International Journal of Engineering, Management & Sciences (IJEMS) ISSN-2348 –3733, Volume -2, Issue-3, March 2015
Thermal Conductivity Enhancement in Oxide Nanofluids –a Mathematical Model Khagendra Kumar Upman Abstract— The study investigated the impact of the nanoparticle size which has been suggested to be an important factor the results were found to be in concord with the experimental observations. The values of the thermal conductivity for different nanofluid combinations were calculated using the expression developed in this study and they agreed with published experimental data. From the study, it was observed that Brownian motion is significant only when the volume fraction is less than 1 % in case of TiO2&ZnO and 4 % in case of Al2O3. The combination of the base fluid and nanoparticles to from nanoclusters is expected provide better heat transfer solution than the conventional fluids , Hence it is concluded that adding nanosized materials to base fluids enhances thermal properties and makes them more suitable to heat exchanger applications as well as for many industrial applications also .
Index Terms— Thermal brownian motion, nanofluids.
conductivity,
because of these present limitations for better understanding the heat transfer mechanism and effect of different parameters on thermal conductivity of nanofluids more studies have to be carried out. In this Paper, new models have been developed to measure the thermal conductivity of Al2O3- water and ZnO and TiO2-water nano fluids. Models have been developed by considering the fact thatthermal conductivity of nanofluid is depends on so many parameters like effect of temperature , volume fraction , size of nano particles , particle density , viscosity , thermal conductivity of particle and as well as base fluid . Rem (Modified Reynolds number) which is a dimensionless quantity based on Boltzmann constant. II. PRESENT MODEL FOR THERMAL CONDUCTIVITY Thermal conductivity of a nanofluid, knf, is given by:-
nanoparticle,
I. INTRODUCTION So far no general mechanisms to have been formulated to understand the strange behavior of the nano fluids including the highly improved effective thermal conductivity, this technology isstill limited for commercial use because there is yet no proven standardized design process for accurately predicting important heat transfer properties. Developing a reliable fundamental model for the thermal conductivity of nanofluids has always been a challenging task for researchers. Early attempts to explain this behavior have made use of the classical model of Maxwell. This model is generally applicable to dilute suspensions with micro particles but when applied to nanofluids the models predicted lower thermal conductivity enhancement as compared to the experimental observations. Several authors extended the Maxwell’s theory such as Bruggeman (1935); Jeffrey (1973); Yu and Choi (2003); Koo and Kleinstreuer (2004); Xie et al. (2005) are some theoretical models and Chon et al. (2005); Li and Peterson (2006); Mintsa et al. (2009) and Teng et al. (2010) are some empirical models. These models are not so accurate and stable against a wide range of experimental data. So
Manuscript received March 24, 2015. Khagendra Kumar Upman, M.Tech Scholar, Department of Mechanical Engineering, AIET Jaipur
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(1) For development of thermal conductivity model first we have to analyze the Brownian motion of nanoparticle. The particles suspended in the liquid are very small, Brownian movement of the particles is quite possible.The root-mean-square velocity (vN) of a Brownian particle can be defined as (2) It can be written as:
(3) (mpis particle mass =
)
Now we consider the effect of the convection of theliquid near the particles due to their Brownian movement. The Reynolds number based on vN given by Eq. (4) can bewritten as: (4) These variables in equation (1) can be expressed in non-dimensional terms as: Knf=f[
, Rem, φ]
(5)
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Thermal Conductivity Enhancement in Oxide Nanofluids –a Mathematical Model
