International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 3- April 2016
Mathematical Modeling and Analysis of Free
Cooling using PCM Filled Air Heat
Exchanger for Energy Efficiency in Building
ThambiduraiMuthuvelana, Karthikeyan. Sb, Krishna Mohan Na, VelrajRc*
a
b
Department of Mechanical Engineering, Annamalai University, Chidambaram 608 002
Department of Mechanical Engineering, Arunai Collage of Engineering, Thiruvanamalai, India
c
Institute for Energy Studies, Anna University, Chennai 600 0025
Abstract — Power shortage and unstable power
supply remain serious problems. Conventional
cooling technologies that utilize harmful refrigerants
consume more energy and cause peak loads leading
to negative environmental impacts. As the world
grapples with the energy and environmental crisis,
there is an urgent need to develop and promote
environmentally
benign sustainable
cooling
technologies. Thermal storage systems are essential
to overcome the disadvantage of the intermittent
nature of energy and variation in cooling demand.
The advantages of LHTES in comparison with
sensible storage are a greater density of stored
energy and a narrow operational temperature
range. In this work empirical relationships were
developed to estimate the PCM temperatures during
charging and discharging by incorporating
independent heat exchanger parameters (mass flow
rate of air, air inlet temperature, charging and
discharging time) using statistical design of
experiments (DoE) concept. A central composite
rotatable design with three factors and five levels
was chosen to minimize the number of experimental
conditions. Within the scope of the design space, the
mass flow rate and the charging and discharging
time appeared to be the most significant two
parameters affecting the PCM temperature among
the three investigated heat transfer parameters.
Further, ANOVA analysis also proved that all the
considered factors are significant and these models
can be effectively used for prediction purpose.
Keywords — Free cooling; Heat transfer; PCMHS29, Design of Experiments; ANOVA analysis.
I. INTRODUCTION
Thermal Energy Storage [TES] technologies are
very important in various fields of engineering
applications. The most common TES technologies
are sensible heat storage and latent heat storage.
Latent heat thermal storage units have received
greater attention by the scientists and engineers
working in the areas of solar heating/ cooling,
energy management in buildings and waste heat
recovery applications due to their high energy
storage capacity and isothermal behaviour.
ISSN: 2231-5381
Mathematical modeling of phase change
phenomenon involves greater complexity as the
solid and liquid domains change continuously with
respect to time. There are several methods being
adopted to model and solve for the temperature
variation in the domain of interest. Many researchers
have investigated experimentally and theoretically
the transient behaviour of the PCM encapsulated in
different geometries and PCM based storage systems
in different configurations particularly shell and tube,
packed bed with capsule of different geometries.
II. PROBLEM DEFINITION
The rate of PCM temperature change in the
LHTES depends on the air flow rate and the
temperature conditions at the storage inlet and the
temperature difference between the air and the
paraffin, respectively. This rate is important during
the selection of an apparent heat capacity function,
which should be determined based on measurements
at approximately the same heating or cooling rate. A
comparison of the numerical and experimental
results showed that the apparent heat capacity, capp
(T), a parameter that describes how the latent heat of
the PCM evolves over the temperature range, should
also include an additional influential parameter a
heating or cooling rate [1].
The charging and discharging processes were
numerically simulated by one dimensional
governing equation for heat transfer fluid (HTF) and
PCM using implicit finite difference method. The
effect of inlet fluid temperature, mass flow rate of
HTF, diameter of the capsule and porosity of the
packed bed on the solidification and temperature
distribution of the packed bed were analyzed [2].
The temperature of freezing/melting is one of the
primary considerations when selecting a PCM. As an
application guideline, a temperature difference of
3 - 4 0C is needed between the cooling medium, in
this case cooling water and the freezing temperature
of the PCM were considered. For melting, a similar
temperature difference is required. Thus, combining
both, the cooling water freezing the PCM needs to
be approximately 8 ºC cooler than the cooling water
melting the PCM. Most of the published literature
mainly focused on effect of heat transfer mode,
design of heat exchanger, effect of phase change
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International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 3- April 2016
materials on heat transfer, effect of experimental
parameters, and comparison on experimental and
numerical model results [3].
