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Title: Analytical modeling of PCM solidification in a shell andtube finned thermal storage for air conditioning systems
Authors: A.H. Mosaffa, F. Talati, H. Basirat Tabrizi, M.A.Rosen
PII: S0378-7788(12)00147-8DOI: doi:10.1016/j.enbuild.2012.02.053Reference: ENB 3650
To appear in: ENB
Received date: 14-12-2011Accepted date: 21-2-2012
Please cite this article as: A.H. Mosaffa, F. Talati, H.B. Tabrizi, M.A. Rosen, Analyticalmodeling of PCM solidification in a shell and tube finned thermal storage for airconditioning systems, Energy and Buildings (2010), doi:10.1016/j.enbuild.2012.02.053
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Analytical modeling of PCM solidification in a shell and tube finned
thermal storage for air conditioning systems
A.H. Mosaffa1*, F. Talati
1, H. BasiratTabrizi
2, M.A. Rosen
3
1Faculty of Mechanical Engineering, University of Tabriz, Iran
2Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran
3Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, Oshawa,
ON, L1H 7K4, Canada
*Corresponding author: Tel: +98 411 3392498, Fax: +98 411 3354153
E-mail address: [email protected] (A.M. Mosaffa), [email protected] (F. Talati),
[email protected] (H. Basirat Tabrizi), [email protected] (M.A. Rosen)
Abstract:
Due to the advantages offered by latent heat thermal storages, phase change materials (PCM)
are used in numerous applications including building air conditioning systems. In this study,
the development is reported of an approximate analytical model for the solidification process
in a shell and tube finned thermal storage. A comparative study is presented for solidification
of the PCM in cylindrical shell and rectangular storages having the same volume and heat
transfer surface area. The PCM solidification rate in the cylindrical shell storage is found to
exceed that for the rectangular storage. The effects are investigated of heat thermal fluid
(HTF) inlet temperature and flow rate on thermal storage performance.
Keywords: cylindrical shell thermal storage, PCM, solidification, analytical modeling, air
conditioning
Nomenclature
Ac cross section area, m2
c specific heat, J kg-1
K-1
*Manuscript
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D tube diameter, m
h convective heat transfer coefficient, W m-2
K-1
k thermal conductivity, W m-1
K-1
l length, m
L latent heat of fusion, J kg-1
m mass flow rate, kg s-1
Nu mean Nusselt number
Pr Prandtl number
R radius, m
Re Reynolds number
t time, s
T temperature, C
u velocity, m s-1
Z distance of solid-liquid interface in z-direction, m
Greek symbols
thermal diffusivity, m2 s
-1
half thickness of fin, m
dimensionless fluid temperature, (TwT)/(TwT,inlet)
dimensionless fin temperature, (TfTm)/(TTm)
density, kg m-3
distance of solid-liquid interface in r-direction, m
Subscripts
c cell
f fin
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heat thermal fluid
s solid
w wall
1. Introduction
Thermal energy storage (TES) facilitates the utilization of renewable energy sources and the
improvement of the energy efficiency. Latent heat thermal storage (LHTS), using phase
change materials (PCMs) to store thermal energy, has many uses. Important LTHS
applications and advances in LHTS materials and heat transfer have recently been reviewed
[1-4]. Due to rising energy costs, thermal storages systems designed for the heating and
cooling of buildings are becoming increasingly important [5-7].
Heat transfer in PCM storages is a transient, non-linear phenomenon with a moving solid-
liquid interface. The non-linearity is the principle challenge in moving boundary problems
and analytical solution for these problems (Stefan problems) are known only for a one-
dimensional domain with simple boundary conditions. Some analytical approximations of
moving boundary problems have been reported [8]. Solomon and Wilson [9] solved the
Stefan problem in a slab with a convective boundary condition end using a quasi-stationary
approximation. Vakilatojjar and Saman [10] developed a semi-analytical method for phase
change in a rectangular storage for air conditioning applications, and investigated the effect
of slab thickness on storage performance. Reviews of various mathematical methods have
been reported by Dutil et al. [11] and Verma et al. [12]. Hawala et al. [13] introduced a phase
change processor method for solving the one-dimensional phase change problem with a
convective boundary which takes into account the sensible effects during the overall process
of PCM melting and solidification. Costa et al. [14] and Sharma et al. [15] employed an
enthalpy formulation developed by Voller [16] to model a PCM slab one- and two-
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dimensionally. Tan and Leong [17] experimentally investigated PCM solidification under
constant heat rate conditions, and found that the enclosure with lower height to width ratio
has higher solidification rate. Liu et al. [18] investigated experimentally PCM solidification
in a vertical annulus energy storage; they obtained radial temperature distributions and
determined that the PCM temperature variation is insensitive to Reynolds number. Agkun et
al. [19] experimentally studied PCM melting and solidification in a shell and tube heat
exchanger. Kalaiselvam et al. [20] analyzed the melting and solidification processes for a
PCM encapsulated in a cylindrical enclosure, and examined the effect of Stefan number on
the time for complete solidification.
