International Journal of Heat and Mass Transfer 68 (2014) 78–84

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Uncertainties in modeling thermal conductivity of laminar forcedconvection heat transfer with water alumina nanofluids

0017-9310/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.09.018

⇑ Tel.: +40 723455071; fax: +40 232213575.E-mail address: aminea@tuiasi.ro

Alina Adriana Minea ⇑Technical University ‘‘Gheorghe Asachi’’ from Iasi, Bd. D. Mangeron no. 63, Iasi 700050, Romania

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 June 2013Received in revised form 9 August 2013Accepted 11 September 2013

Keywords:NanofluidAluminaLaminar flowThermal conductivityHeat transfer coefficient

At this stage of nanofluids development, their thermal conductivity it is not yet known precisely and thejudgment of their true potential is difficult. This fact was illustrated by analyzing their heat transfer per-formance for laminar fully developed forced convection in a tube with two zones: one adiabatic and onewith uniform wall heat flux. Forced convective of a nanofluid that consists of water and Al2O3 in horizon-tal tubes has been studied numerically. Three different models from the literature are used to express thethermal conductivity in terms of particle loading and they led to different qualitative and quantitativeresults in a classical problem of replacement of a simple fluid (water) by a nanofluid in a given situation.In particular, the heat transfer coefficient of water-based Al2O3 nanofluids is increased by 3.4–27.8%under fixed Reynolds number compared with that of pure water. Also, the enhancement of heat transfercoefficient is larger than that of the effective thermal conductivity at the same volume concentration.Moreover, the effect of uncertainties in modeling nanofluids properties was noticed.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

With an ever-increasing thermal load due to trends towardsmaller devices, greater power output for engines, cooling of suchdevices and related systems is a very important problem in high-tech industries. The conventional approach for increasing coolingand heating rates is the use of extended surfaces such as fins andmicrochannels. However, new designs have already expanded thisapproach to its limits. Therefore, there is an urgent need for newand innovative concepts to achieve ultra-high performance pro-cesses. Choi [1] has pioneered ultra-high-thermal conductivity flu-ids, called nanofluids, by suspending nanoparticles in conventionalcoolants.

Dispersing solid particles into liquids to improve the physicalproperties of liquids is hardly new, since the idea can be tracedback to James Clerk Maxwell’s theoretical work [2]. Despitenumerous studies for more than a century on the thermal conduc-tivity of traditional solid/liquid suspensions containing nano- ormicrometer-sized particles, the rapid settling of these particles influids has been a major barrier to developing suspensions for prac-tical applications. In contrast, two significant features – well-sus-pended particles and high thermal conductivities far above thoseof traditional solid/liquid suspensions – make nanofluids strongcandidates for the next generation of coolants for thermal systems.

Nanofluids offer theoretical challenges because the measuredthermal conductivity of a nanofluid containing a low concentrationof nanoparticles is one order of magnitude greater than that pre-dicted by existing theories [3]. This discovery clearly suggestedthat conventional heat conduction models for solid-in-liquid sus-pensions are inadequate. Although liquid molecules close to a solidsurface are known to form layered structures [4], little is knownabout the connection between this nanolayer and the thermalproperties of solid/liquid suspensions. Yu and Choi [5] propose thatthe solid-like nanolayer acts as a thermal bridge between a solidnanoparticle and a bulk liquid and so is key to enhancing thermalconductivity. From this thermally bridging nanolayer idea, theysuggested a structural model of nanofluids that consists of solidnanoparticles, bulk liquid and solid-like nanolayers. The thermalconductivity of the nanolayer on the surface of the nanoparticleis not known. However, because the layered molecules are in anintermediate physical state between a bulk liquid and a solid, thesolid-like nanolayer of liquid molecules would be expected to leadto a higher thermal conductivity than that of the bulk liquid. Basedon this assumption, they have modified the Maxwell equation forthe effective thermal conductivity of solid/liquid suspensions to in-clude the effect of this ordered nanolayer, as it will be shown lateron in this study.

