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HUNGARIAN JOURNAL 
OF INDUSTRIAL CHEMISTRY 

VESZPRÉM 
Vol. 40 (2) pp. 93–99 (2012) 

OPTIMAL DESIGN OF HIGH-TEMPERATURE THERMAL ENERGY STORE 
FILLED WITH CERAMIC BALLS 

T. BORBÉLY1  

1University of Pannonia, Institute of Mechanical Engineering, 10 Egyetem Street, H-8201 Veszprém, HUNGARY 
E-mail: borbelyt@almos.vein.hu 

The momentary amount of the available solar energy and the demand usually are not equal during the usage of solar 
energy for heating and electric power supply. So it is necessary to store the heat energy. This article shows optimal 
design of a new construction, sensible heat store filled with solid heat storage material. The planned heat store has 
cascade system formed a spiral flow-path layout. This is a conceptual model, worked out in case of packed bed with 
ceramic balls. The aim of the special layout is to realize better overall efficiency than regular sensible heat stores have. 
The new construction would like to get higher overall efficiency by long flow-way, powerful thermal stratification and 
spiral flow-path layout which can ensure lower heat loss. The article shows the calculation method of the simulation of 
the charge and discharge and the calculation method of the overall efficiency using the results of the simulations. The 
geometric sizes and operating parameters of the thermal energy store with the best overall efficiency were calculated 
using genetic algorithm (GA). The results of the calculation tasks show that a thermal energy store with long flow-way, 
with cascade system formed spiral flow-path layout has higher overall efficiency than an one-duct, short flow-way 
thermal energy store which has equal mass of solid heat storage material as the long flow-way one, mentioned before. 

Keywords: solar energy, heat storage, solid charge, sensible heat, optimization 

Introduction 

The possible thermal energy storing methods are: 
sensible heat storage, latent heat storage, sorption heat 
storage and chemical energy storage [1–8]. 

The simplest way is the storage of sensible heat, by 
heating a heat storage material without phase changing. 
The energy density of the sensible heat storage will be 
high if the specific heat and the density of the heat 
storage material are great as well [9]. 

Out of the materials which can be found in the 
environment in large quantity, the water has the greatest 
volumetric heat capacity (~4.18 MJ/m3K), but water can 
be applied at atmospheric pressure up to 100ºC only. 
The heat transport media of the concentrated solar 
power systems can be used as heat storage liquids as 
well. The melt of the solar salt (60% NaNO3 + 40% 
KNO3) is used out of these materials in concentrated 
solar power plants as heat storage material (operating 
temperature range 260–550ºC, volumetric heat capacity 
~2.84 MJ/m3K [11]). It is not flammable, not toxic, and 
not too expensive. 

The volumetric heat capacity of some solid materials 
(magnesite, corundum) – because of their higher density 
– come near to the volumetric heat capacity of the water 
with much higher upper temperature limits (magnesite 
3.77 MJ/m3K, corundum 3.3 MJ/m3K, cast iron 
4.1 MJ/m3K [10]). 

Screened pebble stone, cracked stone (1.5–
2.5 MJ/m3K), concrete (0.8–1.8 MJ/m3K), wet soil 

(3.56 MJ/m3K) [10] are used as sensible heat storage 
materials, because they are inexpensive. 

The sensible heat stores are typical regenerative 
heat-exchangers. These are instationary thermal state 
heat-exchangers. The regenerators are long ago applied, 
great heat capacity heat stores with solid fill and with 
short charge-discharge cycle time (10–7200 s). 

My aims were to study the possible interior structure 
of the long-term heat stores, the charge-discharge 
process, to calculate the optimal geometric sizes and 
operating parameters of those. 

Comparison of short (L/D<10) and long (10<L/D) 
heat stores 

The temperature-place function of the heat transport 
medium is similar to the temperature-place function of 
the solid heat storage material at a moment (the 
temperature of the heat transport medium tf is higher at 
charge, lower at discharge than the ts temperature of the 
solid heat storage material), so it is enough to show the 
temperature-place functions of the solid heat storage 
material. 

Charge 

The hot heat transport medium gives a part of its heat 
content to the solid heat storage material by flowing 



 

 

94

through the heat store, which is cold at the beginning of 
the charge period. 

In case of short heat store the outlet temperature of 
the heat transport medium and the solid heat storage 
material are increasing soon after the beginning of the 
charging (Figure 1), in case of long heat store they start to 
increase only at the end of the charge period (Figure 2). 