Due to the relatively low thermal conductivity of PCMs, many investigations have been
performed to improve the heat transfer in LHTS [21-23]. One method is to increase the heat
transfer surface area by employing finned surfaces. Numerous investigations have been
reported of the effect of fins with rectangular cross-sections on the rate of PCM
melting/solidification. For instance, Zhang and Faghri [24] studied the heat transfer
enhancement in a LHTS system using a finned tube, solving the phase change problem via
the temperature transforming method proposed by Cao and Faghri [25]. Stritih [26] studied
numerically phase change in a rectangular LHTS with a finned surface for thermal storage
applications in buildings. Inaba et al. [27] studied numerically a finned rectangular LHTS
with constant-temperature boundary conditions, and found that fin pitch influences on
solidification. Baure [28] developed an approximate analytical model based on the effective
method properties of the PCM-fin in order to predict total solidification time of a PCM with
constant wall temperature. His method is applicable for a cell aspect ratio (the ratio of the
half height of the cell to fin length) smaller than 0.5.
In this article, we present an approximate analytical solution for the two-dimensional
solidification process of a PCM in a shell and tube geometry with radial fins (see Fig. 1). The
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main objective is to provide a convenient analysis and design tool for a finned LHTS that is
reasonably accurate, convenient and physically meaningful. During solidification, heat
transfers by conduction from the solid PCM and the fins’ influence on the solidification is
more than melting [29]. The PCM is cooled and solidified using atmospheric air as the HTF,
flowing on tube side. The outer side is insulated.
2. Model description
The heat transfer in PCM storages with internal fins cannot be determined analytically for a
two-dimensional case. A simplified analytical model is introduced to determine the location
of the solid-liquid interface during solidification by dividing the storage into two regions as
shown in Fig. 2. In region 1, the only heat sink is the HTF and the fin does not influence the
solidification process. Heat is transferred from the wall in r-direction. In region 2, heat is
released by the fin. The main heat transfer mode during solidification is conduction.
Although natural convection is significant, it has a negligible effect on the solid-liquid
interface position compared to conduction [30,31].
The heat transfer problem is difficult to address due to its non-linearity and unsteady nature.
The following assumptions are made to render the problem more tractable:
The liquid PCM and the fin are initially at the melting/solidification temperature, Tm.
The temperature distribution of the thin fin, due to its shape and high conductivity, is
considered one-dimensional.
The physical properties of the PCM and fin are constant.
The PCM is homogeneous, and solidification occurs isothermally.
The temperature variation in the HTF normal to the flow direction is negligible.
The quasi-stationary assumption is applied to convective heat transfer in the fluid
passage, i.e., transient convection is considered as a series of steady-state steps.
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The air velocity profile is assumed to be fully developed. It has been shown that the
inlet air velocity profile has a little influence on the outlet air temperature [10].
3. Mathematical formulation
3.1 Finned cylindrical shell storage
In region 1, the conduction equation for the solid PCM, and the boundary and initial
conditions, can be written as follows:
0),(,11
ttrR
t
T
r
Tr
rrin
s
s
s
(1)
insss RrTtrTH
r
trT
at,0),(
),( (2)
)(at,),( trTtrT ms
(3)
0,,),( tRrRTtrT outinm
(4)
)(at,),()(
trr
trTk
dt
tdL ss
(5)
The solution of Eqs. (1)-(5) is found to be
)/ln()/(1
1)()(
)()()()()(
)exp(ln
2010
0000
1
22
inins
mm
mmmm
m m
ms
m
ms
RRHIJIY
JrYYrJN
t
a
r
TT
TT
(6)
Here J0 and Y0 are Bessel functions of zero order of the first and second kind respectively. The
values of the parameters m are the roots of the following transcendental equations:
0)()( 0000 mm JWYU (7)
where
)()(
)()(
010
010
inmsinmm
inmsinmm
RYHRYW
RJHRJU
(8)
A method to determine the roots of Eq. (7) was presented by Carslaw and Jaeger [32]. The
norm N(m), Ī1 and Ī2 can be expressed as follows:
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)()(1
)(ln
)()(1
)(ln
)()(2)(
0022122
0022121
2
0
2
0
222
0
2
inmm
m
inm
inm
in
inmm
m
inm
inm
in
m
mmsm
RYYRYR
RI
RJJRJR
RI
U
JHUN
(9)
The transcendental equation in Eq. (7) converges very slowly for small values of time or
when Rin. Thus, a Laplace transform is applied to the time variable [33].
