing

the utilization of these three models the continuous solid phase model is more

convenient than the concentric dispersion model and more accurate than the single

phase model, thus it has been extensively used to study the thermal performance of the

packed bed systems [20].

Although several studies are reported on a thermocline system packed with

sensible filler materials, the literature on the performance of a latent thermocline energy

storage system is relatively few; paraffin is often used as a phase change material, given

its ease of handling and adoption to low temperatures building applications. Moreover,

the heat transfer mechanisms that govern the charging and discharging processes at high

temperature are still under development. Previous studies lack in high temperature

phase change process applications and a clear knowledge about the heat transfer

coefficient between the HTF and PCM capsules during charging and discharging

processes. Accordingly, this study aims at developing a two dimensional two phase

packed bed model (continues solid phase) to analyze the dynamic behavior of a packed

bed latent thermal energy storage (LTES) system for high temperature applications. The

LTES system is filled with spherical capsules. Sodium nitrate and Therminol 66 are

used as PCM and HTF respectively. The developed model is used to predict the heat

transfer coefficient between the HTF and the PCM capsules (based on the system

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configuration) and to study the influence of capsule size, fluid temperature (Stefan

number), tank size (length and diameter), HTF flow rate, and insulation layer thickness

on the performance of the system.

2. Model description

2.1 Governing equations

Fig. 1(a) shows the schematic of a lab-scale thermal storage tank; height L and

radius R, packed with PCM encapsulated spherical capsules. The average porosity of

the tank is defined as εavg = VHTF/Vtank. During the charge (discharge) process, the hot

(cold) thermal oil enters the storage tank at inlet, exchanges the heat with the PCM

encapsulated capsules, and leaves the tank via the outlet with a lower (higher)

temperature. The two phase model is developed based on the following assumptions:

(1) The thermo-fluid flow is assumed as symmetrical about the axis, thus, the

governing equations for heat transfer and fluid flow within the storage tank become

two-dimensional.

(2) The PCM capsules behave as continuous, homogeneous and isotropic porous

medium.

(3) The flow inside the tank is incompressible, and the radiation heat transfer

between the capsules is negligible.

(4) The rhombic packing is assumed. Hence, the distribution of spherical

capsules inside the tank is defined by the porosity (ε) and it varies along the radial

direction. A monochromatic exponential expression is used to define the porosity along

the radial direction as given below [18]

−−

−+=

d

rRr

avgavg 5exp1

87.01)(

εεε (1)

where d is diameter of the capsule. Since the Reynolds number is low (less than

380), the flow is assumed as laminar. Most of the low Reynolds number flows inside the

packed bed of spherical capsules are modeled as laminar and validated with

experimental results e.g. [18]. Furthermore, Xia et al. [20] compared the two models,

with and without turbulence; there is no significant difference between the results even

for the Reynolds number of 8300. Based on the foregoing assumptions, the governing

equations for the heat transfer and fluid flow inside the tank are given as follows; the

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axial velocity along the radial direction is obtained by solving the extended Brinkman

equation;

∂∂

∂∂+−−−−=

∂∂

r

ur

rrru

dBru

dA

z

P effηρε

εηε

ε 2323

2

)(1

)()1(

(2)

where η, ρ, P, u and ηeff are dynamic viscosity, density, pressure, velocity and

the effective viscosity respectively. Due to the wall friction, the velocity gradient near

the wall is a function of effective viscosity which is calculated by the following

equation [21]

Re)exp(baeff =η

η (3)

where Re is Reynolds number and the coefficients of A, B, a, and b are obtained

from [18]. By fixing the boundary conditions at axial symmetry ( 0/ =∂∂ ru ) and at the

wall (u = 0), equation (2) is solved and the radial distribution of axial velocity is

obtained. The coupled heat transfer equations for heat transfer fluid and the PCM are

given below [18, 22]

)(1

)()(2

2

2

2

fspff

frf

fzf

fpf

fp TTaUr

T

rr

T

z

T

z

Tuc

t

Tc −+

∂∂

+∂

∂+

∂∂

=

∂∂

+∂

∂λλρρε (4)

)(1

))(1(2

2

2

2

fspss

ss

ss

sp TTaUr

T

rr

T

z

T

t

Tc −−

∂∂

+∂∂

+∂∂

=∂

∂− λλρε (5)

where Tf and Ts represent the temperature of heat transfer fluid and PCM

capsules respectively, cp is specific heat capacity, λ is thermal conductivity, U is overall

heat transfer coefficient, and ap is superficial particle area per unit bed volume which is

given by

( )d

a p

ε−= 16 (6)

In eq. (4) and (5), the last term on the right-hand side, accounts for the heat

transfer between the PCM capsules and HTF, plays a vital role in the thermal

performance of the system. Hence, the overall heat transfer coefficient is calculated

according to the packed-bed configuration, and the thermal resistance concept, which is

described in Appendix A. The thermal conductivities of HTF in the radial (λfr) and axial

(λfz) directions are calculated based on the literature correlations [23-24].

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2.2 Material properties and boundary conditions

In this investigation, the spherical capsule is encapsulated by sodium nitrate and

the crust is made up of two layers as shown in Fig. 1(b); the inner coating is made up of

polymer (thickness = 0.15 mm) and the outer coating is made up of nickel (thickness =

0.15 mm). The temperature dependent thermo-physical properties of sodium nitrate are

given in Table 1. The thermal resistance caused by the crust is included in the model;

which is calculated by using the thickness and the thermal conductivity of the crust.

Thermal conductivities of nickel and polymer are 16.74 and 0.25 Wm-1K-1 respectively.

Therminol 66 is used as HTF and the thermo-physical properties of HTF are given in

Table 2. The boundary conditions are given in Table 3.

