+Comparative-review-of-turbulent-heat-transfer-of-nanofluids_2012_International-Journal-of-Heat-and-Mass-Transfer

International Journal of Heat and Mass Transfer 55 (2012) 5380–5396

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International Journal of Heat and Mass Transfer

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

Comparative review of turbulent heat transfer of nanofluids

Wenhua Yu a,⇑, David M. France b, Elena V. Timofeeva a, Dileep Singh c, Jules L. Routbort a

a Energy Systems Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USAb Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 West Taylor Street (m/c 251), Chicago, IL 60607, USAc Nuclear Engineering Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA

a r t i c l e i n f o

Article history:Received 23 November 2011Received in revised form 4 June 2012Accepted 11 June 2012Available online 4 July 2012

Keywords:NanofluidConvective heat transferTurbulent flowHeat transfer comparisonHeat transfer predictionHeat transfer enhancement

0017-9310/$ - see front matter � 2012 Published byhttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.06

⇑ Corresponding author. Tel.: +1 630 252 7361; faxE-mail addresses: [email protected] (W. Yu), dfrance@u

a b s t r a c t

Three essential aspects of the turbulent-flow, convective heat transfer of nanofluids relevant to theirapplications are comparatively reviewed in detail based on both theoretical analyses and experimentaldata. These aspects are: (a) selection – the comparison criteria of the thermophysical property-relatedheat transfer performance of nanofluids and their base fluids, (b) design – the predictions of the heattransfer coefficients of nanofluids based on homogeneous fluid models by using nanofluid effective ther-mophysical properties, and (c) effectiveness – the enhancements of the heat transfer coefficients of nano-fluids over their base fluids. This review, including research from the inception of nanofluids to date,quantifies the accuracy of bases for future nanofluid evaluation.

� 2012 Published by Elsevier Ltd.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53812. Effective thermophysical properties of nanofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53813. Selection: comparison criteria of nanofluid heat transfer performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5382

3.1. Comparison bases: Reynolds number, flow velocity, and pumping power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53823.2. Figures of merit: Nusselt number, heat transfer coefficient, and Mouromtseff number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53833.3. Pumping power effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53833.4. Mass-specific heat capacity effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53843.5. Temperature effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53843.6. Thermophysical property-based figure-of-merit ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53843.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5384

4. Design: predictions of nanofluid heat transfer coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5386
4.1. Experimental database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53864.2. Component thermophysical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53864.3. Heat transfer coefficient comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53864.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5389
5. Effectiveness: enhancements of nanofluid heat transfer coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5390
5.1. Base fluid heat transfer coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53925.2. Nanofluid heat transfer coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53935.3. Heat transfer coefficient enhancement comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53935.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5394
6. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5394Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5395References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5395

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: +1 630 252 5568.ic.edu (D.M. France), [email protected] (E.V. Timofeeva), [email protected] (D. Singh), [email protected] (J.L. Routbort).

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Nomenclature

cp mass-specific heat capacityd diameterf Fanning friction factorf⁄ resistance coefficientFOM figure of merith heat transfer coefficientk thermal conductivityL lengthm massMo Mouromtseff NumberNu Nusselt numberP pumping powerPr Prandtl numberq heat_q heat flow rater figure-of-merit ratioRe Reynolds number

T temperatureu velocityV volumevp particle volume concentration

Greek symbolsl dynamic viscosityq density

Subscriptse nanofluidf fluidin inletm base fluid matrixout outletp particlew wall

1.1

1.2

1.3

1.4

1.5

Den

sity

ratio

ρe/ ρ

m

ρp/ρ

m=2

ρp/ρ

m=4

ρp/ρ

m=8

W. Yu et al. / International Journal of Heat and Mass Transfer 55 (2012) 5380–5396 5381

1. Introduction

Nanofluids have become increasingly closer to an engineeringreality starting from their initial vision originated more than adecade ago [1]. This trend is clearly reflected and well documentedin numerous original research papers and in extensive state-of-the-art review articles in the engineering literature that cover awide range of nanofluid-related research and development includ-ing nanofluid preparation and characterization [2–22], nanofluideffective thermophysical property investigation [2–49], nanofluidheat transfer performance [4–6,9,10,13–15,17–19,22,24,28,29,31,32,35,36,38–40,46–48,50–55], and nanofluid application explora-tion [5,8,13–15,19,22,29,40,46,56,57].

Accordingly, the main focus of nanofluid research has graduallyshifted from effective thermophysical properties to heat transfercoefficients, an area connected more directly to nanofluid applica-tions. This trend is especially true for one of the most commonnanofluid cooling and heating applications: the turbulent-flow,convective heat transfer of nanofluids. However, while a greatnumber of original research papers concerning nanofluid heattransfer are available in the engineering literature, confusion existsin the nanofluid research community regarding appropriate nano-fluid heat transfer coefficient comparison, nanofluid heat transfercoefficient prediction, and nanofluid heat transfer coefficientenhancement. It is the purpose of the present study to compara-tively review three essential aspects of the turbulent-flow, convec-tive heat transfer of nanofluids relevant to their applications basedon both theoretical analyses and experimental data including (a)selection – the comparison criteria of the thermophysicalproperty-related heat transfer performance of nanofluids and theirbase fluids, (b) design – the predictions of the heat transfer coeffi-cients of nanofluids based on homogeneous fluid models by usingnanofluid effective thermophysical properties, and (c) effective-ness – the enhancements of the heat transfer coefficients of nano-fluids over their base fluids. It is expected that this in-depth reviewwill provide a framework for assessing the current status and forspecifying the future direction of convective heat transferresearches and applications of nanofluids in turbulent flow.

1.00 1 2 3 4 5

Particle volume concentration vp (%)

Fig. 1. Density ratio as a function of particle volume concentration.

2. Effective thermophysical properties of nanofluids

Because of their essential roles in forming the basis for analyz-ing the turbulent-flow, convective heat transfer of nanofluids, a

brief discussion of nanofluid effective thermophysical propertiessuch as density, mass-specific heat capacity, dynamic viscosity,and thermal conductivity is appropriate before the main resultsof this study are presented.

For a nanofluid denoted by the subscript e consisting of parti-cles denoted by the subscript p suspended in a base fluid denotedby the subscript m with a mass of me = mm + mp and a volume ofVe = Vm + Vp, the effective density qe can be calculated from thecomponent densities and the particle volume concentration basedon the physical principle of the mixture rule related to itsdefinition

qe ¼me

Ve¼ mm þmp

Ve¼

Vmqm þ Vpqp

Ve¼ ð1� vpÞqm þ vpqp ð1Þ

where vp = Vp/Ve is the particle volume concentration. Fig. 1 showsthe nanofluid-to-base fluid density ratio qe/qm as a function of theparticle volume concentration vp for three particle-to-base fluiddensity ratios of qp/qm = 2, 4, and 8. It can be seen from Fig. 1 thatthe nanofluid-to-base fluid density ratio increases substantially atlower particle volume concentrations for larger values of qp/qm.

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0 1 2 3 4 5

Mas

s-sp

ecifi

c he

at c

apac

ity ra

tio c

pe/c

pm

Particle volume concentration vp (%)

ρp/ρ

m=2

ρp/ρ

m=4

ρp/ρ

m=8

cpp

/cpm

=0.2

cpp

/cpm

=0.3

cpp

/cpm

=0.1

Fig. 2. Mass-specific heat capacity ratio as a function of particle volumeconcentration.

