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2. Theoretical model for the effective thermal conductivity

of nanofluids

Scientists and engineers have been perplexed by the

anomalous thermal conductivity behavior of nanofluids[6–11]. Traditional conductivity theories of solid/liquidsuspensions, such as the Maxwell [13] and other macro-scale approaches [14] cannot explain that nanofluids haveanomalously high thermal conductivity at very low vol-ume fraction of nanoparticles, size-dependent conductiv-ity, and three fold higher critical heat flux than that of the base fluid. Recently, Jang and Choi [12] have con-structed a theoretical model based on Brownian motion[15], kinetic theory [16], and Kapitza resistance [17].The following analysis is based on Jang and Choi’smodel [12]. The thermal conductivity of nanofluidsinvolves four modes of energy transport in nanofluids

as follows.

2.1. First mode (thermal diffusion of base fluid)

The first mode is the thermal diffusion of a base fluid,which is given by

J U ¼ k BF

dT

d z ð1 f Þ ð2Þ

where f , J U , k BF, and T are the volume fraction, net energyflux across a plane at z axis, thermal conductivity of a base

fluid, and temperature, respectively.

2.2. Second mode (thermal diffusion in nanoparticles)

The second mode is the thermal diffusion in nanoparti-cles in fluids, which is given by

J U ¼ k nanodT dZ f ð3Þ

where k nano is the effective thermal conductivity of a sus-pended nanoparticle which is related to the thermal con-ductivity of a solid nanoparticle and the Kapitza thermalresistance at the solid–liquid interface. Chen [18] investi-gated the thermal conductivity of a single particle whosesize is smaller than the mean free path of the energy carrierand developed a theoretical model for the thermal conduc-tivity of the single particle k particle given by

k particle ¼ k bulk

0:75d nano

lnano

0:75d nano

lnano

þ 1ð4Þ

where k bulk, d nano and l nano are the thermal conductivity of bulk material, characteristic length of nanoparticles, andmean free path of nanoparticles, respectively. The Kapitzaresistance per unit area RK can be calculated as [16]

R K ¼ 1

4 bC V va 1

ð5Þ

where

bC V , v, and a are the heat capacity per unit volume,

mean speed of free electron or lattice wave, and averaged

transmission probability, respectively. The latter is given by

Nomenclature

Ar aspect ratio, Ar ¼ H W C

C p specific heat (kJ/kg K)C mean speed of base fluid molecules (m/s)

C R:M random motion velocity of nanoparticles (m/s)C T translation speed of nanoparticles (m/s) bC V heat capacity per unit volume (J/m3 K)d diameter (m)D characteristic length (m)D0 diffusion coefficient (m2/s) f volume fractionh heat transfer coefficient (W/m2 K)H channel height (m)J U net energy flux (W/m2)k thermal conductivity (W/m K)k b Boltzmann constant (J/K)m mass (kg)Nu Nusselt number p pressure (Pa)PP pumping power (W)Pr Prandtl numberq heat generation (W)

Re Reynolds numberT temperature (C)u velocity along the x-direction (m/s)

v mean speed of free electron or lattice wave (m/s)W width of a channel and a side wall (m)W C channel width (m)

Greek symbols

b constant related to Kapitza resistancee porosity, e ¼ W C

W

l viscosity (N s/m2)h thermal resistance (K/W)q density (kg/m3)

Subscripts/superscripts

BF base fluid

bulk bulk materialseff effective propertyf fluidnano nanoparticle

2458 S.P. Jang, S.U.S. Choi / Applied Thermal Engineering 26 (2006) 2457–2463

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a ¼ 4Z 1Z 2

ðZ 1 þ Z 2Þ2 Oð101Þ; Z ¼ qv ð6Þ

where q is the density. By using Eqs. (5) and (6), we esti-mate the order of RK as

R K Oð107Þ ð7Þ

This value of Kapitza resistance per unit area is very small.However, we need to consider the Kapitza resistance todetermine the effective thermal conductivity of nanoparti-cles suspended in fluid, because the thermal resistance perunit area of the nanoparticle given by Eq. (8) is smallerthan Kapitza resistance per unit area when the characteris-tic length of the nanoparticle is at nanoscale:

Rnanoparticle ¼ d nano

k particle

ð8Þ

By using the series thermal resistance model,

R K þ

d nano

k particle ¼

d nano

k nano ð9Þ

we can obtain the effective thermal conductivity of nano-particles k nano given by

k nano

k particle

¼ d nano

d nano þ k particle R K ¼ b Oð102Þ ð10Þ

Based on Eq. (10), Eq. (3) can be expressed as

J U ¼ bk particle

dT

dZ f ; b ¼ 0:01 ð11Þ

where b is a constant related to the Kapitza resistance. b is

on the order of 0.01, which is consistent with the experi-mental results by Huxtable et al. [19].