2.3 Melt fraction and energy storage formulation

Generally, ideal materials melt (phase change) at a constant temperature, but most

materials do not melt at constant temperature, but over a range of temperature due to

impurities, especially paraffin and salt hydrates [18,30]. Though the range is small (Tm±

2 K), it may significantly influence the performance of the system; the influence of

melting temperature range on the performance of the system was investigated and found

that the bed having PCM melting in a temperature range reaches the complete melting

earlier than the PCM with fixed melting temperature [19]. In this investigation, the

melting of the PCM is assumed between Tsol= 305.8 ºC and Tliq = 306.8 ºC (mushy

zone). Hence, the melt fraction, γ, at each element of the PCM domain is given by

>

≤≤−−

<

=

liqs

liqssolsolliq

sols

sols

TTif

TTTifTT

TT

TTif

1

0

γ (7)

Thus, the molten volume of the packed bed is calculated by

∫∫∫ −= dVVmol )1( εγ (8)

Therefore, the melt fraction of the bed is calculated using equation (9)

molVF

tot

VM

V= (9)

The total volume of the PCM domain is given by ���� = �1 − ��, where V is

the total volume of the tank.

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To express the heat stored in the PCM domain, the hypothesis of local thermal

equilibrium is used. Three stages are considered to express the energy storage in the

PCM medium; (1) Sensible heat storage, when the temperature of the solid PCM rises

from its initial temperature to Tsol, (2) Latent thermal energy storage during phase

change process, from Tsol to Tliq, and (3) Sensible heat storage, when the temperature of

the liquid PCM rises from Tliq to the fluid temperature. The total energy stored in the

PCM domain of the tank is given by

dVEE storedtotal )1( ε−= ∫∫∫ (10)

where Estored is calculated using equation (11)

( )

( ) ( ) ( )

( ) ( ) ( )

>+++

≤≤−−

++

<

=

∫ ∫ ∫

∫ ∫

∫

sol

init

liq

sol

s

liq

sol

init

s

sol

s

init

T

T

liqs

T

T

T

Tliqpfusmmpsp

T

T

liqssol

T

T

solssolliq

fusmmpsp

T

T

solssp

stored

TTifdTcLdTcdTc

TTTifTTTT

LdTcdTc

TTifdTc

E

ρρρρ

ρρρ

ρ

)((11)

All the given equations are solved using finite element method based software

Comsol multiphysics 4.2 [31]. Grid dependent tests were carried out in the preliminary

calculations, four various grid sizes were considered viz.: 2945 (normal), 6732 (fine),

13291 (extra fine) and 18351 (extremely fine). Simulations were performed for these

grid sizes and found that the extra fine (13291) and extremely fine (18351) mesh cases

were producing the same results of fine mesh (6732). Hence, the fine mesh was chosen

to carry out the heat transfer analysis of the TES system.

2.4 Model validation

In order to validate the model, the numerical results are compared with the

reported experimental data [18]. The reported TES system was developed by a

cylindrical tank, 0.34 m diameter and 1.52 m height; it was filled with RT20 paraffin

encapsulated spherical capsules. Air was used as HTF, the diameter of the capsules and

the average porosity of the packed bed were 50 mm and 0.388 respectively. The

numerical model developed in this study is applied to the reported TES system and

simulations are performed for the operating parameters given in the reported paper [18].

Experimentally measured and numerically predicted PCM temperatures during charging

and discharging processes are compared in Fig. 2. The temperatures of the PCM

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capsules at the axis of the 16th row and the 35th row are shown in Fig. 2 (a) and (b)

respectively for two different flow rates. A good agreement is found between the

present numerical results and the experimental measurements. Similarly, the

experimental results of Nallusamy et al. [32] and Benmansour et al. [33] are used to

validate the present model. Simulations are carried out for the operating parameters as

given in Ref. [32] and [33]. Fig. 2 (c) shows the measured [32] and predicted

temperature of the PCM at two axial positions during discharge process, for a Reynolds

number value of 1120. A comparison of measured [33] and predicted temperatures of

PCM at two axial locations during the charge process is shown in Fig. 2 (d). It is seen

from both cases that the predicted values of temperature are in good agreement with the

experimental values during the whole process.

3. Results and discussion

In this investigation, a cylindrical storage tank of length 1.5 m and radius 0.352

m is initially considered. The total (latent and sensible heat) thermal storage capacity of

the packed bed (PCM domain) is 50 kWh when the εavg = 0.388 and, ∆T= 25 K. In order

to study the effect of capsul32e size, HTF temperature, fluid flow rate and the

configuration (length to diameter ratio) of the tank on the performance of the packed

bed thermal storage system, simulations are performed for various cases as shown in

Table 4.

3.1 Temperature distribution of the packed bed

Fig. 3 shows the temporal variation of temperature distribution of the packed

bed during charging and discharging modes for case 1. In each plot, the left side

represents the fluid domain and the right side represents the PCM domain. During

charge mode, the hot HTF enters into the fully discharged (cold) thermal storage tank at

inlet and transfers heat to the PCM capsules, consequently melts the PCM, then the

cooled HTF leaves the tank through the outlet. Similarly during discharge process, the

cold HTF solidifies the PCM and obtains thermal energy from the hot capsules and

leaves the storage system. It is observed that the heat transfer between the PCM

capsules and the fluid is rapid except around the melting point (mushy zone), i. e. during

charge (discharge) mode, the PCM (HTF) absorbs heat from the HTF (PCM) rapidly

until the phase change process, due to sensible heat. Once the phase change process

started, the heat transfer between the encapsulated capsules and the HTF gradually

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decreases due to the thermal resistance caused by the molten (solidified) layer growth

and the latent heat process. After reaching complete melting (solidification), again the

heat transfer between the capsules and the fluid is increased due to the sensible heat. To

emphasize this behavior, the mushy zone is marked by the contour lines in the PCM

domain.

3.2 Effect of capsule size

To investigate the effect of capsule size on the performance of the system, four

different capsule sizes are considered. The fluid flow rate and the inlet HTF temperature

are fixed and the simulations are performed for various cases (1-4) as shown in Table 4;

the tank wall is assumed as perfectly insulated, i.e. the heat losses through the wall are

considered. The initial state of the system is assumed as fully discharged state for the

charging process and fully charged state for the discharging process as described below.

