International Journal of Mechanical & Mechatronics Engineering IJMM E-IJENS Vol:10 No:01
7
Temperature Profile and Thermocline
Thickness Evaluation of a Stratified Thermal
Energy Storage Tank
Joko Waluyo1) and M Amin A Majid2)
continuous profile of temperature distribution. Difficulty in
determining thermocline thickness arises for the case of
discrete temperature data, since the profile formed could not
be used to estimate the thermocline thickness. This paper
discusses a practical method for formulation of thermocline
thickness of stratified thermal energy storage. Curve fitting by
iterative method was adopted to identify the functions which
could represent the S -curve of temperature distribution.
Based on the functions, thermocline thickness was formulated
using functional relationship of temperature profile. Results
identified two functions which could represent S -curve of
temperature distribution, namely sigmoid dose response (S DR)
and four parameter sigmoid (FPS ) functions. Both functions
were observed to well fit the temperature distributions having
coefficient determination more than 0.99. Based on evaluations
the formulations were capable to be utilized for evaluation of
thermocline thickness of the stratified TES . The methods offer
an advantage to obtain an exact value of thermocline thickness.
Index Term-- temperature distribution profile, thermocline
thickness, S tratified thermal energy storage.
I.
INT RODUCT ION
Thermal Energy Storage (TES) systems are useful for
maximizing the thermal energy efficiency for meeting the
fluctuating cooling demands by shifting energy use from on
peak to off-peak hours. This is achieved by charging the
TES tank during the off-peak hours and discharging it later
during the peak hours. Many studies have been undertaken
related to stratified TES tanks. An important parameter in
evaluating performance of charging and discharging of a
TES tank is the thickness of the thermocline being formed.
A thinner thermocline is desired since a thicker thermocline
indicates larger degradation of stratification [1]. The
thickness of the thermocline indicates the extent of mixing
occurred due to inflow streams during the cycles . This
factor influences the degradation of stratification, beside
heat transfer losses from the tank [2].
Thermocline thickness is determined based on water
temperature distribution inside the tank. The water
temperature distribution profile formed could move either
upward or downward during charging or discharging cycles
[3, 4]. Many researches on thermocline thickness parameter
in relation to stratified TES tank performance have been
undertaken based on continuous profile of water
temperature distribution.
1)
Mechanical engineering department, Universiti T eknologi
PET RONAS, Bandar Seri Iskandar, Tronoh, Perak, Malaysia. (phone:
+60165943445), e-mail: [email protected]).
2)
Mechanical engineering department, Universiti T eknologi
PET RONAS, Bandar Seri Iskandar, Tronoh, Perak, Malaysia. (e-mail:
[email protected]).
Using continuous profile, thermocline thickness is
accurately identified as asymptote regions with limit points
located on the edge of profiles [5, 6, 7, 8, 9]. The
difficulties of measuring thermocline thickness arise if the
temperature is available as a discrete data resulted from long
interval-distance of temperature sensors. The un-continuous
profiles that are formed could not be used to determine the
thermocline thickness due to its ambiguity in defining the
limit points. A method to determine thermocline thickness
from discrete data of temperature distribution is investigated
in this study. The study focuses on developing an approach
of establishing the profile by adopting fitting method and
using the profile to formulate the thermocline thickness.
Data acquired from an operating TES tank equipped with
long interval-distance sensors recorded based on hourly
basis, was used in this study.
II.
M ETHODOLOGY
Normally water temperature distribution in the stratified
TES tank consists of 3 regions with warm water at the top,
cool water at the bottom and thermocline region in the
middle. The water temperature forms S-Curve profile
consisting of two asymptote curves as shown in Fig. 1.
Average cool and warm water temperature is formed by the
asymptote values of Tc and Th . Position of the thermocline,
C, defines the boundary line of cool and warm water in the
tank. It also can be interpreted as the cool water depth
occupied in the tank. Thermocline thickness, WTC, is
determined as the region limited by the edges of asymptote
curve.