It is important to note that the actual amount of experimentaldata regarding the nanofluid properties, in particular thermal con-ductivity, remains quite limited. Therefore, in order to estimatesuch properties, researchers often turned to available formulaseither derived from the classical theory of two-phase mixtures or

Nomenclature

c specific heatD hydraulic diameterg gravitational accelerationh heat transfer coefficientk thermal conductivityL channel lengthNu Nusselt numberp pressurer radiusR tube radiusPr Prandtl numberRe Reynolds numberT temperatureq wall heat fluxx, y Cartesian coordinatesv velocity

Greek symbols

a thermal diffusivityb ratio of the nanolayer thickness to the original particle

radiusu volume fraction of particlesq densityl fluid dynamic viscosity

Subscripts0 refers to the reference (inlet) conditionbf refers to base-fluideff effectivef fluidnf refers to nanofluid propertyp particler refers to ‘‘nanofluid/base-fluid’’ ratiow wallm meanexit exit

A.A. Minea / International Journal of Heat and Mass Transfer 68 (2014) 78–84 79

based on semi-empirical models. Most often, the latter were pro-posed for liquid suspensions containing larger (i.e. millimeterand micrometer) size particles. Such approximations induceimportant discrepancies in the determination of nanofluids ther-mal properties (see in particular, [6]) and can cause considerableerrors when assessing the performance of nanofluids (heat transferenhancement) in various thermal applications. This is especiallytrue when a search for optimized operational conditions is pursued(see for example [7]).

As a conclusion, there are a lot of models to predict thermalconductivity of nanofluids and their equations are obtained on the-oretical or experimental background. In this article three commonmodels were selected and the present work is an investigation ofthe effect of these models used to predict nanofluid thermal con-ductivity for laminar forced convection in a two zone tube: onewith an isothermal domain and another one with uniform heat fluxat the wall. The nanofluid under study is water–cAl2O3 mixture.

1.1. Thermal conductivity models

1.1.1. Maxwell equationTo show a new connection between the nanolayers and thermal

conductivity increase in nanofluids, Yu and Choi [5] assumed thatthe thermal energy transport in nanofluids is diffusive. This al-lowed them to use classical models to show the effect of the nano-layer. The Maxwell equation takes into account only the particlevolume concentration and the thermal conductivities of particleand liquid. Other classical models include the effects of particleshape [8], particle distribution [9], and particle/particle interaction[10]. However, all of these models predict almost identicalenhancements at the low volume concentrations (<4%) of interestin this nanofluid study. Therefore, the Maxwell model is used inthis study as representative of all classical models. Particle sizeand the nanolayer have not been accounted for in any classicalmodels.

Based on Maxwell’s work [2], the effective thermal conductivityof a homogeneous suspension can be predicted as

knf

kbf¼

kp þ 2kbf þ 2u kp � kbf� �

kp þ 2kbf �u kp � kbf

� � ð1Þ

where kp is the thermal conductivity of the dispersed particles, kbf isthe thermal conductivity of the dispersion liquid, and u is the par-ticle volume concentration of the suspension.

1.1.2. Nanolayer impactMany studies have focused on the effect of a solid/solid inter-

face on effective thermal conductivity [11–13]. Because of theimperfect contact of the solid/solid interface, the interface resis-tance is a barrier to heat transfer and lowers the overall effectivethermal conductivity. In contrast, this solid/solid contact resistancephenomenon is not dominant at the solid/liquid interface of parti-cle-in-liquid suspensions. In order to include the effect of the liquidlayer, Yu and Choi [5] considered a nanoparticle-in-liquid suspen-sion with monosized spherical particles of radius r and particle vol-ume concentration u. Based on the above discussion, the Maxwellequation (1) was modified into

knf

kbf¼

kp þ 2kbf þ 2u kp � kbf

� �ð1þ bÞ3

kp þ 2kbf �u kp � kbf� �

ð1þ bÞ3ð2Þ

where b is the ratio of the nanolayer thickness to the original parti-cle radius. The nanolayer impact is significant for small particles, asYu and Choi [5] demonstrated. In this study, the value of b has beenfixed to 0.1.

1.1.3. Hamilton and Crosser model and its modificationsIn his study, Maiga et al. [14] introduced Eq. (3) that have been

obtained using the model proposed by Hamilton and Crosser [8]and this, assuming spherical particles. Such a model, which wasfirst developed based on data from several mixtures containing rel-atively large particles i.e. millimetre and micrometer size particles,is believed to be acceptable for use with nanofluids, although itmay give underestimated values of thermal conductivity. Thismodel has been adopted in this study because of its simplicity aswell as its interesting feature regarding the influence of the particleform itself.

knf

kbf¼

kp þ ðn� 1Þkbf � ðn� 1Þu kbf � kp� �

kp þ ðn� 1Þkbf þu kbf � kp� � ð3Þ

In the equation n is the shape factor and is equal to 3 for sphericalnanoparticles. Zhang et al. [15] have shown that this correlationaccurately predicts the thermal conductivity of nanofluids.