 

 
Figure 1: The temperature-place functions of the solid 

heat storage material during charge, in case of short heat 
store 

 

 
Figure 2: The temperature-place functions of the solid 

heat storage material during charge, in case of long heat 
store 

Discharge 

In the discharge period the cold heat transport medium 
flows through the hot heat store in opposite flow 
direction of the charge. 

In case of short heat store the outlet temperatures of 
the heat transport medium and the solid heat storage 
material are decreasing soon after the beginning of the 
discharging (Figure 3), in case of long heat store they 
start to decrease only at the end of the discharge period 
(Figure 4). 
 

 
Figure 3: The temperature-place functions of the solid 
heat storage material during discharge, in case of short 

heat store 
 

 
Figure 4: The temperature-place functions of the solid 
heat storage material during discharge, in case of long 

heat store 

Basic idea of the cascade system heat store 

During the charging and discharging of the long heat 
store the thermocline zone is located only in a part of 
the length of the heat store. It is plausible solution to 
divide to sections the heat store and allow knocking-off 
the sections from the flow-path of the heat transport 
medium. Let’s call these sections as ducts. In case of the 
cascade system heat store the heat transport medium 
must flow through only the ducts where the thermocline 
zone is going along. The transport power demand of the 
heat transport medium can be reduced by this solution. 

The heat-loss of the heat store into the environment 
will be small if the heat store has small specific surface. 
From the prismatic bodies the cylinder with H/D=1 ratio 
has the smallest specific surface, followed by the 
regular n-sided prism with H/S=1 ratio (assuming that 
the heat-loss flux is approximately equal in all sides of 
the body). 

The higher heat-loss of the long heat store (because 
of its greater specific surface) can be reduced by using 
cascade system of the ducts formed a spiral flow-path 
layout (see later on Figure 5). 

Out of the regular n-sided prisms the six-sided can 
be built from cylinders with the best space utilization. 



 

 

95

The geometry and operating of the heat store filled 
with ball particles 

The ducts of the heat store filled with ball particles are 
cylinders made of metal sheet with outer thermal 
insulation (Fig. 5). The metal shell holds the bed of the 
particles, and the heat transport medium. The thermal 
insulation supports the thermal stratification in radial 
direction. To choose cylindrical shape for the ducts is 
necessary – because of the pressure of the packed bed of 
bulk particles. 

The outer geometry of the heat store is regular 
hexagonal prism with H/St≈1 ratio and cascade system 
of the ducts formed a spiral flow-path layout. 

 

 
Figure 5: The construction layout of the heat store filled 

with ball particles and arrangement in the cascade 
system with spiral connection 

St – side distance of the heat store, H – bed height of a 
duct, δi – thickness of the outer thermal insulation 
 
The number of ducts Nj is an odd one in case of full 

filling, but leaving out the last duct we get an even 
number, so it is possible to lock out or join in the heat 
transport medium flow into any duct-pairs (duct-pair 
means a pair of ducts, one downwards and another 
upwards). This way the place of the last duct could be 
used e.g. for service purpose. 

The hot heat transport medium is put in at the top of 
the middle duct at the beginning of charge and it flows 
downwards, it turns in the return band flows into the 
next duct (the lower connecting of the ducts is signed 
with dashed arrow) and flows trough that upwards. The 
heat transport medium coming out from the second duct 
can be led to the next pair of ducts. The heat transport 
medium must flow through only the ducts where the 
thermocline zone is going along. In the discharge period 
the cold heat transport medium flows through the hot 
heat store opposite to the flow-direction of the charge 
(from outside to middle). 

The main sizes of a duct of the heat store can be seen 
in Figure 6. 

 
Figure 6: Top view of a duct 

DSj – outside diameter of the insulated duct, Dj – inside 
diameter of the duct, ds – whole thickness of the thermal 

insulation between two ducts 

Basic differential equations of the heat transport in 
the heat store filled with ball particles 

K. A. R. Ismail, and R. Stuginsky Jr. [126] have 
reported an excellent comparative analysis of several 
models for description of the heat transfer in the heat 
transport medium and the solid heat storage material. 

For both media I have chosen a bit simpler model 
than the general, one-dimensional model. The heat-loss 
boundary condition is not included in the differential 
equations, because it takes effect only at the outer ducts, 
the effect of it will be calculated separately from the 
calculation of the temperature-place functions. 

