ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 322 (2010) 698–704

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials

0304-88

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jmmm

Thermomagnetic natural convection of thermo-sensitive magnetic fluids incubic cavity with heat generating object inside

Hiroshi Yamaguchi a, Xin-Rong Zhang b,a,n, Xiao-Dong Niu a, Keisuke Yoshikawa a

a Energy Conversion Research Center, Department of Mechanical Engineering, Doshisha University, Kyoto 630-0321, Japanb Department of Energy and Resources Engineering, College of Engineering, Peking University, 100871, PR China

a r t i c l e i n f o

Article history:

Received 3 March 2009

Received in revised form

13 October 2009Available online 31 October 2009

Keywords:

Nature convection

Magnetic fluid

53/$ - see front matter & 2009 Elsevier B.V. A

016/j.jmmm.2009.10.044

esponding author.

ail address: xrzhang@coe.pku.edu.cn (X.-R. Zh

a b s t r a c t

Experiment and numerical study were conducted to investigate the heat transfer characteristics

of a thermo-sensitive magnetic fluid (TSMF) filled in a cubic container with a heat generating

square cylinder stick inside and under a uniform magnetic field. The experimental results show

that, regardless of the heat generating object sizes, the heat transfer characteristic of the TSMF is

enhanced when the magnetic field is applied to the TSMF. However, the heat transfer of the TSMF

becomes poor as the size of the inside heat generating object increases because the space where the

fluids go through becomes narrower and the flow is obstructed when the heat generating object size

becomes bigger. Numerical simulation based on the Lattice Boltzmann method confirmed the

experimental findings, and disclosed more flow details of the natural convection of the TSMF inside

cavity.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

Magnetic fluids are super paramagnetic fluids formed by astable colloidal suspension of ferromagnetic nanoparticles dis-persed in carrier liquids such as water or kerosene [1]. Thebehaviors of the magnetic fluids are characterized by the complexinteractions of their hydrodynamic and magnetic properties withmagnetic forces. With the deep understanding of the magneticfluids, numerous applications have been put into practices inindustry in the past decades, i.e., magnetism seal in rotationalbearings, magnetism ink in printers and sound speakers indifferent home electric appliances [2].

Among various magnetic fluids, there is a so-called thermo-sensitive magnetic fluid (TSMF), whose magnetization stronglydepends on temperature. Because of this thermomagneticproperty and its easily being operated in the room temperaturerange, the TSMF is considered as a promising fluid in energyconversion and heat transport systems with small length scales orunder microgravity environments [2,3]. Due to the significant roleof convection mechanism in the energy conversion and heattransportation, a thorough understanding of the relationsbetween the applied magnetic field and the resulting convectionis necessary.

Various studies of thermomagnetic convections of the TSMFhave been carried out in the past years [4–8]. Finlayson [4] first

ll rights reserved.

ang).

studied thermomagnetic convection of the TSMF and showed thatthere exists a critical stability parameter beyond which thethermomagnetic convection occurs. Schwab et al. [5] conductedan experimental investigation of the convective instability in ahorizontal layer of the TSMF and characterized the influence ofthe magnetic Rayleigh number on the Nusselt number. Krakovand Nikiforov [6] addressed the influence of the relativeorientation of the temperature gradient and magnetic field onthermomagnetic convection in a square cavity. Yamaguchi et al.[7,8] performed experiments and numerical analyses in a squareenclosure and characterized the heat transfer in terms of amagnetic Rayleigh number.

However, in most of practical situations, a container with heatgenerating objects is often encountered, and studying theconvection of such a case is scarce but nontrivial in bothacademics and engineering. For this reason, in the present study,the nature convection of the TSMF in a cubic container with a heatgenerating square cylinder object inside under a uniformmagnetic field is investigated experimentally and numerically.Particular attention is focused to examine the effect of themagnetic field on the heat transfer characteristics and flowbehaviors.

The rest of the paper is organized as follows. In Section 2, wegive a detail description of experiments carried in the presentstudy. Section 3 briefs the numerical simulation based on theLattice Boltzmann method (LBM) [9]. Section 4 presents all resultsobtained in experiments and numerical simulations. Discussionsare also given in this section in terms of analyses of the obtainedresults. A conclusion is given in Section 5.

