A New Thermal Conductivity Model for Nanofluids

7/27/2019 A New Thermal Conductivity Model for Nanofluids

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A new thermal conductivity model for nanofluids

Junemoo Koo and Clement Kleinstreuer*

Department of Mechanical and Aerospace Engineering, Campus Box 7910, Broughton Hall 4160, Raleigh,NC 27695-7910, USA; *Author for correspondence (Tel.: +1 919-515-5261; Fax: +1 919-515-7968;E-mail:[email protected])

Received 20 May 2004; accepted in revised from 8 September 2004

Key words: nanofluids, effective thermal conductivity, apparent thermal conductivity, Brownian motion,interparticle potential, modeling and simulation

Abstract

In a quiescent suspension, nanoparticles move randomly and thereby carry relatively large volumes ofsurrounding liquid with them. This micro-scale interaction may occur between hot and cold regions,resulting in a lower local temperature gradient for a given heat flux compared with the pure liquid case.Thus, as a result of Brownian motion, the effective thermal conductivity, keff, which is composed of theparticles conventional static part and the Brownian motion part, increases to result in a lower temperaturegradient for a given heat flux. To capture these transport phenomena, a new thermal conductivity model fornanofluids has been developed, which takes the effects of particle size, particle volume fraction and tem-

perature dependence as well as properties of base liquid and particle phase into consideration by consid-ering surrounding liquid traveling with randomly moving nanoparticles.

The strong dependence of the effective thermal conductivity on temperature and material properties ofboth particle and carrier fluid was attributed to the long impact range of the interparticle potential, whichinfluences the particle motion. In the new model, the impact of Brownian motion is more effective at highertemperatures, as also observed experimentally. Specifically, the new model was tested with simple thermalconduction cases, and demonstrated that for a given heat flux, the temperature gradient changes signifi-cantly due to a variable thermal conductivity which mainly depends on particle volume fraction, particlesize, particle material and temperature. To improve the accuracy and versatility of the keff model, moreexperimental data sets are needed.

Introduction

As first demonstrated by Choi (1995), highlyconductive nanoparticles of very low volumefractions distributed in a quiescent liquid (callednanofluids), may measurably increase the effec-tive thermal conductivity of the suspension whencompared to the pure liquid (Figure 1). Specifi-cally, aluminum and copper-oxide spheres andcarbon-nanotubes of an average diameter of 30nm, in volume concentrations between 0.001%and 6% generated thermal conductivities knanofluid 3kcarrierfluid (Choi et al., 2001; Eastman et al.,

2001; Patel et al., 2003). Thus, the use of nanofl-

uids, for example in heat exchangers, may result inenergy and cost savings and should facilitate thetrend of device minituarization. More exoticapplications of nanofluids can be envisioned inbiomedical engineering and medicine in terms ofoptimal nano-drug targeting and implantablenano-therapeutics devices.

In any case, of major interest in this paper is toexplore why the presence of low-concentrationmetallic nano-spheres in water increases the ther-mal conductivity. Traditional theories, such asMaxwell (1904) or Hamilton and Crosser (1962),

Journal of Nanoparticle Research (2004) 6:577588 Springer 2005

DOI: 10.1007/s11051-004-3170-5


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cannot explain this thermal phenomenon. Thus,new assessments and mathematical models of the

new apparent (or effective) thermal conductivityhave been proposed. For example, Xuan and Li(2000) summarized previous experimental obser-vations and concluded that keff was a function ofboth the thermal conductivities of the nano-materialand carrier fluid, in terms of particle volumefraction, distribution, surface area, and shape.Keblinski et al. (2002) listed four possibleexplanations for the cause of an anomalous in-crease of thermal conductivity: Brownian motionof the nanoparticles, molecular-level layering ofthe liquid at the liquid/particle interface, the nat-ure of heat transport in the nanoparticles, and the

