Thermal Conductivity Enhancement in Oxide Nanofluids –a

International Journal of Engineering, Management & Sciences (IJEMS)ISSN-2348 –3733, Volume -2, Issue-3, March 2015

28 www.alliedjournals.com

Abstract— The study investigated the impact of the

nanoparticle size which has been suggested to be an importantfactor the results were found to be in concord with theexperimental observations. The values of the thermalconductivity for different nanofluid combinations werecalculated using the expression developed in this study and theyagreed with published experimental data.

From the study, it was observed that Brownian motion issignificant only when the volume fraction is less than 1 % in caseof TiO2&ZnO and 4 % in case of Al2O3. The combination of thebase fluid and nanoparticles to from nanoclusters is expectedprovide better heat transfer solution than the conventional fluids, Hence it is concluded that adding nanosized materials to basefluids enhances thermal properties and makes them moresuitable to heat exchanger applications as well as for manyindustrial applications also .

Index Terms— Thermal conductivity, nanoparticle,brownian motion, nanofluids.

I. INTRODUCTIONSo far no general mechanisms to have been formulated to

understand the strange behavior of the nano fluids includingthe highly improved effective thermal conductivity, thistechnology isstill limited for commercial use because there isyet no proven standardized design process for accuratelypredicting important heat transfer properties. Developing areliable fundamental model for the thermal conductivity ofnanofluids has always been a challenging task for researchers.Early attempts to explain this behavior have made use of theclassical model of Maxwell. This model is generallyapplicable to dilute suspensions with micro particles but whenapplied to nanofluids the models predicted lower thermalconductivity enhancement as compared to the experimentalobservations. Several authors extended the Maxwell’s theorysuch as Bruggeman (1935); Jeffrey (1973); Yu and Choi(2003); Koo and Kleinstreuer (2004); Xie et al. (2005) aresome theoretical models and Chon et al. (2005); Li andPeterson (2006); Mintsa et al. (2009) and Teng et al. (2010)are some empirical models. These models are not so accurateand 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

because of these present limitations for better understandingthe heat transfer mechanism and effect of different parameters

on thermal conductivity of nanofluids more studies have to becarried out. In this Paper, new models have been developed tomeasure the thermal conductivity of Al2O3- water and ZnOand TiO2-water nano fluids. Models have been developed byconsidering the fact thatthermal conductivity of nanofluid isdepends 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 basefluid . Rem (Modified Reynolds number) which is adimensionless quantity based on Boltzmann constant.

II. PRESENT MODEL FOR THERMAL CONDUCTIVITY

Thermal conductivity of a nanofluid, knf, is given by:-

(1)

For development of thermal conductivity model first we haveto analyze the Brownian motion of nanoparticle. The particlessuspended in the liquid are very small, Brownian movementof the particles is quite possible.The root-mean-squarevelocity (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 nearthe particles due to their Brownian movement. The Reynoldsnumber based on vN given by Eq. (4) can bewritten as:

(4)

These variables in equation (1) can be expressed innon-dimensional terms as:

Knf=f[ , Rem, φ] (5)

Thermal Conductivity Enhancement in OxideNanofluids –a Mathematical Model

Khagendra Kumar Upman

Thermal Conductivity Enhancement in Oxide Nanofluids –a Mathematical Model


Therefore

(6)Where c,p,q& r are constantsTaking log both side for making eqn

6 linear

(7)According to linear regression analysis:A general form of a multiple linear regression model is givenby

(8)

Where

This is supposed to hold for each observation. The modelcorresponding to each observation in the data set would haveto write:

(9)

(10)

(11)