Of the various PCM integration and applications
in buildings, free cooling is one of the novel concept
in which the cool energy is stored in an external
PCM based air- heat exchanger. A detailed review
on phase change material storage for free cooling of
buildings was carried out [6–13].The free cooling
potential for different climatic locations in Europe
was studied and reported that the PCM with wider
phase change temperature range (12ºC) was found to
be most efficient. The optimal size of the LHTS for
free cooling of building is between 1 and 1.5 kg of
PCM per m3/hr of fresh ventilation air [14]. The free
cooling potential for Bangalore and Pune cities was
analysed and the appropriate technologies to provide
comfort conditions within the temperature range of
25 ± 3ºC was suggested in order to reduce the
energy requirement of the building [15-18].
This paper focuses on developing a mathematical
relationship to predict the PCM temperature during
charging and discharging. Further to develop a
mathematical relationship also to estimate room
temperature to cool the room.
III. MATERIALS AND METHODS
To simulate the weather conditions of Pune, the
present experiments were conducted from 1.30 a.m.
in the early mornings during the month of November
and December as this duration matches well with the
major weather conditions prevailing in Pune city.
Considering the climatic conditions of Pune city, a
suitable PCM - HS 29 PCM (commercial grade
PCM)with phase transition temperature of
approximately 28°C to 30°C was chosen for the
present study.
IV. EXPERIMENTAL DESIGN MATRIX
An initial step in the design of experiments is to
select parameters for the experimental parameter.
For economic (time requirements) and theoretical
reasons (interdependence of parameters) it is not
possible to control all possible parameter variations.
From the literature [4 and 5] the predominant factors
which are having more influence on PCM
temperature were identified. They are as follows:
1. Mass flow rate of heat transfer fluid
2. Inlet air temperature during charging
3. Inlet air temperature during discharging.
4. Charging time.
5. Discharging time.
A large number of experimental trials were
conducted to identify the feasible working limit of
the heat exchanger to transfer the heat from fluid
medium to phase change material (PCM)
solidification temperature and PCM to fluid medium
to reduce the temperature of the room. From the
trials the following parameters were identified for
the solidification of the PCM and given in Table 1
ISSN: 2231-5381
and for melting of the PCM and also room cooling
temperature are given in Table 2.
V. DEVELOPING THE EXPERIMENTAL DESIGN
MATRIX
By considering all the above conditions, the
feasible limits of the parameters were chosen in such
a way that heat transfer experiments were carried out
by solidification, melting of PCM and reduction of
room temperature. Central composite rotatable
design of second order was found to be the most
efficient tool in response surface methodology to
establish the mathematical relation of the response
surface using the smallest possible number of
experiments without losing its accuracy. Due to wide
range of factors, it was decided to use three factors,
five levels and central composite design matrix to
develop mathematical models for the experimental
conditions.
Central composite design is the most popular
response surface method design, which has three
groups of design points such as, factorial points,
axial or star points and center points. In the factorial
part of the design consists of all possible
combinations of the upper (+1) and lower levels (-1)
of the factors. The star points have all of the factors
set to 0, the midpoint, except one factor, which has
the value +/- Alpha (+/- 1.682) for both rotatability
and orthogonality of blocks. Center points, as
implied by the name, are points with all levels set to
coded level 0 -the midpoint of each factor range.
The center points (In this investigation, 15-20) are
usually repeated 4-6 times to get a good estimate of
experimental error (pure error). To summarize,
central composite designs require 5 levels of each
factor: -Alpha, -1, 0, 1, and +Alpha. One of the
commendable feature of the central composite
design is that its structure lends itself to sequential
experimentation. Table 3 shows the 20 sets of coded
conditions used to form the design matrix.