effects of nanoparticle clustering. They ruled outthe possibility of the Brownian motion effect bycomparing the time scales of Brownian motionand the thermal response, a point revisited in theResults section. Xue (2003) proposed a thermalconductivity model based on Maxwells theoryand average polarization theory to take care of theinterface (i.e., liquid nano-layer) effect. He mat-ched the predicted thermal conductivity valueswith the observed ones by changing the nano-layerthickness. Yu and Choi (2003, in press) modifiedthe Maxwell equation and HamiltonCrosser

relation for the effective thermal conductivity ofsolid/liquid suspensions to include the effect of

ordered nano-layers around the particles. Theyalso matched the model with observed conductiv-ities by adjusting the nano-layer thickness andconductivity. Bhattacharya et al. (2004) investi-gated the effect of particle Brownian motion byusing a molecular dynamics type approach whichdoes not consider the motion of fluid moleculesand requires two experimentally determinedparameters. Jang and Choi (2004) suggested aneffective thermal conductivity model consideringthe particles Brownian motion. They focused onthe heat transfer between particles and carrierfluid, neglecting the mixing due to random particle

motion. Furthermore, the validity of their thermalboundary layer thickness, which they defined as3dBF/Pr, where Pr


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micro- and macroscopic energy transfer mecha-nisms in nanofluids. A few illustrative examples of1-D heat conduction with nanofluids are provided.

Theory

It is postulated that the enhanced thermal con-ductivity of a nanofluid, when compared toconventional predictions, is mainly due toBrownian motion which produces micro-mixing.This effect is additive to the thermal conductivityof a static dilute suspension, i.e.,

keff kstatic kBrownian; 1where for example, according to Maxwell (1904)

kstatic

kc 1 3

kdkc 1ad

kdkc 2 kd

kc 1ad

; 2

where ad is the particle volume fraction, kc, thethermal conductivity of the carrier fluid and kd isthat of the particles. Since the speed of thermalwave propagation is much faster than the particleBrownian motion, the static part cannot beneglected.

Concerning temperature-dependence, Patelet al. (2003) performed experiments with suspen-sions of very small volume fractions, i.e.,ad < 0:001%, where the static part of the thermalconductivity is negligible. They observed a mea-surable increase of thermal conductivity even whenthe volume fraction was as low as 0.00013%. Theyalso found that the thermal conductivity of thenanofluids increases with fluid temperature, whichsupports the importance of the Brownian motioneffect (see Eq. (1)). Das et al. (2003) also observedthe temperature dependence of the thermal con-ductivity for different type of nanofluids.

For the derivation of kBrownian, we consider twonanoparticles with translational time-averagedBrownian motion, "v, in two different temperaturefields (or cells) of extent l, where l is the averagedistance for a particle to travel along onedirection without changing its direction due to theparticle Brownian motion (see Figure 2). Itshould be noted that l is different from the in-terparticle distance due to the intervening fluidand interaction between particles and fluid mole-cules. For a gas, l is the same as the interparticledistance since there is no intervening medium

between gas molecules. Hence, the travel time isDt l="v; where (Probstein, 2003)

"v ffiffiffiffiffiffiffiffiffiffiffiffiffi18jTpqdD

3s : 3

Here, "v is the translational time-averaged speeddue to the Brownian effect, j, the Boltzmannconstant, T the fluid temperature, qd the particledensity, and D is its diameter. Table 1 lists typicalvalues of particle traveling speeds and corre-sponding Reynolds numbers.

Defining p as the probability for a particle totravel along any direction, and assuming that eachof the two particle cells are in thermal equilibriumat temperatures of T1 and T2, respectively, these

particles moving to neighboring cells will carryenergy across the interface, i.e.,

qnet DQADt

% pNmdcvT1 T2ADt

pNmdcv"vDTll

A"vdt

4a

Table 1. Particle speed due to the Brownian effect andcorresponding particle Reynolds number of Cu-nanoparticle-

water suspensions

Particle size (nm) Traveling speed

(m/sec)

ReD ()

10 1.63 1.87 102100 5.15 102 5.90 1031000 1.63 103 1.87 103

Figure 2. Schematic illustrating the particles Brownian-motion

effect on micro-mixing.