The differential equation for the description of the 
heat transfer in the heat transport medium is 

 )tt(a
x
t

w
t

c fspf
f

f
f

ff −α=⎟
⎠
⎞

⎜
⎝
⎛

∂
∂

+
τ∂

∂
ρε , (1) 

where: 
 tf – temperature of the flowing heat transport 
medium, 

 ts – temperature of the solid heat storage material, 
 ρf – density of the heat transport medium,  
 cf – specific heat of the heat transport medium, 

wf –average velocity of the heat transport medium in 
the flow-channels, 

 ap – superficial particle area per unit bed volume, 
 αf – heat transfer coefficient between the flowing 
heat transport medium and the solid heat storage 
material, 

 ε –void fraction of the bed. 
 

The superficial ball particle area per unit bed volume 
ap is 

 )1(
d
6

a
b

p ε−= , (2) 

where: db – diameter of the ball particle. 



 

 

96

The differential equation of the heat transfer in the 
heat transport medium was discretized by applying 
explicit forward difference scheme in time and upwind 
difference scheme in space [13, 14]. 

The differential equation for the description of the 
heat transfer in the solid heat storage material is 

 )tt(a
x
tt

c)1( sfpf2
s

2

seffx
s

ss −α+∂
∂

λ=
τ∂

∂
ρε− , (3) 

where: 
ρs – density of the solid heat storage material, 
cs – specific heat of the solid heat storage material, 
λseffx – effective axial thermal conductivity of the 
solid heat storage material. 
 
The effective axial thermal conductivity of the solid 

heat storage material λseffx is 

 

sf

seffx )1(
1

λ
ε−

+
λ
ε

λ , (4) 

where: 
λf – thermal conductivity of the heat transport 
medium, 
λs – thermal conductivity of the solid heat storage 
material. 
 
The differential equation of the heat transfer in the 

solid heat storage material was discretized by applying 
explicit forward difference scheme in time and centred 
difference scheme in space (FTCS) [13, 14]. 

The heat transfer coefficient between the flowing 
heat transport medium and the solid heat storage 
material was calculated according to [15]. 

The design variable 

It is practical to choose the diameter of the ball particle 
db for optimization design variable: x1=db. The flow 
velocity of the heat transport medium wf has the greatest 
influence on the charging of the heat store, but it 
depends on the void fraction ε only (at given inside 
diameter of the duct Dj). In case of the bed filled with 
uniform ball particles the void fraction ε is independent 
from the diameter of the particle db. 

Definition of the restrictions 

Geometric restrictions 

The lower limit of the diameter of the ball particle db is 
restricted by the manufacturing technology and the 
pressure drop, and the upper limit is also restricted by 
the minimal way-length of the heat transfer in the heat 
storage material. 

Limits for x1 design variable are 

 m1.0m01.0 ≤≤ 1x . (5) 

For the minimal specific surface the best value of the 
geometric ratio could be 

 1
S
H

t

≈ . (6) 

Pressure drop restriction 

The pressure drop of the heat transport medium flowing 
through the packed bed in case of incompressible 
medium, according to [16] is 

 
⎥
⎥
⎦

⎤

⎢
⎢
⎣

⎡ ρ
ε
ε−

+
νρ

ε
ε−

=Δ 2f
b

f
f2

b

ff
2

2

w
d

1
75,1w

d
)1(

150L'p , (7) 

where: 
νf – kinematic viscosity of the heat transport 
medium. 

 
The pressure drop of the heat transport medium must 

be restricted in order not to be necessary to install 
pressure resistant shell, not to get high transport work 
demand, can be negligible the compressibility of the gas 
and may be sufficient to use ventilator instead of blower 
or compressor in case of gas heat transport medium. 

The upper pressure drop limit for L=2H flow-way 
length according to the previous requirements is 

 bar1.0Pa10000'p H2 =≤Δ . (8) 

Composition of the objective function 

The objective function is the overall efficiency of the 
heat storage, which is suitable to compare the variants 
of the heat stores. The optimal sizes and operating 
parameters could be got from the maximum-point of the 
objective function. 