Details and discussion regarding the procedure of computingthe physical properties of nanofluids considered have been pre-sented elsewhere [14,16]. It is important to mention that the dataemployed for the nanofluids considered were obtained at fixedreference temperatures, that is to say that the influence of the

Table 1Thermophysical properties of base fluid and nanoparticles at 293 K.

Property Base fluid (water) Nanoparticle (Al2O3)

Specific heat (J/kg K) 4179 765Density (kg/m3) 1086.27 3880Thermal conductivity (W/m K) 0.613 42.64Viscosity (kg/ms) 9.93 � 10�4 –

80 A.A. Minea / International Journal of Heat and Mass Transfer 68 (2014) 78–84

temperature on fluid thermal properties has not been clearlyestablished to date. Finally, for most of the nanofluids of engineer-ing interest including the ones considered in the present study, theamount of experimental data providing information on their phys-ical properties remain, surprisingly, rather scarce if not to say qua-si-non-existing for some. Hence, much more research works willbe, indeed, needed in this field.

knf

kbf¼ 4:97u2 þ 2:72uþ 1 ð4Þ

Eq. (4) have been obtained by Maiga [14–16] using the well-knownmodel proposed by Hamilton and Crosser [8]. It is very interestingto note that such a model, although originally being derived for amixture with millimeter and micrometer size particles, appearsappropriate for use with nanoparticles.

2. Problem statement

The problem under consideration consists in a numerical studyof steady, forced flow and heat transfer of a nanofluid inside a twozone tube: one zone is isothermal and the next one is uniformlyheated, as seen in Fig. 1.

The flow is assumed fully developed and was considered lami-nar. From a practical viewpoint, most nanofluids used for heattransfer enhancement are constituted of very fine particles, usuallyunder 40 nm. Because of such reduced dimensions, it has been sug-gested that they may be easily fluidized and consequently, can beconsidered to behave like a fluid. Furthermore, by assuming a uni-form distribution of the nanoparticles within the base fluid as wellas negligible motion slip and thermal equilibrium conditions be-tween the solid particles and the continuous liquid phase, theresulting mixture can be considered as a conventional homoge-neous single-phase fluid, even if there are studies that consideredthe mixture model. Its effective thermophysical properties arefunctions of those of the constituents as well as of their respectiveconcentration [17]. As a consequence, an extension from the con-ventional fluid to the nanofluid appears quite realistic, and allthe equations for a conventional single-phase fluid can then be ap-plied to a nanofluid as well. Although more data will likely beneeded in order to definitely accept this assumption, it seems tobe validated, through the recent experimental works by Pak andCho [18] and Li and Xuan [19], in which correlations of the formsimilar to that of the well-known Dittus–Boelter formula [20] havebeen proposed to characterize the heat transfer of nanofluids. Inthe present study, in concurrence with the arguments statedabove, the ‘single-phase fluid’ approach in order to be able to studythe thermal behavior of nanofluids have been adopted. For theapplication considered here, we also assume that the nanofluid isNewtonian, incompressible with constant properties that areevaluated at a reference state.

Fig. 1. Tube geometry and

2.1. Physical properties of the nanofluids

The thermophysical properties of the base fluid (water) and thesolid nanoparticles (Al2O3) used in the present study are specifiedin Table 1.

By assuming the nanoparticles are well dispersed within thebase fluid, the effective physical properties of the mixtures studiedcan be evaluated using some classical formulas.

qnf ¼ uqp þ ð1�uÞqbf ð5Þ

cnf ¼uqpcp þ ð1�uÞqbf cbf

qnfð6Þ

lnf

lbf¼ 123u2 þ 7:3uþ 1 ð7Þ

Eq. (5) is a general relationship used to compute the density for aclassical two-phase mixture. Specific heat of nanofluids are calcu-lated by using the formulas summarized by Buongiorno [21] andpresented as Eq. (6). The dynamic viscosity of nanofluids has beencalculated through Eq. (7), which is obtained, by Maiga et al. [16]performing a least-square curve fitting of some experimental dataavailable for the considered nanofluid.