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WithDTl ! rT;N is the total particle number in a

cell, md the particle mass, qd is its density, and cv isits specific heat, md qdVd; Vd is the particle vol-ume, and A is the cross-sectional area of the sys-tem normal to the x-direction, A"vDt V;and NVd=V ad, we obtain

qnet % padqdcv"vDrT 4bqnet kBrownianrT: 4c

Here, kBrownian is the added thermal conductivitydue to Brownian motion of a given nano-sphere.Thus, with Eq. (3)

kBrownian ffiffiffiffiffi

18

p

rpadqdcv

ffiffiffiffiffiffiffiffiffiffiffiffiffijT

qdD

l

D

s: 5

Clearly, as the nanoparticles randomly move, aportion of the surrounding fluid is affected, i.e.,fluid motion and interaction occurs, leading tomicro-scale mixing and heat transfer. For aquantitative assessment, we can assume steadyflow in the Stokes regime (see Table 1) so that forthe axial velocity field surrounding a sphere ofradius R we can write (Koo, 2004).

vx

"v1 1 3

2Rr 1

2Rr

3" # cos2 hh 1 3

4

R

r 1

4

R

r

3" #sin2 h

6

Equation (6) can be employed to estimate the re-gion of influence of a creeping sphere, i.e., with the99% criterion of vanishing impact, the affectedfluid volume is (Figure 2):

Vf

p

6a2b;

7

where a % 37:5D is the short axes and b % 75D isthe long axes of a spheroidal body of fluid. Theshape and size of affected fluid volume dependsalso on the particle shapes.

As a result, not just nanoparticles move arounddue to the Brownian effect but actually signifi-cantly larger fluid bodies, which may interact andlead to vigorous micro-mixing. Thus, Eq. (5) hasto be multiplied by a

D2 b=D 105; 470 in order

to encapsulate the extended Brownian motioneffect. Therefore, Eq. (5) turns into

kBrownian 37:52 75 ffiffiffiffiffi18p

rpbadqdcv

ffiffiffiffiffiffiffiffiffiffiffiffiffijTqdD

l

D

s:

8Only a fraction ofVf (Eq. (7)) will travel with theparticles due to the interaction among affectedfluid volumes for cases, where the interparticledistance is not long enough for neighboring af-fected volumes to travel independently. Here, brepresents the fraction of the liquid volume Vf,which travels with a particle. It will decrease withvolume fraction ad due to the viscous effect of

moving particles.Equation (8) is a result obtained from basickinetic theory without considering the interparticleinteraction, Referring to Deen (1998), it has beenreported that an elementary model indicatingkBrownian $ T1=2 does remarkably well in predictingthe main features of the transport properties ofgases, although real gas conductivities showstronger dependences on the temperature, i.e.,kgas $ T, instead of

ffiffiffiffiT

p. He reported this discrep-

ancies are largely corrected by the rigorous kinetictheory of Chapman and Enskog (Chapman &Cowling, 1951), which considers in some detail the

effect of intermolecular potential energy on theinteractions between colliding molecules.For liquidsolid suspensions, the effect of

interactions between solid particles are muchstronger than between gas molecules, since theinterparticle potential is proportional to 1/d,where d is the interparticle distance, while it is1/d6 for intermolecular potential, which decaysvery fast with distance. As it is observed in ga-ses, the thermal conductivity dependence ontemperature should be greater than what ispredicted from elementary kinetic theory. Basedon this statement, a factorial function f (T, ad,

fluid and particle properties, etc.) is added to Eq.(5) to result in

kBrownian 37:52 75ffiffiffiffiffi

18

p

rpbadqdcv

ffiffiffiffiffiffiffiffiffiffiffiffiffijT

qdD

l

D

sfT; ad; etc.:

9

The function fT; ad; etc. should depend onproperties of the intervening fluid, and henceparticle interactions. Traditionally, the interparti-cle potential can be used to take the interparticle

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interaction into consideration, i.e., the interparti-cle potential is given as

wd AR6d

; 10

where A is the Hamaker constant, R, the particleradius, and dis the surface distance. The Hamakerconstant A for two identical phases 1 interactingacross medium 3 can be described as (Israelachvili,1992):