(12)Where n = no. of variablesRegression analysis helps in understanding the relationshipbetween variables.Linear regression constructs linearmodels.In simple regression it is assumed that the dependentvariable Y is related to a single variable X.But in practice, a

dependent variable may depend on more than oneindependent variable. Insuch a situation, a multiple regressionequation with more than one independent variable isused. Theconstants in a multiple regression equation can be computedwith the help of the“normal equations”. Computation of theestimators of the standard error of the regressioncoefficientsbecomes complex in case of multiple regression and usuallycomputers are usedfor this purpose. As the degree ofcorrelation between the independent variables increases,theregression coefficients become less reliable, i.e. although theindependent variables maytogether explain the dependentvariable, but because of multicollinearity the coefficients ofthe explanatory variables may be rejected. So in case of amultiple regression equation, thecoefficient of multiplecorrelations should also be computed. Partial correlationcoefficientshelp in finding out the extent to which thevariation in the dependent variable is explained byoneindependent variable if all other independent variables arekept constant.Equation 6 is general form of model of thermal conductivityof 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:

For Al2O3Table 1 Parametric values of Al2O3

kp( nanoparticle thermalconductivity )

0.628 W/m-k

kf ( base fluid thermalconductivity )

35 W/m-k

f( density of base fluid ) 997.1 kg/m3

p(density of nanoparticle ) 3970 kg/m3

f(dynamic viscosity of water ) 7.98 x 10-4N.s/m2

dp(particle size ) 40 nm.T (temperature ) 300 KKb(boltzman constant ) 1.38x10-23m2 kg/s2KVolume fraction 1 to 4 %

The above valueswere incorporated into the model describedby Equation 13 and the experimental values by the variousresearchers. The columns with the researchers names containthe data of the thermal conductivities ratio by variousexperiments

Table 2 experimental data (Al2O3)

Volume concentration (In %) 1 2 3 4ResearchersTang et. Al (2010 ) 1.01 1.02 1.035 1.5

Thermal conductivity

ratio(knf/kf)

Das et. Al (2003 ) 1.12 1.14 1.17 1.25Lee et. Al 1.025 1.05 1.07 1.1Lee and choi 1.024 1.045 1.065 1.09Eastman et. Al 1.04 1.08 1.1 1.25Timmofeva et. Al 1.028 1.05 1.075 1.1J.K.E.Goodson(2008) 1.04 1.075 1.125 1.145B.S.A. Shin 1.06 1.09 1.12 1.14Murshad et. Al 1.122 1.142 1.17 1.2Thomsan 1997 1.03 1.12 1.18 1.24



Wang et. Al (2003 ) 1.1 1.125 1.14 1.17Oh. Et. Al 1.045 1.07 1.1 1.14ThakleawYiamsawasd 1.025 1.065 1.09 1.12

Dynamic viscosity

kinematic viscosity

Table 3 calculation of Viscosity (Al2O3)

Reynolds number

Table 4 calculation of Reynolds number (Al2O3 )

volume fraction Reynolds number0.01 0.0593180330.02 0.0578180110.03 0.056354330.04 0.05491311

The constants in the above equation (6) are obtained by usingexperimentaldata (given in table) .Using nonlinear regressionanalysis an empirical correlation topredict the k of Al2O3 +H2O( eqn 13)nanofluids is developed as :

(13)

Figure 1Graphical representation of model (Al2O3 with water)

For ZnOTable 5 Parametric values of ZnOnanofluid


13 W/m-k

kf ( base fluid(water) thermalconductivity )

0.597 W/m-k



f(dynamic viscosity of water)

7.98 x 10-4N.s/m2

dp(particle size ) 30 nmT (temperature ) 300 KKb(boltzman constant ) 1.38x10-23m2 kg/s2KVolume fraction 0.01 to 0.1 %


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)

kef

f(W/m-k)keff/kf

0.01 0.6 0.602 0.601 0.601 0.599 0.6006 1.006030.02 0.605 0.603 0.603 0.602 0.602 0.603 1.010050.03 0.611 0.61 0.609 0.608 0.607 0.609 1.02010.04 0.614 0.614 0.614 0.611 0.612 0.613 1.02680.05 0.619 0.618 0.618 0.617 0.619 0.6182 1.035510.06 0.625 0.626 0.625 0.626 0.624 0.6252 1.04723

volumefraction

Dynamic viscosityof nanofluid ( nf )

in N s/m2

Kinematic viscosityof nanofluid ( ) in

m2/s.0.01 0.000818304 2.06122E-070.02 0.00083934 2.11421E-070.03 0.00086114 2.16912E-070.04 0.000883741 2.22605E-07