dependent variable may depend on more than one independent variable. Insuch a situation, a multiple regression equation with more than one independent variable isused. The constants in a multiple regression equation can be computed with the help of the“normal equations”. Computation of the estimators of the standard error of the regressioncoefficients becomes complex in case of multiple regression and usually computers are usedfor this purpose. As the degree of correlation between the independent variables increases,the regression coefficients become less reliable, i.e. although the independent variables maytogether explain the dependent variable, but because of multicollinearity the coefficients of the explanatory variables may be rejected. So in case of a multiple regression equation, thecoefficient of multiple correlations should also be computed. Partial correlation coefficientshelp in finding out the extent to which the variation in the dependent variable is explained byone independent variable if all other independent variables are kept constant. Equation 6 is general form of model of thermal conductivity of nanofluid. Using experimental data forAl2O3, ZnO& TiO2water nanofluids for a wide rangeof volume fraction, particle size and temperature. The parametric values for this analysis are asfollows:
Therefore (6) Where c,p,q& r are constants Taking log both side for making eqn 6 linear
(7) According to linear regression analysis: A general form of a multiple linear regression model is given by (8)
Where
For Al2O3
Table 1 Parametric values of Al2O3
kp( nanoparticle thermal conductivity ) kf ( base fluid thermal conductivity ) f( density of base fluid ) p(density of nanoparticle ) f(dynamic viscosity of water ) dp(particle size ) T (temperature ) Kb(boltzman constant ) Volume fraction
This is supposed to hold for each observation. The model corresponding to each observation in the data set would have to write: (9) (10) (11)
0.628 W/m-k 35 W/m-k 997.1 kg/m3 3970 kg/m3 7.98 x 10-4N.s/m2 40 nm. 300 K 1.38x10-23m2 kg/s2K 1 to 4 %
The above valueswere incorporated into the model described (12) by Equation 13 and the experimental values by the various Where n = no. of variables Regression analysis helps in understanding the relationship researchers. The columns with the researchers names contain between variables.Linear regression constructs linear the data of the thermal conductivities ratio by various models.In simple regression it is assumed that the dependent experiments variable Y is related to a single variable X.But in practice, a Table 2 experimental data (Al2O3) 1
2
3
4
1.01 1.12 1.025 1.024 1.04 1.028 1.04 1.06 1.122 1.03
1.02 1.14 1.05 1.045 1.08 1.05 1.075 1.09 1.142 1.12
1.035 1.17 1.07 1.065 1.1 1.075 1.125 1.12 1.17 1.18
1.5 1.25 1.1 1.09 1.25 1.1 1.145 1.14 1.2 1.24
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Thermal conductivity ratio(knf/kf)
Volume concentration (In %) Researchers Tang et. Al (2010 ) Das et. Al (2003 ) Lee et. Al Lee and choi Eastman et. Al Timmofeva et. Al J.K.E.Goodson(2008) B.S.A. Shin Murshad et. Al Thomsan 1997
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International Journal of Engineering, Management & Sciences (IJEMS) ISSN-2348 –3733, Volume -2, Issue-3, March 2015 1.1 1.045 1.025
Wang et. Al (2003 ) Oh. Et. Al ThakleawYiamsawasd
1.125 1.07 1.065
1.14 1.1 1.09
1.17 1.14 1.12
Dynamic viscosity
kinematic viscosity
volume fraction 0.01 0.02 0.03 0.04
Dynamic viscosity of nanofluid ( nf ) in N s/m2 0.000818304 0.00083934 0.00086114 0.000883741
Kinematic viscosity of nanofluid ( ) in m2/s. 2.06122E-07 2.11421E-07 2.16912E-07 2.22605E-07 Figure 1Graphical representation of model (Al2O3 with water)
Table 3 calculation of Viscosity (Al2O3)
For ZnO Table 5 Parametric values of ZnOnanofluid Reynolds number Table 4 calculation of Reynolds number (Al2O3 ) volume fraction 0.01 0.02 0.03 0.04
Reynolds number 0.059318033 0.057818011 0.05635433 0.05491311
The constants in the above equation (6) are obtained by using experimentaldata (given in table) .Using nonlinear regression analysis an empirical correlation topredict the k of Al2O3 + H2O( eqn 13)nanofluids is developed as :
