International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 4, Issue 5, May 2015
ISSN 2319 - 4847
Correction of the Effective Thermal
Conductivity of Two Phase System using
Resister Model with Cylindrical Inclusions
1
Harish Kumar Sublania ,2K. J. Singh ,3R.Singh
1
Scholor, MGS University of Bikaner (Rajasthan)
2
Govt. College, Suratgarh, Sri Ganganagar (Rajasthan)
3
University of Rajasthan, Jaipur
ABSTRACT
The study of heat transfer through two phase materials from the values of thermal conductivity of the constituent phases and
there volume fractions. The resister model has been applied to determine the effective thermal conductivity (ETC) in this
arrangement has been divided in to unit cells, each of which contains a cylinder. The random packing of Phases, non-uniform
shape of particles and the flow of the heat flux lines not restricted to be parallel, we replace physical volume fraction of solid
phases by porosity correction term F. In this paper, we calcute the effectivity thermal conductivity using a Jagjiwan & Singh
model with correction of this model. Comparison of the predicted values of the correlations with experimental results is also
made. The predictions of effective thermal condivity of two phase materials match well with the good experimental results.
Keywords:-Efective thermal conductivity; Correction term; Unit cell approach; Cylindrical inclusions; Two phase
systems.
1.INTRODUCTION
The study of thermal parameters of these two phase systems is also valuable for the explosive industry, nuclear reactors
and in missile technology. The importance of two phase materials like ceramics, granular materials, emulsions and
metal foams like in their applications in high emulsions and metal foam lies in their applications in high performance
cryogenic insulations, packed beds, composite materials power generation. In this literature we finds several efforts [49] in which the situation has been simplified by assuming that the particles are of specific shape and arranged in a
particular geometries within the continuous phases. A recent advancement in there estimation of the effective thermal
conductivity specifically for metallic foam saturated with a fluid utilizing a geometrical estimate was developed by
Calmidi and Mahajan [2]and Boomsma and poulikakos [3]. R. Singh & H.S. Kasana [11] independently developed
models utilizing geometrical estimate for calculation of ETC for matelic foam saturated with a fluid.
In this work, the term simulation is used for a Jagjiwan& R.Singh based particle deposition which created two
dimensional structures [18-20]. These simulated structures function as a master for the modeling of the thermal
conductivity. The term modeling here, refers to the mathematical description of the thermal conductivity of porous
media, predominantly using the resister model. We have tried to fill the space arrangement of cells of equal size with
the minimal surface energy and a theoretical model has been proposed to predict ETC of the two phase systems with
cylindrical inclusions.
2.THEORY
To solve this problem we take there some assumptions:
(a) The contact resistance between the fluid and solid phase is negligible, (b) the mixture is homogeneous throughout
and no transfer of heat occurs by way of convection or radiation and (c) the heat flows along the x-axis and the flux
lines remain parallel during the heat flow. Let the grains of the solid phase be 3-D cubic geometry principal axis 2a, 2c
and 2a (a<c) .Suppose the grains located of a simple cube of side 2b each. The distribution of two dimensions is shown
in figure 1 (i). The geometry of a unit cell is shown in fig. (ii). Assume that the origin of coordinate axis be located at
the center of the 3-D cubic geometry. We have divided the unit cell in to eight parts. This is further sub divided in to
rectangular bars shown in figure 1(iv). Length is bar of b, cross section of area is dxdz. The shaded portion of the
element represents the solid phase and non shaded portion are fluid phase.
Volume 4, Issue 5, May 2015
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 4, Issue 5, May 2015
ISSN 2319 - 4847
Fig-1(i-iv). The resister model for two phase systems with cylindrical particles.
The volume fraction of the solid phase of the bar w be
(cdxdz)/(bdxdz) = c/b
(1)
Similarly the volume fraction of fluid phase be
[(b-c) dxdz]/bdxdz = (1-c/b)
(2)
The thermal conductivity of bar will be
λ’ = λ1(c/b) + λ2(1-c/b)
(3)
Where λ1 and λ2 are the thermal conductivities of solid and fluid phase respectively. With refrence to the fig 1(iii).