0.07 0.632 0.631 0.63 0.629 0.628 0.63 1.055270.08 0.635 0.635 0.634 0.634 0.634 0.6344 1.062640.09 0.64 0.64 0.638 0.639 0.64 0.6394 1.071020.1 0.645 0.648 0.645 0.646 0.645 0.6458 1.08174

Dynamic viscosity

Table 7 calculation of viscosity (ZnO)

volumefraction

Dynamic viscosityof nanofluid ( nf )

in N s/m2

Kinematicviscosity of nanofluid

( )in m2/s.

0.0001 0.0007982 1.42536E-070.0002 0.000798399 1.42571E-070.0003 0.000798599 1.42607E-070.0004 0.000798799 1.42643E-070.0005 0.000798998 1.42678E-070.0006 0.000799198 1.42714E-070.0007 0.000799398 1.4275E-070.0008 0.000799598 1.43E-070.0009 0.000799798 1.43E-070.001 0.000799998 1.43E-07

Reynolds number

Table 8 calculation of Reynolds number (ZnO)

volume fraction Reynolds number0.0001 8.34E-020.0002 8.34E-020.0003 0.0833370.0004 8.33E-020.0005 8.33E-020.0006 8.33E-020.0007 8.33E-020.0008 8.32E-020.0009 8.32E-020.0010 8.32E-02

The constants in the above equation (6) are obtained by usingexperimentaldata (given in table) .Using nonlinear regressionanalysis an empirical correlation to predict the k of ZnO +H2O( eqn 14)nanofluids is developed as :

(14)

Figure 2Graphical representation of model (ZnOwith water)

For TiO2Table 9 parametric values of TiO2nanofluid


8.48 W/m-k

kf ( base fluid(water) thermalconductivity )

0.597 W/m-k



f(dynamic viscosity of water ) 7.98 x 10-4N.s/m2

dp(particle size ) 25 nmT (temperature ) 300 KKb(boltzman constant ) 1.38x10-23m2 kg/s2KVolume fraction 0.1 to 1 %


Table 10 Experimental data of TiO2nanofluid

Conc(%)

k1(W/m-k)

k2(W/m-k)

k3(W/m-k)

k4(W/m-k)

k5(W/m-k)

keff(W/m-k)

keff/kf

0.1 0.599 0.603 0.609 0.606 0.605 0.6042 1.012060.2 0.613 0.623 0.619 0.62 0.62 0.6187 1.036340.3 0.624 0.632 0.628 0.626 0.625 0.6265 1.049410.4 0.634 0.635 0.64 0.639 0.638 0.637 1.0670.5 0.645 0.644 0.647 0.65 0.65 0.6465 1.08291



0.6 0.667 0.658 0.653 0.659 0.658 0.6592 1.104180.7 0.679 0.674 0.682 0.681 0.68 0.671 1.123950.8 0.682 0.679 0.681 0.683 0.682 0.6802 1.139360.9 0.689 0.68 0.69 0.687 0.686 0.6865 1.149911 0.69 0.689 0.688 0.694 0.695 0.6921 1.15929

Dynamic viscosity

kinematic viscosity

Table 11 calculation of viscosity (TiO2 )

volumefraction

Dynamic viscosity ofnanofluid ( nf ) in N

s/m2

Kinematic viscosityof 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-070.005 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-070.010 0.000818304 1.96001E-07

Reynolds number

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

The constants in the above equation (6) are obtained by usingexperimentaldata (given in table) .Using nonlinear regressionanalysis an empirical correlation topredict the k of TiO2 +H2O( eqn 15)nanofluids is developed as :

(15)

Figure 3Graphical representation of model For TiO2 withwater

III. DISCUSSION OF RESULTS AND COMPARISON WITH OTHERMODELS

It describes the comparison of results obtained from thedeveloped mathematical model with the results publishedfrom the experimental data. The experimentaldata was obtained from various relevant researches so as tovalidate the model for various nanofluids combinations. Themathematical model was then compared with other modelsdeveloped to understand and compare the proximity of theresults.