During the charge mode, the tank kept at a fully discharged state; the HTF and the

PCM capsules kept at lower than melting temperature, Tinit= Tliq - ∆T (256.8 ºC for case

1, where Tliq=306.8 ºC, ∆T= 50 ºC). At t > 0, the hot HTF introduced at the inlet, Tf=

Tliq+∆T, which exchanges the heat between the hot HTF and the cold encapsulated

PCM, consequently the PCM melts and store the latent heat by phase change process

until complete melting. Then, the sensible heat storage process continues until thermal

equilibrium between the PCM and the HTF.

During the discharge mode, the HTF and the PCM capsules temperatures initially

kept at higher than the melting temperature corresponding to a fully charged state, Tinit=

Tsol+∆T (355.8 ºC for case 1, Tsol=305.8 ºC, ∆T=50 ºC). At t>0, the cold HTF

introduced at the inlet of the tank, Tf= Tsol-∆T, which extracts heat from the PCM

capsules, consequently solidifies the PCM and releases the stored energy by phase

change process until complete solidification. Then, the sensible heat transfer process

continues until the thermal equilibrium between the PCM and HTF.

Fig.4 shows the PCM temperature distribution of the axial symmetry at various

time instants during charge and discharge process for different capsule radii. The heat

transfer at each element of the system generally undergoes any one of the following

three regions: (1) The cold region or constant low temperature zone (CLTZ), when the

temperature of the element is at the initial temperature (Ts= Tinit), D to outlet zone,

which is marked in the plot at 20 minutes during charging mode of case 1. (2) The hot

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region or constant high temperature zone (CHTZ), when the temperature of the element

is equal to HTF temperature (Ts= Tf), inlet-A zone, (3) the thermocline region which

consists of constant melt temperature zone (CMTZ); when the temperature of the

element is between the mushy zone, B-C zone, and heat exchange zone (HEZ); when

Tinit<Ts<Tsol and Tliq<Ts<Tf , two intermediate zones C-D and A-B. Three regions (four

zones) at 20 minutes during charge process of case 1 are apparently seen in the Figure.

In the constant high and low temperature zones; the HTF and the PCM are in thermal

equilibrium, in the CMTZ; latent energy transfer between the PCM and HTF takes

place, and in the HEZ; sensible heat transfer between the HTF and the PCM occurs.

During the initial period, capsules close to the inlet are being charged whereas capsules

close to the outlet of the bed are still at the initial bed temperature. The CLTZ is clearly

observed at 7 minutes during charging mode of case 1. At 38 minutes, it is observed that

the temperature of the PCM at the outlet of the tank is increased from 256.8 to 268º C,

all capsules are under charge process.

The effect of capsule size on the charging and discharging behavior is clearly seen

in the Figure. It is noticed that the thermocline region (CMTZ and HEZ) is gradually

increasing when increasing the capsule size. An increase in capsule radius is

accompanied by a decrease in surface to volume ratio, consequently the heat transfer

area decreases and eventually the conduction resistance within the PCM capsules

increases, resulting in the slower heat exchange between the HTF and the PCM

capsules. During the initial stage, when the temperature of the capsules at the inlet

exceeds the melting temperature (at 7 min. of case 1; packed bed filled with small size

capsules), the temperature of the capsules at the outlet does not increased; which is at

the initial temperature. In case 4, when the temperature of the capsules at the inlet

exceeds the melting temperature (at 28 min. of case 4; packed bed filled with large size

capsules), the temperature of the capsules at the outlet significantly increased. The same

effect is observed during the discharge process. This is due to the heat transfer area

between the working fluid and capsules which is low for case 4. To emphasize this

effect, the marked region (yellow color) is given in the Figure. It is also observed that

the thermocline thickness (CMTZ and the HEZ) is higher during discharge mode than

charge mode. This is because of heat overall transfer coefficient, which is low during

solidification than the melting process due to natural convection. Here, the natural

convection effect during melting is taken into account by using the effective thermal

conductivity of liquid PCM.

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To compare the heat transfer rate between the axis and boundary regions, the

instantaneous temperature distribution at positions P2 and P3 are compared in Fig. 5.

The point P2 is located on the axis at Z= 0.38 m and the point P3 is close to the wall at

the same height, Z=0.38 m. It is noticed that the temperature near the boundary region

significantly deviates from the axis region. This difference is already noticed in Fig. 3,

the cross sectional temperature distribution is uniformly distributed except the boundary

region. Taking the PCM temperature at 0.2 h as an example, the temperature at P2 is

290 ºC whereas which is increased about 307 ºC at P3. The significant difference

between these regions is due to the fluid velocity which is higher close to the boundary

region.

Fig. 6(a) shows the melt fraction of the bed as a function of time while charging

and discharging for different capsule radii. Since the velocity of the HTF close to the

wall of the cylinder is high, melting rate is high in near wall region than the axis region

(as was seen in Figs. 3 and 5). Hence, the melting is initially started close to the inlet-

wall region and gradually spread throughout the tank. This effect is observed in the

Figure. During initial stage, the melt fraction rate is low (until 0.4 h for case 4) because

only a few capsules close to the inlet-wall region are melted; the rest of the capsules are

under sensible heat storage and phase change process. Once the capsules which are

close to the inlet-axis region (at P1) are completely melted, the rate of the melting is

increased. This transition is clearly observed in the Figure (at 0.4 h for case 4). It is

noticed that the time for complete melting and solidification is gradually increasing with

increasing the capsule size due to the heat transfer area between the working fluid and

the capsules which is large for small size capsules. It is also observed that the complete

solidification time is longer than the melting time. This is because of the overall heat

transfer coefficient which is low during solidification than the melting process. Since

the natural convection effect in the molten PCM of small size capsules does not have

significant influence on the charge rate, there is no significant difference between the

complete melting and solidification time. But, the natural convection effect in the

molten PCM is gradually increasing when increasing the capsule size, consequently the

difference between the complete melting and solidification time of the storage tank is

increasing. This result is similar to the reported model results [19] whereas this effect is

not noticed in some reported models [18, 34-35] because in these models the overall

heat transfer coefficient is not calculated according to the state of the PCM (fully solid,

fully liquid, or phase change process).