2.2. Test cases

For the purposes of the present study three combinations ofrelations for the calculation of the nanofluid properties were con-sidered, as illustrated in Table 2. Density, specific heat and viscos-ity remained the same for each model and the variable is thethermal conductivity that was selected from the most popular rela-tions used through latest research studies [17–19,21,22]. Case B isthe one considered by Mansour et al. [23] and identified as theBMGN combination after the authors. Also, Case C is similar withthe one considered by Manca et al. [24] in his research work.

As one can observe in Fig. 2, these three models give substan-tially different results for the nanofluid thermal conductivity, knf,especially if it compares case B with cases A and C. The substantialdifferences between the predictions of these different expressionsfor thermophysical properties can be attributed to the fact thatnone (except Eq. (4)) was specifically developed for nanofluids.Furthermore they are all based on the assumption that nanoparti-cles are uniformly distributed throughout the base fluid while inreality there are considerable uncertainties with respect to their

boundary conditions.

Table 2Case studies.

Case A Case B Case C

cnf ¼uqpcpþð1�uÞqbf cbf

qnfcnf ¼

uqp cpþð1�uÞqbf cbf

qnfcnf ¼

uqpcpþð1�uÞqbf cbf

qnf

qnf ¼ uqp þ ð1�uÞqbf qnf ¼ uqp þ ð1�uÞqbf qnf ¼ uqp þ ð1�uÞqbf

knf

kbf¼ kpþ2kbf�2u kbf�kpð Þ

kpþ2kbfþu kbf�kpð Þknf

kbf¼ kpþ2kbfþ2u kp�kbfð Þð1þbÞ3

kpþ2kbf�u kp�kbfð Þð1þbÞ3knf

kbf¼ 4:97u2 þ 2:72uþ 1

lr ¼lnf

lbf¼ 123u2 þ 7:3uþ 1 lr ¼

lnf

lbf¼ 123u2 þ 7:3uþ 1 lr ¼

lnf

lbf¼ 123u2 þ 7:3uþ 1

Fig. 2. Comparison of thermal conductivity models.

A.A. Minea / International Journal of Heat and Mass Transfer 68 (2014) 78–84 81

concentration. Finally, these expressions do not account for the ef-fects of the size disparity between the nanoparticles.

2.3. Numerical method

The single-phase model is employed in the simulation and ananofluid composed of water and Al2O3 nanoparticles flowing ina two zone tube with uniform heating at the wall boundary ofthe second zone is considered.

The set of nonlinear differential equations was solved by controlvolume approach. Control volume technique converts the governingequations to a set of algebraic equations that can be solved numer-ically. For the convective and diffusive terms, a second order upwindmethod was used. Pressure and velocity were coupled using SemiImplicit Method for Pressure Linked Equations ⁄⁄⁄[SIMPLE]. In thecase studies, the selected grid for the present calculations is consid-ered for 100 and 180 nodes for r-direction and x-direction, respec-tively. Other combinations of nodes were also tested in the presentwork, but all of them gave similar values of velocity and temperatureas the outlet (the differences were maximum 5%). Therefore thementioned nodes were accepted as the optimal ones. In order to ob-tain the required accuracy with minimum number of nodes, thenodes are concentrated at the entrance region and near the tubewalls where temperature gradients are high.

3. Results and discussions

3.1. Validation of the present discussion

The case studies presents the hydrodynamic and thermalbehaviors of laminar forced convective flow of a conventional fluid

inside a circular tube with two zones: an isothermal zone and azone with constant heat flux. As was stated before, the tube hasa diameter of 0.12 m and a length of 8.64 m. The fluid enters thetube with a constant inlet temperature of 300 K and with uniformaxial velocity. The Reynolds number was varied from 500 to 2300.A uniform heat flux of 10000 W/m2 was subjected to the walls inthe second zone.

Before the application of the numerical solution to the case ofnanofluid heat transfer, the numerical solution was verified byconsidering the flow of pure water inside the flow configurationdescribed in Fig. 1. In order to demonstrate the validity and alsoprecision of the model and the numerical procedure, a comparisonwith the previously published traditional expressions have beendone.

