Its dimensionless form is:

2
**2
()
sCR s s
sf
rss
Hk
CHU
tz
θ
θ
θθ
τρ
∂∂
=− − +
∂∂
(27) Fig. 9. Comparison of dimensionless energy storage in the ‘washer’, normalized by the ideal
maximum energy change in the ‘washer’, due to different methods of solution
(
(/2)/
washer i s
Bi h d k= =3.0; / 6.0
eq i
Dd

hh
k
Bi w
CHU
ρ
⎛⎞
⎜⎟
⎛⎞
⎜⎟
=
⎜⎟
+
⎜⎟
⎝⎠
+
⎜⎟
⎝⎠
(28)
For most thermal storage materials, such as rocks, molten salts, concrete, soil, and sands, the
value of
/( )
sss
kCHU
ρ
is very small (in the order of
6
110
−
×
); while other terms in Eq.(27)

increases from 0 to 1.0, while for
solidification it decreases from 1.0 to 0.
Considering the invariant of the temperature of the material during a phase change process,
the energy balance equation for HTF is:

2
()
ff
s
mf
ff
TT
hS
TT U
tz
CR
ρεπ
∂
∂
−= +
∂
∂
(30)
Equation (29) and (30) can be reduced to dimensionless equations by introducing the same
group of dimensionless parameters:

**
1
()
ff

TTC
ψ
=
−Γ is introduced. Since the phase
change temperature is known, Eqs. (31) and (32) can be solved separately.
5.3 Numerical methods and solution to governing equations
5.3.1 Solution for the case of no phase change
A number of analyses and solutions to the heat transfer governing equations of a working
fluid flowing through a packed-bed have been presented in the past (Schumann, 1929;
Shitzer & Levy, 1983; McMahan, 2006; Beasley, 1984; Zarty & Juddaimi, 1987). As the
pioneering work, Schumann (Schumann, 1929) presented a set of equations governing the
energy conservation of fluid flow through porous media. Schumann’s equations have been
widely adopted in the analysis of thermocline heat storage utilizing solid filler material
inside a tank. His analysis and solutions were for the special case where there is a fixed fluid
temperature at the inlet to the storage system. In most solar thermal storage applications this
may not be the actual situation. To overcome this limitation, Shitzer and Levy (Shitzer &
Levy, 1983) employed Duhamel’s theorem on the basis of Schumann’s solution to consider a
transient inlet fluid temperature to the storage system. The analysis of Schumann, and
Shitzer and Levy, however, still carry with them some limitations. Their method does not
consider a non-uniform initial temperature distribution. For a heat storage system,
particularly in a solar thermal power plant, heat charge and discharge are cycled daily. The
initial temperature field of a heat charge process is dictated by the most recently completed

heat discharge process, and vice versa. Therefore, non-uniform and nonlinear temperature
distribution is typical for both charge and discharge processes. To consider a non-uniform
initial temperature distribution and varying fluid temperature at the inlet in a heat storage
system, numerical methods have been deployed by researchers in the past.
To avoid the long mathematical analysis necessary in analytical solutions, numerical
methods used to solve the Schumann equations were discussed in the literature by
McMahan (McMahan, 2006, 2007), and Pacheco et al. (Pacheco et al., 2002), and

1
Dt
D
fr
r
*
f
θθ
τ
θ
−=
(33)
Separating and integrating along the characteristic, the equation becomes:

∫∫
−=
*
fr
r
f
dt)(
1
d
θθ
τ
θ
(34)
Similarly, Eq.(12) for the energy balance for the filler material is reposed along the
characteristic
*

, as the filler material must have the capacity to store the energy
being delivered to it, or vice versa. Finally, separating and integrating along the characteristic
for Eq.(35) results in:

∫∫
−−=
*
fr
r
CR
r
dt)(
H
d
θθ
τ
θ
(36)
There are now two characteristic equations bound to intersections of time and space. A
discretized grid of points, laid over the time-space dimensions will have nodes at these
intersecting points. A diagram of these points in a matrix is shown in Fig. 10. In space, there
are
i = 1, 2, . . . ,M nodes broken up into step sizes of
*
z
Δ
to span all of
*
z
. Similarly, in

numerically as:

