International Journal of Refrigeration 30 (2007) 323e328www.elsevier.com/locate/ijrefrig

Soret and Dufour effects on convective instabilitiesin binary nanofluids for absorption application

Jake Kim a, Yong Tae Kang b,*, Chang Kyun Choi a

a Seoul National University, School of Chemical and Biological Engineering, Gwanak-gu, Seoul, Republic of Koreab Kyung Hee University, School of Mechanical and Industrial Systems Engineering, Yong-in, Kyung-gi 449-701, Republic of Korea

Received 13 October 2005; received in revised form 21 April 2006; accepted 26 April 2006

Available online 10 July 2006

Abstract

Thermodiffusion (Soret effect) and diffusionthermo (Dufour effect) effects on convective instabilities in nanofluids havebeen theoretically investigated. Thermodiffusion implies that mass diffusion is induced by thermal gradient, which is so-calledthe Soret effect. Diffusionthermo implies that heat transfer is induced by concentration gradient, which is so-called the Dufoureffect. By using the linear stability theory under one-fluid model, a characteristic dimensionless parameter was newly obtained.From the instability analysis with given conditions, it is found that the convective motion in nanofluids sets in easily as the Soretand Dufour effects and the initial concentration of nanoparticles increase.� 2006 Elsevier Ltd and IIR. All rights reserved.

Keywords: Refrigeration; Absorption system; Colloidal suspension; Particle; Diffusion; Heat transfer; Thermal conductivity; Nanofluid

Nanofluides binaires utilises dans les applications a absorption:effets de Soret et de Dufour sur l’instabilite de la convection

Mots cles : Refrigeration ; Systeme a absorption ; Suspension collo€ıdale ; Particule ; Diffusion ; Transfert de chaleur ; Conductivite

thermique ; Nanofluide

1. Introduction

Nanofluid is the promising heat and mass transfer me-dium which nanoparticles are well dispersed in the basefluid.It is widely accepted that stably distributed nanoparticles

* Corresponding author. Tel.: þ82 31 201 2990; fax: þ82 31 202

8106.

E-mail address: ytkang@khu.ac.kr (Y.T. Kang).

0140-7007/$35.00 � 2006 Elsevier Ltd and IIR. All rights reserved.

doi:10.1016/j.ijrefrig.2006.04.005

cause to increase the thermal conductivity of nanofluid ab-normally [1] and forced convective heat transfer coefficientin a tube [2]. Recently, Kim et al. [3] investigated the effectof nanoparticles on the convective instability and convectiveheat transfer characteristics of nanofluids by introducinga new factor and showed that nanofluids could be analyzedsimply under one-fluid model. For a special purpose, it ispossible to use a binary mixture instead of a pure liquid asa basefluid to make a nanofluid (binary nanofluid). In a binarymixture, it is well-known that the thermodiffusion effect

324 J. Kim et al. / International Journal of Refrigeration 30 (2007) 323e328

Nomenclature

a dimensionless wave numberC concentration (mol/m3)Cp heat capacity (W/kg K)D diffusivity (m2/s)DSr Soret coefficient (mol/K s m)d fluid layer depth (m)F dimensionless factorf addition factor of nanofluidg! gravitational accelerator vector (m/s2)g gravitational accelerator (m/s2)J flux (quantity/m2s)K diffusivity ratiok thermal conductivity (W/m2 K)P pressure (N/m2)Ra Rayleigh numberT temperature (K)U!

velocity vector (m/s)W vertical velocity (m/s)w dimensionless vertical velocityZ vertical coordinate (m)z dimensionless coordinate

Greeksa thermal diffusivity (m2/s)aDf Dufour coefficient (K m5/mol s)

bS solutal gradient (m3/mol)bT temperature gradient (K�1)g ratio of thermal conductivity

�¼knp=kbf

�d1 ratio of density

�¼rnp=rbf

�d2 ratio of capacity

�¼�rCp

�np=�rCp

�bf

�

fv volume fractionfw weight fractionf1 dimensionless concentrationl effective volumer density (kg/m3)t dimensionless timej separation ratio DbS=ðDSrbTÞ

Subscripts0 basic stateb bottomDf Dufouri initial stateR reference stateS solutal quantitySr Soretu upper

Superscript* amplitude

(Soret effect) as well as the concentration gradient can in-duce mass diffusion. Ryskin et al. [4] investigated the Soreteffect of nanoparticles on the convective instability in ferro-fluids where ferromagnetic nanoparticles are contained.They showed that nanoparticles made the fluid more unsta-ble than the pure fluid with the same thermal gradient. Thebinary nanofluid is applicable as a working fluid in absorp-tion refrigeration, as a solution in electro or electroless plat-ing and as a transfer medium in medical treatment. Inabsorption systems with H2O/LiBr or NH3/H2O solutionwith nanoparticles (binary nanofluid solution), the metalnanoparticles such as Fe and Cu can enhance the heat trans-fer performance by the high effective thermal conductivityand can also improve the mass transfer performance by thehigh mixing rate.