5382 W. Yu et al. / International Journal of Heat and Mass Transfer 55 (2012) 5380–5396

Similarly, for a thermal equilibrium nanofluid with a tempera-ture change of DT = DTm = DTp = DTe under a heat input ofqe = qm + qp, the effective mass-specific heat capacity cpe can be cal-culated from the component densities, the component mass-spe-cific heat capacities, and the particle volume concentration basedon the physical principle of the mixture rule related to itsdefinition

cpe ¼qe

meDT¼ mmcpm þmpcpp

mm þmp¼

Vmqmcpm þ Vpqpcpp

Vmqm þ Vpqp

¼ð1� vpÞqmcpm þ vpqpcpp

ð1� vpÞqm þ vpqpð2Þ

Fig. 2 shows the nanofluid-to-base fluid mass-specific heat capacityratio cpe/cpm as a function of the particle volume concentration vp forvarious particle-to-base fluid density ratios and particle-to-basefluid mass-specific heat capacity ratios. It can be seen in Fig. 2 thatcpe/cpm is considerably more sensitive to the density ratio qp/qm

than to the specific heat ratio cpp/cpm in the range of interest.The situations are more complicated for the effective dynamic

viscosity and the effective thermal conductivity. In addition tothe particle volume concentration and the corresponding compo-nent properties such as the base fluid dynamic viscosity, the basefluid thermal conductivity, and the particle thermal conductivity,the effective dynamic viscosity and the effective thermal conduc-tivity also depend on many other component- and application-re-lated nanofluid parameters. These parameters include the particlesize and distribution, the particle shape and orientation, the parti-cle aggregation and cluster, the particle Brownian motion, the basefluid-particle interfacial layer and nature, the nanofluid pH andadditive, and the nanofluid temperature. To take the effects ofthese influence parameters into account, many physicalmechanisms and mathematical models have been proposed basedon theoretical analyses and experimental data for the dynamic vis-cosity and for the effective thermal conductivity [10,16,27,33,35,37,38,42,44]. For the effective dynamic viscosity le, many ofthe developed mathematical equations can generally be viewedas a modification to the Einstein equation [58]

le ¼ ð1þ 2:5vpÞlm ð3Þ

which is a first-order approximation as a function of the base fluiddynamic viscosity lm and the particle volume concentration vp ob-tained by assuming that the disturbance of the flow pattern of thebase fluid caused by a given particle does not overlap with the flow

disturbance caused by the presence of a second suspended particle.For the effective thermal conductivity ke, many of the developedmathematical equations can generally be viewed as a modificationto the Maxwell equation [59]

ke ¼ km þ 3vpkp � km

2km þ kp � vpðkp � kmÞkm ð4Þ

which is a first-order approximation, based on the effective mediumtheory, as a function of the base fluid thermal conductivity km, theparticle thermal conductivity kp, and the particle volume concentra-tion vp. However, while they result in improvements to the abovefirst-order approximate equations for predicting the effective dy-namic viscosity and the effective thermal conductivity of nanofl-uids, many of these developed mathematical models are of anempirical nature with the empirical parameters to be determinedfrom the experimental data and therefore do not predict nor explainall the experimental data well. Comprehensive physics-based math-ematical models, which include the major influence parameters andcan predict experimental data well, have yet to be developed. Cur-rently, the most accurate and reliable approach for obtaining theeffective dynamic viscosity and the effective thermal conductivityof a nanofluid is through experimental measurements conductedunder the same conditions as its practical applications.

3. Selection: comparison criteria of nanofluid heat transferperformance

In selecting a nanofluid for a specific application, it is necessaryto analyze the forced single-phase convective heat transfer perfor-mance of a nanofluid relative to its base fluid. For this purpose, twoprimary factors must be chosen: figures of merit and comparisonbases. Figures of merit are quantities used to characterize theforced single-phase convective heat transfer performance and todetermine the utility of a nanofluid relative to its base fluid whilecomparison bases specify the conditions under which figures ofmerit are evaluated.

3.1. Comparison bases: Reynolds number, flow velocity, and pumpingpower

Three bases, Reynolds number, flow velocity, and pumpingpower have been used frequently in the engineering literaturefor comparing the forced single-phase convective heat transferperformance of a nanofluid relative to its base fluid. One of themost important focuses of heat transfer research is to develop pre-diction equations for the heat transfer coefficients based on theo-retical analyses or experimental data or both. In order for thoseequations to be used under various configuration and applicationconditions, non-dimensional parameters are generally used forrepresenting the heat transfer coefficients, the fluid thermophysi-cal properties, the geometric configurations, etc. For forced sin-gle-phase convective heat transfer, the non-dimensionalparameters include the Nusselt number Nu, the Prandtl numberPr, the Reynolds number Re, etc.

Due to this common practice in the thermal science and heattransfer research community, the Reynolds number is a naturalchoice of the comparison basis for comparing the forced single-phase convective heat transfer performance of a nanofluid relativeto its base fluid as witnessed by the large percentage of researchpapers in the engineering literature. Unfortunately, while it is com-monly used for the dimensional analysis of fluid dynamics, for thedetermination of dynamic similitude between different experi-mental cases, and for the characterization of different flow re-gimes, the Reynolds number is an inappropriate basis forcomparing the forced single-phase convective heat transfer perfor-


mance of a nanofluid relative to its base fluid for the following rea-sons [60]. First, using the constant Reynolds number comparisonbasis requires that the flow velocity for the more viscous nanofluidbe higher than that for the less viscous base fluid, which meansthat the net result for the constant Reynolds number comparisonis a combination of the nanofluid thermophysical property effectand the flow velocity effect. Second, in contradiction to commonsense, the constant Reynolds number comparison indicates thatthe nanofluid convective heat transfer coefficient enhancement in-creases as its dynamic viscosity increases, the reason for which isagain rooted in the flow velocity effect. Third, the constant Rey-nolds number comparison completely ignores the pumping powerpenalty associated with pumping the more viscous nanofluid at ahigher flow velocity to reach the same Reynolds number as the lessviscous base fluid. Therefore, the constant Reynolds number basisshould not be used for comparing the forced single-phase convec-tive heat transfer performance of a nanofluid relative to its basefluid because it distorts the physical situation. If increased pump-ing power is required for the nanofluid (to maintain the constantReynolds number with the base fluid), it must also be availableto pump the base fluid. Then, at increased Reynolds number, thebase fluid often outperforms the nanofluid although a constantReynolds number comparison would erroneously show otherwise.

For the constant flow velocity comparison, the flow velocities ofthe nanofluid and the base fluid are the same. The constant flowvelocity comparison eliminates the flow velocity effect that isinherently present in the constant Reynolds number comparisonand therefore is an appropriate basis for comparing the forced sin-gle-phase convective heat transfer performance of a nanofluid rel-ative to its base fluid. It should be noted that, under the constantflow velocity comparison, the pumping power for the more viscousnanofluid is still higher than that for the less viscous base fluid.However, from a practical application viewpoint, when the flowsystem under consideration is designed for a certain maximumflow rate and when the flow pumping power is only a small portionof the total power consumed, the constant flow velocity compari-son provides a quite accurate picture of the forced single-phaseconvective heat transfer performance of a nanofluid relative to itsbase fluid [60]. In many applications the pumping power differencebetween a nanofluid and its base fluid is small at the constantvelocity.

For the constant pumping power comparison, the forced single-phase convective heat transfer performance of a nanofluid is com-pared to that of its base fluid under the condition that the nano-fluid and the base fluid use the same amount of pumping power,which means that the flow velocity for the more viscous nanofluidis slightly lower than that for the less viscous base fluid. If a certainamount of energy is available to pump a nanofluid, it must be con-sidered to be available to pump the base fluid as well. Therefore,while it gives the most conservative criterion for a beneficial nano-fluid, the constant pumping power, as the logical choice of thecomparison basis, represents the most unambiguous comparisonbasis and should be used for comparing the forced single-phaseconvective heat transfer performance of a nanofluid relative to itsbase fluid whenever the necessary information is available [60].

3.2. Figures of merit: Nusselt number, heat transfer coefficient, andMouromtseff number

Simple figures of merit for comparing the forced single-phaseconvective heat transfer performance of a nanofluid relative to itsbase fluid include the Nusselt number, the heat transfer coefficient,and the Mouromtseff number.

For the same reason as the Reynolds number being used as thecomparison basis, the Nusselt number is a natural choice of thesimple figure of merit for comparing the forced single-phase con-

vective heat transfer performance of a nanofluid relative to its basefluid. However, the Nusselt number is an inappropriate figure ofmerit for comparing the forced single-phase convective heat trans-fer performance of two or more different fluids. According to itsdefinition, the ratio of the convective heat transfer coefficient tothe conductive heat transfer coefficient of a fluid, the Nusselt num-ber is very useful for analyzing the heat transfer mechanisms andfor correlating forced single-phase convective heat transfer coeffi-cients. When comparing the forced single-phase convective heattransfer performance of two or more fluids with different thermalconductivities, it is the forced single-phase convective heat transfercoefficient but not the convective-to-conductive heat transfer coef-ficient ratio that is important. Using the Nusselt number for com-paring forced single-phase convective heat transfer performanceof different fluids gives misleading information. For example alow Nusselt number might be caused by a high thermal conductiv-ity rather than a low forced single-phase convective heat transfercoefficient.