∫∫
−=
2,2
1,1
2,2
1,1
*
fr
r
f
dt)(
1
d
ϑ
ϑ
ϑ
ϑ
θθ
τ
θ
(37)
The numerical integration of the right hand side is performed via the trapezoidal rule and
the solution is:

*
ffrr
r
ff

1,1
ϑ
, and
2,2
f
θ
is the value of
f
θ
at
2,2
ϑ
, and similarly so
for
r
θ
.
The integration for Eq. (36) along
*
z
=constant is:

∫∫
−−=
22
12
22
12
,
,

1,22,2
Δ
θ
θ
θ
θ
τ
θθ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
+
−=−
(40)
Equations (38) and (40) can be reposed as a system of algebraic equations for two
unknowns,
2,2
f
θ
and
2,2
r
θ

⎟
⎠
⎞
⎜
⎜
⎝
⎛
−+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
⎥
⎥
⎥

*
CR
f
r
*
r
r
*
f
r
f
r
*
CR
r
*
CR
r
*
r
*
2
tH
1
2
tH
2
t
2
t

Δ
τ
Δ
τ
Δ
τ
Δ
(41)
Cramer’s rule (Ferziger, 1998) can be applied to obtain the solution efficiently. It is
important to note that all coefficients/terms in Eq.(41) are independent of
*
z ,
*
t ,
f
θ
, and
r
θ
, thus they can be evaluated once for all. Therefore, the numerical computation takes a
minimum of computing time, and is much more efficient than the method applied in
references (McMahan, 2006, 2007). Fig. 10. Diagram of the solution matrix arising from the method of characteristics
From the grid matrix in Fig.10 it is seen that the temperatures of the filler and fluid at grids
1,i
ϑ
are the initial conditions. The temperatures of the fluid and filler at grid
1,1

the march of time
*
t
Δ
has no limitation.
The above numerical integrations used the trapezoidal rule; the error of such an
implementation is not straightforwardly analyzed but the formal accuracy is on the order of
)t(O
2*
Δ
for functions (Ferziger, 1998) such as those solved in this study.
5.3.2 Solutions for the case with phase change
For the governing equations of the phase change case, the adopted convention of having the
z-direction coordinate always follow the flow direction is preserved, such that for heat

charging, z=0 is for the top of a tank, and for heat discharging, z=0 is for the bottom of a
tank. The two governing equations (Eq. (31) and Eq.(32)) for the phase change process can
be discretized using finite control volume methodology:

** * ** **
**
() () () ( 1)
()
**
1
()
tt t tt tt
fi fi fi fi
tt
mfi

−Φ
−−=
Δ
(43)
From Eq.(42) the fluid temperature
**
()
tt
fi
θ
+
Δ
can be solved, which is then used in Eq. (43) to
solve for the fusion ratio
**
tt
i
+
Δ
Φ .
The procedures for finding the solution of phase change problem are as follows:
1.
Solve the non-phase-change governing equation analytically using Eq.(12) for the phase
change material for the inlet point.
2.
Monitor the temperature at each time step as given by Eq.(12), and see if the
temperature at a time step is greater than the fusion temperature, if yes, the solution for
that and subsequent time steps are to be solved using the phase change equation (Eq.
(43)
3.

H for the example problem, are summarized in
Table 7.
The numerical computation started from a discharge process assuming initial conditions of
an ideally charged tank with the fluid and rocks both having the same high temperature
throughout the entire tank, i.e. 1
fs
θ
θ
=
= . After the heat discharge, the temperature
distribution in the tank is taken as the initial condition of the following charge process. The
discharge and charge time were each set to 4 hours. The fluid mass flow rate was
determined such that an empty (no filler) tank was sure to be filled by the fluid in 4 hours.