In the calculation of the overall efficiency I relate the 
part of the extractable heat quantity which can be used 
for heating and electric power production to the sensible 
heat storage capacity of the heat store as it is here: 

 
cap

trlhid
o Q

QQQ −−
=η , (9) 

 
where: 

Qhid – extractable heat quantity from the heat store 
during a charge-discharge cycle without heat-loss, 
Ql – heat-loss to the environment through the 
boundary surfaces during a charge-discharge cycle, 
Qtr – heat-equivalent of the transport work demand 
during a charge-discharge cycle, 
Qcap – sensible heat storage capacity of the heat store 
between the inlet and outlet temperature of the heat 
transport medium at the charge. 
The optimal value of the design variable can be 

searched by optimization using the calculation of the 



 

 

97

temperature-place functions during the whole length of 
the charge-discharge cycle.  

The heat quantity Qhid 

The heat quantity Qhid is the difference between the 
heat-content of the heat store after charge and after 
discharge without heat-loss. 

It is necessary to know the temperature-place 
functions of the heat store at the end of the charge and at 
the end of the discharge. 

The heat quantity Qhid is 

( )∫ τ+τ−τ
π

ε−ρ=
HN

0
dcscs

2
j

sshid

j

dx),x(t),x(t
4

D
)1(cQ , (10) 

where: 
τc – term of charge, 
τd – term of discharge. 

 
The final temperature-place function of the charge of 

the heat store is the initial condition of the discharge. In 
the discharge period the heat transport medium flows 
through the heat store in opposite flow direction of the 
charge. 

The heat quantity Ql 

The heat quantity Ql is the heat-loss into the environment 
through the boundary surfaces during a charge-
discharge cycle is 

 lblslrl QQQQ ++= , (11) 

where: 
Qlr – heat-loss to the environment through the roof 
surface, 
Qls – heat-loss to the environment through the side 
surfaces, 
Qlb – heat-loss to the environment through the bottom 
and the ambient ground. 

 
The calculation of the heat-loss has taken into 

account the temperature of the heat store changing in 
place and time. 

The heat quantity Qtr 

The heat quantity Qtr is the heat-equivalent of the 
transport work demand during a charge-discharge cycle. 
The heat quantity Qtr is approximated 

 
oh

H2ff
.

dHd2jcHc2j
tr

'pm)NN(
Q

η

Δρτ+τ
= , (12) 

where: 
Nj2Hc – average number of duct-pairs which are 
simultaneously used during the charge, 

Nj2Hd – average number of duct-pairs which are 
simultaneously used during the discharge, 

f
.

m  – mass flow rate of the heat transport medium, 
Δp’2H – pressure drop of the heat transport medium 
on L=2H flow-way length, 
ηoh – overall efficiency of the electric power 
production in a heat power station. 

The heat quantity Qcap 

The heat quantity Qcap is the sensible heat storage 
capacity of the heat store between the inlet and outlet 
temperature of the heat transport medium at the charge: 

)tt(cm)tt(cmQQ cs,sce,ssscco,fci,fff
.

cf

.

cap −=τ−=τ= ,(13) 

where: 

f

.
Q  – heat current during the charge, 
tf,ci – inlet temperature of the heat transport medium 
at charge, 
tf,co – outlet temperature of the heat transport 
medium at charge, 
ms – mass of the heat storage material, 
ts,cs – (homogeneous) temperature of the solid heat 
storage material at the start of the charging, 
ts,ce – (homogeneous) temperature of the solid heat 
storage material at the end of the charging. 

Basic data of the optimization task 

I have made the calculations during the optimization 
with the following main data: 

τc=63 day=1512 h=5 443 200 s, f
.

Q =2 MW, tf,ci=400°C, 
ts,cs=100

 oC, ds=0.2 m. 
τd=58 day=1392 h=5 011 200 s. 
The solid heat storage material is magnesite, its physical 
properties are at ts,mid [10]: 
ts,mid=(ts,cs+ts,ce)/2=(100°C+400°C)/2=250°C  
λs=23.26 W/mK, ρs=3500 kg/m

3, cs=1077.5 J/kgK. 
The required mass of the heat storage material for an 
ideal heat store: ms=33 678 t, ε=0.3. 
The heat transport medium is nearly ambient pressure 
air, its physical properties are at tf,mid [10]: 
tf,mid=(tf,ci+tf,co)/2=(400°C+100°C)/2=250°C 
λf=0.0425 W/mK, ρf=0.6715 kg/m

3, cf=1038.5 J/kgK, 
νf=4.1525·10

-5 m2/s. 
The final temperature-place function of the charge of 

the heat store is the initial condition of the discharge. 
The mass flow rate of the heat transport medium is 

constant during the charge-discharge process. 
The inlet temperature of the heat transport medium 

during discharge is: tf,di=100°C. 
Data of the outer thermal insulation: 

λi=0.0468 W/mK, δi=1 m. 
ηoh=0.3. 