The objective of the present study is to investigate theSoret and Dufour effects on the convective instabilities inthe binary nanofluids. In the previous studies, the convec-tive instabilities have been carried out for the normal nano-fluids [3], binary mixtures [4] and for thermodiffusioneffect in binary nanofluids [5]. This study expands the con-vective instability analysis from the normal nanofluids tothe binary nanofluids where the binary fluids are used asbase ones and investigates the various effects such as theSoret and Dufour effects on the thermal fluid characteristicsof nanofluids.

2. Mathematical formulation

When a quiescent, horizontal nanofluid layer is heatedfrom below, the buoyancy-driven convection, so-called Ray-leigheBenard convection will set in with exceeding a certainthermal gradient as shown in Fig. 1. This well-known phe-nomenon has been theoretically and experimentally investi-gated under linear and non-linear stability theories. Basicmathematical formulation is carried out in detail inRef. [3]. The linear temperature gradient causes the linear

heating

d

Tu

Tb

Benard-Rayleigh cells temperature profile

Fig. 1. Schematic diagram.

325J. Kim et al. / International Journal of Refrigeration 30 (2007) 323e328

concentration distribution of nanoparticles by expanding thevolume of basefluid as follows:

C0 ¼ Cif1� bTDTðd� ZÞg: ð1Þ

From Eq. (1), it is inferred that the solutal gradient,DC(¼Ci�C0) is directly related to the thermal one, DT.

The thermal and solutal gradients imposed through nano-fluids led to simultaneous heat and mass transfer. Inmixtures, temperature and concentration gradients inducemass flow, called the Soret effect [5]. On the other hand con-centration gradient causes heat flow, called the Dufour ef-fect. With the above effects, heat and mass flows can beexpressed as follows [6]:

�Jh ¼ kVTþ rCpaDfVC ð2Þ�Jm ¼ DVCþDSrVT: ð3Þ

Thermodynamically, Dufour coefficient, aDf and Soretcoefficient, DSr are defined with thermodynamic proper-ties, but for convenience they are expressed as simplesymbols [7].

The governing equations with the Soret and Dufoureffects and the linear equation of the state are given by [8]

V$U!¼ 0; ð4Þ

rR

D

DtU!¼�VPþ mV2 U

!þ rg;! ð5Þ

D

DtT ¼ aV2T þ aDfV

2C; ð6Þ

D

DtC¼ DV2CþDSrV

2T; ð7Þ

r¼ rR½1� bTðT� TRÞ þ bSðC�CRÞ�: ð8Þ

where the total time derivative D/Dt represents v=vt þ U!

$V.In the one-fluid model based on the volume average

concept, the dependent variables, W, T and C can besubstituted by the volume average form without any changeof equation forms expect those of physical properties frompure ones to mixture. It is similar to the Boussinesq approx-imation [9].

By using the linear stability theory under the principle ofthe exchange of stability, the main stability equation ofdimensionless vertical velocity amplitude is obtained as

�D2 � a2

�3w� ¼ Ra a2w�: ð9Þ

where the derivative D represents for d($)/dz. It is remark-able that Eq. (9) is the same as the result from the linear sta-bility theory [9]. Therefore by using Ra the convectivestability of nanofluids can be analyzed in the same manneras for the normal fluid.

Here the representative Rayleigh number Ra is defined as

Ra¼ Ra�1þFS þFSrþFDf

�½1�K��1; ð10Þ

where

Ra¼ gbTDTd3

an; ð11Þ

FS ¼bSCia

D; ð12Þ

FSr ¼bSDSr

bTD; ð13Þ

FDf ¼bTCiaDf

D; ð14Þ

K ¼ aDfDSr

aD: ð15Þ

Ra means the Rayleigh number of a basefluid based on tem-perature gradient and physically the ratio of gravitationaltime scale to diffusive time scale. FS, FSr and FDf representthe solutal, Soret and Dufour effects, respectively. It isnoticeable that they are scaled by the solutal diffusivetime. K means the ratio of Dufour and Soret coefficients tosolutal and thermal ones. If K is unity, heat and mass flowsdisappear due to Dufour and Soret flows. Eq. (10) obtainedhere expresses the stability condition of nanofluids. It is veryinteresting that the main effects characterizing a system arelinearly added. It means that the effects independently act asthe driving forces to set in the convective motion in nano-fluids. If there is no Dufour effect, Eq. (10) is reduced tothe previous results [5].

3. Effect of nanoparticles

In order to investigate the effect of nanoparticles on theconvective instability in binary nanofluid, the equations ofstate of binary nanofluid as a function of the volume andweight fractions of nanoparticle are constituted.