A traditional expression for calculation of heat transfer in fullydeveloped laminar flow in smooth tubes is that recommended byShah, for constant wall heat flux [25]:

Nu ¼ 1:953 RePrDL

� �13

ð8Þ

The equation is valid for fully developed laminar flow in tubes forfluids with RePr D

L

� �P 33:3.

Fig. 3 displays the comparison of Nusselt Number from Shahcorrelation and computed values from the present study for water,with Prandtl number of 6.77.

The figure shows that the numerical results of the present anal-ysis and the predictions of the Shah solution are in completeagreement.

Fig. 3. Comparison of Nu number from Shah formula and computed values for water.

Table 3Fully developed Nusselt number and heat transfer coefficient values obtained fromthe numerical solution for pure water and Al2O3/water nanofluid with differentparticle volume fractions for Re = 500 and test case A.

Fluid Nu Nu enhancement h h enhancement(Nunf/Nubf) (hnf/hbf)

Water 7.13 – 36.4436 –1% 7.17592 1.00644039 37.7208 1.0350462% 7.26524 1.01896774 39.2883 1.0780583% 7.39705 1.03745442 41.0994 1.1277544% 7.51121 1.05346564 42.9569 1.178723

82 A.A. Minea / International Journal of Heat and Mass Transfer 68 (2014) 78–84

3.2. Effect of uncertainties in thermal conductivity on heat transferenhancement

After confirming that the computational model is generatingcorrect results, nanofluids were analyzed at various Reynolds num-bers. The case study presents the hydrodynamic and thermalbehaviors of forced convective flow of a nanofluid inside a circulartube with two zones: an isothermal one and a constant heat fluxone. The nanofluid consists of Al2O3 nanoparticles with an averagediameter of 38 nm. The tube has a diameter of 0.12 m and a totallength of 8.76 m. The isothermal zone is at fluid entrance and isof 5.76 m, followed by the heating zone of 2.88 m. The fluid entersthe tube with a constant inlet temperature of 300 K and with uni-form axial velocity. The Reynolds number was varied from 500 to2300. The heat transfer performance of flowing nanofluids wasdefined in terms of the following convective heat transfercoefficient (h) and the Nusselt number (Nu):

h ¼ qTw � Tm

ð9Þ

Table 4Summary of heat transfer enhancement for the numerical tests.

Nanofluid volume fraction, % Heat transfer enhancement for case A

Re = 500 Re = 1000

0 1 11 1.035 1.0362 1.078 1.0803 1.128 1.1324 1.179 1.189

Heat transfer enhancement for case B0 1 11 1.042 1.0422 1.092 1.0933 1.150 1.1534 1.209 1.218

Heat transfer enhancement for case C0 1 11 1.034 1.0352 1.076 1.0793 1.127 1.1314 1.177 1.187

Nu ¼ hDknf¼ hð2RÞ

knfð10Þ

where q is the heat flux, Tw is the pipe wall temperature at a givenlocation along the pipe and Tm is the mean temperature in the pipeat the location where Tw is defined, D is the tube diameter, knf is thefluid thermal conductivity. For this study, the Nu number and the

Re = 1500 Re = 2000 Re = 2300

1 1 11.086 1.041 1.0401.133 1.082 1.0771.189 1.141 1.1251.248 1.199 1.182

1 1 11.093 1.048 1.0461.147 1.095 1.0901.211 1.162 1.1461.278 1.228 1.210

1 1 11.085 1.040 1.0391.132 1.081 1.0751.187 1.140 1.1241.246 1.198 1.180

Table 5Thermal conductivity enhancement depending on selected cases.

Nanofluid volume fraction, u [%] Thermal conductivity enhancement, kr

Case A Case B Case C

1 1.029 1.039 1.0282 1.059 1.079 1.0563 1.088 1.119 1.0864 1.120 1.161 1.117

Fig. 4. Variation of heat transfer coefficient with nanofluid volume fraction at different Re numbers.

Fig. 5. Variation of wall temperature on tube exit for case B.