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Fig. 6(b) presents the temporal variation of total (sensible and latent) and latent

energy stored in the system during charge and discharge processes for various capsule

radii. The total energy stored in the system is calculated using Eq. 10. At a given time,

the total energy stored in the tank varies according to the particle size. The total energy

storage capacity of the tank at fully charged state is almost constant. However,

negligible amount of storage capacity is gradually decreasing when increasing the

capsule size; the total storage capacity of the tank filled with 0.010, 0.015, 0.020 and

0.025 m radius capsules is 67.70, 67.10, 66.51 and 65.92 kWh respectively. This is due

to the total volume of the PCM capsules in the tank. In order to define the porosity

(distribution of capsules) along the radial direction, the monochromatic exponential

expression is used [18]. Thus, the total volume of the PCM capsules inside the tank is

0.351, 0.348, 0.345 and 0.342 m3 for 0.010, 0.015, 0.020 and 0.025 m radius capsules

respectively.

3.3 Influence of Stefan number

In order to study the effect of heat transfer fluid temperature on the performance

of the packed bed thermal energy storage system, simulations are performed for various

Stefan numbers, cases 1, 5-10 as shown in Table 4. Fig. 7 shows the PCM temperature

distribution at axial symmetry of the storage tank at various time instants while charging

and discharging for various Stefan numbers (∆T). The heat transfer between the HTF

and the PCM is increasing when increasing the difference between them (∆T). Thus, the

thermocline region (CMTZ and the HEZ) is decreasing when increasing the ∆T and this

effect is clearly seen in the Figure. It is also noticed that the charge mode CMTZ and

HEZ are lower than the discharge mode due to the natural convection effect during

charge process.

The melt fraction of the storage system during charge and discharge processes is

shown in Fig. 8 (a) for various ∆T (Stefan numbers). As expected, the heat transfer rate

is increasing when increasing the temperature difference between the PCM and HTF

(∆T), as a result the complete melting time is decreased when the ∆T is increased. The

complete melting times of the storage tank for cases 1 (∆T= 50 K) and 5 (∆T= 5 K) are

1.65 and 7.00 h respectively. It can be seen that when the ∆T is decreased about 90%,

from 50 to 5 K, the complete charging time of the bed is 77% increased.

The total (sensible and latent) energy stored in the packed bed during charge and

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discharge processes is shown in Fig. 8 (b) as a function of time for various ∆T. As was

seen in Fig. 6, the latent heat storage capacity of the tank is 33.11 kWh when the storage

tank is in fully charged state. The total energy storage capacity of the tank is increased,

about 3.33 kWh when the ∆T = 5 K, due to the sensible heat storage. When the ∆T is

increased about 90%, from 5 to 50 K, the total energy storage c43apacity of the tank is

increased around 45%, from 36.44 to 67.00 kWh, due to the sensible heat storage.

3.4 Effect of length/diameter ratio of the tank

In this section, performance of the packed bed thermal energy storage system is

investigated as a function of L/D (length/diameter) aspect ratio of the tank. By keeping

the volume of the tank as constant, three L/D ratios (configurations) are considered; 1.5,

2.13 and 2.50. Simulations are performed for the cases 1, 11-12 as shown in table 4, and

the temperature distribution of the tank at 50 minutes during charge mode is shown in

Fig. 9, to show the configuration of the tank for different aspect ratios.

The melt fraction of the packed bed during charge and discharge processes is

shown in Fig. 10 for various L/D aspect ratios. The diameter of the tank is decreased

when the L/D aspect ratio is increased and consequently the velocity of the tank is

increased. Hence, the melt process is started at 5 and 6 minutes for L/D = 1.5 and L/D =

2.5 respectively. The charging/discharging process varies according to the L/D ratio;

which is shown in Fig. 10 by zoom over the melt fraction profiles. However, there is no

significant difference in the complete melting/solidification time for the given operating

parameters.

3.5 Influence of HTF flow rate

The effect of HTF flow rate on the thermal performance of the storage system is

investigated in this section. Capsule size and HTF temperature are fixed and simulations

are performed for various flow rates, cases 1, 13-15 as shown in Table 4. Instantaneous

PCM temperature distribution of the system at axial symmetry while charging and

discharging is shown in Fig. 11 for various HTF flow rates. As shown in Fig. 12, the

velocity of the fluid inside the tank is low when the HTF flow rate is 1 m3/h,

consequently the heat transfer process is started at the inlet and gradually spread

throughout the tank. Since the velocity of the fluid is high when the flow rate is 4 m3/h,

the heat transfer process is started at inlet and rapidly spread throughout the tank. As a

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result, the thermocline region (CLTZ and HEZ) is increased when the fluid flow rate is

increased. This effect is apparently seen in Fig.11. It is also noticed that the heat transfer

between the HTF and the PCM is increasing when increasing the HTF flow rate.

The melt fraction and the stored energy of the packed bed during charge and

discharge processes are shown in Fig. 13 (a) and (b) respectively as a function of time

for various HTF flow rates. As expected, the complete melting time of the tank is

decreased when the fluid flow rate is increased, due to the heat transfer rate. It is noticed

that the difference between the complete melting and solidification time is not

significantly influenced by the fluid flow rate since the capsule size is small. Using

these results, the fluid flow rate can be fixed according to the actual operation time. It

can be seen that when the fluid flow rate is increased about 50%, from 1 to 2 m3/h, the

complete discharging time of the bed is 47% decreased. Similarly, when the flow rate is

increased about 33 and 25 % in subsequent cases, the discharging time is decreased

about 29 and 23 % respectively.