The concentration of binary nanofluids is expressed as

Cs ¼ ð1�fwÞC; ð16Þ

Cnp ¼ fwC; ð17Þ

where fw denotes the weight fraction of nanoparticle inbinary nanofluids. It is related to the volume fraction ofnanoparticle in binary nanofluids fv as follows:

fw ¼fvd1

ð1�fvÞ þfvd1

; ð18Þ

where d1

�¼rnp=rbf

�denotes the ratio of the density of

nanoparticles to that of a basefluid.The relationship of the density of a nanofluid rnf is

obtained from mass conservation for a constant volume.

rnf

rf

¼ ð1�fvÞ þfvd1; ð19Þ

326 J. Kim et al. / International Journal of Refrigeration 30 (2007) 323e328

If the nanofluid is in the thermal equilibrium state, thefollowing relationship is valid.

�rCp

�nf�

rCp

�f

¼ ð1�fvÞ þfvd2; ð20Þ

where d2ð¼�rCp

�np=�rCp

�bfÞ denotes the ratio of the heat

capacity based on the volume of nanoparticles to that ofa basefluid. It is interesting that rnf and (rCp)nf are linearlyrelated with the volume fraction of nanoparticles fv. Forthe viscosity m, the Brinkman model has been recommended[10].

mnf

mbf

¼ 1

ð1�fvÞ2:5: ð21Þ

For the colloidal suspension, the authors proposed thefollowing relationship for the thermal conductivity of nano-fluid knf modifying the Bruggeman model where the meanfield approach was applied [3].

knf

kbf

¼ ð3lfv � 1Þgþf3ð1� lfvÞ � 1gþffiffiffiffiffiffiDB

p

4; ð22Þ

where

DB ¼ ½ð3lfv � 1Þgþf3ð1� lfvÞ � 1g�2þ8g: ð23Þ

The effective volume l means physically the increasingvolume quantity by the chaotic motion of nanoparticlessuch as the Brownian motion.

Inserting Eqs. (16) and (17) into Eq. (7) produces

D

Ds

¼ ð1�fwÞ þfwd3: ð24Þ

where d3

�¼Dnp=Ds

�denotes the ratio of the diffusion coef-

ficient of nanoparticles in binary nanofluids to that of solutein binary basefluid.

While the relationships of the density and the heat capac-ity are expressed in a linear form with respect to the volumefraction of nanoparticles, the relation of the diffusion coeffi-cient appeared as the function of the weight fraction ofnanoparticles.

If there is no Dufour effect, the following relationshipcan be induced for the separation ratio j.

j¼ ð1�fwÞ þfwd4

ð1�fwÞ þfwd3

jbf ; ð25Þ

where d4ð¼ðDSrÞnp=ðDSrÞsÞ denotes the ratio of thepseudo Soret coefficient of nanoparticles in binary nano-fluids to that of solute in binary basefluid. As stated earlierthe separation ratio j implies the influence of the Soreteffect on the convective instability in binary nanofluids.

Using the above mentioned simple relationships, the con-vective instability and the heat transfer characteristics of a

binary nanofluid can be analyzed as a function of theproperties of a binary basefluid and nanoparticles, and thevolume fraction of nanoparticles. The Rayleigh numberfor a binary nanofluid which is the most important parameterto measure the stability is expressed as follows:

Ranf ¼ gRabf ð26Þ

Here the addition factor g for binary nanofluids is definedas

g¼1þjnf þ

anf

Dnf

ðbSÞnfðCs;iÞnf

1þjbf þabf

Dbf

ðbSÞbfðCs;iÞbf

f ð27Þ

where an addition factor f of normal nanofluids is defined as[3]

f ¼ Ranf

Rabf

: ð28Þ

They play the key role to measure the effect of nanopar-ticles on the convective instability in binary nanofluids aswell as in normal (single component) nanofluids. For exam-ple, if g has the value of 2, RayleigheBenard convection ina binary nanofluid sets in more easily than that of a binarybasefluid by a half of temperature difference.

4. Results and discussion

The base conditions for numerical analysis are summa-rized with data in Table 1. Fig. 2 shows the effect of

Table 1

Base data for numerical calculation

D (m2/s) bT (K�1) Cs,i (kg/m3) fv d1 d2 jbf

10�6 10�3 102 10�1 8.96 0.83 �0.5

-100 -50 0 50 100-2000

-1000

0

1000

2000

stable

unstable

00.050.1

Ra

unstable

ψbf

φv

Fig. 2. Ra vs jbf for various fv’s.