From the reference paper of Jagjiwan& R.Singh [20] we find this eqution will
be1/λe=[(a/b)/{(λ1-λ2)(πac)}/(4b2)+λ2]+ [(1-a/b)/ λ2 ]
(4)
Therefore
λe = λ2[(λ1-λ2){(πac)/(4b2)} + λ2] / [(1-a/b)( λ1-λ2){( πac ) / (4b2)}] + λ2
(5)
The unit cell contains one cylinder that lies inside. Therefore, fractional volume of the solid phase will be
Ф1 = πa2c/8b3
(6)
If c=b then we get
Ф1 = (π/8) (a2/b2)
Therefore
Eq. 5 & 7 find the following relations
For the cubic packing of cylindrical inclusions the maximum value of the packing fraction will be less than 0.785 (
a<b). Then eqn. 8 valid for 0< ф1<0.785.The effective thermal conductivity depends upon various characteristics of the
system. We have to modify the expression given in eqn. (8) by change some correction term. Considering random
packing of phases, non-uniform shape of particles and the flow of the heat flux lines not restricted to be parallel, we
here replace physical volume fraction of solid phase by porosity correction term F. The function F is a physical volume
fraction of solid phase and the ratio of the thermal conductivity of the constituent phases. Therefore Eqn. 8 may be
written as:
Rearranging the Eqn. (9), we get
Volume 4, Issue 5, May 2015
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Volume 4, Issue 5, May 2015
AF + BF1/2 + C = 0
ISSN 2319 - 4847
(10)
Where
Result and discussion
The theoretical model discussed above on two phase systems, for which the characteristics of the constituent phases,
including thermal conductivities of solid phase, fluid phase, porosity and the experimental results for the ETC have
been cited in the literature [18-20]. First of all, function F is calculated from a large number of experimental data
reported in the literature, by putting the value of thermal conductivity of constituent phases and as in Eq. (9) . A curve
has been plotted between F1/2and φ12/3*exp [λ2/(λ1+λ2)] The plot of φ12/3*exp[λ2/(λ1+λ2)] versus F1/2 are shown in fig.
1-2 . It is found that F1/2 (for Solid-air , emulsion, suspension, granular, and solid-solid two phase systems) increases
roughly linearly with increasing φ12/3*exp[λ2/(λ1+λ2)] . The linear expression
F1/2 = C1φ12/3*exp [λ2/(λ1+λ2)] +C2
(11)
Is suggested, C1 and C2 are constant. These constants are calculated for different type of samples and we found that, the
values of these constants for Solid-air, emulsion, suspension, granular, and solid-solid two phase systems are 0.83379
and 0.0393 respectively.
On putting (8) as the porosity correction term in (11) we have calculated values of effective thermal conductivity for a
large number of samples reported in the literature. Tables show a comparison of experimental results of effective
thermal conductivity and calculated values from (9). The average deviation is 4.08% for solid-air, emulsion,
suspension, granular, and solid-solid two phase systems shown table-1.
Table – 1 Comparison of ETC value for two phase system using Eq. (9). The thermal conductivity is in W m-1K-1.
Ref.[18-20].
S.N Type of sample
φ1
λ1
λ2
λtheo
λexp
Error
0.012
80.3505085
25
1
Cu/solder
398
78.1
79.8
0.69
4
4
0.013
80.5049783
2
Cu/solder25
398
78.1
80
0.63
6
2
0.050
85.4133555
3
Cu/solder25
398
78.1
85.2
0.25
7
9
0.099
92.5732783
4
Cu/solder25
398
78.1
92.4
0.19
6
8
0.019
81.2649358
5
Cu/solder25
398
78.1
80.8
0.58
5
3
0.026
82.1468155
6
Cu/solder25
398
78.1
81.7
0.55
3
6
0.028
82.4477715
7
Cu/solder25
398
78.1
82
0.55
6
8
0.102
93.0897677
8
Cu/solder25
398
78.1
92.7
0.42
9
6
0.237
118.907894
115.