With the current configuration, after five discharge and charge cycles the results of all
subsequent discharge processes were identical—likewise for the charge processes. It is
therefore assumed that the solution is then independent of the first-initial condition. The
data presented in the following portions of this section are the results from the cyclic
discharge and charge processes after 5 cycles.

ε

r
τ

CR
H
H R
t
0.25 0.0152 0.3051 14.6 m 7.3 m

Filler material (granite rocks) properties:
s
ρ

= 2630 kg/m3;
C
s
=775 J/(kg K); k
s
=2.8 W/(m K); d
r
= 0.04 m;
Table 7. Dimensions and parameters of a thermocline tank (Van Lew et al., 2011) Fig. 11. Dimensionless fluid temperature profile in the tank for every 0.5 hours
Shown in Fig. 11 are the temperature profiles in the tank during a discharge process, in
which cold fluid enters into the tank from bottom of the tank. The location of
*
0z = is at the
bottom of a tank for a discharge process. The temperature profile evolves as discharging
proceeds, showing the heat wave propagation and the high temperature fluid moving out of

the storage tank. The fluid temperature at the exit (
*
1z
=
) of the tank gradually decreases
after 3 hours of discharge. At the end of the discharge process, the temperature distribution
along the tank is shown in Fig. 12. At this time the fluid and rock temperatures,

is used to denote
s
θ
, as rocks are used as the storage material in the example). Fig. 13. Dimensionless temperature distribution in the tank after time t
*
= 4 of charge

(Here
r
θ
is used to denote
s
θ
, as rocks are used as the storage material in the example)
A heat charge process exhibits a similar heat wave propagation scenario. The temperature
for the filler and fluid along the flow direction is shown in Fig. 13 after a 4 hour charging
process. During a charge process, fluid flows into the tank from the top, where
*
z is set as
zero. It is seen that for the bottom region (
*
z
from 0.7 to 1.0) the temperatures of the fluid
and rocks decrease significantly. A slight temperature difference between heat transfer fluid
and rocks also exists in this region.
*
1t =
. At
*
2.5t =
, or t=2.5 hours, the fluid temperature
f
θ
starts to drop. This is because
the energy from the rock bed has been significantly depleted and incoming cold fluid no
longer can be heated to
f
θ
= 1 by the time it exits the storage tank.
The above numerical results agree with the expected scenario as described in section 4. To
validate the above numerical method, analytical solutions were obtained using a Laplace
Transform method by the current authors (Karaki, et al, 2010), which were only possible for
cases with a constant inlet fluid temperature and a simple initial temperature profile. Results
compared in Fig. 15 are obtained under the same operational conditions—starting from a

fully charged initial state and run for 5 iterations of cyclic discharge and charge processes.
The fluid temperature distribution along the tank (
*
z =0 for bottom of the tank) from numerical
results agrees with the analytical results very well. This comparison essentially proves the
effectiveness and reliability of the numerical method developed in the present study. Fig. 15. Comparison of numerical and analytical results of the temperature distribution in
the tank after time t

100, and 1000) for
*
z . The high level of accuracy of the current numerical method, even with
only 20 nodes, demonstrates the accuracy and stability of the method with minimal
computing time.
6.2 Comparison of modeling results with experimental data from literature
6.2.1 Temperature variations in charge processes
The authors have conducted experimental tests (Karaki at al., 2011). The test conditions for a
given heat charge process are listed in Table 8, which also shows the dimensionless
parameters. As shown in Fig. 17, at the initial time the thermocline tank has a uniform
temperature equal to room temperature. The temperature readings from the thermocouples
at the top of the tank provide the inlet fluid temperatures in a charge process.