A 34jT

1 31

3

2 3hme

16ffiffiffi2p n

21 n232

n21

n23

3=2

; 11

where is the electric dipole constant, n, therefractivity, h, the Planck constant, and me is thefrequency where the dielectric medium has thestrong absorption peak(see Table 4). For dilutecases, the thermal conductivity of the nanofluids isexpected to show a relatively low dependence onparticle interactions, due to the retardation effectof the interparticle potential, i.e., it is dispersionenergy that suffers retardation. The complexitiesof the involved phenomena, make it difficultto obtain the function f theoretically if notimpossible. Therefore, we decided to determine the

function f from experimental data, and checkwhether the experimentally determined functionshows the same trend with the theory, to confirmthe validity of the new model. Furthermore,molecular dynamics simulations considering theparticle-particle, and particle-liquid interactionsshould be performed to determine the averagetraveling distance without changing its directionl/D. Here, it is assumed that l/D1 and the exper-imentally determined function fcontains its effect.

Now, with the expectation value of particles tomove in one direction being, 0.197 (Koo, 2004),Eq. (5) can be rewritten as

kBrownian 5 104badqlclffiffiffiffiffiffiffiffiffijTqdD

sfT; ad; etc.:

12

Determination of the functions b and f

Since b is related to the particle motion, it shoulddepend not only on volume fraction but alsotemperature, particle shape and material proper-

ties of particles and carrier fluid. Here we assumedthat b is a function of volume fraction only, sinceall other dependencies can be covered with thefunction f. The function b can be determined withease under the assumption that f is unity for thegiven conditions, i.e., temperature, material prop-erties of particles and carrier fluid, and particleshape, because most experimental data available inthe literature have been obtained at a fixed tem-perature. Figure 3 and Table 2 show the resultobtained for water-based nanofluids. For the casesof small volume fractions, ad < 1%, where theparticle interaction effect is relatively less signifi-

cant due to both long interparticle distance and thepotential retardation effect, the function b isindependent of the type of particle. In contrast, thefunction b depends on the type of particle in highconcentration cases, ad < 1%, which is found to bethe result of interparticle interactions. As was ex-pected, the function b decreased with the volumefraction ad in most cases. It will be discussed inmore detail in Results and Discussion section.

There are not many data sets available fordetermining the function f. Das et a1. (2003) andPatel et al. (2003) observed that their experimen-tally measured thermal conductivity increased with

temperature. Specifically, the thermal conductivityenhancement increased by 7% for the 1%waterAl2O3 nanofluid, and by 15% for the 4%waterAl2O3 nanofluid for the temperature rise of30 K. It increased by 20% for the 1 and 4%waterCuO nanofluid for the same temperaturerise. Patel et al. (2003) did experiments with very lowconcentration nanofiuids and they observed about5% increase of thermal conductivity enhancementfor 30 K temperature rise. Specifically, an expression

for f (T; ad, properties of particles and carrier fluid)in Eq. (12) was obtained from the CuO-nanofluidsexperimental data of Das et al. (2003). The function

was assumed to vary continuously with the particlevolume fraction because data for two volumeconcentration cases were available.

fT; ad6:04ad 0:4705T 1722:3ad 134:63:

13

It should be noted that the function is a linearequation because of the Taylor series truncation;but, that functional dependence can be confirmedwith Eqs. (10) and (11) as well. Eq. (13) is valid forthe given experimental conditions in the ranges

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1% < ad < 4% and 300 < T< 325[K]. Thechange of kBrownian with ad is given as

dkBrownian

dad1:9318qlcl

ffiffiffiffiffiffiffiffiffijT

qdD

s0:2728a0:7272d

6:04a 0:4705T 1722:3ad 134:63 a0:2728d 6:04T 1722:3g:

14The last term in the equation becomes negativeand increases with temperature, which explains thethermal conductivity reduces near x

1 m in

Results and Discussion section.We investigated other transport mechanisms,

such as thermophoresis and fluid thermal energy,

potentially contribution to enhanced thermalconductivities of nanofluids, and found that they

are negligible (Koo, 2004). The results for typicalnanoparticles (CuO, D10 nm, ad1%) sus-pended in water are given in Table 3.