From the viewpoint of practical applications, the heat transfercoefficient h is a direct expression of the heat transfer capabilityof a fluid in forced single-phase convective heat transfer. It is anappropriate simple figure of merit for comparing the forced sin-gle-phase convective heat transfer performance of a nanofluid rel-ative to its base fluid.

In additional to the forced single-phase convective heat transfercoefficient, which can be used for the comparison of both theoret-ical predictions and experimental data of different fluids, there isanother simple figure of merit, called the Mouromtseff Number,which is primarily used for the comparison of theoretical predic-tions of different fluids. The dimensional Mouromtseff NumberMo, introduced in 1942 by Mouromtseff [61] to represent the effecton the forced single-phase convective heat transfer coefficient h ofthe combined fluid thermophysical properties of density q, mass-specific heat capacity cp, dynamic viscosity l, and thermal conduc-tivity k, essentially is an alternative expression of the forced single-phase convective heat transfer coefficient. It is defined as

Mo ¼ qa1 ca2p la3 ka4 ð5Þ

where the exponents a1, a2, a3, and a4 are determined from a heattransfer coefficient equation for the appropriate heat transfer mode.The ratio of Mouromtseff Numbers for a nanofluid to its base fluid isequivalent to the ratio of their heat transfer coefficients at the con-stant velocity. As discussed above, this comparison is quite accuratefor nanofluids.

3.3. Pumping power effects

The concept of the pumping power increase is often consideredas a negative effect in contrast with the concept of the heat transfercoefficient enhancement as a positive effect for comparing theforced single-phase convective heat transfer performance of differ-ent fluids. Therefore, the heat transfer coefficient h can be com-bined with the pumping power P to form a combined figure ofmerit, which is usually used in terms of the ratio of the heat trans-fer coefficient enhancement to the pumping power increase ratherthan the ratio of the heat transfer coefficient to the pumping power[62]. For the comparison between a nanofluid denoted by the sub-script e and its base fluid denoted by the subscript m, this com-bined figure of merit can be expressed as

FOM ¼ ðhe=hmÞ=ðPe=PmÞ ð6Þ

which reduces to the heat transfer coefficient ratio as a simple fig-ure of merit under the condition of the constant pumping powercomparison basis. A higher figure of merit value indicates more ofa gain in the heat transfer enhancement compared to the pumping


power increase. It should be noted that, in practical applications,the pumping power effect may not have the simple relationshipto the heat transfer coefficient effect as suggested in the above fig-ure of merit and therefore can be better taken into account throughthe constant pumping power being used as the comparison basis.

3.4. Mass-specific heat capacity effects

For comparing the forced single-phase convective heat transferperformance of different fluids, another very important thermo-physical property is the capacity to store thermal energy, whichis usually described by the mass-specific heat capacity. A combinedfigure of merit was suggested by Yeh and Chu [63] to capture thecombination effect of the fluid thermal energy storage capacitycharacterized by the mass-specific heat capacity cp, the heat trans-fer capability characterized by the heat transfer coefficient h, andthe hydraulic performance characterized by the pumping power P

FOM ¼ cph=P ð7Þ

It should be noted that (a) the way that the above figure of meritcombines the three primary parameters is arguable; (b) the mass-specific heat capacity difference between a nanofluid and its basefluid depends on the component thermophysical properties andthe particle volume concentration of the nanofluid and generallyis not very large; and (c) as previously pointed out, the pumpingpower effect can be better taken into account through the constantpumping power being used as the comparison basis.

3.5. Temperature effects

The ultimate cooling or heating effects of a fluid can be assessedby wall or fluid temperature changes. Moreover, practical heattransfer systems are often confined to certain wall or fluid temper-ature limits. Therefore, for heat transfer system design purposes,those temperature requirements naturally form comparison bases,and, under these comparisons, the pumping power is usually em-ployed as the figure of merit. Two commonly-used temperature-related comparison bases are: the constant fluid temperaturechange Tfin � Tfout for cooling with the wall cooler than the bulkfluid or Tfout � Tfin for heating with the wall hotter than the bulkfluid and the constant wall-fluid temperature difference Tf � Tw

for cooling or Tw � Tf for heating [64]. The pumping power as thefigure of merit under the constant fluid temperature change is usedunder the condition of the fluid temperature change much largerthan the wall-fluid temperature difference and involves the effectof the fluid mass-specific heat capacity; the pumping power as afigure of merit under the constant wall-fluid temperature differ-ence is used under the condition of the wall-fluid temperature dif-ference much larger than the fluid temperature change and isequivalent to comparing the pumping power under the constantheat transfer coefficient basis. Both comparisons are dependenton the heat transfer mode and the corresponding heat transferequation.

3.6. Thermophysical property-based figure-of-merit ratios

Considering one of the most commonly-encountered engineer-ing heat transfer problems – a forced single-phase internal flowthrough a circular tube with the flow channel diameter of d andthe flow channel length of L, one can derive the mathematicalexpressions of the figures of merit discussed above. One set ofthese figures of merit can be obtained as functions of the thermo-physical properties for hydrodynamically and thermally fully-developed turbulent flow under the presumptions that the heattransfer coefficient h is estimated from the Dittus–Boelter equationas [65]

h ¼ 0:023Re4=5Pr3=10ðk=dÞ ð8aÞ

for cooling or

h ¼ 0:023Re4=5Pr2=5ðk=dÞ ð8bÞ

for heating, and that the Fanning friction factor f necessary forpumping power calculations is estimated from the classical equa-tion as

f ¼ 0:046Re�1=5 ð9aÞ

or from the Blasius equation as [66]

f ¼ 0:0791Re�1=4 ð9bÞ

The figure-of-merit ratios, thus obtained, for a nanofluid to its basedfluid under various comparison bases are listed in Table 1 wherer > 1 indicates that the nanofluid has better heat transfer perfor-mance than the base fluid, and the larger the figure-of-merit ratior is, the better the nanofluid performs compared to its base fluid.

Several characteristics can be seen from Table 1 about figure-of-merit ratios for a nanofluid and its based fluid in fully-developedinternal turbulent flow: (a) for the constant flow velocity compar-ison with nanofluids generally having Pe/Pm > 1 and cpe/cpm < 1, onewould always have r1 > r2 > r3, r9 > r10 > r11, r17 > r18 > r19, andr25 > r26 > r27; (b) for the constant pumping power comparisonwith Pe/Pm = 1 and cpe/cpm < 1 for most nanofluids, one would haver4 = r5 > r6, r12 = r13 > r14, r20 = r21 > r22, and r28 = r29 > r30; (c)because the heat transfer coefficient ratio for the constant flowvelocity comparison is unrelated to the Fanning friction factor,one would always have r1 = r9 and r17 = r25; (d) the differences ofthe figure-of-merit ratios between cooling and heating andbetween the Fanning friction factor being estimated from theclassical equation or from the Blasius equation are insignificantfor the same comparison basis, which can be verified by comparingthe exponents of their property ratios; and (e) the figure-of-merit

ratios r7 = r23 = (qe/qm)2(cpe/cpm)14/5(le/lm)�1/5 and r15 = r31 = (qe/

qm)2(cpe/cpm)11/4(le/lm) �1/4 for the constant fluid temperature

change comparison are unrelated to the thermal conductivity ratioand are generally less than unity, which means that this compari-son basis is unsuitable for comparing the heat transfer perfor-mance of a nanofluid to its base fluid.

The density ratio qe/qm and the mass-specific heat capacityratio cpe/cpm, as pointed out previously, can be theoretically calcu-lated for a certain nanofluid, and therefore the figure-of-merit ra-tios in Table 1 can be translated into relationships between thedynamic viscosity ratio le/lm and the thermal conductivity ratioke/km by letting r = 1 for various comparison bases. Some of thecommonly-used figure-of-merit ratios are shown for le/lm andke/km relationships for both cooling and heating applications inFig. 3 where the curves represent the criteria r = 1. The regionson the left of the curves (above the curves) in Fig. 3 representthe improved heat transfer performance for a nanofluid, and theregions on the right of the curves (below the curves) representthe deteriorated heat transfer performance for a nanofluid. Oneof the interesting features in Fig. 3 is that, on the le/lm and ke/km

plane, the criteria of r = 1 for the heat transfer coefficient ratio un-der the constant pumping power comparison and for the pumpingpower ratio under the constant heat transfer coefficient compari-son are the same curve, which implies that they are interchange-able for judging a beneficial nanofluid. In fact, one has thefollowing relationships: r7

4 ¼ r28, r55

12 ¼ r1616, r7

20 ¼ r224, and r55

28 ¼ r1632.