Tank Length 0.65(m)
Initial temperature 21.9 (
o
C)
Tank inner diameter 0.241 (m)
High temperature 79.82 (
o
C)
Rock nominal diameter 0.01 (m)
Low temperature 21.9 (
o
C)
Oil flow rate 1.0 (Liter/min.)
Porosity 0.324
Density of rocks 2632.8 kg/m
3
/( / )

Temperature (
o
C)
TC-1
TC-2
TC-3
TC-4
TC-5
TC-6
TC-7
TC-8
TC-9
TC-1 0
TC-1 1
TC-1 2
TC-1 3
TC-1 4

Fig. 17. Temperatures in the center of the tank along the height of 65 cm for a charging
process (Thermocouples were set every 5cm)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7

7.25
z
∗

θ
avgFig. 18. The temperature distribution along the height in the tank at different time points
(Solid lines are from simulation results and dashed lines are from experimental tests)
Based on the temperature measurements, the temperature in the tank increases gradually
and at the end of the charging process, the temperatures at all the locations are sufficiently
high. The temperature distribution along the height in the tank at different times is shown in
Fig. 18. Obviously at the end of the charge, the temperature at the top of the tank (at z*=0) is
high. Using the initial temperature distribution and the inlet fluid temperature, together
with the properties listed in Table 8, numerical simulation results were obtained and are
shown in Fig. 18 for the average temperature of the fluid and rocks. The real time and the
dimensionless time are listed in Table 8. The agreement between the experimental data and
the modeling simulation is very satisfactory.
The thermal storage performance test results of a thermocline tank reported in the literature
(Pacheco et al., 2002) was also referenced to validate the current modeling work. The
experimental tests used eutectic molten salt (NaNO
3
-KNO
3
, 50% by 50%) as the heat transfer
fluid and quartzite rocks and silica sands as the filler material. Thermocouples in the test
apparatus were imbedded in the packed-bed. Temperatures in the tank at different height
locations were recorded during a two-hour heat discharge process after the tank was
charged for the same length of time. The storage tank dimensions, packed-bed porosity, and

f
ρ
=1733 kg/m
3
;
C
f
=1550 J/(kg K);
k
f
=0.57 W/(m K);
m

=7.0 kg/s;
f
μ
= 0.0021 Pa s⋅ ;
Quartzite rocks/sands mixture properties:
s
ρ

= 2640 kg/m3;
C
s
=1050 J/(kg K); k
s
=2.5 W/(m K); d
r
= 0.015 m;
Table 9. Dimensions and parameters of a thermocline tank for the test in literature (Pacheco

CR
H
0.4210
Time (s) 0 476.2 952.5 1428.7 1905.0 2381.2
Dimensionless time t* 0 1.61192 3.22384 4.83576 6.44768 8.0596
Table 10. Conditions of a discharge test

0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
z*=z/H
θ
avg
t*=0 Modeling
t*=0.767 Modeling
t*=1.534 Modeling
t*=2.300 Modeling
t*=3.067 Modeling
t*=0.0 Test
t*=0.767 Test
t*=1.534 Test

TC-7
TC-8
TC-9
TC-10
TC-11
TC-12
TC-13
TC-14 (top-out)

Fig. 20. Temperatures in the center of the tank along the height of 65 cm for a discharging
process (Thermocouples were set every 5cm)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z
*
θ
f
(Solid lines are from simulation results and dashed lines are from experimental tests)

The temperature distribution along the height in the tank at different times is shown in Fig.
21. At the end of the discharge, the temperature on top of the tank (at z*=1) becomes low,
which means that stored energy has been discharged. Using the measured initial
temperature distribution and the inlet fluid temperature (at z*=1) together with the
properties listed in Table 10, numerical simulation results were obtained and are compared
with the test results in Fig. 21. The real time and the dimensionless time are listed in Table
10. Again, the agreement between the experimental data and the modeling simulation is
satisfactory.
6.3 Correlation of energy delivery effectiveness to dimensionless parameters
Based on the above discussion and the dimensionless governing equations obtained in
section 5.2.3, the energy delivery effectiveness,
η
, is a function of four dimensionless
parameters,
/
cd
ΠΠ,
d
Π
,
r
τ
, and
CR
H . Solutions of the dimensionless governing
equations for energy charge and discharge allow us to develop a database so that a series of
charts and diagrams for
(/, ,, )

η
τ
r1

τ
r2

τ
r3

τ
r4

τ
r5

Π
c
/
Π
d

1.0
0.0
H
CR1

H
CRn


exactly 1.0 if the energy charge period is equal to or larger than that of the discharge
(
/1.0
cd
ΠΠ≥
). For any non-ideal thermocline system
η
can only approach 1.0.