2.3. Third mode (collision between nanoparticles)

The third mode is the collision of nanoparticles witheach other by translational motion of nanoparticles. Thisphysical meaning indicates that flux of energy is trans-ported by collision of nanoparticles with each other.

J U ¼ 1

3lC V C T f

dT

d z ¼ k BML

dT

d z ð12Þ

where C T, k BML and l are translational speed of a nanopar-ticle, the effective thermal conductivity of the third modeand mean collision length of a nanoparticle during thetranslation motion, respectively. By kinetic theory, thetranslational speed of a nanoparticle can be calculated

1

2mC 2T ¼

3

2k bT ! C T ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi18k bT

qpd 3nano

s ð13Þ

where d nano, k b = 1.3807 · 1023 J/K, and q are diameterof nanoparticles, Boltzmann constant and density, respec-tively. By an order-of-magnitude analysis of Eq. (12), wehave found that this mode is much smaller than the other

modes as given by

k BML Oð105Þ ð14Þ

So we can neglect the effect of the third mode.

2.4. Last mode (nanoconvection due to Brownian motion)

The last mode is the convection effect of Brownian

motion of nanoparticles which is due to the thermallyinduced fluctuations. The Brownian motion causes nano-particles to vary in direction many millions of times persecond, as presented by Einstein [15].

J U ¼ hðT nano T BFÞ f ¼ 3C 1d BF

d nano

k BF Re2d nano Pr

dT

d z ð15Þ

Red nano ¼C R:Md nano

v ; C R:M ¼

2 D0

lBF

ð16Þ

where d BF, d nano are the equivalent diameters of a base fluidmolecule and a nanoparticle, respectively. Red nano

and Pr

are the Reynolds number and the Prandtl number and

C 1 = 6 · 106, C R:M, D0, l BF, and v are an empirical con-stant, random motion velocity of nanoparticles, diffusioncoefficient presented by Einstein [15], mean free path andkinematic viscosity of base fluid, respectively. The meanfree path of base fluid [16] is defined by

lBF ¼ 3k BF bC V C ð17Þ

where C is the mean speed of base fluid molecules.From Eqs. (2)–(17), we can theoretically derive the fol-

lowing expression for the thermal conductivity of nano-fluids, k nanofluids:

k nanofluids ¼ k BFð1 f Þ þ bk particle f

þ 3C 1d BF

d nano

k BF Re2d nano Pr ð18Þ

In order to validate this new model for the effective ther-mal conductivity of nanofluids, Eq. (18), we compare theprevious experimental results [6] with results predicted fromthis new model. The experimental results match closelywith results from this new model as shown in Fig. 1(a).Fig. 1(b) shows the predicted thermal conductivity of water-based nanofluids containing 6 nm copper-nanoparti-cles and 2 nm diamond-nanoparticles. The effective thermalconductivity of these nanofluids will be used for numericalinvestigation on the cooling performance of a microchannelheat sink. It should be noted that there is a fundamental dif-ference between metal/diamond and metal oxide in terms of thermal performance. One of the questions we want to raiseis whether the difference is due to the conductivity of thenanoparticles themselves or due to some other mechanism.Although the thermal conductivity of metal/diamond ismore than an order of magnitude higher than that of metaloxide, the thermal conductivity of nanofluids is only weaklydependent on that of nanoparticles when the nanoparticleconductivity reaches a threshold level. Jang and Choi haveshown that a key mechanism governing the effective thermal

conductivity of nanofluids is not material properties of

S.P. Jang, S.U.S. Choi / Applied Thermal Engineering 26 (2006) 2457–2463 2459

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nanoparticles but Brownian motion of nanoparticles [12].The Jang and Choi model shows that nanoparticle size isthus the major parameter. Because the model can predictstrongly size-dependent conductivity, it can be used to pre-dict the effective thermal conductivity of nanofluids con-

taining smaller nanoparticles such as 2 nm diamond and6 nm copper or larger metal oxides such as 30 nm Al2O3

and 20 nm CuO. However, because metal/diamond parti-cles are much smaller than metal oxides, the data fromthe metal oxides in Fig. 1(a) cannot be simply extrapolatedinto the performance of the ultra-small Cu or diamond par-ticles in Fig. 1(b).

