Influence of Non-Uniform Distributions of Filler Porosity on the
Thermal Performance of Thermocline Storage Tanks
YUCHAO HUA, LINGAI LUO
CNRS, Laboratoire de thermique et énergie de Nantes
Nantes Université
La Chantrerie, Rue Christian Pauc, 44300 Nantes
FRANCE
Abstract: -Thermal energy storage is of critical importance for the highly-efficient utilization of renewable
energy sources. Over the past decades, the single-tank thermocline technology has attracted much attention
owing to its high cost-effectiveness. In the present work, we investigate the influence of the filler porosity’s
non-uniform distribution on the thermal performance of the packed-bed sensible heat thermocline storage tanks,
using the analytical model obtained by the Laplace transform. Our analyses prove that the different porosity
distributions can result in the significantly different behaviors of outlet temperature and thus the varied
charging and discharging efficiencies, when the total amount of filler materials (i.e., the integration of porosity)
is fixed. The results indicate that a non-uniform distribution of the fillers with the proper design can improve
the heat storage performance without changing the total amount of the filling materials, which may provide a
new way to optimize the thermocline storage tanks.
Key-Words: - Energy storage, Heat thermocline tank, Packed bed, Transient thermal analysis, Analytical
model, Laplace transform, Solar energy
Received: August 15, 2021. Revised: June 19, 2022. Accepted: July 17, 2022. Published: August 5, 2022.
1 Introduction
The utilization of renewable energy sources serves
as the key solution for the problems from carbon
emission to environment pollution. In this regard,
the solar energy has shown its feasibility for various
industrial and domestic applications [1]. Due to the
strong time dependence of solar irradiation, the
integration of thermal energy storage (TES)
modules plays the crucial role for the solar plants,
which allows the power output be flexible and stable
[2][3]. Therefore, the performance of TES is of
critical importance for the overall efficiency of solar
plants [4].
Among the existing thermal energy storage
technologies, the single-tank thermocline tank [5],
where both the hot and cold fluids are contained in a
single tank, has received much attention due to its
high cost-effective approach compared to the
conventional two-tank storage systems. For a
thermocline tank, some inexpensive solid materials
are usually filled into it to reduce the volume of
expensive heat transfer fluid (HTF) required for
storage and improve the degree of thermal
stratification [6]. The research on the influence of
the filler’s properties, like material types, porosity
and thermal conductivity etc., has been extensively
investigated in literature [6–9]. Nevertheless, the
filler distribution within the thermocline tank is
usually assumed to be uniform. Few research papers
have well discussed the effect of non-uniform
distributions of filler on the performance of
thermocline storage tanks.
The present work is to study the packed-bed
sensible heat thermocline tanks with the filler of
non-uniform distribution. An analytical model is
derived by the Laplace transform, which is capable
of considering the non-uniform distribution of filler
porosity. Based on this model, it is found that the
different porosity distributions can led to the
significantly different behaviors of outlet
temperature of HTF when the total amount of filler
materials is fixed.
2 Problem Formulation
Fig. 1 illustrates a typical thermocline heat storage
tank with height H and diameter 2R. Here, different
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DOI: 10.37394/232012.2022.17.18
Yuchao Hua, Lingai Luo
E-ISSN: 2224-3461
161
Volume 17, 2022
from the previous work, the porosity distribution of
filler is non-uniform and assumed to be dependent
on the height of tank.
Fig. 1 Schematic of the thermocline heat storage
tank during the discharging process.
To construct the governing equations, a set of
assumptions are adopted as follows:
(a) The distributions of fluid flow and solid filler
are assumed to be uniform in radial direction,
and thus the problem becomes one dimensional
along the axil direction;
(b) The heat conduction within packed bed along
axil direction is neglected;
(c) The flow is assumed to be incompressible and
laminar.
Therefore, the transient thermal transport process
within a specific packed-bed sensible heat
thermocline storage tank for discharging is governed
by the 1D equations for the heat transfer fluid and
the solid filler respectively,
()
+()()
=
()()()+() (1)
1()
=
()()()+() (2)
in which ε(z) is the porosity of the packed-bed
fillers, () is the temperature of fluid “f” and solid
“s”, () is the density, () is the specific heat,
() is the fluid velocity, is the effective heat
transfer coefficient (HTC) for heat loss to the
ambient of temperature , and is the area per
unit length for the ambient heat loss. The heat
transfer surface area of fillers per unit length is
()=61()
(3)
with the equivalent diameter of filler particles d.
Moreover, the HTC, (), between fluid and solid
in porous
media is given by [9],
= 0.191 f
./(4)
where is the mass flow rate, Pr is the Prandtl
number, and the Reynolds number (Re) is modified
for porous media as [6],
=4
4(1)(5)
with the fluid dynamic viscosity .
Here, the porosity ε is set as a function varying
with the height of the storage tank (z), which
corresponds to the non-uniform distributions of
filling materials. The average porosity of the
fillers is calculated as,
=()
(6)
Then, the average fluid velocity is given by,
=
.(7)
For clarity, the governing equations (1) & (2) are
converted to be dimensionless:
=1
()
+1
(),(8)
=HCR
()HCR
(),(9)
with the dimensionless numbers as below,
=
,=
,
=
,=
,=
,
=
, =
,
=
(1).
where is the inlet temperature of fluid for
charging, while is that for discharging.