Figure 23 shows four charts of
η
versus /
cd
Π
Π at
d
Π
=4.0. Each chart is for a specific
r
τ

with multiple curves for different
CR
H . All the data of energy delivery effectiveness were
obtained based on several cyclic operations of the energy charge and discharge, and the
results are consequently independent of the number of cycles. More charts with wide range
of
d
Π ,
r

(c)
0.1
r
τ
= (d)
0.2
r
τ
=

Fig. 23. Multiple graphs from modeling results for energy storage effectiveness versus
/
cd
ΠΠat
d
Π =4.0
Observing the above four graphs one can easily draw the following conclusions:
1.
The energy delivery effectiveness never reaches 1.0 if /1.0
cd
ΠΠ< . This proves that
only for an ideal thermocline storage tank can 1.0
η
=
at /1.0
cd
Π

when compared to ( )
f
C
ρ
, and therefore the energy storage capability is improved, and
η
can approach 1.0 easier. On the other hand, when the void fraction in a packed bed
approaches 1.0, it will make
CR
H →∞, and the thermal storage effectiveness can
approach that of an ideal case. However, in most practical applications, a low void
fraction in a thermocline tank is required for the purpose of using less heat transfer
fluid, and therefore a smaller
CR
H value is practical and preferable.
4.
For cases where
η
could never approach 1.0, even at large
/
cd
Π
Π
values, it is
obviously attributable to the fact that the storage tank is too small, and reselection of a
larger storage tank is needed.
Designers for a thermal storage system often need to calibrate or confirm that a given
storage tank can satisfy an energy delivery requirement. Under such a circumstance, the
dimensions of the storage tank and the power plant operational conditions are known,
which means the values of

decided in step (1). Using these dimensions, the parameters—
d
Π
,
r
τ
,
CR
H for a
thermocline tank with filler material, can be evaluated, where
d
Π
is determined based
on the required operational time.
3.
Look up the design charts (such as those in Fig. 23) and see if an energy delivery
effectiveness of 1.0 can be achieved. Often the energy delivery effectiveness will not
approach 1.0 for the first trial design. This is because the first trial uses a minimum
volume. However, with the results from the first trial one can predict the required
height or volume of the tank necessary to decrease
r
τ
and
d
Π
in the same proportion.
A couple of trail iterations may be needed to eventually satisfy the criterion of
η

closing to 1.0.

d
Π decreases accordingly since it depends on the height of the tank.
Occasionally, a calibration analysis requires a designer to find a proper time period of
energy charge that can satisfy the needed operation time of a power plant. The known
parameters will be the tank volume,
r
τ
, as well as
CR
H at a required operation period of
d
Π .
The first step of the calibration should be the examination of the criterion given in Eq. (3),
from which a minimum tank volume can be chosen. If the minimum tank volume is
satisfied, the second step of calibration will be to find a proper
/
cd
Π
Π that can make the
energy delivery effectiveness approach 1.0. Graphs including curves at the required
CR
H
and the given
r
τ
and
d
Π
must be looked up. Conclusions can be easily made depending on
whether the energy delivery effectiveness can approach 1.0 for a particular value of

CR
H ,
d
Π
, and
r
τ
are found to be 0.451,
3.03, and 0.0227, respectively. Given in Fig. 24 is a chart for
d
Π
=3 and
r
τ
=0.0227 at
various values of
CR
H and /
cd
Π
Π . It is seen that on the curve of
CR
H =0.45, there is
no time ratio
/
cd
Π
Π that allows the energy delivery effectiveness to be close to 1.0.
Therefore, the ideal volume chosen will not satisfy the energy storage need.
2.