In region 2, the energy balance for the fin, and the boundary and initial conditions, can be
written as follows:
0,,11
tRrR
t
T
Z
TT
k
k
r
Tr
rroutin
f
f
fm
f
sf
(10)
inff
fRrTtrTH
r
trT
at,0),(
),( (11)
out
fRr
r
trT
at,0
),( (12)
0,,),( tRrRTtrT outinmf (13)
Introducing the following dimensionless variables
inour
outout
inour
inin
inour
inour
f
s
inour
f
m
mf
RR
R
RR
R
RR
r
Z
RR
k
k
RR
t
TT
TT
,,
)(,
)(,
2
2
allows the solution for the temperature of the fin to be written as
dN
tr m
m
mm
m
out
in
),()(),(exp)(
1),( 0
1
0
22
(14)
where () is the solution for the steady-state case involving Eqs. (10)-(13) and
)()()()(),( 10100 outmmoutmmm YJJY (15)
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The value of the parameter m and the norm N(m) can be obtained from Eqs. (7)-(9) by
replacing the variables , Rin and Hs with out, in and Hf respectively. Details on solving the
integral in Eq. (14) are presented by Korenev [34].
3.2 Heat transfer fluid
The energy balance governing heat transfer in the thermal fluid is as follow:
z
TcmTThA airpwc
)()( (16)
A solution of the above equation can be found for laminar flow with a fully developed
velocity profile using a correlation given by Kays et al. [35]:
NuzTT
TTz
inletw
w *
,
* 4exp)(
(17)
where z*=(2z/D)/(Re Pr). The heat transfer coefficient in the entry length for a fully
developed velocity profile is expressed in the form of an eigenvalue solution as follows:
)exp(8ln
2
1 *2
2*z
G
zNu n
n
n
(18)
where Gn and n are constants and eigenvalues respectively.
Since the wall temperature of each thermal storage cell varies in the flow direction, Eq. (17)
cannot be used directly. But the problem can be solved by superposition method, resulting in
the following expression:
n
i
iwiwiinletoutlet TTzlTT1
1,,
**
,, )(1 (19)
where i is the cell number and l*=(2z/D)/(Re Pr).
4. Results and discussion
To select appropriately a PCM for a building application requires a knowledge of the
melting/solidification temperature relevant to the application. In building applications, PCMs
with a phase change temperature of 18-30C are preferred to meet the need for thermal
comfort. The PCM used in this investigation is calcium chloride hexahydrate, CaCl2.6H2O,
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and the fins are aluminum with a constant thickness of 1 mm. Thermophysical properties of
the solid PCM and aluminum are listed in Table 1.The storage system specifications for the
analysis are given in Table 2.
Fig. 3 compares the solid-liquid interface location obtained by the present analytical model
and a two-dimensional numerical method in one storage cell. The enthalpy method with a
finite difference scheme is used to simulate the solid-liquid interface location in the cells.
Zivkovic and Fujii [38] used this method in a one-dimensional PCM storage. Talati et al. [39]
developed the enthalpy method for a two-dimensional finned storage. The results indicate
that the solid-liquid interface locations obtained by the analytical model and numerical
method are in good agreement.
Fig. 4 shows the fractions of solidified PCM, as determined with the derived analytical model
for three different cell aspect ratios (lc/lf), viz. 0.5, 1 and 1.6,where the cell volume and heat
transfer surface area are each held fixed. The geometry of the storage affects the rate of
solidification. When cell aspect ratio is large, heat flows mainly through the wall from the
solid-liquid interface to the HTF. When cell aspect ratio is small, the fin has a major role in
heat transfer. It is observed in this figure that, in each configuration, the enclosure with lower
cell aspect ratio has a higher solidification rate.
The fraction of solidified PCM encapsulated in finned cylindrical shell and finned rectangular
storages are compared in Fig.5 for cell aspect ratios less than unity. The variation of fraction
of solidified PCM with time for the finned rectangular storage was presented by Mosaffa et
al. [40]. Assuming the length of the geometrically different cells to be equal, other relevant
dimensions of the cells are determined on the basis of equal encapsulated PCM volume and
equal heat transfer surface area. As can be seen in Fig. 5, the solidification rate of the PCM
encapsulated in the finned cylindrical shell storage exceeds that in the finned rectangular
storage.
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The variation of air temperature with time and distance from the entrance is shown in Fig.6.
When the thickness of solid PCM increases, so does the thermal resistance. Therefore, the
heat extraction decreases and the distribution temperature of air in the flow direction
decreases as time passes.
The effect of air velocity on the variation of solid fraction of the PCM with time is shown in
Fig. 7. As expected, the solid fraction increases with increasing air flow rate. Since increasing
flow rate results in a higher Reynolds number and heat transfer from the PCM to air
increases.