3.6 Effect of insulation layer thickness

To study the influence of insulation layer thickness on the heat losses through

the wall of the storage tank, calculations are made for various insulation layer

thicknesses, cases 16-21 as shown in Table 4. Calcium silicate insulation layer is

assumed in this study since it is commonly used high temperature insulation layer. The

convective heat flux boundary condition is applied at the wall of the tank as given in

Table 3. The overall wall heat transfer coefficient (Uw) is calculated according to the

thermal resistance caused by the insulation layer thickness. The atmospheric air is

assumed around the tank at 300 K. The natural convection correlation proposed by

Churchill and Chu [36] is used to calculate the heat transfer coefficient at the external

surface as given below,

( )

2

27/816/9

6/1

Pr599.01

387.06.0

+

+=Ra

Nu

(12)

The melt fraction and the heat transfer rate at the wall of the bed during charge

and discharge modes are shown in Fig. 14 as a function of time for various insulation

layer thicknesses. As can be seen in the Figure, the charging and discharging behaviors

of these cases are predicted. During charge (discharge) mode, the heat loss through the

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wall is gradually increasing (decreasing) according to the charged volume of the tank.

When it reached fully charged (discharged) and steady state, the heat loss through the

wall is almost constant. In all cases, the energy loss through the wall of the tank is less

than 1% of the inlet energy at steady state. Since the heat loss through the wall is low,

the complete melt/solidification time is not significantly changed between these cases.

As expected, the heat loss through the wall is increasing when decreasing the insulation

layer thickness.

3.7 Cyclic charging and discharging

The dynamic behavior and performance of the thermal energy storage system,

subjected to cyclic charge and discharge process, is required to design and optimize the

storage tank. The storage tank is initially assumed in a fully discharged state. To charge

the tank, the hot HTF is introduced at the top boundary of the tank, thus charge process

progresses from the top to bottom of the tank whereas to discharge the tank, the cold

HTF is introduced at the bottom boundary of the tank. The cyclic process differs from

the single charge and discharge processes as considered in the previous sections. In the

cyclic process, the initial condition for subsequent charge and discharge processes are

not fully discharged or charged state, instead, the final state of the previous cycle. The

performance of the LTES system is analyzed until a periodic state is established.

Generally in CSP plants, the HTF leaving the tank during discharge process is

fed into the power block for superheated steam generation. Since the power block cycle

process depends on the HTF temperature, it requires the termination of discharge

process when the HTF exit temperature reaches minimum cut-off temperature, Td, [37].

Similarly the maximum cut-off temperature, Tc, assigned for the HTF exit temperature

during charge mode due to the receiver temperature limitations [37]. The minimum and

maximum cut-off temperatures are determined by the application of interest, in this

investigation Tc and Td are assumed to be 301.8 and 311.8 ºC respectively. To quantify

the amount of useful energy that a storage tank can deliver during the discharge process,

the cyclic total utilization, EUtl, is introduced, which is defined as the ratio between the

useful energy that can be recovered in a cycle to the total energy storage capacity of the

tank.

Temporal variation of the PCM and the HTF temperature distributions at the

axial symmetry of the tank during charging and discharging cycles are shown in Fig. 15.

Parameters correspond to case 1 are used for this cyclic process. The storage tank is

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initially in a fully discharged state with the HTF and the PCM capsules kept at 256.8 ºC.

At t > 0, the hot HTF is introduced at the top boundary of the tank at 356.8 ºC. As

charge process continues, the hot HTF exchanges energy with the cold PCM capsules

and melts the PCM. As illustrated in the previous sections, the temporal variation of

CMTZ, HEZ, CHTZ and CLTZ are predicted as shown in the Figure. During the

operation, the melt fraction of the tank is predicted and shown in Fig.16 as a function of

time. Charging process continues until the exit temperature of the HTF reaches the

maximum cut-off temperature, 301.8 ºC, by that time, the total melt fraction of the tank

corresponds to 0.65. In this cyclic operation, the HTF and the PCM temperature

distributions at the final state of charge process becomes the initial condition for the

subsequent discharge process. Then at t > 0, the cold HTF is introduced at the bottom

boundary of the tank at 256.8 ºC. Hence, the HTF extracts the thermal energy from the

hot PCM capsules and solidifies the PCM. The discharging process continues until the

exit temperature of the HTF reaches minimum cut-off temperature, Td. Temporal

variation of the HTF and the PCM temperatures at the axial symmetry of the tank are

predicted during discharge processes and shown in the Figure. The charge and discharge

times to reach the cut-off temperature in the first cycle are 62 and 41 minutes

respectively. To represent the total utilization of the system EUtl, the shaded region

(yellow color marked region) is given in the Figure. The total utilization of the system

for this cycle is about 44.50%.

In the operation of cycle 2, the HTF and the PCM temperature distributions at

the final state of discharge process becomes the initial condition for the subsequent

charge process of cycle 2. The hot HTF is introduced at the top boundary of the tank

and the charge process is carried out until the exit HTF is reached the cut-off

temperature, Tc. Since the initial condition for this cycle corresponds to partially

discharged system, compared to the fully discharged system in cycle 1, the total charge

time of this cycle is less than the cycle 1, about 22 minutes. Again, the discharge

process is carried out using the final state of the charge process as the initial condition.

Discharge process is performed until the exit HTF is reached the cut-off temperature,

and the total utilization of this cycle is 42.49%, about 2.01% lower than the cycle 1.

Subsequent charge and discharge processes are carried out until the difference

between the total utilizations is less than 0.5% in the subsequent cycles (periodic state)

[37]. It is noticed that the difference of total utilization of the system between cycle 2

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and 3 is 0.16%, and the total charge and discharge times are not varied. Hence, three

cycles of operations are required to achieve the periodic state for case 1.