9
Cu/solder25
398
78.1
3.04
7
1
4
0.084
10
Cu/solder25
398
78.1
90.3117015
90.2
0.12
8
0.158
102.535947
11
Cu/solder25
398
78.1
102
0.53
6
7
0.251
122.237610
12
Cu/solder25
398
78.1
118
3.59
6
1
0.289
132.157780
13
Cu/solder25
398
78.1
125
5.73
4
7
132.607389
14
Cu/solder25
0.291
398
78.1
125
6.09
7
0.61
0.16
0.25701995
0.23
15
cellosize/flexol26
0.3
9.37
6
1
2
5
0.60
0.18
0.23742732
0.26
10.7
16
Water/Oil solvent27
0.2
4
2
2
6
4
0.23650649
0.23
27
17
cellosize/polypropyllene glycol
0.3
0.55
0.15
1.07
9
4
Volume 4, Issue 5, May 2015
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 4, Issue 5, May 2015
18
Water/mineral Oil27
0.4
19
Selenium/poly. Glycol29
0.1
20
Water/Oil solvent27
0.2
21
Water/Oil solvent27
0.4
22
Water/mineral Oil27
0.2
23
Water/mineral Oil27
0.4
24
cellosize/polypropyllene glycol26
0.1
25
cellosize/polypropyllene glycol26
0.1
26
cellosize/polypropyllene glycol26
0.1
27
cellosize/polypropyllene glycol26
0.1
28
Lead powder/Si rubberg23
0.05
0.61
1
5.20
8
0.60
5
0.60
7
0.61
1
0.61
1
0.55
1
0.57
7
0.55
1
0.57
7
34.7
2
29
Bi. powder/Si rubberg23
0.05
8.33
30
Bi. powder/Si rubberg23
0.16
8.33
31
Bi. powder/Si rubberg23
0.24
8.33
32
Silica powder/dimethyl vnyle28
0.1
1.68
33
Silica powder/dimethyl vnyle28
0.15
1.68
34
Silica powder/dimethyl vnyle28
0.25
1.68
0.14
9
0.14
0.18
2
0.17
3
0.14
9
0.14
9
0.15
0.15
4
0.15
0.15
4
0.38
5
0.38
5
0.38
5
0.38
5
0.17
6
0.17
4
0.17
4
0.29974871
9
0.19950752
3
0.23750313
7
0.32636385
3
0.20251287
2
0.29974871
9
0.17189389
9
0.17684265
6
0.17189389
9
0.17684265
6
0.49343737
5
0.46312929
1
0.61578663
1
0.76621905
3
0.22315232
1
0.24632654
3
0.31124182
5
Error = 4.08
ISSN 2319 - 4847
0.29
2
0.18
0.26
7
0.31
2
0.23
4
0.29
3
0.18
2
2.65
10.8
4
11.0
5
4.60
13.4
6
2.30
5.55
0.18
1.75
0.18
2
5.55
0.18
1.75
0.46
3
0.43
3
0.59
1
0.73
4
0.23
1
0.25
2
0.29
6.57
6.96
4.19
4.39
3.40
2.25
7.32
Table -2 Comparison of ETC values for two phase systems using Eq. (9) and R.Singh model .The thermal conductivity
is in W m-1 K-1. Ref.[18-20].
λth(our
J
S.N Type of sample
φ1
λ1
λ2
λexp
Error
Error
&R.singh
model)
25
1
Cu/solder
0.0124 398
78.1
79.8
80.351
0.690
79.731
0.10
2
Cu/solder25
0.0136 398
78.1
80
80.505
0.631
79.952
0.10
3
Cu/solder25
0.0507 398
78.1
85.2
85.413
0.250
86.837
1.90
25
4
Cu/solder
0.0996 398
78.1
92.4
92.573
0.188
95.784
3.60
5
Cu/solder25
0.0195 398
78.1
80.8
81.265
0.575
81.052
0.30
6
Cu/solder25
0.0263 398
78.1
81.7
82.147
0.547
82.325
0.70
7
Cu/solder25
0.0286 398
78.1
82
82.448
0.546
82.754
0.90
8
Cu/solder25
0.1029 398
78.1
92.7
93.090
0.420
96.393
3.90
9
Cu/solder25
0.2377 398
78.1
115.4 118.908 3.040
123.52
7.00
10
Cu/solder25
0.0848 398
78.1
90.2
90.312
0.124
93.67
3.20
11
Cu/solder25
0.1586 398
78.1
102
102.536 0.525
106.969
4.80
25
12
Cu/solder
0.2516 398
78.1
118
122.238 3.591
126.685
7.30
13
Cu/solder25
0.2894 398
78.1
125
132.158 5.726
135.779
8.60
14
Cu/solder25
0.291
398
78.1
125
132.607 6.086
136.181
8.90
15
cellosize/flexol26
0.3
0.616 0.161 0.235 0.257
9.370
0.276
17.60
16
Water/Oil solvent27
0.2
0.604 0.182 0.266 0.237
10.742 0.258
2.80
17
cellosize/polypropyllene
0.3
0.55
0.15
0.234 0.237
1.071
0.256
9.50
Volume 4, Issue 5, May 2015
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Volume 4, Issue 5, May 2015
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
glycol27
Water/mineral Oil27
Selenium/poly. Glycol29
Water/Oil solvent27
Water/Oil solvent27
Water/mineral Oil27
Water/mineral Oil27
cellosize/polypropyllene
glycol26
cellosize/polypropyllene
glycol26