Th
Tank Height, H
Abstract— Determination of thermocline thickness requires a
WTC
C
Tc =
Th =
WTC =
C =
Average cool water temperature
Average warm water temperature
Thermocline thickness
Midpoint of thermocline
Tc
Fig. 1. S-curve of temperature profile [7]
The temperature profile formed could be represented as a
function (1) with one variable of x and four parameters of
Tc, Th, C and S.
(1)
T ( x)  f (Tc , Th , C, S , x)
The temperature distribution in equation (1) was used as a
basis for determining thermocline thickness in this study.
The analysis steps are as follows:
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International Journal of Mechanical & Mechatronics Engineering IJMM E-IJENS Vol:10 No:01
Case I (flow rate 393 m3/hr)
16
18.00
19.00
14
20.00
Temperature ( C)
12
o
S-curve profile using iterative method.
Identifying the temperature profile functions.
Determining the parameters in the functions.
Formulation of thermocline thickness.
Evaluation.
Temperature data of the TES system of a district cooling
plant were acquired for the study. The TES system consists
of two 1,250 tons of refrigeration (RT) of steam absorption
chillers (SACs) and four 325 RT electric chillers (ECs) and
one 5,400 m3 storage TES tank with designed capacity of
10,000 RTh. Inlet nozzle is made from 20” NPS located at
elevation 3.4 m height, while outlet nozzle is 12” NPS at
elevation 12.3 m. Both nozzles are provided with diffuser on
its end-connection in the storage tank. Overflow line is
connected at elevation of 14.025 m. The entire tank is
externally insulated. The tank is equipped with 14
temperature sensors, installed at approximately 1 m vertical
interval, to measure the water temperatures. The lowest
temperature sensor is located at 0.51 m height. All
temperatures are hourly recorded with acquisition data
system. The schematic flow diagram of the system is shown
in Fig. 2.
21.00
10
22.00
23.00
8
24.00
6
01.00
02.00
4
0
2
4
6
8
10
12
14
Sensor Elevation (m )
(a)
Case II (flow rate : 524 m3/hr)
16
18.00
19.00
14
20.00
12
21.00
o
iv.
v.
vi.
vii.
profile. This profile move upward from hour 18.00 to the
final condition at 02.00. Higher movement of the profile of
case II is occurred than of that for case I. These movements,
therefore, reveal a more temperature distribution span of
case II compare to case I.
Temperature ( C)
i. Acquiring temperature data from an operating TES.
ii. Plotting the temperature distribution.
iii. Fitting of the functions that could represent the
8
22.00
10
23.00
8
24.00
01.00
6
02.00
4
0
2
4
6
8
10
12
14
Sensor Elevation (m)
(b)
Fig 3. T emperature distributions for (a) case I and (b) case II
Fig. 2. Schematic flow diagram of charging cycle
TES tank is charged by the ECs during off-peak hours.
Normally, the charging is served by three or four of ECs.
For the purpose of this study, hourly temperature records
during charging period of 9th September 2008 and 15th April
2009 were used.
The charging cycle was operated
continuously from 18.00 hours to 02.00 hours of the
following day. These two charging cycles were served by 3
and 4 units of ECs, and this is represented as case I and case
II, respectively. The flow rates of the charging cycles for
case I and case II were 393 m3 /hr and 524 m3 /hr,
respectively.
III. RESULT AND DISCUSSION
Temperature distribution data
The plot temperature distribution for case I and II are
depicted in Fig. 2 (a) and (b), respectively. The hourly
temperature distribution within charging periods are
presented with respect to sensor elevation in the TES tank.
Each hourly charging course form a continues S-curve
Fitting of the functions
Data as depicted in Fig 3 (a) and (b) were used to fit
some possible functions to represent the S-curve
temperature profile. The functions consist of parameters Tc,
Th , C and S. Fitting was done by utilizing commercial
software of Sigmaplot [10] using non linear regression by
iterative method. Two functions were identified that could
represent the S-curve: sigmoid dose response (SDR) and 4
parameters sigmoid (FPS) function. The first function was
formed as a modification from the sigmoidal dose response
(variable slope) function and the second function was
obtained from the modification of 4 parameters sigmoid
function.