AL2O3 – 40 NM WITH WATER BASE FLUIDThe experimental values were used to plot the effectivethermal conductivity v/s volume fraction. The plot is shown inFigure 17 and it clearly indicates that the thermal conductivityincreases with an increase in the volume fraction ofnanoparticles, the developed model is in good agreement withthe experimental data.



Figure 4 Comparison of the thermal conductivity model ofAl2O3nanofluid models with experimental data

Using the empirical correlation (eqn 13) obtained, figures18,19 & 20 are drawn to show the effect of variablesparameter on thermal conductivity of nanofluid .

Effect of volume concentrationFigure 18shows the effect of particle diameter on the thermalconductivity of nanofluid atvarious volume fractions. Itindicates that with increasing particle diameter thethermalconductivity enhancement decreases. Further, it shows thatwith increasingvolume fraction the effective thermalconductivity of a nanofluid increases. The rateof increase ofthe k value is found to be less at higher volume fractionscomparedto lower fractions.

Figure 5 Effect of Vol. fraction on thermal conductivity ofAl2O3nanofluid

Effect of temperatureFigure 19 shows the effect of temperature on the thermalconductivity of a nanofluid. It indicates that with increasingtemperature the k value of the nanofluidincreases. Further, wecan conclude from the graph that the effect is moredominantin the small-sized particles than with large-sizedones.

Figure 6 Effect of temp. on thermal conductivity of Al2O3nanofluid

Effect of particle diameterFigure 20 shows the effect of particle diameter on the k valueof nanofluid. It indicates that with increase in particle size thethermal conductivity effect decreases.

Figure 7 Effect of Particle size on thermal conductivity ofAl2O3 nanofluid

TIO2 – 25 NM WITH WATER BASE FLUIDThe experimental values were used to plot the effectivethermal conductivity v/s volume fraction. compare presentcorrelation with experimental data and other thermalconductivity model. The plot is shown in Figure 21 and itclearly indicates that the thermal conductivity increases withan increase in the volume fraction of nanoparticles. The trendof predictions obtained using new developed model isalmostparallel to the experimental data.



Figure 8Comparison of present model with experimental dataand existing model (TiO2 )

This is slightly unusal since the models thatwere previouslyunder predicting the experimental data. Thismay be a result ofexperimental error or (more likely) a result of otherphenomenon such as atmospheric condition or type ofnanoparticle with the fluid which result in these deviations.Using the empirical correlation (eqn 14) obtained, figures22,23 & 24 are drawn to show the effect of variablesparameter on thermal conductivity of nanofluid .

Effect of volume concentrationFigure 22 shows the effect of particle diameter on the thermalconductivity of nanofluid atvarious volume fractions. Itindicates that with increasing particle diameter thethermalconductivity enhancement decrease that’s why nano sizedparticle are so important. Further, it shows that withincreasingvolume fraction the effective thermal conductivityof a nanofluid increases. The rateof increase of the k value isfound to be less at higher volume fractions comparedto lowerfractions.

Figure 9 Effect of volume concentration on thermalconductivity of TiO2nanofluid

Effect of temperatureFigure 23 shows the effect of temperature on the thermalconductivity of a nanofluid. It indicates that with increasingtemperature the k value of the nanofluidincreases. Further, wecan conclude from the graph that the effect is moredominantin the small-sized particles than with large-sizedones.

Figure 10 Effect of Temperature on thermal conductivity ofTiO 2 nanofluid

Effect of particle diameter

Figure 24 shows the effect of particle diameter on the k valueof nanofluid. It indicates that with increase in particle size thethermal conductivity effect decreases.



Figure 11 Effect of Particle dia. on thermal conductivity ofTiO 2 nanofluid

ZNO – 30 NM WITH WATER BASE FLUIDThe experimental values were used to plot the effectivethermal conductivity v/s volume fraction. Compare presentcorrelation with experimental data and other thermalconductivity model. The plot is shown in Figure 25 and itclearly indicates that the thermal conductivity increases withan increase in the volume fraction of nanoparticles.