A.A. Minea / International Journal of Heat and Mass Transfer 68 (2014) 78–84 83

convective heat transfer coefficient were evaluated at the tube exit.The mean exit temperature was calculated with:

Tmexit ¼R R

0 vTð2prÞdrR R0 vð2prÞdr

¼R R

0 vrTdrR R0 vrdr

ð11Þ

In Table 3, fully developed Nusselt numbers and associated heattransfer coefficients are listed for nanofluids with particle volumefractions ranging between 1.0% and 4.0% (Re = 500). When the tableis examined, it is seen that Nusselt number increases with the addi-tion of nanoparticles to the working fluid. The reason of this enhance-

84 A.A. Minea / International Journal of Heat and Mass Transfer 68 (2014) 78–84

ment can be the flattening in the radial temperature profile which isthe result of the variation of thermal dispersion in radial direction.

Table 4 is the summary of numerical results for heat transferenhancement for all three cases and for five Reynolds numbers.The heat transfer enhancement was estimated with:

hr ¼hnf

hbfð12Þ

Table 5 contains the thermal conductivity enhancement of nanofl-uids compared with the base fluid on considered cases. One cansee that the results for cases A and C are similar and case B is con-sidering a higher thermal conductivity enhancement. Moreover, if itcompares with Table 4 one can notice that the heat transferenhancement is higher than the conductivity enhancement.

Thermal conductivity enhancement was calculated as:

kr ¼knf

kbfð13Þ

If it looks at Tables 4 and 5 one can notice that the heat transfercoefficient and thermal conductivity are increasing with the nano-fluid volume fraction. Moreover, a thermal conductivity enhance-ment of 2.9–16.1% can be observed along with a heat transferenhancement that varies between 3.4–27.8%.

Fig. 4 illustrates the development of the fluid heat transfer coef-ficient profile comparing the three considered cases for two Rey-nolds numbers. The profiles shown clearly indicate the discusseddifferences in fluid thermal performances and the similarities be-tween cases A and C.

As one can see from Fig. 4, case B offers better thermal perfor-mances for both illustrated Re numbers and the increase of Renumber and/or nanofluid volume fraction generates a higher heattransfer coefficient.

As with the wall temperature on tube exit, Fig. 5 shows compar-atively the nanofluids for the case B. As is clearly seen, the walltemperature on exit obtained in this case is considerably loweras the volume fraction and Re number increases.

Although the use of nanoparticles in traditional cooling fluidscan be expected to increase the shear stresses and pressure lossesinside any cooling application [26], it is believed that the heattransfer benefits of such fluids in certain engineering applicationswill outweigh this side-effect, especially in micro-sized applica-tions in which high heat transfer, low temperature tolerancesand small component size are required.

4. Conclusion

In this paper, the effect due to the uncertainty in the values ofthe physical properties of water–cAl2O3 nanofluid on their thermo-hydraulic performance for laminar fully developed forced convec-tion in a two zones tube was investigated. In order todemonstrate the validity and also precision of the model and thenumerical procedure, comparison with the previously publishedtraditional expressions has been done. Nusselt numbers from thepresent numerical analysis for forced convection flow are com-pared with the equations given by Shah formula.

In addition this article clearly presents that the nanoparticlessuspended in water enhance the convective heat transfer coeffi-cient in the thermally fully developed regime, despite low volumefraction between 1% and 4%. Analyzing the three cases presented inTable 2 one can notice that case B have the most increased valuesfor heat transfer coefficient, compared with cases A and C that offersimilar results. In particular, the heat transfer coefficient of water-based Al2O3 nanofluids is increased by 27.8% at case B and 4 vol%under the fixed Reynolds number (Re = 1500) compared with thatof pure water. The smallest increase in heat transfer coefficient

(3.4%) was noticed for case C at Re = 500 for 1% volume fraction.Also, the convective heat transfer coefficient of water-basedAl2O3 nanofluids is increased with volume fraction of Al2O3 nano-particles and the enhancement of the heat transfer coefficient islarger than that of the effective thermal conductivity at the samevolume concentration for all considered cases.

Finally, by analyzing a classical problems of replacement of asimple fluid by a nanofluid in a given installation, it have illus-trated that the heat transfer efficiency vary significantly with thethermal conductivity of the nanofluid. Since the effects of certainnanofluid characteristics (such as average particle size and spatialdistribution of nanoparticles) on this property is not presentlyknown precisely, it is quite difficult to conclude on the presumedadvantages of nanofluids over conventional heat transfer fluids.

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