Results and discussion

Higher thermal conductivities due to Brownianmotion effect

In a quiescent suspension, nanoparticles moverandomly and thereby carry relatively large vol-

umes of surrounding liquid with them. This micro-scale interaction may occur between hot and cold

Table 2. Comparison of augmented thermal conductivities

Type of particles b Remarks

Au-citrate, Ag-citrate

and CuO

0.0137 (100ad)0.8229 ad1%

Al2O3 0.0017 (100ad)0.0841 ad>1%

Table 3. Comparison of augmented thermal conductivity

parameters

Parameter type* Increase (%)

kBrownian/kwater 1950

kTP/kwater 0.07

kPTE/kwater 0.80

*kBrownian is given with Eq. (5), kTP is due to thermophoresis,

and kPTE is due to particle thermal energy.

= 0.0137(100d)-0.8229

= 0.0011(100d)-0.7272

= 0.0017(100d)-0.0841

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01

volume fraction [%]

Xuan & Li (2003) 100 nm Cu in waterMasuda et al. (1993) 13 nm Al2O3 in waterLee et al. (1999) 23.6 nm CuO in waterWang et al. (2003) 50 nm CuO in waterPak & Cho (1998) 13 nm Al2O3 in water

Pak & Cho (1998) 27 nm TiO2 in waterDas et al. (2003) 28.6 nm CuO in waterPatel et al. (2003) 15 nm Au-citrate in waterPatel et al. (2003) 70 nm Ag-citrate in waterCurve fit lowCurve fit high CuOCurve fit high Al2O3Power curve fit for a < 1 %Power curve fit for CuO a > 1 %Power curve fit for Al2O3 a > 1 %

Figure 3. Comparison ofb-functions obtained from experimental data.

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regions, resulting in a lower local temperaturegradient for a given heat flux compared with thepure liquid case. Thus, as a result of Brownianmotion, the effective thermal conductivity, keff,increases, which results in a lower temperaturegradient for a given heat flux (see Figure 4). Fur-thermore, at elevated suspension temperatures, theBrownian motion effect increases and higher keff-values arise (see Figures 4 and 6) as was experi-mentally observed by Patel et al. (2003) and Daset al. (2003). As stated in the Introduction, Ke-blinski et al. (2002) argued that the thermal dif-fusion time scale is much smaller than Brownian

diffusion time scale by an order of 102

for 10 nmparticle suspensions. However, considering theeffective complex particle-plus-liquid volume to beb37.52 75 Vd and taking a typical value of bas 0.01 (see Figure 3), the two time scales haveabout the same order of magnitude, and henceBrownian motion becomes a very importantmechanism for augmented heat transfer.

Additional confirmation of the importance ofthe Brownian effect can be deduced when differentcarrier fluids and nano-material have been used.For example, Lee et al. (1999) found thermalconductivity improvement in ethylene glycol based

nanofluids when compared to water based ones.

Xie et al. (2002) observed that the thermal con-ductivity augmentation ratio decreases with anincrease in base liquid thermal conductivity. Xieet al. (2003) compared the effective thermalconductivity of distilled-water (DW)-, ethylene-glycol(EG)-, and decene (DE)-based carbonnanotube-suspensions and observed the followingresult.

kDW < kEG < kDE 15Lee et al. (1999) also observed that the ethyleneglycol based nanofluids showed higher thermalconductivity than the water based ones.