4. Conclusions

A transient two-dimensional numerical model for thermal energy storage system

packed by spherical capsules is developed to investigate the thermal performance of the

system. The model is validated using the reported experimental results, and then which

is used to analyze the effect of capsule size, Stefan number, L/D ratio of the tank, HTF

flow rate and insulation layer thickness on the performance of the system during charge

and discharge processes. The important results obtained from this study are summarized

as follows:

a) The complete melting time is shorter compared to the solidification time due to

the high heat transfer coefficient during melting.

b) The natural convection effect in the molten PCM is gradually increasing when

increasing the capsule size, consequently the difference between the complete

melting and solidification times of the storage tank is gradually increasing with

increasing the capsule size.

c) The charging and discharging rates of small size capsules are significantly

higher than the large size capsules. The thermocline region is increasing when

increasing the capsule size.

d) The thermocline region (CMTZ and HEZ) is decreased when the Stefan number

is increased; as a result the effective discharge time and the total utilization are

increased.

e) Significant difference is not found in the complete melting/solidification time of

the system when the L/D ratio of the tank varied from 1.5 to 2.5 for the given

conditions.

f) The thermocline region is increasing when increasing fluid flow rate,

consequently the effective discharge time and the total utilization are decreased.

The heat loss through the wall is increased when the insulation layer thickness is

decreased.

Acknowledgement

The research grant provided by E-ON Company through research project

entitled “Innovative Latent Thermal Energy Storage System for Concentrating Solar

Power Plants (PROJECT CODE: CC - EIRI - 14 – 2010)” is gratefully acknowledged.

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Appendix A

Determination of overall heat transfer coefficient (U)

The overall heat transfer coefficient between the capsules and heat transfer fluid,

U, plays a vital role in the performance of the thermal storage system. Various

correlations have been developed to calculate the heat transfer coefficient based on

experimental results, and have been used in numerical models [38]. The difference

between various correlations and their influence on the performance of the system has

been studied by Chao et al. [38]. In this investigation, the overall heat transfer

coefficient is calculated by the following correlation [34-35, 39]

+=

g

f

ff udfe

dU

ηρλ

33.0Pr (A1)

where, the coefficients e, f and g are usually obtained from the experimental results. In

this study, these coefficients are predicted by one dimensional conduction model as

described below.

During charging/discharging mode, the overall heat transfer coefficient, U, of

each capsule depends on the state of the PCM capsule (fully solid, fully liquid, or phase

change process). If the capsule is in fully solid (discharged) or liquid (charged) state, the

overall heat transfer coefficient can be calculated by the thermal resistance theory as

given below,

1

2

1111111

4

1111−

+

−+

−

=

++

=

oocnikcipolponikpolp hrrrrrARRRAU

λλπ (A2)

where Ro is the thermal resistance due to convection on the external surface, Rnik

is the thermal resistance due to nickel coating layer, Rpol is the thermal resistance due to

polymer coating layer, and Ap is the surface area of capsule. If the capsule is in phase

change process, the overall heat transfer coefficient can be calculated by:

1

2

1111111111

4

11

11

−

+

−+

−+

−

=

+++

=

oocnikcipolimPCMp

onikpolinp

hrrrrrrrA

RRRRAU

λλλπ

(A3)

where Rin(t) is the resistance due to the solidified/molten PCM layer inside the

capsule. Since the Rin(t) depends on the solid–liquid interface position, which is

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obtained by using the one dimensional heat conduction model derived by [40, 30] and

[41] for solidification and melting respectively.

According to the previous studies on melting and solidification of the PCM,

thermal conduction is the major mechanism of heat transfer during solidification

whereas natural convection plays a vital role during melting [42-43]. Most of the studies

only considered the thermal conduction, due to the complexity in describing the natural

convection, during melting of the PCM which resulted in a large deviation from the

experimental results. In this model, the convective effects present in the liquid region

during melting is included by using the concept of effective thermal conductivity. Since

the natural convection depends on the Rayleigh number, which is evaluated in terms of

characteristic length (Lα= ri-rm) as the thickness of liquid layer, where rm is the interface

position (where temperature is equal to melting point). The ratio of effective thermal

conductivity to the thermal conductivity of the liquid has been correlated as follows

[44],

( ) ( )

4/1

55/75/74

4/1

)(

2

861.0Pr

Pr74.0

+

−

+=

−−ommo

Lmo

liq

eff

dddd

Radd

λλ

(A4)

When the phase change takes place in a range of temperatures, the enthalphy

formulation model has been found as more convinient method for numerical solution

because the govering equations need not to be descretized for solid and fluid domain

according to the phase chage interface. The governing equation and boundary

conditions are given below

∂∂+

∂∂=

∂∂

r

T

rr

T

t

Tc p

22

2

λρ

(A5)

i

c

i

pol

ic

oc

co

nik

i

o

i

f rrat

r

rrr

rr

rrr

hr

r

TT

t

T =

−+

−+

−=

∂∂−

λλ

λ22

1

(A6)

00 ==∂∂

ratr

T

(A7)

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The heat transfer coefficient h between the HTF and the spherical capsules in the

packed bed is calculated using the correlation given by [45]

( )[ ] sNuNu ε−+= 15.11 (A8)

( ) ( )

−++++=

++=

−

2

3/21.0

8.023/15.0

2,

2,min,

1PrRe443.21

PrRe037.0PrRe664.02

turbslamsss NuNuNuNu

Before carrying out the heat transfer analysis, the developed 1D model was validated

by simulating the problem of [30], where paraffin was used as PCM and good

agreement was found. Using this 1D model, charging and discharging behaviours of a

sodium nitrate encapsulated capsule at P1(as shown in Fig. 1) are predicted. Fig. A1

shows the solid-liquid interface position (rm) of the capsule during charging and

discharging modes as a function of time for different capsule sizes. The convective heat

flux at the outer surface of the capsule is calculated according to the thermo-fluid flow

at P1. As expected, the complete melting/solidification time is decreased when the

capsule size is decreased. Here, the overall heat transfer coefficient (Eq. A3) is

calculated according to the thickness of the liquid/solid layer (Lα= ri-rm). In continious

phase model, this method is complex and computationally expensive due to the

prediction of the liquid/solid layer thickness of each capsule inside the tank. Hence, the

correlation given in Eq. A1 is used to calculate the overall heat transfer coefficient. In

order to obtain the e, f and g coefficients, various values are assumed and simulations

are performed using the continuous phase model and the complete solidification/melting

time of the capsule at P1 is predicted. Then, the appropriate coefficients are obtained by

comparing the 1D model results (complete melting/solidification time).