3.7. Summary

The mass-specific heat capacity difference between a nanofluidand its base fluid usually is not very large due to its low particle

Table 1Figure-of-merit ratios for nanofluids in fully-developed internal turbulent flow.

Flow Comparison basis Figure-of-merit ratio

Fully-developed internal turbulent flow for cooling withthe classical equation for the Fanning friction factor

h¼ 0:023Re4=5Pr3=10ðk=dÞ

¼ 0:023u4=5

d1=5

q4=5c3=10p k7=10

l1=2

¼ 0:0487P2=7

d3=7L2=7

q4=7c3=10p k7=10

l39=70

f ¼ 0:046Re�1=5

P ¼ ðp=2ÞdLu3qf

¼ 0:0723d4=5Lu14=5q4=5l1=5

¼ 0:1421L _q14=5

d24=5ðTfin � TfoutÞ14=5

l1=5

q2c14=5p

¼ 39159d3=2Lh7=2 l39=20

q2c21=20p k49=20

Flow velocity ur1 ¼ he

hm¼ Moe

Mom¼ qe

qm

� �4=5 cpe

cpm

� �3=10 lelm

� ��1=2kekm

� �7=10

Flow velocity ur2 ¼ he=hm

Pe=Pm¼ cpe

cpm

� �3=10 lelm

� ��7=10kekm

� �7=10

Flow velocity ur3 ¼ cpe he=Pe

cpmhm=Pm¼ cpe

cpm

� �13=10 lelm

� ��7=10kekm

� �7=10

Pump power Pr4 ¼ he

hm¼ qe

qm

� �4=7 cpe

cpm

� �3=10 lelm

� ��39=70kekm

� �7=10

Pump power Pr5 ¼ he=hm

Pe=Pm¼ he

hm¼ qe

qm

� �4=7 cpe

cpm

� �3=10 lelm

� ��39=70kekm

� �7=10

Pump power Pr6 ¼ cpe he=Pe

cpmhm=Pm¼ cpehe

cpmhm¼ qe

qm

� �4=7 cpe

cpm

� �13=10 lelm

� ��39=70kekm

� �7=10

Temperature change Tfin � Tfout r7 ¼ PmPe¼ qe

qm

� �2 cpe

cpm

� �14=5 lelm

� ��1=5[64]

Temperature difference Tf � Tw r8 ¼ PmPe¼ qe

qm

� �2 cpe

cpm

� �21=20 lelm

� ��39=20kekm

� �49=20

Fully-developed internal turbulent flow for cooling with theBlasius equation for the Fanning friction factor

h ¼ 0:023Re4=5Pr3=10ðk=dÞ

¼ 0:023u4=5

d1=5

q4=5c3=10p k7=10

l1=2

¼ 0:0422P16=55

d23=55L16=55

q32=55c3=10p k7=10

l63=110

f ¼ 0:0791Re�1=4

P ¼ ðp=2ÞdLu3qf

¼ 0:1242d3=4Lu11=4q3=4l1=4

¼ 0:2414L _q11=4

d19=4ðTfin � TfoutÞ11=4

l1=4

q2c11=4p

¼ 53193d23=16Lh55=16 l63=32

q2c33=32p k77=32


hm¼ Moe

Mom¼ qe

qm

� �4=5 cpe

cpm

� �3=10 lelm

� ��1=2kekm

� �7=10


Pe=Pm¼ qe

qm

� �1=20 cpe

cpm

� �3=10 lelm

� ��3=4kekm

� �7=10

Flow velocity ur11 ¼ cpe he=Pe

cpm hm=Pm¼ qe

qm

� �1=20 cpe

cpm

� �13=10 lelm

� ��3=4kekm

� �7=10

Pump power Pr12 ¼ he

hm¼ qe

qm

� �32=55 cpecpm

� �3=10 lelm

� ��63=110kekm

� �7=10

Pump power Pr13 ¼ he=hm

Pe=Pm¼ he

hm¼ qe

qm

� �32=55 cpe

cpm

� �3=10 lelm

� ��63=110kekm

� �7=10

Pump power Pr14 ¼ cpe he=Pe

cpm hm=Pm¼ cpehe

cpmhm¼ qe

qm

� �32=55 cpe

cpm

� �13=10 lelm

� ��63=110kekm

� �7=10

Temperature change Tfin � Tfout r15 ¼ PmPe¼ qe

qm

� �2 cpecpm

� �11=4 lelm

� ��1=4

Temperature difference Tf � Tw r16 ¼ PmPe¼ qe

qm

� �2 cpe

cpm

� �33=32 lelm

� ��63=32kekm

� �77=32

Fully-developed internal turbulent flow for heating with theclassical equation for the Fanning friction factor

h ¼ 0:023Re4=5Pr2=5ðk=dÞ

¼ 0:023u4=5

d1=5

q4=5c2=5p k3=5

l2=5

¼ 0:0487P2=7

d3=7L2=7

q4=7c2=5p k3=5

l16=35

f ¼ 0:046Re�1=5

P ¼ ðp=2ÞdLu3qf

¼ 0:0723d4=5Lu14=5q4=5l1=5

¼ 0:1421L _q14=5

d24=5ðTfout � TfinÞ14=5

l1=5

q2c14=5p

¼ 39159d3=2Lh7=2 l8=5

q2c7=5p k21=10


hm¼ Moe

Mom¼ qe

qm

� �4=5 cpe

cpm

� �2=5 lelm

� ��2=5kekm

� �3=5


Pe=Pm¼ cpe

cpm

� �2=5 lelm

� ��3=5kekm

� �3=5

Flow velocity ur19 ¼ hecpe=Pe

hmcpm=Pm¼ cpe

cpm

� �7=5 lelm

� ��3=5kekm

� �3=5

Pumping power Pr20 ¼ he

hm¼ qe

qm

� �4=7 cpe

cpm

� �2=5 lelm

� ��16=35kekm

� �3=5

Pumping power Pr21 ¼ he=hm

Pe=Pm¼ he

hm¼ qe

qm

� �4=7 cpe

cpm

� �2=5 lelm

� ��16=35kekm

� �3=5

Pumping power Pr22 ¼ hecpe=Pe

hmcpm=Pm¼ cpehe

cpmhm¼ qe

qm

� �4=7 cpe

cpm

� �7=5 lelm

� ��16=35kekm

� �3=5

Temperature change Tfout � Tfin r23 ¼ PmPe¼ qe

qm

� �2 cpe

cpm

� �14=5 lelm

� ��1=5

Temperature difference Tw � Tf r24 ¼ PmPe¼ qe

qm

� �2 cpe

cpm

� �7=5 lelm

� ��8=5kekm

� �21=10

Fully-developed internal turbulent flow for heating with theBlasius equation for the Fanning friction factor

h ¼ 0:023Re4=5Pr2=5ðk=dÞ

¼ 0:023u4=5

d1=5

q4=5c2=5p k3=5

l2=5

¼ 0:0422P16=55

d23=55L16=55

q32=55c2=5p k3=5

l26=55

f ¼ 0:0791Re�1=4

P ¼ ðp=2ÞdLu3qf

¼ 0:1242d3=4Lu11=4q3=4l1=4

¼ 0:2414L _q11=4

d19=4ðTfout � TfinÞ11=4

l1=4

q2c11=4p

¼ 53193d23=16Lh55=16 l13=8

q2c11=8p k33=16


hm¼ Moe

Mom¼ qe

qm

� �4=5 cpe

cpm

� �2=5 lelm

� ��2=5kekm

� �3=5[60]


Pe=Pm¼ qe

qm

� �1=20 cpe

cpm

� �2=5 lelm

� ��13=20kekm

� �3=5

Flow velocity ur27 ¼ hecpe=Pe

hmcpm=Pm¼ qe

qm

� �1=20 cpe

cpm

� �7=5 lelm

� ��13=20kekm

� �3=5

Pumping power Pr28 ¼ he

hm¼ qe

qm

� �32=55 cpe

cpm

� �2=5 lelm

� ��26=55kekm

� �3=5[60]