kp( nanoparticle thermal conductivity ) kf ( base fluid(water) thermal conductivity ) f( density of base fluid ) p(density of nanoparticle ) f(dynamic viscosity of water ) dp(particle size ) T (temperature ) Kb(boltzman constant ) Volume fraction
13 W/m-k 0.597 W/m-k 997.1 kg/m3 5600 kg/m3 7.98 x 10-4N.s/m2 30 nm 300 K 1.38x10-23m2 kg/s2K 0.01 to 0.1 %
The above valueswere incorporated into the model described by Equation 15 and the experimental values by the various researchers. The columns with the researchers names contain the data of the thermal conductivities ratio by various experiments
(13) Table 6 Experimental data of ZnOnanofluid Conc(%)
k1 (W/m-k)
k2 (W/m-k)
k3 (W/m-k)
k4 (W/m-k)
k5 (W/m-k)
f(W/m-k)
kef
keff/kf
0.01
0.6
0.602
0.601
0.601
0.599
0.6006
1.00603
0.02
0.605
0.603
0.603
0.602
0.602
0.603
1.01005
0.03
0.611
0.61
0.609
0.608
0.607
0.609
1.0201
0.04
0.614
0.614
0.614
0.611
0.612
0.613
1.0268
0.05
0.619
0.618
0.618
0.617
0.619
0.6182
1.03551
0.06
0.625
0.626
0.625
0.626
0.624
0.6252
1.04723
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Thermal Conductivity Enhancement in Oxide Nanofluids –a Mathematical Model
0.07
0.632
0.631
0.63
0.629
0.628
0.63
1.05527
0.08
0.635
0.635
0.634
0.634
0.634
0.6344
1.06264
0.09
0.64
0.64
0.638
0.639
0.64
0.6394
1.07102
0.1
0.645
0.648
0.645
0.646
0.645
0.6458
1.08174
Dynamic viscosity
(14) Table 7 calculation of viscosity (ZnO) volume fraction 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001
Dynamic viscosity Kinematic of nanofluid ( nf ) viscosity of nanofluid in N s/m2 ( ) in m2/s. 0.0007982 1.42536E-07 0.000798399 1.42571E-07 0.000798599 1.42607E-07 0.000798799 1.42643E-07 0.000798998 1.42678E-07 0.000799198 1.42714E-07 0.000799398 1.4275E-07 0.000799598 1.43E-07 0.000799798 1.43E-07 0.000799998 1.43E-07
Figure 2Graphical representation of model (ZnOwith water) For TiO2 Table 9 parametric values of TiO2nanofluid
Reynolds number
kp( nanoparticle thermal conductivity ) kf ( base fluid(water) thermal conductivity ) f( density of base fluid ) p(density of nanoparticle ) f(dynamic viscosity of water ) dp(particle size ) T (temperature ) Kb(boltzman constant ) Volume fraction
Table 8 calculation of Reynolds number (ZnO) volume fraction 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010
Reynolds number 8.34E-02 8.34E-02 0.083337 8.33E-02 8.33E-02 8.33E-02 8.33E-02 8.32E-02 8.32E-02 8.32E-02
8.48 W/m-k 0.597 W/m-k 997.1 kg/m3 4175 kg/m3 7.98 x 10-4N.s/m2 25 nm 300 K 1.38x10-23m2 kg/s2K 0.1 to 1 %
The above valueswere incorporated into the model described by Equation 14 and the experimental values by the various researchers. The columns with the researchers names contain the data of the thermal conductivities ratio by various experiments
The constants in the above equation (6) are obtained by using experimentaldata (given in table) .Using nonlinear regression analysis an empirical correlation to predict the k of ZnO + H2O( eqn 14)nanofluids is developed as : Table 10 Experimental data of TiO2nanofluid Conc (%) 0.1 0.2 0.3 0.4 0.5
k1 (W/m-k) 0.599 0.613 0.624 0.634 0.645
k2 (W/m-k) 0.603 0.623 0.632 0.635 0.644
k3 (W/m-k) 0.609 0.619 0.628 0.64 0.647
k4 (W/m-k) 0.606 0.62 0.626 0.639 0.65
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k5 (W/m-k) 0.605 0.62 0.625 0.638 0.65
keff (W/m-k) 0.6042 0.6187 0.6265 0.637 0.6465
keff/kf 1.01206 1.03634 1.04941 1.067 1.08291
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International Journal of Engineering, Management & Sciences (IJEMS) ISSN-2348 –3733, Volume -2, Issue-3, March 2015 0.6 0.7 0.8 0.9 1
0.667 0.679 0.682 0.689 0.69
0.658 0.674 0.679 0.68 0.689
0.653 0.682 0.681 0.69 0.688
0.659 0.658 0.6592 1.10418 0.681 0.68 0.671 1.12395 0.683 0.682 0.6802 1.13936 0.687 0.686 0.6865 1.14991 0.694 0.695 0.6921 1.15929 The constants in the above equation (6) are obtained by using experimentaldata (given in table) .Using nonlinear regression analysis an empirical correlation topredict the k of TiO2 + H2O( eqn 15)nanofluids is developed as :
Dynamic viscosity
kinematic viscosity
Table 11 calculation of viscosity (TiO2 ) volume fraction
Dynamic viscosity of nanofluid ( nf ) in N s/m2
Kinematic viscosity of nanofluid ( ) in m2/s.