The coefficients of e and f are fixed at 2 and 1.1 respectively [34], and

simulations are carried out for various g coefficients. Fig. A2 shows the complete

melting and solidification times of the capsule at P1 as a function of g coefficient. The

complete melting/solidification time predicted by the 1D model is marked in the Figure

for comparison. It is obviously observed that, during charge process of case 1, the

coefficient g = 0.479 is producing almost the same result as 1D model. Hence, this

coefficient is used to calculate the overall heat transfer coefficient. Similarly, the

appropriate coefficients are predicted for all cases.

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Figure captions

Fig. 1. Schematic of (a) thermal storage tank and (b) spherical capsule

Fig.2. Experimentally measured and predicted PCM temperature distribution during

discharge (a, c) and charge (b, d) processes for various operating conditions.

Fig. 3. Temperature distribution of the tank at (a) 5 min, (b) 20 min, (c) 40 min, (d) 60

min, (e) 80 min, (f) 100 min, during charging and discharging modes for case1.

Fig. 4. PCM temperature distribution of the axial symmetry at various time instants

during charge and discharge processes for different capsule radii

Fig. 5. Instantaneous temperature distributions of the PCM and HTF at P2 (close to the

axis) and P3 (close to the wall) during charge and discharge processes for case 1

Fig. 6. Temporal variation of (a) melt fraction and (b) stored energy of the packed bed

during charge and discharge processes for different capsule radii

Fig. 7. The PCM temperature distribution at axial symmetry of the storage tank as a

function of time during charge and discharge processes for various Stefan numbers (∆T)

Fig. 8. (a) Melt fraction and (b) total stored energy of the packed bed as a function of

time during charging and discharging modes for various Stefan numbers (∆T)

Fig. 9. Temperature distribution of the system at 50 minutes during charge mode for

various configurations (L/D ratios) of the tank

Fig. 10. Melt fraction of the packed bed as a function of time during charging and

discharging modes for various L/D aspect ratios

Fig. 11. Instantaneous PCM temperature distribution of the system at axial symmetry

while charging and discharging for various HTF flow rates

Fig. 12. Velocity of the HTF as a function of radius of the tank for various HTF flow

rates

Fig. 13. (a) Melt fraction and (b) stored energy of the packed bed as a function of time

during charge and discharge processes for various HTF flow rates.

Fig. 14. Melt fraction and heat transfer rate at the wall of the packed bed as a function of

time during charging and discharging modes for various insulation layer thicknesses.

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Fig. 15. The PCM and HTF temperature distributions at the axial symmetry of the

system as a function of time during charging and discharging cycles

Fig. 16. Melt fraction of the storage tank as a function of time during cyclic charge and

discharge processes

Fig. A1. The solid-liquid interface position (rm) as a function of time during charging

and discharging modes for different capsule sizes.

Fig. A2. The charge (completely molten) and discharge (completely solidified) times of

the capsule at P1 as a function of g coefficient for various cases. The complete

charge/discharge time predicted by the 1D model is marked in the figure for

comparison.

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Nomenclature

ap superficial particle area per unit bed volume (m-1)

Ap surface area of capsule (m2)

cp specific heat (J/kg·K)

d diameter of the capsule (m)

D diameter of the tank (m)

Etotal total energy stored in the PCM domain (W·s)

EUtl cyclic total utilization (%)

h heat transfer coefficient (W/m2 ·K)

L length of the tank (m)

Lfus latent heat of fusion (J/kg)

MVF melt fraction of the bed

Nu Nusselt number

P pressure (Pa)

Pr Prandtl number

r radial coordinate (m)

rc outer radius of inner layer coating (m)

ri inner radius of the capsule (m)

rm solid-liquid interface (m)

ro outer radius of the capsule (m)

R storage tank radius (m)

Ra Rayleigh number

Re Reynolds number

Rpol thermal resistance of polymer layer (K/W)

Rnik thermal resistance of nickel layer (K/W)

Ro thermal resistance due to convection (K/W)

Rin thermal resistance due to molten/solidified layer (K/W)

Ste Stefan number

t time (s)

T temperature (K)

u velocity (m/s)

U overall heat transfer coefficient (W/m2 ·K)

Uw overall wall heat transfer coefficient (W/m2 ·K)

V volume of the tank (m3)

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Vtot volume of the PCM (m3)

Vmol molten volume of the bed (m3)

z axial coordinate (m)

Greek symbols

γ melt fraction at each element

ε porosity of the tank

η dynamic viscosity (Pa·s)

ηeff effective viscosity (Pa·s)

λ thermal conductivity (W/m·K)

ρ density (kg/m3)

Subscripts

avg average

c charging cutoff

d discharging cutoff

eff effective

f fluid

fr radial direction

fz axial direction

i inner

init initial

liq liquid phase

m melting

nik nickel

o outer

pol polymer

s solid

sol solid phase

Abbreviations

CHTZ constant high temperature zone

CLTZ constant low temperature zone

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CMTZ constant melt temperature zone

CSP concentrated solar power

HEZ heat exchange zone

HTF heat transfer fluid

LTES latent thermal energy storage

NREL national renewable energy laboratory

PCM phase change material

TES thermal energy storage

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Tables Table 1. Thermophysical properties of sodium nitrate

Properties Sodium Nitrate Refs.