3. Microchannel heat sink with nanofluids

In the paper the problem of forced convective flowthrough a microchannel heat sink of water-based nanofl-

uids containing 1 vol.% of 6 nm copper-nanoparticles

or 1 vol.% of 2 nm diamond-nanoparticles is consid-ered. Fig. 2 shows physical and numerical domains.The top surface is insulated and the bottom surface isuniformly heated. A coolant passes through the micro-channel heat sink and takes heat away from a heat-dissi-pating component attached below. The material of amicrochannel heat sink is silicon (k s = 150 W/m K). Inanalyzing the problem, the flow can be assumed to belaminar and both hydrodynamically and thermally fullydeveloped because the hydraulic diameter of the micro-channel heat sink is sufficiently small and the Reynoldsnumber is on the order of 100 or less. All thermophysicalproperties are assumed to be constant. Specifically, thethermal conductivity of nanofluids is calculated by usingEq. (18) and the effective viscosity of nanofluids is consid-ered by the Einstein model [15] applicable to dilutesuspensions.

leff ¼ lf ð1 þ 2:5 f Þ for f < 0:05 ð19Þ

0.00 0.01 0.02 0.03 0.04 0.05

1.00

1.05

1.10

1.15

1.20

1.25

k e f f / k b a s

e f l u i d

Volume Fraction

Base Fluid: Water

Al2O

3, Experimental Results [6]

CuO, Experimental Results [6]

Al2O

3, New Model

CuO, New Model

0.000 0.005 0.010 0.015 0.020 0.025

0.5

1.0

1.5

2.0

2.5

k e f f / k b a s e f l u i d

Volume Fraction

Theoretical Resutls

Water

Water-based Copper (6nm)

Water-based Diamond (2nm)

(a)

(b)

Fig. 1. Thermal conductivity of nanofluids normalized to that of the basefluid as a function of nanoparticle volume fraction: (a) comparison of model predictions with experimental thermal conductivity data for copperoxide-in-water and aluminum-in-water nanofluids, (b) prediction forthermal conductivity of 6 nm copper-in-water and 2 nm diamond-in-water.

q

y

z

x C w

W

H

0 z

2

W

y

H

2

C W

(a)

(b)

Fig. 2. Schematic of a microchannel heat sink: (a) physical domain, (b)numerical domain.


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In addition, the effective specific heat and density of nano-fluids are calculated by mixing theory [20,21].

C p ;eff ¼ C p ;f ð1 f Þ þ C p ;particle f ð20Þ

qeff ¼ qf ð1 f Þ þ qparticle f ð21Þ

where C p,f , C p,particle, C p,eff , lf , leff , qf , qparticle, and qeff are

specific heats of base fluid, nanoparticles, effective specificheat of nanofluids, viscosity of base fluid, effective viscosityof nanofluids, density of base fluid, nanoparticles, and theeffective density of nanofluids, respectively.

In order to evaluate the cooling performance of themicrochannel heat sink with nanofluids, the momentum

equation for the fluid and the energy equation for boththe fin and the fluid should be solved. The governing

Table 1Comparison between the thermal resistance of Min et al. [5] and that of present model

Microchannel heatsink with water [5]

Microchannel heatsink with water(present model)

Pumping power (W) 2.27 2.27

Porosity, e = W C/W 0.58 0.58Aspect ratio, Ar = H /W C 8.2 8.2Thermal resistance, h (C/W) 0.06019 0.06018

Fig. 3. Temperature contours for cross-sectional area of microchannel heat sinks: (a) water, (b) nanofluid: water + copper (6 nm), (c) nanofluid:

water + diamond (2 nm).


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momentum and energy equations and boundary conditionsare given as

X-momentum equation

1

leff

d p

d xþo

2u

o y 2þo

2u

o z 2 ¼ 0

u ¼ 0 at y ¼ 0; y ¼ H

ou

o z ¼ 0 at z ¼ 0; u ¼ 0 at z ¼

W C

2

ð22Þ

Energy equation

qeff C p ;eff uoT

o x ¼ k eff

o2T

o y 2 þ

o2T

o z 2

T ¼ T W at y ¼ 0;

oT

o y ¼ 0 at y ¼ H

oT

o z ¼ 0 at z ¼ 0; z ¼

W

2

ð23Þ

where p, T , T W, u, W , and W C are pressure, temperature,wall temperature, x-velocity, width of a channel and a sidewall, and channel width, respectively. The governing equa-tions are solved by the control-volume-based finite differ-ence method. The cooling performance of a microchannelheat sink with nanofluids is evaluated by the thermal resis-tance, h, defined as

h ¼T max T inq

ð24Þ

where q, T in, and T max, are heat generation, temperature of an inlet coolant and the maximum temperature at the bot-

tom surface of the microchannel heat sink, respectively.The thermal resistance is numerically calculated underthe fixed pumping power, PP = DP Æ Q where DP and Q