Pumping power Pr29 ¼ he=hm

Pe=Pm¼ he

hm¼ qe

qm

� �32=55 cpe

cpm

� �2=5 lelm

� ��26=55kekm

� �3=5

Pumping power Pr30 ¼ he cpe=Pe

hmcpm=Pm¼ cpe he

cpm hm¼ qe

qm

� �32=55 cpecpm

� �7=5 lelm

� ��26=55kekm

� �3=5

Temperature change Tfout � Tfin r31 ¼ PmPe¼ qe

qm

� �2 cpe

cpm

� �11=4 lelm

� ��1=4

Temperature difference Tw � Tf r32 ¼ PmPe¼ qe

qm

� �2 cpe

cpm

� �11=8 lelm

� ��13=8kekm

� �33=16


1.0

1.2

1.4

1.6

1.8

2.0

1.0 1.5 2.0 2.5 3.0

Ther

mal

con

duct

ivity

ratio

ke/ k

m

Dynamic viscosity ratio μe/μ

m

r >1

r<1

r1=r

9=1

r4=1 & r

8=1

r12

=1 & r16

=1(a) cooling

1.0

1.2

1.4

1.6

1.8

2.0

1.0 1.5 2.0 2.5 3.0

Ther

mal

con

duct

ivity

ratio

ke/ k

m

Dynamic viscosity ratio μe/μ

m

r >1

r<1

r17

=r25

=1

r20

=1 & r24

=1

r28

=1 & r32

=1(b) heating

Fig. 3. Criteria for a beneficial nanofluid (an alumina-in-water nanofluid at 25 �C).


volume concentration, and therefore the comparison criterionchoice of the forced single-phase convective heat transfer of ananofluid and its base fluids can generally be viewed as a balancebetween its heat transfer enhancement and its pumping power in-crease. While all of the comparison criteria discussed have beenadopted to some extent by nanofluid research groups worldwide,these comparison criteria can basically be classified into two maincategories based on the following principles: to achieve the highestheat transfer coefficient under the condition of the constant pump-ing power or to reach the lowest pumping power under the condi-tion of the constant heat transfer coefficient.

4. Design: predictions of nanofluid heat transfer coefficients

The ability to predict the heat transfer coefficient of a fluid isessential for successfully designing heat transfer systems usingthe fluid. As a suspension of a base fluid and particles, a nanofluidis a two-phase and sometime multi-phase mixture. As an alterna-tive to a rigorous mathematical treatment of this complicated flowregime, engineering researchers of the nanofluid community havebeen adopting the approach that treats a nanofluid as a homoge-neous fluid of effective thermophysical properties and predictsits heat transfer coefficients with standard single-phase equationsfor pure fluids. The figure-of-merit ratio criteria presented in Table

1 are also based on this approach by using the Dittus–Boelter equa-tion for its simplicity. In order to assess the accuracy of this ap-proach, comparisons were made with experimental data of theturbulent-flow convective heat transfer coefficients of nanofluidsfound in the engineering literature.

4.1. Experimental database

While a great number of experimental data of the turbulent-flow convective heat transfer coefficients of nanofluids exist inthe engineering literature, not all of them are suitable for examin-ing their predictability using the standard single-phase equationsdeveloped for pure fluids. According to the purpose of the presentstudy, the chosen experimental database, as listed in Table 2, islimited to convective heat transfer coefficients of nanofluids underthe condition of internal turbulent flow through a circular tubewhere nanofluid effective thermophysical properties, nanofluidtemperatures, nanofluid flow velocities, and other necessaryparameters are obtainable. The chosen experimental database iscomprised of base fluids of water (H2O) and ethylene glycol/watermixture with a 50% volume concentration (50/50 C2H6O2/H2O) andparticles of alumina (Al2O3), copper (Cu), cupric oxide (CuO), dia-mond, multiwalled carbon nanotube (MWCNT), silicon carbide(SiC), silica (SiO2), titania (TiO2), and zirconia (ZrO2).

4.2. Component thermophysical properties

In addition to the effective dynamic viscosity and the effectivethermal conductivity, other thermophysical properties of the com-ponents of the nanofluids are also essential for comparing experi-mental data with predictions. As part of the present study, accuratesources, listed in Table 3, were compiled for the thermophysicalproperties of density, mass-specific heat capacity, dynamic viscos-ity and thermal conductivity.

4.3. Heat transfer coefficient comparisons

For the purpose of comparing experimental data with predic-tions of the turbulent-flow convective heat transfer coefficientsof the nanofluids, three commonly-used single-phase equationsfor pure fluids were used in the present study: the Dittus–Boelterequation [65], the Petukhov–Popov equation [106]

h ¼ ðf �=8ÞRePrðk=dÞ1þ 3:4f � þ ð11:7þ 1:8Pr�1=3Þðf �=8Þ1=2ðPr2=3 � 1Þ

ð10Þ

and the Gnielinski equation [107]

h ¼ ðf �=8ÞðRe� 1000ÞPrðk=dÞ1þ 12:7ðf �=8Þ1=2ðPr2=3 � 1Þ

ð11Þ

where the friction coefficient f⁄ is calculated using the followingequation [108]

f � ¼ ð1:82 log Re� 1:64Þ�2 ð12Þ

The comparison results between the experimental data and the pre-dictions of the heat transfer coefficients of the nanofluids are plottedaccording to their component materials in Figs. 4–13. The compari-sons can generally be divided into the following four groups: (a) allthree equations predict most of the experimental data within the±20% range, which includes the alumina-in-water nanofluids [67–69], the titania-in-water nanofluids [67,78,79], and the zirconia-in-water nanofluids [68] shown in Figs. 4, 11, and 12, respectively;(b) two equations predict most of the experimental data withinthe ±20% range, which includes the copper-in-water nanofluids[70] and the silicon carbide-in-50/50 ethylene glycol/water nanofl-uids [80] shown in Figs. 5 and 13, respectively; (c) one equation pre-

Table 2Experimental database for nanofluid heat transfer coefficients.

Reference (author and year) Nanofluid Concentration (%) Channel(mm)

Note

Pak and Cho 1998 [67] 13-nm Al2O3 in H2O (1) 1.34 and (2) 2.78 10.66 Measured le and ke

Williams et al. 2008 [68] 46-nm Al2O3 in H2O (3) 0.9, (4) 1.8, and (5) 3.6 9.398 Fitted le and ke Size fromDLS and TEM

Torii 2010 [69] 33-nm Al2O3 in H2O (6) 0.1, (7) 1.0, and (8) 5.0 3.96 Measured l and k ratiosXuan and Li 2003 [70] <100-nm Cu in H2O (9) 0.3, (10) 0.5, (11) 0.8, (12) 1.0, (13) 1.2,

(14) 1.5, and (15) 2.010 Measured le and ke

Fotukian andEsfahany 2010 [71]

30–50-nm CuO in H2O (16) 0.015, (17) 0.031, (18) 0.039, (19) 0.078, (20) 0.118, (21)0.157, and (22) 0.236

5 Einstein equation for le Maxwellequation for ke

Torii 2010 [69] 47-nm CuO in H2O (23) 0.1, (24) 1.0, and (25) 5.0 3.96 Measured l and k ratiosTorii and Yang 2009 [72] 2–10-nm diamond in H2O (26) 0.1, (27) 0.4, and (28) 1.0 3.96 Measured l and k ratiosTorii 2010 [69] 2–10-nm diamond in H2O (29) 0.1, (30) 1.0, and (31) 5.0 3.96 Measured l and k ratiosLiao and Liu 2009 [73] 10–20-nm � 1–2-lm

MWCNT in H2O(32) 0.243, (33) 0.486, and (34) 0.978 1.02 Measured l and k ratios

Yu et al. 2009 [74] 170-nm SiC in H2O (35) 3.7 2.2733 Fitted le and ke [75] Size from DLSand SAXS

Timofeeva et al. 2010 [76] 16-nm SiC in H2O (36) 4.1 2.2733 Measured le and ke Size from BETTimofeeva et al. 2010 [76] 29-nm SiC in H2O (37) 4.1 2.2733 Measured le and ke Size from BETTimofeeva et al. 2010 [76] 66-nm SiC in H2O (38) 4.1 2.2733 Measured le and ke Size from BETTimofeeva et al. 2010 [76] 90-nm SiC in H2O (39) 4.1 2.2733 Measured le and ke Size from BETFerrouillat et al. 2011 [77] 22-nm SiO2 in H2O (40) 2.33, (41) 7.95, and (42) 18.93 4 Measured le and ke

Pak and Cho 1998 [67] 27-nm TiO2 in H2O (43) 0.99, (44) 2.04, and (45) 3.16 10.66 Measured le and ke

Duangthongsuk andWongwises 2009 [78]

21-nm TiO2 in H2O (46) 0.2 8.13 Measured le and ke [79]

Duangthongsuk andWongwises 2010 [79]

21-nm TiO2 in H2O (47) 0.2, (48) 0.6, (49) 1.0, (50) 1.5, and (51) 2.0 8.13 Measured le and ke

Williams et al. 2008 [68] 60-nm ZrO2 in H2O (52) 0.2, (53) 0.5, and (54) 0.9 9.398 Fitted le and ke Size fromDLS and TEM

Timofeeva et al. 2011 [80] 16-nm SiC in 50/50C2H6O2/H2O

(55) 4.0 2.2733 Measured le and ke Size from BET






(58) 1.0 and (59) 4.0 2.2733 Measured le and ke Size from BET

Table 3Thermophysical properties for nanofluid components.