0.001
0.000799998
1.91616E-07
0.002
0.000802004
1.92097E-07
0.003
0.000804017
1.92579E-07
0.004
0.000806
1.93E-07
0.005
0.000808063
1.93548E-07
0.006
0.000810097
1.94035E-07
0.007
0.000812138
1.94524E-07
0.008
0.000814186
1.95015E-07
0.009
0.000816242
1.95507E-07
0.010
0.000818304
1.96001E-07
(15)
Figure 3Graphical representation of model For TiO2 with water
Reynolds number
III. DISCUSSION OF RESULTS AND COMPARISON WITH OTHER MODELS
Table 12 Calculation of Reynolds number (TiO2 ) volume fraction
Reynolds number
0.001
7.87E-02
0.002
7.85E-02
0.003
7.83E-02
0.004
7.81E-02
0.005
7.79E-02
0.006
7.77E-02
0.007
7.75E-02
0.008
7.73E-02
0.009
7.71E-02
0.010
7.69E-02
It describes the comparison of results obtained from the developed mathematical model with the results published from the experimental data. The experimental data was obtained from various relevant researches so as to validate the model for various nanofluids combinations. The mathematical model was then compared with other models developed to understand and compare the proximity of the results. AL2O3 – 40 NM WITH WATER BASE FLUID The experimental values were used to plot the effective thermal conductivity v/s volume fraction. The plot is shown in Figure 17 and it clearly indicates that the thermal conductivity increases with an increase in the volume fraction of nanoparticles, the developed model is in good agreement with the experimental data.
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Thermal Conductivity Enhancement in Oxide Nanofluids –a Mathematical Model
Figure 4 Comparison of the thermal conductivity model of Al2O3nanofluid models with experimental data
Figure 6 Effect of temp. on thermal conductivity of Al2O3 nanofluid
Using the empirical correlation (eqn 13) obtained, figures 18,19 & 20 are drawn to show the effect of variables parameter on thermal conductivity of nanofluid .
Effect of particle diameter Figure 20 shows the effect of particle diameter on the k value of nanofluid. It indicates that with increase in particle size the thermal conductivity effect decreases.
Effect of volume concentration Figure 18shows the effect of particle diameter on the thermal conductivity of nanofluid atvarious volume fractions. It indicates that with increasing particle diameter thethermal conductivity enhancement decreases. Further, it shows that with increasingvolume fraction the effective thermal conductivity of a nanofluid increases. The rateof increase of the k value is found to be less at higher volume fractions comparedto lower fractions.
Figure 7 Effect of Particle size on thermal conductivity of Al2O3 nanofluid
Figure 5 Effect of Vol. fraction on thermal conductivity of Al2O3nanofluid
TIO2 – 25 NM WITH WATER BASE FLUID The experimental values were used to plot the effective thermal conductivity v/s volume fraction. compare present correlation with experimental data and other thermal conductivity model. The plot is shown in Figure 21 and it clearly indicates that the thermal conductivity increases with an increase in the volume fraction of nanoparticles. The trend of predictions obtained using new developed model isalmost parallel to the experimental data.
Effect of temperature Figure 19 shows the effect of temperature on the thermal conductivity of a nanofluid. It indicates that with increasing temperature the k value of the nanofluidincreases. Further, we can conclude from the graph that the effect is more dominantin the small-sized particles than with large-sized ones.
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International Journal of Engineering, Management & Sciences (IJEMS) ISSN-2348 –3733, Volume -2, Issue-3, March 2015
Figure 8Comparison of present model with experimental data and existing model (TiO2 ) This is slightly unusal since the models thatwere previously under predicting the experimental data. Thismay be a result of experimental error or (more likely) a result of other phenomenon such as atmospheric condition or type of nanoparticle with the fluid which result in these deviations .Using the empirical correlation (eqn 14) obtained, figures 22,23 & 24 are drawn to show the effect of variables parameter on thermal conductivity of nanofluid .
Figure 9 Effect of volume concentration on thermal conductivity of TiO2nanofluid Effect of temperature Figure 23 shows the effect of temperature on the thermal conductivity of a nanofluid. It indicates that with increasing temperature the k value of the nanofluidincreases. Further, we can conclude from the graph that the effect is more dominantin the small-sized particles than with large-sized ones.