The thermal conductivity due to Brownian mo-tion is not explicitly related to the fluid thermalconductivity, while the static part of the effectivethermal conductivity is strongly dependent on it(see Eq. (2)). The thermal conductivity of the threeliquids are 0.611, 0.252, and 0.145 (W/mK),respectively. Moreover, their thermal capacities(qlcl), to which the thermal conductivities due toBrownian motion are proportional, are 4153,2674, and 1700 (kJ/m3K), respectively. Using thefact that

kBrownian

kstatic $qlcl

kc:

16

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

0.0001 0.001 0.01 0.1 1 10

volume fraction [%]

k

eff/kc

(1) Patel et al. (2003) 15 nm Au-citratemodel prediction for dataset (1)(2) Wang et al. (2003) 50 nm CuOmodel prediction for dataset (2)(3) Das et al. (2003) 28.6 nm CuOmodel prediction for dataset (3)(4) Lee et al. (1999) 23.6 nm CuOmodel prediction for dataset (4)(5) Matsuda et al. (1993) 13 nm Al2O3model prediction for datasets (5) and (6)(6) Pak & Cho (1998) 13 nm Al2O3(7) Patel et al. (2003) Ag-citratemodel prediction for dataset (7)

Figure 4. Comparison of model predictions with experimental data sets.

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The ratio among the three suspensions is

kBrownian

kstatic

DW

:kBrownian

kstatic

EG

:kBrownian

kstatic

DE

0:58 :0:91 :1;17

which explains their observations and merits theeffect of Brownian motion.

Interparticle interactions for high concentrations,i.e., ad > 1%, cases

As mentioned in Theory section, the intermolecu-lar interaction becomes more important for sus-

pensions with high particle concentrations. Theinterparticle potentials for suspensions of nano-particles of d > 10 nm, the non-retarded part ofthe potential is most important (Israelachvili,1992). Table 4 shows the Hamaker constantsof the non-retarded term(cf. Eq. (11), where

A0 34 kT 1313 2

). Clearly, the b-function, when

ad > 1% (see Figure 3) for the four particlematerials, i.e., Cu, Al2O3, CuO, and TiO2, encap-

sulates the same relative magnitute trend as A0 (seeTable 4). This supports our explanation that theeffective thermal conductivity for ad>1% is mainlyaffected by interparticle interactions. As shownwith Eq. (11), interparticle interaction becomesmore important when the difference in electric di-pole constants between particle and carrier fluidincreases. It also depends on the volume fractionad and the particle size (cf. Eq. (10)). Compared tothe potential between (gas) molecules which is in-versely proportional to d6, the cluster potentialimpacts long range. Figure 5 shows the effect ofthe exponent in the potential on the impact dis-

tance; lower exponent cases show longer impactdistance. Especially, the attractive part is inversely

Table 4. Parameter values for different materials (see Eq. (11))

Phase material n A0

Cu inf 3.11 1021Al2O3 4.5 1.761 2.48 1021CuO 18.1 2.6 1.24 1021TiO2 86 2.6 4.06 1024Water 80.4 1.33

Ethylene glycol 37.0

Decene 2.2 2.2

*A0 is the first term in Eq. (11).

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0 0.5 1 1.5 2 2.5 3 3.5

non-dimensionalized distance

non-dimensiona

lizedpotential

w(d)~1

d

w(d)~1

d3

w(d)~1

d6

Figure 5. Comparison of effect of the exponent in the intermolecular potential on impact distance.

Figure 6. Comparisons of temperature, temperature gradient,

and effective thermal conductivity profiles.

c

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0 0.4 0.6 0.81

x[m]

300

305

310

315

320

325

330

T[K]

pure water k = const.

pure water k = f(T)

nanofluid (28.6 nm CuO 1 %) k = kstatic


+kBrownian



+kBrownian

qx=const.

x[m]

20

22

24

26

28

30

dT__dx

[K/m]


pure water k = f(T)



+kBrownian



+kBrownian

0 0.2 0.4 0.6 0.8 1

x[m]

0.60

0.65

0.70

0.75

0.80

0.85

0.90

keff

[W/mK]


pure water k = f(T)



+kBrownian



+kBrownian

0.2

0 0.4 0.6 0.810.2 1

(a)

(b)

(c)

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proportional to the surface distance itself. Israe-lachvili (1992) reported that large distance contri-butions to the interaction disappear only forvalues of the exponent in the potential greater than3. But for the exponent smaller than 3, the con-tribution from more distant molecules will domi-nate over that of near by molecules. The measured

gas thermal conductivities show stronger temper-ature dependence (keT) compared to the pre-dicted values using kinetic theory. More rigoroustreatment of kinetic theory including the effect ofintermolecular potential energies can correct thediscrepancies for gases (Deen, 1998). Strongertemperature dependence compared to gases is ex-pected in liquids due to long-reaching potential.