 

 

98

I have applied the genetic optimization algorithm of 
the Matlab software in order to find the optimal 
geometric sizes and operating parameters of the thermal 
energy store with the best overall efficiency.  

Optimal sizes and operating parameters of the 
thermal energy store with the best overall efficiency 

The optimization process has been executed with number 
of ducts Nj=1, 6, 18. The results are summarized in 
Table 1 and Figure 7. 
 
Table 1: Optimal sizes and overall efficiencies with 
several number of ducts 

Nj [-] 1 6 18 
x1 [m] 0.045 0.1 0.1 
H [m] 27.5 29.6 28.8 
L [m] 27.5 177.6 518.4 

DSj [m] 25.5 10.1 6.0 
St [m] 27.5 29.6 28.8 

db [mm] 45 100 100 
wf [m/s] 0.064 0.412 1.203 
Qcap [PJ] 10.89 10.89 10.89 
Qhid [PJ] 8.45 9.38 9.98 
Ql [PJ] 0.92 1.11 1.01 
Qtr [PJ] 0.01 0.25 2.87 
ηo [-] 0.6908 0.7363 0.5601 

 

 
Figure 7: The optimal overall efficiency against the 

number of ducts 

Main results 

The best overall efficiency can be reached with six 
ducts. It is better than the overall efficiency of one-duct 
type and eighteen-duct type. The expected significant 
increasing of the overall efficiency was ruined by the 
increasing transport work demand of the heat transport 
medium and the increasing heat-loss. 

The advantage of the six-duct type against the one-
duct type is the smaller inside diameter of a duct Dj. It is 

easier to distribute the stream of the heat transport 
medium along a smaller flowing cross-section than 
along a larger one. 

The overall efficiency decreases with increasing of 
the number of ducts, the main reason of it is the 
powerful increasing of the transport work demand. 

In case of liquid heat transport medium the transport 
work demand would be much smaller, so better overall 
efficiency could be reached with greater number of 
ducts. The upper limit of the overall efficiency would be 
restricted by the heat-loss. 

The void fraction of a packed bed, filled with 
uniform balls, in case of exact space-filling ε=0.26 is, in 
practice ε≈0.3 is! The void fraction of the volume of the 
heat store would have to fill with heat transport liquid. 

This large liquid volume brings up the idea of the 
establishment of a heat store with liquid heat storage 
medium, built from cylindrical tanks, in cascade system 
formed a spiral flow-path layout. The thermal 
stratification of the heat store with liquid heat storage 
material would be sharper, the overall efficiency would 
be higher than with solid heat storage material. 

Conclusions 

I have worked out the constructional and the 
mathematical model of the long flow-way, sensible heat 
store with cascade system. The temperature-place 
functions and the overall efficiency can be calculated by 
using the mathematical model. 

I have used genetic optimization algorithm in order 
to find the optimal variant of the thermal energy store 
with the best overall efficiency. 

The chargeable and the dischargeable heat quantity 
of the multi-duct, long flow-way heat store is more than 
of the one-duct, short flow-way thermal energy store 
with equal mass of solid heat storage material. The 
temperature level of the outgoing heat transport medium 
is more advantageous in case of the multi-duct heat 
store than in case of the one-duct type during the whole 
charge-discharge cycle. 

The heat-loss can be reduced by using heat store 
with small specific surface and by allowing charge from 
middle to outside and discharge from outside to middle. 

The transport power demand of the heat transport 
medium can be reduced by making the flow of the 
transport medium only through the ducts where the 
thermocline zone is going along. 

According to the results of the optimization higher 
overall efficiency can be reached in case of six-duct 
type heat store than in case of one-duct type, filled with 
ceramic balls. 

The overall efficiency decreases with increasing of 
the number of ducts – because of the strong increasing 
of the transport work demand. 



 

 

99

Acknowledgements 

This work was supported by the research project 
TÁMOP-4.1.1.C-12/1/KONV-2012-0017. 
 

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