Density (kg/m3)

solid phase 2130 [25] mushy zone Linear interpolation liquid phase 1908 [25]

Dynamic viscosity (Pa s) 0.0119 − 1.53�10��� [26] Latent heat of fusion (J/kg) 178000 [25]

Melting temperature (℃) 306.8 [26] Specific heat (J/kg·K) 444.53 + 2.18� [27]

Thermal expansion coef.(K-1) 6.6�10�� [27]

Thermal conductivity (W/m·K) 0.3057 + 4.47�10��� [28]

Table 2. Thermophysical properties of Therminol 66 [29]

Properties Therminol 66 [29]

Density (kg/m3) −0.614254 ∗ �� − 273.15 − 0.000321∗ �� − 273.15� + 1020.62

Specific heat (kJ/kg/K) 0.003313 ∗ �� − 273.15 + 0.0000008970785∗ �� − 273.15� + 1.496005

Thermal conductivity (W/m/K) −0.000033 ∗ �� − 273.15 − 0.00000015∗ �� − 273.15� + 0.118294

Kinematic Viscosity (mm2/s) ��� �!".#$��%��$#.&�'"�.���.�!()*

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Table 3. Boundary conditions. Boundary HTF PCM

Inlet Tf (t) 0/ =∂∂ zTs

Outlet 0/ =∂∂ zT f 0/ =∂∂ zTs

Wall 2116)(

0)/(

−−

=∂∂

casesforTTU

rT

aw

f 0/ =∂∂ rTs

Axial symmetry 0/ =∂∂ rT f 0/ =∂∂ rTs

Table 4. Details of different cases

Study

Capsule radius (m)

∆T = Tf

-Tm (K) Ste

HTF flow rate (m3/h)

Tank (L/D) ratio

Insulation layer thickness (m)

Influence of capsule size

Case 01 0.010 50 0.46948 1 2.13 ∞ (wall assumed as insulated)

Case 02 0.015 50 0.46948 1 2.13 ∞

Case 03 0.020 50 0.46948 1 2.13 ∞

Case 04 0.025 50 0.46948 1 2.13 ∞

Effect of HTF temperature

Case 05 0.01 5 0.04694 1 2.13 ∞

Case 06 0.01 10 0.09389 1 2.13 ∞

Case 07 0.01 20 0.18779 1 2.13 ∞

Case 08 0.01 30 0.28169 1 2.13 ∞

Case 09 0.01 45 0.42253 1 2.13 ∞

Case 10 0.01 65 0.61033 1 2.13 ∞

Influence of L/D ratio of tank

Case 11 0.01 50 0.46948 1 1.50 ∞

Case 12 0.01 50 0.46948 1 2.50 ∞

Effect of HTF flow rate

Case 13 0.01 50 0.46948 2 2.13 ∞

Case 14 0.01 50 0.46948 3 2.13 ∞

Case 15 0.01 50 0.46948 4 2.13 ∞

Effect of insulation layer thickness

Case 16 0.010 50 0.46948 1 2.13 0.01

Case 17 0.010 50 0.46948 1 2.13 0.015

Case 18 0.010 50 0.46948 1 2.13 0.02

Case 19 0.010 50 0.46948 1 2.13 0.03

Case 20 0.010 50 0.46948 1 2.13 0.04

Case 21 0.010 50 0.46948 1 2.13 0.05

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Fig. 1. Schematic of (a) thermal storage tank and (b) spherical capsule

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Fig.2. Experimentally measured and predicted PCM temperature distribution during discharge (a, c) and charge (b, d) processes for various operating conditions.

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Fig. 3. Temperature distribution of the tank at (a) 5 min, (b) 20 min, (c) 40 min, (d) 60

min, (e) 80 min, (f) 100 min, during charging and discharging modes for case1.

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Fig. 4. PCM temperature distribution of the axial symmetry at various time instants

during charge and discharge processes for different capsule radii

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Fig. 5. Instantaneous temperature distributions of the PCM and HTF at P2 (close to the

axis) and P3 (close to the wall) during charge and discharge processes for case 1

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Fig. 6. Temporal variation of (a) melt fraction and (b) stored energy of the packed bed

during charge and discharge processes for different capsule radii

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Fig. 7. The PCM temperature distribution at axial symmetry of the storage tank as a

function of time during charge and discharge processes for various Stefan numbers (∆T)

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Fig. 8. (a) Melt fraction and (b) total stored energy of the packed bed as a function of

time during charging and discharging modes for various Stefan numbers (∆T)

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Fig. 9. Temperature distribution of the system at 50 minutes during charge mode for

various configurations (L/D ratios) of the tank

Fig. 10. Melt fraction of the packed bed as a function of time during charging and

discharging modes for various L/D aspect ratios

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Fig. 11. Instantaneous PCM temperature distribution of the system at axial symmetry

while charging and discharging for various HTF flow rates

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Fig. 12. Velocity of the HTF as a function of radius of the tank for various HTF flow

rates

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Fig. 13. (a) Melt fraction and (b) stored energy of the packed bed as a function of time

during charge and discharge processes for various HTF flow rates.

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Fig. 14. Melt fraction and heat transfer rate at the wall of the packed bed as a function of

time during charging and discharging modes for various insulation layer thicknesses.

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Fig. 15. The PCM and HTF temperature distributions at the axial symmetry of the

system as a function of time during charging and discharging cycles

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Fig. 16. Melt fraction of the storage tank as a function of time during cyclic charge and

discharge processes

Fig. A1. The solid-liquid interface position (rm) as a function of time during charging

and discharging modes for different capsule sizes.

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Fig. A2. The charge (completely molten) and discharge (completely solidified) times of

the capsule at P1 as a function of g coefficient for various cases. The complete

charge/discharge time predicted by the 1D model is marked in the figure for

comparison.

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� Numerical analysis on charging and discharging performance of a TES system � Influence of operating parameters and system configuration on melting and

solidification process � Numerical modeling of the TES system to elucidate its performance � Dynamic behavior of the system subjected to partial charging and discharging

cycles.