The second-order polynomial (regression)
equation is used to represent the response surface Y
is given by,
Y = β0 + ∑βi Xi + ∑βii Xi2+ ∑βijXiXj
(1)
The selected model should include the effects of
main and interaction effects of all the factors. With
the view of considering this criterion, for the three
factors, the selected polynomial can be expressed as,
Y = b0 + b1(A) + b2(B) + b3(C) + b12(AB) + b13(AC)
+ b23(BC) + b11(A2) + b22(B2) + b33(C2)
(2)
where b0 is the average of the responses and b1, b2,
b3,…,b33 are regression coefficients that depend on
respective linear, interaction, and squared terms of
factors. In order to estimate the regression
coefficients, a number of experimental design
techniques are available. In this work, central
composite rotatable design was used which fits the
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International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 3- April 2016
second order response. The fitted regression
equation to predict PCM temperature and room
temperature are given below,
PCM Temperature during discharging = 30.22+
0.38A+0.11B +0.21C-0.089AB+0.013AC-0.013BC0.96A2-1.06B2-0.75C2
(4)
PCM Temperature during charging = 27.53089A+0.68B-0.061C+0.088AB+0.51AC+0.41BC+
0.85A2+1.1B2+0.64C2
(3)
Room Temperature during discharging = 29.160.55A-0.33B+0.89C+.58AB+0.31AC+0.28BC+
0.49A2 + 0.53B2 +0.1C2
(5)
Table 1 Important experimental parameters and their levels for charging
Levels
Notations
Units
-1.682
-1
0
1
No.
Parameters
1
Mass flow rate
A
kg/s
0.05
0.09
0.15
0.21
0.25
2
Inlet air temp.
B
°C
24
25
27
28
30
3
Charging time
C
min
30
84
165
245
300
No.
Table 2 Important experimental parameters and their levels during discharging
Levels
Parameters
Notations
Units
-1.682
-1
0
1
+1.682
+1.682
1
Mass flow rate
A
kg/s
0.02
0.09
0.15
0.21
0.1
2
Inlet air temp.
B
°C
26
27.6
30
32.4
34
3
Discharging time
C
min
10
57
125
193
240
Table 3 Experimental parameters and its response during charging
Sl.
No
Coded Value
Original Value
Response1
Mass flow
rate
Inlet temp.
Charging time
A
B
C
PCM
Temp
(A)
(B)
(C)
(kg/s)
(ºC)
(min.)
(ºC)
1
-1
-1
-1
0.09
25.22
84.73
31.5
2
1
-1
-1
0.21
25.22
84.73
28.4
3
-1
1
-1
0.09
28.78
84.73
31.7
4
1
1
-1
0.21
28.78
84.73
29.2
5
-1
-1
1
0.09
25.22
245.27
29.4
6
1
-1
1
0.21
25.22
245.27
28.6
7
-1
1
1
0.09
28.78
245.27
31.5
8
1
1
1
0.21
28.78
245.27
30.8
9
-1.682
0
0
0.05
27.00
165.00
31.4
10
1.682
0
0
0.25
27.00
165.00
28.4
11
0
-1.682
0
0.15
24.00
165.00
29.4
12
0
1.682
0
0.15
30.00
165.00
31.8
13
0
0
-1.682
0.15
27.00
30.00
29.4
14
0
0
1.682
0.15
27.00
300.00
29.2
15
0
0
0
0.15
27.00
165.00
27.5
16
0
0
0
0.15
27.00
165.00
27.6
17
0
0
0
0.15
27.00
165.00
27.5
18
0
0
0
0.15
27.00
165.00
27.5
19
0
0
0
0.15
27.00
165.00
27.5
20
0
0
0
0.15
27.00
165.00
27.6
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International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 3- April 2016
Table 4 Experimental design matrix and its response during discharging
Expt.
No
Mass flow rate
Inlet air temp.
Discharging time
Response 1PCM Temp.