327J. Kim et al. / International Journal of Refrigeration 30 (2007) 323e328

nanoparticles on convective instability in (Ra, jbf)-space.The convective stability is expressed through the stabilitycurves. The inner region of the curves represents the unsta-ble state and the outer region the stable one. As can be seen,Ra sharply decreases with an increase in the absolute valueof jbf. It means that the Soret effect of solute dissolved ina nanofluid causes the convective motion more easily. Fur-thermore, it is found that with an increase of the volume frac-tion of nanoparticles, the stable region becomes wider forRa< 0 (upper heating system), but there is a switching re-gion where the state of the liquid layer changes from stableto unstable with respect to jbf for Ra> 0 (bottom heatingsystem). It means that for upper heating system nanopar-ticles make the liquid layer stable while for bottom heating

-100 -50 0 50 100-1000

0

1000

δ4=4 δ4=-2 δ4=-5

δ4=0δ4=4

δ4=-1

Ra

ψbf

δ4=0

δ4=-5δ4=-2

Fig. 3. Ra vs jbf for various d4’s.

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Cu

g or

f

φv

g fAg

Fig. 4. Comparison of addition factors g and f between silver nano-

fluids and copper ones.

system nanoparticles can cause a unique convective motion.It is expected that at jbfx�0:3 and Ra¼ 103 the liquidlayer without nanoparticles is stationary while the liquidlayer with nanoparticles of fv ¼ 0:1 is unstable.

Fig. 3 shows the Soret effect of nanoparticles on convec-tive instability in (Ra, jbf)-space for given conditions. Ifd4> 0, both the Soret effect of solute and that of nanopar-ticles reinforces to induce convective motion in binary nano-fluids. On the other hand, if d4< 0, the effect is reversed. Ford4>�1, the fluid becomes unstable with an increase of d4,but for d4<�1, it becomes stable with an increase of the ab-solute value of d4 for given conditions. From these results, itis concluded that d4 plays an important role in the convectiveheat transfer of binary nanofluid, and it can be a good indexto select the best nanoparticleesolute pair for heat transferenhancement.

The addition factors, f and g, are compared between sil-ver based binary nanofluid and copper based ones as shownin Fig. 4. Properties used in numerical calculation are tabu-lated in Table 2. The results show that the binary additionfactor g is always higher than the normal addition factor ffor normal basefluid. It means that the heat transfer enhance-ment by the Soret effect in binary nanofluids is more signif-icant than that in normal nanofluids. Furthermore, theprofiles for silver are always higher than those for copper.

Table 2

Properties of a basefluid (water) and nanoparticles (copper and

silver) [11]

r (kg/m3) Cp (J/kg K) k (W/mK) g d1 d2

H2O 997 4180 0.607

Cu 8933 385 401 661 8.96 0.83

Ag 10,500 235 429 714 10.5 0.59

-1.0 -0.5 0.0 0.5 1.00.0

0.4

0.8

1.2

1.6

2.0

Unstable

Ra

X 10

-3

FSr

FDf

10-2

10-1

FS = 1

K = 10-4

Stable

Fig. 5. Ra vs FSr for various FDf.

328 J. Kim et al. / International Journal of Refrigeration 30 (2007) 323e328

This is because the thermal conductivity of silver is higherthan that of copper.

Fig. 5 shows the Dufour effect on the convective instabil-ity in (Ra, FSr)-space for given conditions. The upper regionof the stability curves falls on the unstable state and thelower one on the stable state. As shown in Fig. 2, Ra de-creases with respect to FSr. It means that the Soret diffusionmakes nanofluids unstable. As can be seen, the stabilitycurves move downward with an increase of FDf. The fact im-plies that the Dufour diffusion destabilizes the nanofluids.

From the definitions of FS and FDf, the effect of Ci is in-cluded in FS and FDf implicitly. Fig. 6 shows the effect of theinitial concentration of nanoparticles on the convective in-stability in (Ra, FSr)-space with given conditions. As shownin Fig. 6, it is found that the denser nanofluids are the moreunstable they become.

5. Conclusions

The Soret and Dufour effects on the convective instabil-ities in nanofluids have been investigated. From the resultsof the present study, the following conclusions are drawn:

1. Both the Soret and the Dufour effects make nanofluidsunstable.

-1.0 -0.5 0.0 0.50.0

0.2

0.4

0.6

0.8

1.0

Ra

X 10

-4

FSr

Ci

102

103

104

FS = 1

FDf = 10-2

K = 10-4

1.0

Fig. 6. Ra vs FSr for various Ci.

2. The heat transfer enhancement by the Soret effect inthe binary nanofluids is more significant than that inthe normal nanofluids.

3. The denser the initial concentration of nanoparticles isthe more unstable the nanofluids become.

The results from the present study are useful to analyzethe convective instability in nanofluids in easy manner andthey can provide a guideline to select the nanoparticleebasefluid pair in nanofluids.

Acknowledgement

This work was supported by Korea Science and Engi-neering Foundation Grant (R01-2004-000-10736-0).

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