SDR function form as the following,
Th  Tc
T  Tc 
1  10( C  X ) S1
FPS function is expressed as,
T  Tc 
Th  Tc
1  e(C  X ) / S2
(2)
(3)
Both SDR and FPS functions relate temperature
distribution to one variable of X and the four parameters.
SDR function has four parameters of Tc, Th , C and S 1 .
While FPS has parameters of Tc, Th , C and S 2 . Parameters
Tc and Th are cool and warm temperatures (o C). X variable
expresses the dimensionless elevation (x.N/H), where x is
the elevation of the temperature sensors (m), H is effective
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International Journal of Mechanical & Mechatronics Engineering IJMM E-IJENS Vol:10 No:01
height of the tank content of water (m) and N number of
stratified layers. Parameter C implies of a dimensionless
elevation unit. S 1 for SDR and S 2 for FPS are constant
parameters related to slope parameter of the functions.
The plotting of fitting profile are presented in Fig 4 (a)
and (b) and Fig 5 (a) and (b) for SDR and FSP, respectively,
covering of case I and II. Referring to Fig 4 (a),(b) and 5
(a),(b), it is indicated that the temperature in the mixing
region fitted to the asymptotes curves, and the remains
approached to the flat line. It can be seen that the adjusted
temperature resulted to more clear demarcation of the
temperature in the mixing region of the asymptotes curve
following the fitting functions. Indicated from Fig. 4 (a) and
5 (a), both SDR and FPS functions have the same curvature
profiles in the fitting of temperature data. Similar curvature
profiles were also noted for 4 (b) and 5 (b) in the fitting of
case II. The coefficient of determination, R2 of both
function, depicted in Table I and II, are greater than 0.99,
indicating that the temperature data were fitted well to the
functions.
Determination of parameters
Parameters of Tc, Th , C and S were obtained from the
fitting of SDR and FPS functions. The parameters of both
functions are tabulated in Table I and II, covering both
case I and II. The values of R2 , coefficient of
determination, for evaluation the goodness of fitting are
also provided.
From Table I and II, it is noted that fitting of case I using
SDR and FPS functions revealed the same values of Tc, Th ,
and C on each hourly charging cycle. On the other hand , it
is observed that the parameters of S 1 and S 2 are different.
Similar observation was found for the case II as well. It is
noted that SDR and FPS have similar profiles and
parameters of Tc, Th , and C in the fitting. In addition, the
values of coefficient of determination, R2 , are equal for SDR
and FPS functions in the fitting of cases I and II. This
indicates that both SDR and FPS function are enable to fit
the S-curve of temperature distribution. Either one of them
can be chosen for the implementation. The different of
these functions is due to usage of the basis number 10 and
natural number, e.
As indicated in Table I and II, the values of Th and Tc
decrease for both case I and case II. The decreased in
values of Tc was due to incoming supply of cooler water at
the bottom of the tank from the ECs. While the decreased in
values of Th , was due to diffusion of thermocline region.
The values of parameter C are also shown in Table I and II.
For both cases the values of C increase with charging time,
with case I smaller than case II. The trends occurred due to
increased cool water depth as a result of more cool water
was generated with charging time. These trends were also
noted by (Nelson et al, 1998) and (Karim, 2009) through
their experimental investigations on stratified tanks.
Formulation of thermocline thickness
By identifying the function of temperature profile,
thermocline thickness could be formulated based on
functional relationship. The concept for determining the
thermocline thickness was S-curve temperature profile as
illustrated in Fig. 1. The formulation the thickness of
thermocline was achieved refer to the function of
temperature profile.
9
Formulation of thermocline thickness using SDR
function was obtained by rearranging equation (2) into
the form
Th  Tc
 1  10(C  X ) S1
T  Tc
(4)
The left term of equation (4) is re-arranged using
dimensionless cut-off temperature =(T-Tc)/(Th -Tc),
Musser (1998b), describing the limit point of the
thermocline thickness.