Material Density (kg/m3) Mass-specific heat capacity (J/kg K) Dynamic viscosity (Pas) Thermal conductivity (W/m K)

H2O Fitted equation [81] Fitted equation [81] Fitted equation [81] Fitted equation [81]50/50 C2H6O2/H2O Data table [82] Data table [82] Data table [82] Data table [82]Al2O3 Data table [83] Data table [83] Data table [83]Cu 8960 [84] Data table [85] Data table [86]CuO 6310 [87] Data table [88] 17.6 [89]Diamond 3520 [90] 509.1 [91] 2200 [92]MWNT 1740 [93] Data plot [94] 3000 [95]SiC Data table [96] Data table [96] Data table [96]SiO2 2200 [97] Data table [98] Data table [99]TiO2 4230 [100] Fitted equation [101] 11.7 [102]ZrO2 5680 [103] Fitted equation [104] 2 [105]


dicts most of the experimental data within the ±20% range, whichincludes the multiwalled carbon nanotube-in-water nanofluids[73], the silicon carbide-in-water nanofluids [74,76], and the sil-ica-in-water nanofluids [77] shown in Figs. 8–10, respectively; and(d) all three equations predict experimental data trends but withsome experimental data outside the ±20% range, which includesthe cupric oxide-in-water nanofluids [69,71] and the diamond-in-water nanofluids [69,72] shown in Figs. 6 and 7, respectively.

All of the predictions shown in Figs. 4–13 are reasonable withthe exception of two sets of data, and the differences in those casesare attributed to inaccuracies in the effective thermophysical prop-erties used in the predictions made in the present study. The heattransfer data for cupric oxide-in-water nanofluids [71] were re-ported without measurements of the effective dynamic viscosityand the effective thermal conductivity. Without this information

other data were excluded from the database of this study; howeveran exception was made for these data because of their very lowparticle concentrations of <0.236%. Because of the very low con-centrations, it is reasonable to use property predictions from theEinstein equation [58] for the dynamic viscosity and the Maxwellequation [59] for the thermal conductivity when making heattransfer coefficient predictions. However, the poor results ofFig. 6 for these data indicate that even at these low particle concen-trations the property predictions were too inaccurate. Also, inobtaining the diamond-in-water experimental data [69], the effec-tive dynamic viscosity measurement reported for the five volumepercent concentration was considerably higher than and inconsis-tent with the rest of the effective dynamic viscosity measurementspresented. This measurement was used in the predictions of Fig. 7and is the likely cause of the poor agreement with the experimen-

0

5000

10000

15000

0 5000 10000 15000

(9)(10)(11)(12)(13)(14)(15)

Expe

rimen

tal d

ata

(W/m

2 K)

Dittus-Boelter prediction (W/m2K)

(a) +20%

-20%

0

5000

10000

15000

0 5000 10000 15000

(9)(10)(11)(12)(13)(14)(15)

Expe

rimen

tal d

ata

(W/m

2 K)

Petukhov-Popov prediction (W/m2K)

(b) +20%

-20%

0

3750

7500

11250

15000

0 3750 7500 11250 15000

(9)(10)(11)(12)(13)(14)(15)

Expe

rimen

tal d

ata

(W/m

2 K)

Gnielinski prediction (W/m2K)

(c) +20%

-20%

Fig. 5. Heat transfer coefficient comparison of copper-in-water nanofluids.

0

10000

20000

30000

40000

0 10000 20000 30000 40000

(1)(2)(3)(4)(5)(6)(7)(8)

Expe

rimen

tal d

ata

(W/m

2 K)


(a) +20%

-20%

0

10000

20000

30000

40000

0 10000 20000 30000 40000

(1)(2)(3)(4)(5)(6)(7)(8)

Expe

rimen

tal d

ata

(W/m

2 K)


(b) +20%

-20%

0

10000

20000

30000

40000

0 10000 20000 30000 40000

(1)(2)(3)(4)(5)(6)(7)(8)

Expe

rimen

tal d

ata

(W/m

2 K)


(c) +20%

-20%

Fig. 4. Heat transfer coefficient comparison of alumina-in-water nanofluids.


tal data. Note that the predictions for the rest experimental data inFig. 7, where no property inconsistencies were noted, are excellentfor all three prediction equations.

When differences between the experimental data and the theo-retical predictions occur in Figs. 4–13, it is observed that the exper-

imental data are usually greater than the theoretical predictions.This situation is consistent with additional effects of particleinteractions on the heat transfer coefficients such as increasedturbulence and reduced boundary layer thickness. From the com-parisons of Figs. 4–13, the following observations were made with

0

10000

20000

30000

0 10000 20000 30000

(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)

Exp

erim

enta

l dat

a (W

/m2 K

)


(a) +20%

-20%

0

10000

20000

30000

0 10000 20000 30000

(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)

Exp

erim

enta

l dat

a (W

/m2 K

)


(b) +20%

-20%

0

10000

20000

30000

0 10000 20000 30000

(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)

Exp

erim

enta

l dat

a (W

/m2 K

)


(c) +20%

-20%

Fig. 6. Heat transfer coefficient comparison of cupric oxide-in-water nanofluids.

0

10000

20000

30000

0 10000 20000 30000

(26)(27)(28)(29)(30)(31)

Exp

erim

enta

l dat

a (W

/m2 K

)


(a) +20%

-20%

0

10000

20000

30000

0 10000 20000 30000

(26)(27)(28)(29)(30)(31)

Exp

erim

enta

l dat

a (W

/m2 K

)


(b) +20%

-20%

0

10000

20000

30000

0 10000 20000 30000

(26)(27)(28)(29)(30)(31)

Exp

erim

enta

l dat

a (W

/m2 K

)

Gnielinski prediction (W/m 2K)

(c) +20%

-20%

Fig. 7. Heat transfer coefficient comparison of diamond-in-water nanofluids.


regard to the accuracy of the single-phase equations in predictingnanofluid heat transfer. With the exceptions of the experimentaldata with inaccurate effective thermophysical properties discussedabove, the Petukhov–Popov equation predicts all of the experimen-tal data very well while the Dittus–Boelter equation and the Gnielin-ski equation predict the majority of the experimental data well.

4.4. Summary

A turbulent-flow nanofluid with a low particle volume concen-tration can normally be treated as a homogeneous fluid becauseits particles and base fluid usually are well mixed. For such a

0

20000

40000

60000

80000

0 20000 40000 60000 80000

(32)(33)(34)

Expe

rimen

tal d

ata

(W/m

2 K)


(a) +20%

-20%

0

20000

40000

60000

80000

0 20000 40000 60000 80000

(32)(33)(34)

Expe

rimen

tal d

ata

(W/m

2 K)


(b) +20%

-20%

0

20000

40000

60000

80000

0 20000 40000 60000 80000

(32)(33)(34)

Expe

rimen

tal d

ata

(W/m

2 K)

Gnielinski prediction (W/m 2K)

(c) +20%

-20%

Fig. 8. Heat transfer coefficient comparison of multiwalled carbon nanotube-in-water nanofluids.