Effect of volume concentration Figure 22 shows the effect of particle diameter on the thermal conductivity of nanofluid atvarious volume fractions. It indicates that with increasing particle diameter thethermal conductivity enhancement decrease that’s why nano sized particle are so important. Further, it shows that with increasingvolume fraction the effective thermal conductivity of a nanofluid increases. The rateof increase of the k value is found to be less at higher volume fractions comparedto lower fractions.
Figure 10 Effect of Temperature on thermal conductivity of TiO 2 nanofluid
Effect of particle diameter Figure 24 shows the effect of particle diameter on the k value of nanofluid. It indicates that with increase in particle size the thermal conductivity effect decreases.
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Thermal Conductivity Enhancement in Oxide Nanofluids –a Mathematical Model
of a nanofluid increases. The rateof increase of the k value is found to be less at higher volume fractions comparedto lower fractions.
Figure 11 Effect of Particle dia. on thermal conductivity of TiO 2 nanofluid ZNO – 30 NM WITH WATER BASE FLUID The experimental values were used to plot the effective thermal conductivity v/s volume fraction. Compare present correlation with experimental data and other thermal conductivity model. The plot is shown in Figure 25 and it clearly indicates that the thermal conductivity increases with an increase in the volume fraction of nanoparticles.
Figure 13Effect of Volume concentrationon thermal conductivity of ZnOnanofluid
Effect of temperature Figure 27 shows the effect of temperature on the thermal conductivity of a nanofluid. It indicates that with increasing temperature the k value of the nanofluidincreases. Further, we can conclude from the graph that the effect is more dominantin the small-sized particles than with large-sized ones.
Figure 12Comparison of present model with experimental data and existing model (ZnO ) Results from Maxwell and murshad were able to explain significant portion of the enhancement but were not thorough enough to explain the unusual thermal conductivity of the nanofluids observed during experimentation. Using the empirical correlation (eqn 14) obtained, figures 26,27 & 28 are drawn to show the effect of variables parameter on thermal conductivity of nanofluid . Effect of volume concentration Figure 26shows the effect of particle diameter on the thermal conductivity of nanofluid atvarious volume fractions. It indicates that with increasing particle diameter thethermal conductivity enhancement decrease that’s why nano sized particle are so important. Further, it shows that with increasingvolume fraction the effective thermal conductivity
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Figure 14 Effect of Temperature on thermal conductivity of ZnOnanofluid
Effect of particle diameter Figure 28 shows the effect of particle diameter on the k value of nanofluid. It indicates that with increase in particle size the thermal conductivity effect decreases.
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International Journal of Engineering, Management & Sciences (IJEMS) ISSN-2348 –3733, Volume -2, Issue-3, March 2015 such as a nanoparticle chain, the effect on thermal conductivitywhen the nanoparticle combine to form various shapes and the different base fluid. Exploring the limitingfactors based on this result can be a topic for future studies. So this advanced technology ofsuspending nanoparticles in base fluids might provide answers to improved thermalmanagement. Improved understanding of complex nanofluids will have an even broader impact. V. FUTURE SCOPE OF WORK
Figure 15Effect of Particle dia. on thermal conductivity of ZnOnanofluid IV. CONCLUSION The mathematical model developed to calculate the thermal conductivity is a function of the thermal conductivities of the fluid and the nanoparticle, clustering effect, the nanolayer, volume fraction, nanoparticle diameter. The developed equation was compared to other models in the literature to understand the proximity of the results. Based on the results obtained and validation, it is found that the Brownian motion play a critical role in the thermal conductivity enhancement of nanofluids . The mathematical model developed lies on par with experimental data which confirms that the assumptions made for the development of mathematical model are accurate and well within the practical limitations. Thermal conductivity of nanofluids increases with increase in volume fraction of Nanoparticles in base fluid, temperature of nanofluids and decrease in size ofnanoparticles, The model developed was found to be applicable for almost all oxides nanolfluid. Evaluation of accuracy of existing theoretical and empirical models for thermal conductivity of metal oxides nanofluids by comparing the predicted versus experimental values have revealed that the existing models provide accuracy. A new thermal conductivity models have been developed using dimensionless analysis,which includes Reynolds number and Boltzmannconstant. Further studies are required to better understand the mechanism of heat conduction throughnanofluids and the influence of different experimental conditions on the thermalconductivity of nanofluids for more accurate modeling. The factors governing the overall enhancement of thermal conductivity is also understood better by solving the mathematical model using the various assumptions. Overall, the model predictions were found to be in good agreement with experimental data. The study can be further scrutinized by varying some of the parameters
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More fundamental study has to be carried out for the effect of different parameters on Thermal conductivity of nannofluids. A standard theoretical model for nanofluid thermal conductivity has to be developed by taking consideration of all the possible mechanisms such as interracial layer, brawnion motion, clustering etc. and effect of all the factors such as size, shape, temperature, volume fraction, ultrasonication time and pH. A standard design of experiment for thermal conductivity of nanofluid can be proposed by doing efficient number of experiments over wide range of variables. More application based testing is required to evaluate the effect of convection Phenomenon.