cellosize/polypropyllene
glycol26
cellosize/polypropyllene
glycol26
Lead powder/Si rubberg23
Bi. powder/Si rubberg23
Bi. powder/Si rubberg23
Bi. powder/Si rubberg23
Silica powder/dimethyl vnyle28
Silica powder/dimethyl vnyle28
Silica powder/dimethyl vnyle28
ISSN 2319 - 4847
0.4
0.1
0.2
0.4
0.2
0.4
0.611
5.208
0.605
0.607
0.611
0.611
0.149
0.14
0.182
0.173
0.149
0.149
0.292
0.18
0.267
0.312
0.234
0.293
0.300
0.200
0.238
0.326
0.203
0.300
2.654
10.838
11.048
4.604
13.456
2.303
0.311
0.189
0.258
0.354
0.215
0.311
6.60
5.30
3.20
13.40
7.70
6.20
0.1
0.551
0.15
0.182
0.172
5.553
0.198
4.40
0.1
0.577
0.154
0.18
0.177
1.754
0.225
7.50
0.1
0.551
0.15
0.182
0.172
5.553
0.18
0.80
0.1
0.577
0.154
0.18
0.177
1.754
0.185
3.00
0.05
0.05
0.16
0.24
0.1
0.15
0.25
34.72
8.33
8.33
8.33
1.68
1.68
1.68
0.385
0.385
0.385
0.385
0.176
0.174
0.174
0.463
0.433
0.591
0.734
0.231
0.252
0.29
0.493
6.574
0.463
6.958
0.616
4.194
0.766
4.390
0.223
3.397
0.246
2.251
0.311
7.325
Error = 4.08
0.467
0.90
0.452
4.40
0.582
1.50
0.683
6.80
0.224
2.80
0.245
2.40
0.248
2.90
Error = 4.735
Fig. 2- Variation of porosity correction term and φ12/3*exp[λ2/(λ1+λ2)].
Fig. 3 comparison between experimental, theoretical and J.& R.Singh (model) values of ETC of the sample no. 1-14.
Volume 4, Issue 5, May 2015
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Volume 4, Issue 5, May 2015
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Fig. 4 comparison between experimental, theoretical and J.& R.Singh (model) values of ETC of the sample no. 15-34.
4. CONCLUSIONS
We have observed from these figures that for our model the average percentage deviation are better than others models.
The empirical model proposed here is capable of predicting effective thermal conductivities close to the experimental
values even for mixtures of higher conductivity ratio and high porosities; whereas one may find that other models give
higher deviations in those situations. This model simple but powerful enough without compromising on the results.
This clearly indicates that the slope of the curve as shown in fig. (1) Strongly depends on the ratio of thermal
conductivity of the constituent phases. The correlation presented here showed that the effective thermal conductivity
strongly depends on the ratio of thermal conductivity of the constituents. Other factors have small effect on the ETC.
The parameters of fluid, such as the size, volume fraction, the thickness of the interfacial layer, are shown to play
important roles in the enhancement of thermal conductivity. The model predictions have been shown to be reasonable
and are in good agreement with the available experimental data. It is expected that the experimentally validated model
will be helpful in the evaluation of the effective thermal conductivity for foam like materials in the whole range of
porosity. This method was therefore adopted to measure and discuss thermal conductivity in the succeeding studies.
Acknowledgements:-The authors would like to thank Dr.Kamaljeet Singh for critical comments and helpful
discussion.
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[20]. Jagjiwanram, R. Singh /Indian journal of pure and applied physics vol.42 (2004), 600-609.
[21]. H.F. Zhang, X. S. Ge and H. Ye, “ Randomly Mixed Model for Predicting The ETC of Moist Porous Media,”
Journal of physics D:Applied physic,Vol. 39, No. 1, 2006, pp.220-226.
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UK, 2006.
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