Figure 12Comparison of present model with experimentaldata and existing model (ZnO )

Results from Maxwell and murshad were able to explainsignificant portion of the enhancement but were not thoroughenough to explain the unusual thermal conductivity of thenanofluids observed during experimentation.

Using the empirical correlation (eqn 14) obtained, figures26,27 & 28 are drawn to show the effect of variablesparameter on thermal conductivity of nanofluid .

Effect of volume concentrationFigure 26shows the effect of particle diameter on the thermalconductivity of nanofluid atvarious volume fractions. Itindicates that with increasing particle diameter thethermalconductivity enhancement decrease that’s why nano sizedparticle are so important. Further, it shows that withincreasingvolume fraction the effective thermal conductivity

of a nanofluid increases. The rateof increase of the k value isfound to be less at higher volume fractions comparedto lowerfractions.

Figure 13Effect of Volume concentrationon thermalconductivity of ZnOnanofluid

Effect of temperature

Figure 27 shows the effect of temperature on the thermalconductivity of a nanofluid. It indicates that with increasingtemperature the k value of the nanofluidincreases. Further, wecan conclude from the graph that the effect is moredominantin the small-sized particles than with large-sizedones.

Figure 14 Effect of Temperature on thermal conductivity ofZnOnanofluid

Effect of particle diameter

Figure 28 shows the effect of particle diameter on the k valueof nanofluid. It indicates that with increase in particle size thethermal conductivity effect decreases.



Figure 15Effect of Particle dia. on thermal conductivity ofZnOnanofluid

IV. CONCLUSION

The mathematical model developed to calculate thethermal conductivity is a function of the thermalconductivities of the fluid and the nanoparticle,clustering effect, the nanolayer, volume fraction,nanoparticle diameter. The developed equation wascompared to other models in the literature to understandthe proximity of the results.

Based on the results obtained and validation, it is found thatthe Brownian motion play a critical role in the thermalconductivity enhancement of nanofluids .

The mathematical model developed lies on par withexperimental data which confirms that the assumptionsmade for the development of mathematical model areaccurate and well within the practical limitations.

Thermal conductivity of nanofluids increases with increasein volume fraction ofNanoparticles in base fluid, temperature of nanofluidsand decrease in size ofnanoparticles,

The model developed was found to be applicable foralmost all oxides nanolfluid. Evaluation of accuracy ofexisting theoretical and empirical models for thermalconductivity of metal oxides nanofluids by comparingthe predicted versus experimental values have revealedthat the existing models provide accuracy.

A new thermal conductivity models have been developedusing dimensionless analysis,which includes Reynoldsnumber and Boltzmannconstant.

Further studies are required to better understand themechanism of heat conductionthroughnanofluids and the influence of differentexperimental conditions on the thermalconductivity ofnanofluids for more accurate modeling.

The factors governing the overall enhancement of thermalconductivity is also understood better by solving themathematical model using the various assumptions.Overall, the model predictions were found to be in goodagreement with experimental data. The study can befurther scrutinized by varying some of the parameters

such as a nanoparticle chain, the effect on thermalconductivitywhen the nanoparticle combine to formvarious shapes and the different base fluid. Exploringthe limitingfactors based on this result can be a topic forfuture studies. So this advanced technologyofsuspending nanoparticles in base fluids might provideanswers to improved thermalmanagement. Improvedunderstanding of complex nanofluids will have an evenbroader impact.

V. FUTURE SCOPE OF WORK

More fundamental study has to be carried out for the effectof different parameters onThermal conductivity of nannofluids.

A standard theoretical model for nanofluid thermalconductivity has to be developed by takingconsideration of all the possible mechanisms such asinterracial layer, brawnion motion, clustering etc. andeffect of all the factors such as size, shape, temperature,volume fraction, ultrasonication time and pH.

A standard design of experiment for thermal conductivityof nanofluid can be proposed by doing efficient numberof experiments over wide range of variables.

More application based testing is required to evaluate theeffect of convectionPhenomenon.

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Thermal Conductivity Enhancement in Oxide Nanofluids –a