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Thermal Conductivity Enhancement in Oxide Nanofluids –a Mathematical Model
[11] Masuda, H., A. Ebata, K. Teramae and N. Hishinuma, 1993, Alternation of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles (dispersion of -Al2O3, SiO2 and TiO2 ultra-fine particles), NetsuBussei(Japan) 4, 227-233. [12] Brownian dynamics simulation to determine the effective thermal conductivity of nanofluids P. Bhattacharya, S. K. Saha, A. Yadav, P. E. Phelan, and R. S. Prasher ,J. Appl. Phys. 95, 6492 (2004). [13] Role of Brownian motion in the enhanced thermal conductivity of nanofluidsSeokPil Jang) and Stephen U. S. Choi APPLIED PHYSICS LETTERS VOLUME 84, NUMBER 21. [14] Model for effective thermal conductivity of nanofluids ,Qing-ZhongXue ,Physics Letters A 307 (2003) 313–317. [15] Contribution of Brownian Motion in Thermal Conductivity of Nanofluids S. M. SohelMurshed and C. A. Nieto de Castro ,Proceedings of the World Congress on Engineering 2011 Vol III . [16] Thermal Conductivity of Nanofluids A. K. Singh ,Defence Science Journal, Vol. 58, No. 5, September 2008, pp. 600-607. [17] A Review on Nanofluids: Preparation, StabilityMechanisms, and Applications Wei Yu andHuaqingXie ,Hindawi Publishing Corporation Journal of Nanomaterials Volume 2012, Article ID 435873, 17 pages. [18] Tang D.W., Liu S., Zheng X. H., Araki N., 2007, Thermal-Conductivity and Thermal-Diffusivity Measurements of Nanofluids by 3ω Method and Mechanism Analysis of Heat Transport, Thermophys, Vol. 28, PP. 1255–1268. [19] Thermal conductivity and particle agglomeration in alumina nanofluids: Experiment and theory Elena V. Timofeeva, Alexei N. Gavrilov, James M. McCloskey, and Yuriy V. Tolmachev, PHYSICAL REVIEW E 76, 061203 2007. [20] J.K.E.Goodson, and J.S. Lee, “ Thermal Conductivity Measurement and Sedimeentation detection of Al2O3 Nanofluids by Using the 3W Method”, Int. J. Heat Fluid Flow, 29, 2008, p 1456. [21] B.S.A. Shin, “Minimum Quantity Lubrication (MQL) Grinding Using Nanofluid”, The University of Mishigan, http://wumrf.engin.umich.edu/research/file/advmach_files/mql. [22] Heat Transfer Enhancement by Nano Fluids , J. of Convective Heat and Mass TransferJ. Thompson, Proc. Symp. Nanophase and Nanocomposite Materials II, Vol. 457, Materials Research Society, Boston (1996), pp. 3–11. [23] Measurement of the thermal conductivity of titania and alumina nanofluidsThakleawYiamsawasda, AhmetSelimDalkilic b, SomchaiWongwises ,ThermochimicaActa 545 (2012) 48– 56. [24] Thermal Conductivity of Nanoscale Colloidal Solutions (Nanofluids) Ravi Prasher , physical review letters 94, 025901 (2005). [25] “Thermal Conductivity of nanofluid” divyaratra ,M.tech thesis , MNIT 2012. [26] Prajapati O.S. (2012) ,“ Al2O3-Water Nanofluids in Convective Heat Transfer” Applied Mechanics and Materials Vols. 110-116 (2012) pp 3667-3672.
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