Response 2 Room Temp.
kg/s
ºC
Min
ºC
ºC
1
0.02
30
125
26.9
31.5
2
0.08
27.6
57
27.6
28.5
3
0.06
30
125
30.2
29.2
4
0.06
34
125
27.4
30.1
5
0.06
30
125
30.2
29.1
6
0.08
27.6
193
28.1
30.4
7
0.06
26
125
27
31.2
8
0.06
30
240
28.5
30.9
9
0.04
27.6
193
27.1
32
10
0.1
30
125
28.2
29.6
11
0.04
32.4
57
27.1
29.1
12
0.08
32.4
193
28.1
31.4
13
0.04
27.6
57
26.7
31.4
14
0.04
32.4
193
27.4
30.8
15
0.06
30
125
30.2
29.1
16
0.06
30
125
30.3
29.2
17
0.06
30
10
27.8
27.9
18
0.06
30
125
30.2
29.2
19
0.06
30
125
30.2
29.1
20
0.08
32.4
57
27.6
28.5
Table 5 ANOVA for response surface quadratic model for PCM temperature during charging
Source
Sum of
Squares
df
Mean
Square
F
Value
p-value
Prob> F
Model
49.19
9
5.47
850.38
< 0.0001
A-Mass flow
rate
10.8
1
10.8
1680.7
< 0.0001
B-Inlet
temperature
6.38
1
6.38
993.16
< 0.0001
C-Charging
time
0.051
1
0.051
7.97
0.0181
AB
0.061
1
0.061
9.53
0.0115
AC
2.1
1
2.1
326.96
< 0.0001
BC
1.36
1
1.36
211.82
< 0.0001
2
10.39
1
10.39
1616.76
< 0.0001
B
2
17.33
1
17.33
2696.58
< 0.0001
C
2
5.85
1
5.85
909.84
< 0.0001
3.82
0.0838
A
Residual
0.064
10
6.43E-03
Lack of Fit
0.051
5
0.01
Pure Error
0.013
5
2.67E-03
Cor Total
49.25
19
Significant
not significant
Std. Dev.=0.08R-Squared = 0.9987 Mean = 29.29Adj R-Squared = 0.9975 C.V. % = 0.27Pred R-Squared = 0.9908
PRESS = 0.45Adeq Precision = 74.999
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International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 3- April 2016
Table 6 ANOVA for response surface quadratic model for PCM temperature during discharging
Source
Sum of
Squares
df
Mean
Square
F
Value
p-value
Prob> F
Model
34.43
9
3.83
28402.42
< 0.0001
2.02
1
2.02
15014.11
< 0.0001
0.15
1
0.15
1120.24
< 0.0001
0.6
1
0.6
4447.54
< 0.0001
AB
0.064
1
0.064
474.25
< 0.0001
AC
1.27E-03
1
1.27E-03
9.44
0.0118
BC
1.27E-03
1
1.27E-03
9.44
0.0118
A2
13.25
1
13.25
98336.04
< 0.0001
B2
16.34
1
16.34
1.21E+05
< 0.0001
59632.83
< 0.0001
0.052
0.9972
A-Mass flow
rate
B-Inlet air
temp.
C-Discharging
time
C
2
8.03
1
8.03
Residual
1.35E-03
10
1.35E-04
Lack of Fit
6.62E-05
5
1.32E-05
Pure Error
1.28E-03
5
2.56E-04
Cor Total
34.43
19
Significant
not significant
Std. Dev. = 0.12 R-Squared = 0.987 Mean = 28.33 Adj R-Squared = 0.99 C.V. % = 0.041 Pred R-Squared = 0.99
PRESS = 2.39E-03 Adeq Precision = 433.647
Table 7 ANOVA for response surface quadratic model of room temperature
Source
Sum of
Squares
df
Mean
Square
F
Value
p-value
Prob> F
Model
27.36
9
3.04
4001.21
< 0.0001
4.13
1
4.13
5437.58
< 0.0001
1.51
1
1.51
1981.93
< 0.0001
10.82
1
10.82
14249.19
< 0.0001
AB
2.7
1
2.7
3557.89
< 0.0001
AC
0.75
1
0.75
987.69
< 0.0001
BC
0.62
1
0.62
818.27
< 0.0001
2
3.51
1
3.51
4624.93
< 0.0001
B
2
3.98
1
3.98
5240.24
< 0.0001
C
2
201.57
< 0.0001
6.01
0.0355
A-Mass flow
rate
B-Inlet air
temp.