0

10000

20000

30000

40000

50000

0 10000 20000 30000 40000 50000

(35)(36)(37)(38)(39)

Expe

rimen

tal d

ata

(W/m

2 K)


(a) +20%

-20%

0

10000

20000

30000

40000

50000

0 10000 20000 30000 40000 50000

(35)(36)(37)(38)(39)

Expe

rimen

tal d

ata

(W/m

2 K)


(b) +20%

-20%

0

10000

20000

30000

40000

50000

0 10000 20000 30000 40000 50000

(35)(36)(37)(38)(39)

Expe

rimen

tal d

ata

(W/m

2 K)


(c) +20%

-20%

Fig. 9. Heat transfer coefficient comparison of silicon carbide-in-water nanofluids.


homogeneous turbulent-flow nanofluid, the standard single-phaseheat transfer coefficient equations will provide quite accurate pre-dictions. Therefore, if the effective thermophysical properties of ananofluid are accurate, the heat transfer coefficient predictionsfrom the standard single-phase equations will be very good.

5. Effectiveness: enhancements of nanofluid heat transfercoefficients

The effectiveness of a nanofluid over its base fluid can generallybe measured by its heat transfer coefficient enhancement specifiedas the ratio of the heat transfer coefficients of the nanofluid to its

0

5000

10000

15000

20000

0 5000 10000 15000 20000

(40)(41)(42)

Expe

rimen

tal d

ata

(W/m

2 K)


(a) +20%

-20%

0

5000

10000

15000

20000

0 5000 10000 15000 20000

(40)(41)(42)

Expe

rimen

tal d

ata

(W/m

2 K)


(b) +20%

-20%

+20%

-20%

0

5000

10000

15000

20000

0 5000 10000 15000 20000

(40)(41)(42)

Expe

rimen

tal d

ata

(W/m

2 K)


(c) +20%

-20%

Fig. 10. Heat transfer coefficient comparison of silica-in-water nanofluids.

0

10000

20000

30000

40000

0 10000 20000 30000 40000

(43)(44)(45)(46)(47)(48)(49)(50)(51)

Expe

rimen

tal d

ata

(W/2 K

)


(a) +20%

-20%

0

10000

20000

30000

40000

0 10000 20000 30000 40000

(43)(44)(45)(46)(47)(48)(49)(50)(51)

Expe

rimen

tal d

ata

(W/m

2 K)


(b) +20%

-20%

0

10000

20000

30000

40000

0 10000 20000 30000 40000

(43)(44)(45)(46)(47)(48)(49)(50)(51)

Expe

rimen

tal d

ata

(W/m

2 K)


(c) +20%

-20%

Fig. 11. Heat transfer coefficient comparison of titania-in-water nanofluids.


base fluid. It should be noted that the concept of the heat transfercoefficient enhancement of a nanofluid must not be confused withthe concept of the heat transfer coefficient prediction of the nano-fluid. The former relates the heat transfer coefficient of the nano-

fluid to that of its base fluid while the latter relates the heattransfer coefficient of the nanofluid to its effective thermophysicalproperties. A good heat transfer coefficient prediction of a nano-fluid does not necessarily indicate a good heat transfer coefficient

0

5000

10000

15000

20000

0 5000 10000 15000 20000

(52)(53)(54)

Expe

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tal d

ata

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0

5000

10000

15000

20000

0 5000 10000 15000 20000

(52)(53)(54)

Expe

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tal d

ata

(W/m

2 K)


(b) +20%

-20%

0

5000

10000

15000

20000

0 5000 10000 15000 20000

(52)(53)(54)

Expe

rimen

tal d

ata

(W/m

2 K)


(c) +20%

-20%

Fig. 12. Heat transfer coefficient comparison of zirconia-in-water nanofluids.

0

5000

10000

15000

20000

25000

0 5000 10000 15000 20000 25000

(55)(56)(57)(58)(59)

Expe

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ata

(W/m

2 K)


(a) +20%

-20%

0

5000

10000

15000

20000

25000

0 5000 10000 15000 20000 25000

(55)(56)(57)(58)(59)

Expe

rimen

tal d

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2 K)


(b) +20%

-20%

0

5000

10000

15000

20000

25000

0 5000 10000 15000 20000 25000

(55)(56)(57)(58)(59)

Expe

rimen

tal d

ata

(W/m

2 K)


(c) +20%

-20%

Fig. 13. Heat transfer coefficient comparison of silicon carbide-in-50/50 ethyleneglycol/water nanofluids.


enhancement. By following the figure of merit approach discussedpreviously, heat transfer coefficient enhancements were obtainedfor the nanofluid experimental database of the present study. Thecomparison bases of the constant flow velocity and the constantpumping power were used.

5.1. Base fluid heat transfer coefficients

For the purpose of enhancement analyses of nanofluids overtheir base fluids, accurate heat transfer coefficients of the

Table 4Prediction equations for base fluid heat transfer coefficients.

Reference (author and year) Base fluid Channel (mm) Prediction equation

Pak and Cho 1998 [67] (B1) H2O 10.66 hm = hPetukhov�Popov

Xuan and Li 2003 [70] (B2) H2O 10 hm = hDittus�Boelter for hm < 5000 W/m2 Khm ¼ ðhDittus�Boelter þ hPetukhov�Popov Þ=2for hm P 5000 W/m2 K

Williams et al. 2008 [68] (B3) H2O 9.398 hm = hPetukhov�Popov

Duangthongsuk and Wongwises 2009 [78] (B4) H2O 8.13 hm = hPetukhov�Popov for hm < 7000 W/m2 Khm = hDittus�Boelter for hm P 7000 W/m2 K

Liao and Liu 2009 [73] (B5) H2O 1.02 hm = 1.10hGnielinski for T = 29 �Chm = hPetukhov�Popov for T = 58 �C

Torii and Yang 2009 [72] (B6) H2O 3.96 hm = hGnielinski

Yu et al. 2009 [74], Timofeeva et al. 2010 [76] (B7) H2O 2.2733 hm = 1.12hGnielinski

Duangthongsuk and Wongwises 2010 [79] (B8) H2O 8.13 hm = 1.13hDittus�Boelter for hm < 7000 W/m2 Khm ¼ ðhDittus�Boelter þ hPetukhov�Popov Þ=2for hm P 7000 W/m2 K

Fotukian and Esfahany 2010 [71] (B9) H2O 5 hm = hDittus�Boelter

Torii 2010 [69] (B10) H2O 3.96 hm ¼ ðhDittus�Boelter þ hGnielinskiÞ=2Ferrouillat et al. 2011 [77] (B11) H2O 4 hm ¼ ðhDittus�Boelter þ hGnielinskiÞ=2 for heating

hm ¼ ðhDittus�Boelter þ hPetukhov�Popov Þ=2 for coolingTimofeeva et al. 2011 [80] (B12) 50/50 C2H6O2/H2O 2.2733 hm = 1.28hGnielinski

0

20000

40000

60000

80000

0 20000 40000 60000 80000

(B1)(B2)(B3)(B4)(B5)(B6)

(B7)(B8)(B9)(B10)(B11)(B12)

Exp

erim

enta

l dat

a (W

/m2 K

)

Prediction (W/m 2K)

Fig. 14. Heat transfer coefficient comparison of base fluids.

Table 5Experimental database for nanofluid heat transfer coefficient enhancements.

Reference (author and year) Nanofluid Concentrationrange (%)

Pak and Cho 1998 [67], Williams et al. 2008[68], Torii 2010 [69]

(E1) Al2O3 inH2O

0.1–5.0

Xuan and Li 2003 [70] (E2) Cu in H2O 0.3–2.0Torii 2010 [69] (E3) CuO in H2O 0.1–5.0Torii and Yang 2009 [72], Torii 2010 [69] (E4) diamond in

H2O0.1–1.0

Liao and Liu 2009 [73] (E5) MWCNT inH2O

0.243–0.978

Yu et al. 2009 [74], Timofeeva et al. 2010[76]

(E6) SiC in H2O 3.7–4.1

Ferrouillat et al. 2011 [77] (E7) SiO2 in H2O 2.33–18.93Pak and Cho 1998 [67], Duangthongsuk and

Wongwises 2009 [78], Duangthongsukand Wongwises 2010 [79]

(E8) TiO2 in H2O 0.20–3.16

Williams et al. 2008 [68] (E9) ZrO2 in H2O 0.2–0.9Timofeeva et al. 2011 [80] (E10) SiC in 50/

50 C2H6O2/H2O1.0–4.0


nanofluids and the base fluids are essential. In many enhancementpredictions, the heat transfer coefficients of the nanofluids areexperimentally measured and the heat transfer coefficients of thebase fluids are theoretically predicted. The best predictions of theheat transfer coefficients of the base fluids in the database of thepresent study were obtained by using the best prediction equationswith each set of data rather than a single equation for all. Those pre-diction equations for the experimental database of the presentstudy are listed in Table 4 where the B designation refers to the basefluid of a particular experiment and the comparisons between theexperimental data and the theoretical predictions are plotted inFig. 14. It can be seen from Fig. 14 that the predictions are consid-ered to be in very good agreement with the experimental data.