1
(5)
 1  10( C X ) S

Distance from C to X express the half- thickness of the
thermocline, and represented as follows:
1
Log (  1)

(6)
CX 
S1
Therefore, thermocline thickness is defined as
1
WTC 
2.Log (
1
 1)

(7)
S1
Using the similar analysis, thermocline thickness for FPS
function is determined as
1
(8)
WTC  2.Ln.(  1).S 2

Equation (7) and (8) were used to determine thermocline
thickness for a predetermined value of . Conceptually, the
dimensionless cut-off ratio  values are in the range of 0 to
0.5 covering minimum and maximum thermocline
thickness. For =0 indicate that the thermocline edges
profile are located at Tc and Th , therefore gives a maximum
thickness. With =0.5, the limit points are at midpoint of
thermocline region resulting zero values of the thickness.
The evaluations were conducted by performing
calculation using varies of  from 0 to 0.5, with parameters
S 1 and S 2 as described in Table I and II, for SDR and FPS
function, respectively. Evaluated results of thermocline
thickness of SDR function are tabulated in Table III and IV,
for case I and II, respectively. In the calculation using FPS
function, the result has similar value with maximum
deviation of 1.5 % from SDR values. This is due to the
usage of rounded value up to 2 digits decimal of the
parameter. If it is required to reduce the deviation, us ing
more decimal digits of the fitting parameters is
recommended.
From Table III to IV, it is noted that higher thermocline
thickness occurred at the lesser values of . Thermocline
has maximum thicknesses at =0.001. In addition, with 
equal to 0 and 0.5, revealed ∞ and 0, respectively.
Therefore, range 0<<0.5 can be chosen for its
implementation, depending on the appropriateness of the
requirement. The smaller value takes advantage of
expressing the real thermocline thickness in the cycles.
It is highlighted that the above method is justified as a
practical method to determine the thermocline thickness.
This is achieved by converting the discrete data of
temperature distribution to a continues profile content of
parameters average cool and warm water temperature,
midpoint of thermocline position and slope gradient.
Thermocline thickness was then determined using
functionally relationship of the temperature profile function.
In the method, the goodness of fitting of the discrete data
has significant role to the result in reflecting the thickness of
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International Journal of Mechanical & Mechatronics Engineering IJMM E-IJENS Vol:10 No:01
thermocline. Determination of thermocline thickness based
on functional relationship takes beneficial in generating
exact values of the parameters rather than estimation.
The above calculation was obtained based on
temperature profile with has coefficient of determination
more than 0.99 to the acquired data. This indicates that the
obtained calculations are reliable to be applied for
evaluation of stratified TES. Confirmation the magnitude
values of the parameters, however, can not be conducted
since the parameters might vary over the wide range depend
on configuration and operating condition of the stratified
TES (Zurigat and Ghajjar, 2002).
Thermocline thickness evaluation of stratified TES
For evaluation of thermocline position during charging
cycle, results from Table I and II indicate that thermocline
position moves upwards from the bottom part to upper part
of TES tank during charging cycle.
With regards to thermocline thickness, for case I it is
lower than that for case II, as seen from Table III and IV.
This indicates that higher flow rate resulted to thicker
thermocline. The finding on the variations of thermocline
thickness for different flow rates as noted in this study were
also reported by (Karim, 2009) and (Musser and Bahnfleth,
1998b).
For thermocline thickness growth evaluation, Table I,II
and III,IV were used to relate parameter WTC and C. Using
observation at =0.1, it indicates that thermocline thickness
change with respect to time. For case I, the maximum
thermocline thickness is 1.35 m, in the charging of hour
20.00 when the midpoint of thermocline position at 3.64 m
elevation. Case II has maximum thermocline thickness of
1.84 m, also charging at 20.00, with midpoint of
thermocline position at 4.32 m elevation. The occurrence of
maximum thermocline thickness at the lower part of the
storage tank are noted, this is due to its position nearby inlet
diffuser where the mixing has more influence.
10
REFERENCE
[1]. Zurigat YH and Ghajar AJ, 2002, Heat Transfer and Stratification in
Sensible Heat Storage System, in Thermal Energy Storage System
and Applications, Eds. Dincer and Rosen, Wiley, New York.
[2]. Dincer I and Rosen MA, 2002, T hermal Energy Storage System and
Applications, John Wiley and Sons.