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Yuchao Hua, Lingai Luo
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For the discharging process, the initial condition
is given by, = 0: ==(),(10)
where () is the initial dimensionless
temperature distributions. The boundary conditions
are, = 0: = 0,
= 0. (11)
Solving Eqs. (8) and (9) with the corresponding
initial and boundary conditions Eqs. (10) and (11)
derives the predictive model for the temperature of
fluid and solid during the discharging process. Due
to the symmetry between discharging and charging
processes [10], the discharge model can also be
employed to predict the charging behavior by the
transform of two variables and as below,
= 1 ,= 1 .(12)
3 Problem Solution
The Laplace transform can be used to solve the
equations above. The Laplace transform of
dimensionless temperature is expressed as,
()= ()exp()
,
()= ()exp()
.(13)
Then, Eqs. (8) and (9) are transformed to,
=
1
+1
,(14)
=
HCR
HCR
.(15)
Solving Eqs. (14) and (15) gives the analytical
models of and ,
(,)=
exp f
f10+f20()
0
,(16)
(,)=
()+
+()
+HCR
+
.(17)
with
=
+1
+1
1
+HCR
+
,
=
1
1
+HCR
+
+1
,
=
1 + 1
1
+HCR
+
.
The analysis on the influence of filler porosity’s
non-uniform distributions on the thermal
performance of the storage tank are conducted using
the analytical model above.
4 Results and Discussions
The performance of a specific heat storage tank is
usually characterized by the discharging ( )
and charging () efficiencies [5],
=,()
{}
,(18)
c=fhf,out()
c
0f{hc}
c
0
.(19)
where is the cut-off time for charging as the
outlet temperature of fluid reaches 20% of the
temperature difference (), and is the
the cut-off time for discharging as the outlet
temperature of fluid decreases to 80% of the
temperature difference ( ). Due to the
symmetry between discharging and charging,
should be equal to .
Following the identical manner to convert Eqs.
(18) and (19) to dimensionless, we have
= ,
=c.(20)
In this sense, we only need to analyze the outlet
temperature of fluid, ,, during discharging for
evaluating the influence of different filler’s porosity
distributions on both the charging and discharging
efficiencies.
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In the numerical experiments, the parameters
setting is as follows: average porosity = 0.5, tank
height = 2 m , tank radius = 2 m , equivalent
diameter of solid particles = 0.01 m, mass flow
rate = 2 kg/s , and =. Moreover, the
properties of fluid (solar salt) and filler material
(quartzite) are given in Tab.1 referring to Ref. [6].
Tab.1 Properties of fluid and filler material [6]
Material Density
kg/m3
Specific
heat
J/kg-K
Dynamic
viscosity
kg/m-s
Thermal
conductivity
W/m-K
Solar
salt
1899 1495 0.00326 0.57
Quartzite
rock
2640 1050 N/A 2.8
Fig. 2 (a) Outlet temperature of fluid varying with
time; (b) Varied porosity distributions along z*.
According to Fig.2, even when the total amount
of filler material, that is, the integration of porosity,
is prescribed, the varied porosity distributions can
lead to the different behaviors of outlet temperature.
Furthermore, in order to evaluate the degree of non-
uniformity, the standard variation of porosity along
the axil direction is calculated,
=(())
.(21)
Apparently, the non-uniformity increases with the
increasing . Fig.2 shows that the cut-off time
for discharging increases as the non-
uniformity, i.e., , is enhanced. A bigger
means a longer valid discharging process, since the
outlet temperature of HTF can maintain at the high
temperature during a longer time range. Thus, the
thermal performance of the storage tank is improved
as well.
4 Conclusion
In the present work, the analytical model for
characterizing the transient thermal transport
process within the packed-bed sensible heat
thermocline storage tanks with a height-dependent
filler porosity is derived using the Laplace
transform. The analyses based on the models
indicate that the different porosity distributions can
result in the significantly-different behaviors of
outlet temperature and thus the varied charging and
discharging efficiencies, when the total amount of
filler materials is fixed, and the thermal performance
will be improved with the increasing non-uniformity
of filler porosity distributions. Our work may
provide a new way to optimize the thermocline
storage tanks in practice.
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0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
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1.0
σ
ε
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ε
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ε
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σ
ε
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σ
ε
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ε
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θ
f,out
t*
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0.0
0.2
0.4
0.6
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1.0
σ
ε
: 0
σ
ε
: 0.82
σ
ε
: 0.61
σ
ε
: 0.32
σ
ε
: 0.31
σ
ε
: 0.42
ε
z*
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Yuchao Hua, Lingai Luo
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Volume 17, 2022
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Yuchao HUA carried out the mathematic derivation
and the simulation, and wrote the draft.
Lingai Luo was responsible for the supervision and
polishing the manuscript.
Sources of funding for research
presented in a scientific article or
scientific article itself
This work is financially supported by Région Pays
de la Loire France within the NExT2Talents
program TOP-OPTIM project (998UMR6607 EOTP
NEXINTERTALENTHUA), and by the French
ANR within the project OPTICLINE (ANR-17-
CE06–0013).
Creative Commons Attribution
License 4.0 (Attribution 4.0
International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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