5.2. Nanofluid heat transfer coefficients

The calculation of nanofluid heat transfer enhancements in-volves the same nanofluid effective thermophysical properties asthe analysis of nanofluid heat transfer coefficient predictions. Con-sequently, the two data sets [69,71] discussed previously with re-spect to property uncertainties were eliminated from the database.

The remaining experimental data were reorganized according tonanofluid component materials and are listed in Table 5 wherethe E designation refers to the nanofluid of a particular experiment.

5.3. Heat transfer coefficient enhancement comparisons

The experimental heat transfer coefficient enhancements of thenanofluids over their base fluids were obtained by dividing theexperimental heat transfer coefficients of the nanofluids by the pre-dicted heat transfer coefficients of their base fluids using the equa-tions listed in Table 4. Two comparison bases were used: theconstant flow velocity and the constant pumping power. For theconstant flow velocity-based comparisons, the heat transfercoefficients of the base fluids were calculated directly from the pre-diction equations listed in Table 4 with the same flow velocities andtemperatures of their corresponding nanofluids. For the constantpumping power-based comparisons, the flow velocities of the basefluids are different from those of their corresponding nanofluids. Inthe constant pumping power comparisons, the pumping power wascalculated as

P ¼ ðp=2ÞdLu3qf ¼ 0:03955pd3=4Lu11=4q3=4l1=4 ð13Þ

0.0

0.5

1.0

1.5

0.0 0.5 1.0 1.5

(E1)(E2)(E3)(E4)(E5)(E6)(E7)(E8)(E9)(E10)

Expe

rimen

tal e

nhan

cem

ent r

atio

Figure-of-merit ratio

(a) flow velocity basis+20%

-20%

O

O'

0.0

0.5

1.0

1.5

0.0 0.5 1.0 1.5

(E1)(E2)(E3)(E4)(E5)(E6)(E7)(E8)(E9)(E10)

Expe

rimen

tal e

nhan

cem

ent r

atio

Figure-of-merit ratio

(b) pumping power basis+20%

-20%

O

O'

Fig. 15. Heat transfer coefficient enhancement comparison of various nanofluids.


which was obtained by calculating the Fanning friction factor fromthe Blasius equation [66]. This approach is supported by the factthat the Fanning friction factor of a nanofluid in turbulent flow fol-lows the Blasius equation very well [67–70,73,77,109]. Also as dis-cussed previously, the heat transfer coefficient ratio of a nanofluidto its base fluid is not significantly affected by using the classicalequation or using the Blasius equation for the Fanning friction fac-tor (<0.012 for the experimental database of the present study).From Eq. (13), the base fluid flow velocity um for the constantpumping power-based comparison (Pm = Pe) can be expressed as afunction of the experimentally-known nanofluid velocity ue [60]

um ¼ ðqe=qmÞ3=11ðle=lmÞ

1=11ue ð14Þ

Therefore, for the constant pumping power-based comparisons, theheat transfer coefficients of the base fluids were calculated from theprediction equations listed in Table 4 with the flow velocities com-puted from Eq. (14). While it is possible to use all the experimentaldata points for comparisons at the constant flow velocity and theconstant pumping power, the experimental heat transfer coefficientenhancement ratios and corresponding figure-of-merit ratios areonly presented in Fig. 15 averaged over the nanofluid componentmaterial and the particle volume concentration. These average re-sults represent the overall nanofluid performance and are the mostimportant criteria for judging the benefit of nanofluid heat transfer.

In Fig. 15, (a) the horizontal unity line separates the regimes ofthe experimental heat transfer coefficient enhancements (above)and deteriorations (below); (b) the vertical unity line separatesthe regimes of the figure-of-merit ratio enhancements (right) anddeteriorations (left); and (c) the line OO0 separates the regimes ofthe experimental heat transfer coefficient enhancement ratios lar-ger (above) and smaller (below) than the corresponding figure-of-merit ratios. It can be seen from Fig. 15 that the experimental heattransfer enhancement ratios agree very well with the correspond-ing figure-of-merit ratios with almost all of the data being within±20% range and most of the data being within ±10% range. The fig-ure-of-merit ratios for all of the data with experimental heat trans-fer coefficient deteriorations are smaller than unity (shadedregime); however, the experimental heat transfer coefficientenhancement ratios for some of the data with figure-of-merit ratiodeteriorations are larger than unity. Both are indicative of addi-tional particle interaction effects on the heat transfer enhance-ments of the nanofluids. This particle interaction improvement isalso observed when the heat transfer coefficient ratios showenhancements (greater than unity). In almost every case whenthere are heat transfer coefficient enhancements, they are largerthan the corresponding figure-of-merit ratios.

It is observed in Fig. 15 that the constant flow velocity-basedcomparisons and the constant pumping power-based comparisonsfollow very similar patterns. However, considering the magnitudesit is observed that the experimental heat transfer enhancement ra-tios and the corresponding figure-of-merit ratios for the constantpumping power-based comparisons are smaller than those forthe constant flow velocity-based comparisons. As discussed previ-ously, the difference is small and is as expected.

5.4. Summary

While there are potential additional particle effects, the domi-nant influence factor on the heat transfer coefficient enhancementof a nanofluid over its base fluid is its effective thermophysicalproperties. These results further validate the treatment of a nano-fluid as a homogeneous fluid with its effective thermophysicalproperties. For such nanofluids the constant velocity and constantpumping power figure-of-merit ratios serve as good predictors ofthe nanofluid heat transfer enhancements.

6. Concluding remarks

With more than a decade of research and development of nano-fluid technologies, the main focus of nanofluid investigations hasgradually shifted from laboratory effective thermophysical prop-erty characterizations to practical heat transfer coefficient mea-surements. This shift in focus is evident in the great number oforiginal research papers on nanofluid heat transfer in the engineer-ing literature. The present comparative review of convective heattransfer of nanofluids concentrates on one of the most com-monly-encountered engineering heat transfer problems – a forcedsingle-phase internal turbulent flow through a circular tube be-cause of its key role in studying, designing, and implementingpractical turbulent-flow nanofluid application systems. The analy-ses of the experimental data from multiple research groups of theconvective heat transfer coefficients of nanofluids in turbulent flowin the engineering literature show that (a) turbulent-flow nanofl-uids with low particle volume concentrations can normally betreated as homogeneous fluids due to the nature of theirwell-mixed particles and base fluids; (b) the additional effects ofparticles on the convective heat transfer coefficients for suchhomogeneous nanofluids in turbulent flow are insignificantcompared to the dominant influence factor – the effective


thermophysical properties of the nanofluids; (c) the convectiveheat transfer coefficients of nanofluids in turbulent flow can, there-fore, be predicted quite accurately with standard single-phaseequations of Dittus–Boelter, Petukhov–Popov, or Gnielinski; and(d) the actual heat transfer coefficient enhancement ratios of nano-fluids over their base fluids can also be predicted quite accuratelywith the heat transfer coefficient ratios of the nanofluids over theirbase fluids calculated with the Dittus–Boelter equation using thethermophysical properties of the nanofluids and their base fluidsfor the specific comparison bases of the constant flow velocity orthe constant pumping power; however, using the Reynolds num-ber as a comparison basis is inappropriate, and using the Nusseltnumber in place of the heat transfer coefficient is misleading fornanofluid enhancements. These findings provide the possibilityand the basis for judging the heat transfer performance of nanofl-uids under practical application conditions through their effectivethermophysical properties. Consequently, the results point to thefuture research directions of engineering nanofluids with the de-sired heat transfer characteristics through the proper combinationsof their thermophysical properties, especially the effective dy-namic viscosity and the effective thermal conductivity, by optimiz-ing particle size, particle shape, particle coating, particle volumeconcentration, base fluid, and fluid additive.

Acknowledgements

This work was sponsored by the Office of Vehicle Technologiesand the Office of Industrial Technology Program of the U.S. Depart-ment of Energy under contract number DE-AC02–06CH11357 atArgonne National Laboratory, managed by UChicago Argonne LLC.

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