C-Discharging
time
A
0.15
1
0.15
Residual
7.60E-03
10
7.60E-04
Lack of Fit
6.51E-03
5
1.30E-03
Pure Error
1.08E-03
5
2.17E-04
Cor Total
27.36
19
Significant
Significant
Std. Dev.= 0.028 R-Squared=0.9997 Mean=29.92 Adj R-Squared =0.9995 C.V. % =0.092 Pred R-Squared=0.998 PRESS= 0.054
Adeq Precision =210.202
All the coefficients were obtained applying
central composite rotatable design using the Design
Expert statistical software package (Version 8.07.1).
The significance of each coefficient was determined
by Student’s t test and p values, which are listed in
Table5-7.In all the cases, A,B, C, AB, AC, BC,
ISSN: 2231-5381
A2,B2 and C2 are significant model terms, values of
―Prob>F‖ less than 0.05 indicates that model terms
are significant. After determining the significant
coefficients (at 95% confidence level), the final
empirical relationships were constructed using only
these coefficients.
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International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 3- April 2016
empirical relationships can be effectively used for
VI. CHECKING ADEQUACY OF THE DEVELOPED
prediction purpose. The normal probability plot of
RELATIONSHIPS
the residual dilution is shown in Fig. 2, which
reveals that the residuals are falling on the straight
Analysis of variance (ANOVA) technique was
line, indicating that the errors are distributed
used to check the adequacy of the developed
normally. Similar results were found for the
empirical relationships. In this investigation, the
developed mathematical model to predict PCM
desired level of confidence was considered to be
temperature and room temperature and the related
95%. The relationship may be considered to be
results are shown in Figs. 3 – 6.
adequate provided that
(a) the calculated value of the F ratio Design-Expert®
of the modelSoftware
Predicted vs. Actual
PCM TEMP
developed should not exceed the standard tabulated
Color points by value of
value of F ratio and
PCM TEMP:
31.8
(b) the calculated value of the R ratio
of the
27.5
developed relationship should exceed the
standard
tabulated value of R ratio for a desired level of
confidence.
It is found that the model is adequate. The value
of probability > F in Tables 5-7 for the empirical
relationships are less than 0.05, which indicates that
the empirical relationships are significant. Lack of fit
42
42
was not significant for all the developed empirical
relationships as desired. The Fisher’s F test with a
very low probability value (P model > F= 0.0001)
Actual
demonstrates a very high significance. The goodness
Fig.1 Correlation graph between experimental actual temperature
of fit of the model was checked by the determination
and predicted temperature
coefficient (R2). The coefficient of determination (R2)
Design-Expert® Software
Normal Plot of Residuals
PCM TEMP This
was calculated to be 0.9987 for response.
implies that 99.87 % of experimental Color
data points
confirms
by value of
99
PCM TEMP:
the compatibility with the data predicted
31.8 by the
95
model, and the model does not explain27.5
only about
90
0.12 % of the total variations. The R2 value is always
80
70
between 0 and 1, and its value indicates aptness of
50
the model.
30
2
For a good statistical model, R value should be
20
2
10
close to 1.0. The adjusted R value reconstructs the
5
expression with the significant terms. The value of
1
the adjusted determination coefficient (Adjusted
2
R =0.9999) also high to advocate for a high
significance of the model. The Predicted R2 (0.99)
-1.66
-0.66
0.34
1.34
2.34
implies that the model could explain 95% of the
Internally Studentized Residuals
variability in predicting new observations. This is in
2
reasonable agreement with the Adjusted R of PCM
Fig.2 Normal probability plot of residuals for PCM temperature
temperature. The value of coefficient of variation is
during charging.