The effect of air velocity on the performance of the thermal storage is shown in Fig.8. The
higher air velocity can be seen to decrease the outlet air temperature. It seems that the effect
of increasing mass flow rate is more significant than increasing the heat extracted from the
storage. Therefore, increasing the air flow rate decreases the difference between the inlet and
outlet air temperatures. Furthermore, the results show that the difference in outlet air
temperature corresponding to two successive air velocities becomes smaller for higher air
velocities. The difference between outlet air temperatures for lower air velocities, i.e. 0.5 and
1 m s-1
, is 1.38C at t= 3600 s and for higher air velocities, i.e. 1 and 1.5 m s-1
, is 0.58C at
t=3600 s. However, it can be concluded that the effect of higher air velocity on thermal
storage performance is not considerable, whereas it increases solidification rate of the PCM.
Fig. 9 depicts the effect of inlet air temperature on the variation of PCM solid fraction with
time. It is evident from this figure that as the inlet air temperature decreases, the PCM
solidification rate increases.
Fig. 10 illustrates the effect of the inlet air temperature on the variation of outlet air
temperature with time. The results show that the effect of inlet air temperature exceeds that of
air velocity on the performance of the thermal storage.
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5. Conclusions
An approximate analytical solution is presented for the solidification process in a shell and
tube thermal energy storage. This analytical solution is compared to that obtained via a two-
dimensional numerical method based on an enthalpy formulation for prediction of the solid-
liquid interface location. The results for the two methods are in good agreement.
Furthermore, the variation PCM solid fraction with time is compared for the cylindrical shell
and rectangular storage arrangements, with the same volume and heat transfer surface area.
The results indicate that the PCM solidifies more quickly in the cylindrical shell storage than
in rectangular storage. In addition the solid fraction of the PCM increases more quickly when
the cell aspect ratio is small. The effects of air velocity and inlet air temperature on the
performance of the thermal storage are analyzed. It is found that the effect of inlet air
temperature is more significant than that of air velocity on the outlet temperature. The
presented analytical model provides a good prediction of the performance of thermal storage
systems in building applications, and is expected to be useful for determining optimum
designs for air conditioning systems for various climatic conditions.
Acknowledgements
The support of the Iranian Fuel Conservation Organization (IFCO) is gratefully
acknowledged.
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Fig. 1. Schematic of the thermal energy storage system
Fig. 2. Schematic of energy storage, showing division into two region and symmetry cell
Fig. 3. Analytical and numerical results of the solid-liquid interface location in one cell (lc=10
mm, lf= 20 mm)
Fig. 4. Variation of PCM solid fraction with time for various cell aspect ratios
Fig. 5. Variation of PCM solid fraction with time for cylindrical shell and rectangular cell
configurations
Fig. 6. Air temperature distribution along the tube passing through the thermal storage system
for a cell aspect ratio of 0.5
Fig. 7. Variation of PCM solid fraction in the storage with air velocity and time for a cell
aspect ratio of 0.5
Fig. 8. Effect of air velocity on the thermal storage system performance for a cell aspect ratio
of 0.5
Fig. 9. Variation of PCM solid fraction in the storage with inlet air temperature and time for a
cell aspect ratio of 0.5
Fig. 10. Effect of inlet air temperature on the thermal storage system performance for a cell
aspect ratio of 0.5
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Table 1: Thermophysical properties of CaCl2.6H2O [36] and aluminum [37].
Property CaCl2.6H2O Aluminum fin
Density,s (kg m-3
) 1710 2770
Heat capacity, cs (J kg-1
K-1
) 1460 875
Thermal conductivity, ks (W m-1
K-1
) 1.088 177
Latent heat of fusion, L (J kg-1
) 187,490
Melting/solidification temperature , Tm (C) 29.7
Table 1
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Table 2: Thermal storage system specifications used in the analysis.
Storage height 10 cm
Volume of cells 50.3 cm3
Inner radius, Rin 10 mm
Fin thickness, 2 1 mm
Inlet air temperature, T,inlet 10C
Air velocity, u 1.5 m s-1
Table 2
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Figure 1
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Figure 2
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Figure 3
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Figure 4
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Figure 5
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Figure 6
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Figure 7
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Figure 8
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Figure 9
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Figure 10
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REASEARCH HIGHLIGHTS
> Simplified analytical model for solidification of PCM in shell & tube finned storage.
> We formulate energy equation in the presence of a heat thermal fluid on the walls.
> We compare solidification time for PCM in cylindrical shell and rectangular storages.
> We investigate the effects of inlet air temperature on thermal storage performance.
> We investigate the effects of air flow rate on thermal storage performance.
*Highlights