Fig. 24. Energy delivery effectiveness versus
/
cd
Π
Π at
d
Π
=3 and
r
τ
=0.0227 Fig. 25. Energy delivery effectiveness versus
/
cd
Π
Π at
d
Π
=2.42 and
r
τ
=0.0181
7.2 Design example 2—a system as shown in Fig. 2(b)
For the same solar thermal power plant and operational conditions as in Example 1, the
thermal storage primary material is molten salt with properties of
3
1680 /k
g

(25), where the correction coefficient for the washer is
c
w = 3.69894 for the ratio of
/1.74
eq i
Dd= . The minimum volume of the tank is used in the first trial of the design.
The dimensionless values of
CR
H ,
d
Π
, and
r
τ
are 0.522, 3.032, and 0.221, respectively.
Given in Fig. 26 is a chart for
d
Π
=3.032 and
r
τ
=0.221 at various values of
CR
H and
/
cd
ΠΠ
. The figure shows that at an
CR
H

=1.444 and
r
τ
=0.105. It is seen that at an
CR
H value of
0.50 (close to 0.522) and a time period ratio,
/
cd
Π
Π , of 1.2, the energy delivery effectiveness
can reach 0.96.
Note that in the finalized storage tank (with a height of 13.5 m), the volume of heat transfer
fluid takes 69% of the volume of an ideal thermal storage tank (with a height of 6.44 m). The
energy delivery effectiveness reached 0.96 if the charging time period was kept 20% longer
than the required discharge time. Obviously, the weak heat transfer between the HTF and

the primary thermal storage material in Example 2 is responsible for the much larger ratio of
the actual volume compared to the volume of the ideal storage tank. In Example 1 the ratio
of the actual volume of the finalized storage tank is only 1.25 times that of the volume of its
corresponding ideal storage tank. Therefore, it is important that in order to improve the
energy delivery efficiency, the heat transfer between the fluid and thermal storage material
must be improved, for example, by using pipes with fins. Nevertheless, this type of thermal
storage system, as in Example 2, still saves 31% of the heat transfer fluid. Fig. 27. Energy delivery effectiveness versus
/
cd
Π

timely event.
Due to space limitations, this chapter did not provide a large number of the developed
charts for thermal storage system sizing and design. However, the authors are currently
working to provide a handbook including a large number of design charts covering a wide
range of design parameters that industry will need. It is hoped that the handbook will be
available to readers in the near future.
9. Acknowledgement
The authors are grateful for the support provided by the US Department of Energy and the
National Renewable Energy Laboratory, under DOE Award Number DE-FC36-08GO18155,
and the US Solar Thermal Storage LLC. The Support from the Idaho National Laboratory
under award number 00095573 for phase change thermal storage is also gratefully
acknowledged.
10. Nomenclature
f
a The cross section area of a storage tank (m
2
)
Bi Biot number ( /
p
s
Lh k
=
)
C Heat capacity ( /
o
Jk
g
C
⋅
)

Lp Characteristic length of ‘particles’ for Biot number.
m

Mass flow rate (kg/s)
N Number of tubes for heat transfer fluid in a storage tank
Pr Prandtl number
Q

Thermal energy involved per unit of time (W)
Q Thermal energy involved ( J )
r Average radius of the filler material (rocks) (m)
R Radius of the storage tank (m)
Re Modified Reynolds number for porous media
Ss Surface area of filler material per unit length of the storage tank ( m )
t Time (sec)
T
H
High temperature of fluid from solar field (°C)
T
L
Low temperature of fluid from power plant (°C)

U Fluid velocity in the axial direction in the storage tank (m/s)
V Volume (
3
m
)
z Location of a fluid element along the axis of the tank (m)
Greek symbols
s

g
m )
θ Dimensionless temperature.
Subscript
c Energy charge process
d Energy discharge process
f Thermal fluid
ref A required reference value
r Rocks
s Filler material (rocks), the primary thermal storage material.
z Location along the axis of the tank
Superscript
* Dimensionless values
11. References
Abdoly, M.A., Rapp, D., 1982, Theoretical and experimental studies of stratified thermocline
storage of hot water, Energy Conversion and Management, Vol. 22, No. 3, 1982, pp.
275-285.
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