[3]. Nelson JEB, Balakrishnan, Srinivasa Murthy S, 1999, Experiments
on Stratified Chilled-Water T anks, International Journal of
Refrigeration 22, pp. 216-234.
[4]. Karim MA, 2009, Performance Evaluation of a Stratified ChilledWater T hermal Storage System, World Academy of Science,
Engineering and T echnology 53, pp. 326-334.
[5]. Bahnfleth WP and Musser A, 1998, T hermal Performance of Full
Scale Stratified Chilled Water Storage Tank, ASHRAE Transaction
104(2), pp. 377-388.
[6]. Musser A and Bahnfleth WP, 1998a, Evolution of T emperature
Distributions in a Full-Scale Stratified Chilled-Water Storage T ank
with Radial Diffusers, ASHRAE T ransactions, Vol. 107(1).
[7]. Musser A and Bahnfleth WP, 1998b, Field-Measured Performance of
Four Full-Scale Cylindrical Stratified Chilled-water T hermal
Storage T anks, ASHRAE T ransaction 105 (2), pp. 218-230.
[8]. Homan K, Sohn C and Soo S, 1996, T hermal Prformance of
Stratified Chilled Water Storage T ank, HVAC&R Research 2(2):
158-170.
[9]. Yoo J, Wildin MW and T ruman CR, 1986, Initial Formation of
T hermocline thickness in Stratified T hermal Storage T ank,
ASHRAE T ransaction 92 (2A): 280-292.
[10]. Systat Software Inc, 2008, SigmaPlot 11 User’s Guide, CA USA.
IV. CONCLUSIONS
Results from the study indicate that there were two
functions which could be used to represent the S-curve of
temperature distribution, namely sigmoid dose response
(SDR) and four parameter sigmoid (FPS) function. Both
functions have similar curvature in the fitting of temperature
distribution. Using these functions, the average cool and
warm water temperature and cool water depth in storage
tank could be ascertained. The method offer an alternative
solution for formulating thermocline thickness based on
functional relationship of temperature profile. It takes
advantage to simplify the determination of thermocline
thickness and eliminate the ambiguities of limit points on
the edge of thermocline profile. This approach to figure out
the temperature profile and determine the thermocline
thickness based for functionally relationship will assist in
the evaluation of performance of TES tank.
A CKNOWLEDGM ENT
The authors would like to acknowledge the support of
Universiti Teknologi PETRONAS for this research.
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International Journal of Mechanical & Mechatronics Engineering IJMM E-IJENS Vol:10 No:01
11
Case I - SDR function
16
Temperature (oC)
14
12
10
18.00
19.00
20.00
21.00
22.00
23.00
24.00
01.00
02.00
8
6
4
0
2
4
6
8
10
12
14
X (dimensionless elevation)
(a)
Case II - SDR Function
16
12
o
Temperature ( C)
14
10
18.00
19.00
20.00
21.00
22.00
23.00
24.00
01.00
02.00
8
6
4
0
2
4
6
8
10
12
14
X (dimensionless elevation)
(b)
Fig. 4. Fitting SDR function for (a) case I and (b) case II
Case I - FPS function
16
Temperature (oC)
14
12
10
18.00
19.00
20.00
21.00
22.00
23.00
24.00
01.00
02.00
8
6
4
0
2
4
6
8
10
12
14
X (dimensionless elevation)
(a)
Case II - FPS Function
16
12
o
Temperature ( C)
14
10
18.00
19.00
20.00
21.00
22.00
23.00
24.00
01.00
02.00
8
6
4
0
2
4
6
8
10
12
14
X (dimensionless elevation)
(b)
Fig. 5. Fitting FPS function for (a) case I and (b) case II
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International Journal of Mechanical & Mechatronics Engineering IJMM E-IJENS Vol:10 No:01
12
T ABLE I.