are pressure drop across a system and volume flow rate,respectively. In our model, we change the entry velocityto keep the pumping power fixed. The condition of thefixed pumping power used in this paper physically meansthat the power required to drive the fluid through the heatsink is fixed. Therefore, the fixed pumping power conditionfor evaluating cooling performance of heat sinks is a phys-ically practical constraint [3]. Before investigating the cool-ing performance of a microchannel heat sink with

nanofluids, we validated the numerical code with resultspresented by Min et al. [5] as shown in Table 1.Based on the numerical results, Fig. 3 shows colored

temperature contours of a cross-sectional area of a micro-channel heat sink with water, water-based nanofluids con-taining Cu (1 vol.%, 6 nm), and water-based nanofluidscontaining diamond (1 vol.%, 2 nm) under the conditionof fixed pumping power, 2.25 W and fixed heat flux,300 W/cm2. The height, aspect ratio, and porosity of microchannel heat sinks are 350 lm, 8.6 and 0.5. Theporosity is defined by

e ¼W C

W ð25Þ

The base size of microchannel heat sinks is 1 cm · 1 cm.Fig. 3(a) shows that, when water is used as the coolant,there is a large region of deep blue, indicating the maxi-mum temperature difference greater than 13 C betweenthe heated microchannel wall and the coolant. However,this deep blue region shrinks for water-based nanofluidscontaining Cu (1 vol.%, 6 nm) as shown in Fig. 3(b), andcompletely disappears for water-based nanofluids contain-ing diamond (1 vol.%, 2 nm) as shown in Fig. 3(c). The col-ored contour maps are a powerful way to visually examinethe cooling performance of different coolants flowing in themicrochannel heat sink. As shown in Fig. 3, the uniformityof temperature in the channel is enhanced by using nanofl-uids because the thermal conductivity of nanofluids is lar-ger than that of water.

Fig. 4 shows that nanofluids reduce the thermal resis-tance, as defined by Eq. (24). The results indicate thatthe cooling performance of a microchannel heat sink atPP = 2.25 W is enhanced by about 10% and 4% for water-based nanofluids containing diamond (1 vol.%, 2 nm) andcopper (1 vol.%) respectively, compared with that of themicrochannel heat sink with water. Thus, when the temper-ature difference between junction temperature and inlet

coolant temperature (ambient temperature) is 80 C, heatflux of up to 1350 W/cm2 can be dissipated by a microchan-nel heat sink with water-based nanofluids containing dia-mond (1%, 2 nm) at PP = 2.25 W. So, we can propose amicrochannel heat sink with nanofluids as a next generationcooling devices for removing ultra-high heat flux.

4. Conclusion

A combination of microchannel heat sink (small charac-teristic length) with nanofluids (enhanced thermal conduc-tivity) has been introduced as a new direction for high

cooling performance. Based on a theoretical model of ther-

1.50 1.75 2.00 2.25 2.50

0.058

0.060

0.062

0.064

0.066

0.068

0.070

0.072

0.074

T h e r m a l R e s i s

t a n c e ( o C / W )

Pumping Power (kW)

Microchannel Heat Sink

Water

Nanofluids: Water+Cu(1%, 6nm)

Nanofluids: Water+Diamond(1%, 2nm)

Fig. 4. Thermal resistances of microchannel heat sinks with water, water-based nanofluid containing copper and water-based nanofluid containingdiamond.


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mal conductivity of nanofluids that accounts for the funda-mental role of Brownian motion, the temperature contoursand thermal resistance of a microchannel heat sink withnanofluids have been numerically investigated. The resultsshow that the cooling performance of a microchannel heatsink with water-based nanofluids containing diamond

(1 vol.%, 2 nm) at the fixed pumping power of 2.25 W isenhanced by about 10% compared with that of a micro-channel heat sink with water. Nanofluids reduce both thethermal resistance and the temperature difference betweenthe heated microchannel wall and the coolant. Finally,the potential of deploying a combined microchannel heatsink with nanofluids as the next generation cooling devicesfor removing ultra-high heat flux is shown. Finally, thepotential of deploying a combined microchannel heat sinkwith nanofluids as the next generation cooling devices forremoving ultra-high heat flux as much as 1350 W/cm2,when the difference between junction temperature and inletcoolant temperature is 80 C, is demonstrated.

Acknowledgements

This work was supported by the Korea Research Foun-dation Grant (KRF-2004-003-D00047). S. Choi was par-tially supported by the U.S. Department of Energy,Office of Transportation Technologies—Office of Free-domCar and Vehicle Technologies, under Contract W-31-109-Eng-38.

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