Examples of heat conduction in nanofluids

The axial conduction heat flux component can beexpressed as

qx keffdTdx

: 18

A sample problem is solved considering a steadystate without internal heat generation, constantheat flux case. Figure 6 shows the profiles oftemperature, temperature gradient, and effectivethermal conductivity in pure water and uniformnanofluids of different particle volume fractionsfor a heat flux of 18 (W/m2)] in a 1-D system. Thetemperature at x1 m is lower by 7 K for the 4%nanofluid case compared to the pure water of

constant thermal conductiviy case. From the datain Table 2, the thermal conductivity due toBrownian effect is proportional to a0:2728d , while thestatic part is proportional to ad. The higher con-tribution of Brownian motion in the nanofluid ofad1% compared to that of the 4% case, makesthe temperature difference very small. Referring to

temperature gradient and thermal conductivitygraphs, the difference between the two nanofluidsbecomes smaller at higher temperatures due to theBrownian effect. The temperature gradients, con-sidering the temperature dependence of the ther-mal conductivity, show decreasing trends since thethermal conductivity increases with temperature asindicated in Eq. (12).

It is possible for particles to have nonuniformdistributions due to the interaction between par-ticles and wall. Particles could be denser eithernear a wall or the center of conduits. Figure 7shows three different particle distribution profiles

with an average volume fraction of 2.5%. Theresulting profiles of temperature, temperaturegradient, and the effective thermal conductivityare shown in Figure 8. The temperature profilesfor all cases are almost the same, while T(x) andkeff(x) change and differ significantly. The thermalconductivity with a concave ad-distribution ishigher near x0, whereas that for the uniform

0 0.2 0.4 0.6 0.8 1x [m]

0

0.01

0.02

0.03

0.04

0.05

volumefraction

uniformconcaveconvex

qx=const.

Figure 7. Volume fraction Profiles.

Figure 8. Effect of volume fraction change on temperature-,

temperature gradient-, and effective thermal conductivity-

profiles.

c

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x[m]

300

305

310

315

320

325

330

T[K]


x[m]

22

23

24

25

26

27

28

dT__dx

[K/m]


0 0.4 0.6 0.8

x[m]

0.7

0.75

0. 8

keff

[W/mK]


0.2 1

0 0.4 0.6 0.80.2 1

0 0.4 0.6 0.80.2 1

(a)

(b)

(c)

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distribution case is highest near x1 m. This isthe impact of the particle volume fraction on theBrownian effect part of the effective thermalconductivity.

Conclusions

A new thermal conductivity model for nanofluidshas been developed, which takes the effects ofparticle size, particle volume fraction and tem-perature dependence as well as properties of baseliquid and particle phase into consideration by

considering surrounding liquid traveling withrandomly moving nanoparticles. The time scalesfor such a complex particle-liquid body creatingmicro-mixing and local heat transfer were found tobe of about the same magnitude, which manifeststhe efficient energy transport due to Brownianmotion. The strong temperature dependence wasattributed to the long impact range of interparticlepotential. In the new model, the Brownian motioneffect was found to become more effective athigher temperatures as observed experimentally.

The effect of interparticle potential was found tobe very important for dense nanofluids, i.e.,

ad > 1% by comparing the experimentallyobtained thermal conductivities with the calculatedinterparticle potentials. The electric dipole constantwas observed to be a very important parameterinfluencing the interparticle potential and henceresulting in a particle-type dependence of the ther-mal conductivity.

The new model was tested with simple thermalconduction cases, and demonstrated that for agiven heat flux, the temperature gradient changessignificantly due to a variable thermal conductiv-ity, which mainly depends on particle volumefraction, particle size, particle material and tem-

perature. To improve the accuracy and versatilityof the keff model, more experimental data sets areneeded.

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A New Thermal Conductivity Model for Nanofluids