P ARAMETERS ON THE TEMP ERATURE DISTRIBUTION OF SDR FUNCTION
SDR
18:00
19:00
20:00
21:00
22:00
23:00
0:00
1:00
2:00
FPS
18:00
19:00
20:00
21:00
22:00
23:00
0:00
1:00
2:00
Case I (Flow rate:393 m3/hr)
Tc
6.94
6.93
6.91
6.91
6.90
6.89
6.87
6.85
Th
13.61
13.58
13.56
13.55
13.54
13.54
13.53
13.51
13.49
6.83
Tc
6.94
6.93
6.91
6.91
6.90
6.89
6.87
6.85
C
1.98
2.71
3.64
4.61
5.55
6.51
7.47
8.44
9.39
Case II (Flow rate:524 m3/hr)
2
S1
R
2.11
1.60
1.41
1.42
1.44
1.48
1.71
1.83
1.83
0.9983
0.9987
0.9994
0.9995
0.9995
0.9995
0.9996
0.9996
0.9997
Tc
7.08
7.06
7.02
7.02
6.95
6.91
6.89
6.86
6.73
Th
13.83
13.74
13.74
13.71
13.63
13.57
13.53
13.38
13.36
C
1.94
3.10
4.32
5.63
6.86
8.13
9.35
10.53
11.79
S1
R2
1.42
1.20
1.04
1.03
1.24
1.31
1.29
1.32
1.30
0.9932
0.9942
0.9965
0.9967
0.9975
0.9983
0.9987
0.9996
0.9993
T ABLE II.
P ARAMETERS ON THE TEMP ERATURE DISTRIBUTION OF FPS FUNCTION
Case I (Flow rate:393 m3/hr)
Case II (Flow rate:524 m3/hr)
2
R
S2
S2
Th
C
Tc
Th
C
6.83

0
0.001
0.1
0.2
0.3
0.4
0.5
13.61
13.58
13.56
13.55
13.54
13.54
13.53
13.51
13.49
1.98
2.71
3.64
4.61
5.55
6.51
7.47
8.44
9.39
0.21
0.27
0.31
0.31
0.30
0.29
0.25
0.24
0.24
0.9983
0.9987
0.9994
0.9995
0.9995
0.9995
0.9996
0.9996
0.9997
7.08
7.06
7.02
7.02
6.95
6.91
6.89
6.86
6.73
13.83
13.74
13.74
13.71
13.63
13.57
13.53
13.38
13.36
1.94
3.10
4.32
5.63
6.86
8.13
9.35
10.53
11.79
T ABLE III.
T HERMOCLINE THICKNESS OF CASE I USING SDR FUNCTION
WTC - Case I (Flow rate : 393 m3/hr) - SDR
18:00 19:00 20:00 21:00 22:00 23:00 0:00
1:00
0.31
0.36
0.42
0.42
0.35
0.33
0.34
0.33
0.33
R2
0.9932
0.9942
0.9965
0.9967
0.9975
0.9983
0.9987
0.9996
0.9993
2:00
∞
∞
∞
∞
∞
∞
∞
∞
∞
2.85
0.91
0.57
0.35
0.17
0
3.74
1.19
0.75
0.46
0.22
0
4.26
1.35
0.85
0.52
0.25
0
4.22
1.34
0.85
0.52
0.25
0
4.16
1.32
0.83
0.51
0.24
0
4.06
1.29
0.82
0.50
0.24
0
3.52
1.12
0.71
0.43
0.21
0
3.14
1.00
0.63
0.40
0.18
0
3.15
1.00
0.63
0.40
0.18
0
T ABLE IV.
T HERMOCLINE THICKNESS OF CASE II USING SDR FUNCTION

0
0.01
0.1
0.2
0.3
0.4
0.5
18:00
WTC - Case II (Flow rate : 524 m3/hr) - SDR
19:00 20:00 21:00 22:00 23:00 0:00
1:00
2:00
∞
∞
∞
∞
∞
∞
∞
∞
∞
4.23
1.35
0.85
0.52
0.25
0
5.01
1.59
1.01
0.61
0.29
0
5.80
1.84
1.16
0.71
0.34
0
5.84
1.86
1.17
0.72
0.34
0
4.85
1.54
0.97
0.59
0.28
0
4.57
1.45
0.92
0.56
0.27
0
4.67
1.48
0.94
0.57
0.27
0
4.55
1.45
0.91
0.56
0.27
0
4.61
1.47
0.92
0.56
0.27
0
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