Design-Expert® Software
also around 0.041 indicates that the deviations
Predicted vs. Actual
PCM Temp.
between experimental and predicted values are low.
2
Color points by value of
Adequate precision measures the signalPCM
to Temp.:
noise
30.3
ratio. A ratio greater than 4 is desirable. In this
26.7
investigation the ratio (433) which indicates an
adequate signal. This model can be used to navigate
the design space and able to predict the responses for
a given input in coded form.
Tables 6 and 7 show the ANOVA analysis for the
models developed to predict the PCM during
discharging temperature and room temperature
shows that the models are significant as desired and
co-efficient of variation, determination coefficient,
and adequate precision are showing good results as
Experimental PCM temp
desired. Further, correlation graphs were drawn
Fig.3 Correlation graph between experimental PCM and predicted
relating experimental values and predicted values as
PCM temperature during discharging.
shown in Fig. 1 and it is found that the developed
31.80
Predicted
30.73
29.65
28.57
27.50
28.57
29.65
30.73
28.13
29.19
31.80
Normal % Probability
27.50
30.30
Predicted
29.23
28.15
27.07
26.00
26.00
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27.06
30.25
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International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 3- April 2016
VII.
olor points by value of
CM Temp.:
30.3
99
95
90
Normal % Probability
26.7
CONCLUSION
Normal Plot of Residuals
esign-Expert® Software
CM Temp.
80
70
50
30
20
10
5
1
-1.63
-0.57
0.49
1.54
2.60
Internally Studentized Residuals
Fig.4 Normal probability plot of residuals during discharging
Predicted vs. Actual
gn-Expert® Software
m Temp.
3 2.1 0
points by value of
m Temp.:
.0
3 1.0 5
Predicted
.9
3 0.0 0
2
2 8.9 5
2 7.9 0
2 7.9 5
2 8.9 7
3 0.0 0
3 1.0 2
3 2.0 5
Experimental room temp
Fig. 5 Correlation graph between experimental and predicted
room temperature.
Normal Plot of Residuals
esign-Expert® Software
oom Temp.
olor points by value of
oom Temp.:
32.0
99
It was experimentally seen that the use of design
of experiments coupled with the statistical fitting
model strategy is a simple, effective, and efficient
way to develop robust, highly efficient and quality
experimental procedure to analyze heat transfer
characteristics. Based on the experimental results,
the following conclusions have been drawn.
 Mathematical model was developed using
response surface methodology to estimate
charging and discharging temperature of the
PCM and room temperature. The models can be
effectively used to predict charging and
discharging temperature of the PCM and room
temperature within the limits of chosen factors.
 Charging temperature of the PCM is influenced
by mass flow rate of air, air inlet temperature
and charging time. Of the three factors
considered, mass flow rate of air has the highest
effect (Fisher ratio = 1680) and inlet air
temperature (Fisher ratio=7.97) has the lowest
effect on the solidification temperature of the
PCM.
 Discharging temperature of the PCM is
influenced by mass flow rate of air, air inlet
temperature and charging time. Of the three
factors considered, mass flow rate has the
highest effect (Fisher ratio = 15014), inlet
temperature (Fisher ratio=1120) has the lowest
effect on the melting temperature of the PCM
(Table 5). However, discharge time (14249) has
a major effect on room cooling temperature as
shown in Table 6 followed by mass flow rate of
air (Fisher ratio=5437).
 Developed empirical relationships can be
effectively used to predict the temperature of
HS 29 PCM during charging and discharging,
and room temperature during discharging within
the range of parameters considered in this
investigation. The established procedures can
possibly be extended to other materials.
Normal % Probability
95
27.9
REFERENCES
90
80
70
50
30
20
10
5
1
-2.12
-0.95
0.21
1.37
2.53
Internally Studentized Residuals
Fig. 6 Normal probability plot of residuals for room
temperature during discharging
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