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H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 698–704 699

2. Experiment

2.1. Test fluid

The TSMF used in the present study is a solvent that the Mg–Znferrite nanoparticles suspend in a kerosene solution. This solventhas strong temperature dependence of the magnetization in thenormal room temperature range under the constant magneticfield strength (see Fig. 1), and thus can be used as an ideal testfluid to study the convection mechanism of the TSMF. Table 1shows the properties of the test TSMF at representativetemperature T0=298.15 K.

2.2. Experimental setup

Schematic diagrams of the experimental setup and testsection in the present experiment are shown in Fig. 2(a) and(b), respectively. The test section includes a cubic containerwith the side length of 7.5 mm and filled with the test TSMF. Thecontainer is sealed with two heat conductive acrylic blocks at itsup and bottom, and insulated acryl plates of thickness 1.0 mmand glass–wool plywood at both sides of it. To ensure theinsulation condition on two side walls of the container, Styrofoam

is further adhered tightly on both sides of the container.A brass square cylinder stick as a heat generating object(Fig. 3) is inserted in the center of the cubic containerand its end is heated through a winded silicon cord heater .The material properties of the brass stick are given in Table 2. The

Fig. 1. Temperature characteristics of magnetization of the test TSMF.

Table 1The TSMF fluid properties used in present study at T0=298.15 K.

Density r0. (kg/m3) 1397�103

Viscosity Z.(Pa s) 1.680�10�2

Thermal conductivity l (W/(m K) 1.750�10�1

Saturation magnetization Ms (A/m) 2.95�105

Gravitational acceleration g (m/s2) 9.8

temperatures on the top and bottom of the container are controlledby an air conditioner right neighboring the heat conductive blocks

. The air conditioner consists of two Peltier elements , a heatpump , two heat sinks and a fan . The Peltier elementsare connected to a Peltier controller . Two electromagnets(Helmholtz coil) are put separately on the above and below ofthe air conditioner, and the uniform magnetic field is generated bya DC current controller . The temperatures in the test section aremeasured through the Thermister sensors and the obtained dataare recorded and processed by a Data processor .

2.3. Experimental measurements and evaluations

To evaluate the heat transfer characteristics of the TSMF insidethe cubic container, the spatial distributions of the magnetic fieldsare first measured and the corresponding magnetic field strengthsH0 are calculated before the TSMF is filled in the container. Thisprocedure also confirmed the uniformity of the magnetic fieldimposed in the container. In the measure process, from the centerof the cavity 15 equal-interval points along the radical and axialdirections are chosen, respectively, and the magnetic fieldstrength is measured using a Gauss meter (GM4002: accuracy72%, ELECTRONIC MAGNETIC INDUSTRIES). As shown in Fig. 4,which displays the distributions of the representative magneticfield along (a) the center of two electromagnets and (b) the radialdirection at central plane of the container, a very good uniformmagnetic field can be obtained with the use of electromagnets inthe present experiment.

After the magnetic field is measured, the TSMF is then sealedin the container and the stick is heated on the end of it. Thetemperatures at the different positions along the center line of theheat conductive blocks, the container and the heat generatingstick are measured by the inserted thermal couples (YamariIndustries, TMB type, resolving power 1/10 K, wire diameterj=5�10�4 m) at the measured magnetic fields (see Fig. 3). Dueto the difficulties of directly measuring the temperatures on thewalls of the container, the temperatures of TU and TL on therespective upper and bottom walls of the container and that of TH

of the stick on one side wall of the container are extrapolatedbased on the measured temperatures.

The evaluation of the heat transfer characteristics of the testTSMF in the container is based on the following parameters:

Gr¼r2

0gbDTd3C

Z20

; Grm¼r2

0m0w0H20d2

C

Z20

Nu� ¼lCbCdSy

1=3

lDT

CI ¼Nu�Grma0 � Nu�Grm ¼ 0

Nu�Grm ¼ 0

;

where DT ¼ TH � ðTUþTLÞ=2 is the representative temperaturedifference, and Gr and Grm are the Grashof and magnetic Grashofnumbers, respectively; Nu� is an effective Nusselt numberconsidering the effect of the stick size; CI is the coefficient ofthe heat transfer increasing ratio; dC and dS are the side lengths ofthe cubic container and the heat generating stick, respectively;y¼ dC

dSis ratio of the side lengths of the container and the stick; lC

and bC are the thermal conductivity and the slope of the

Specific heat Cp.(J/kg K) 1.387�105

Expansion coefficient b (1/K) 6.90�10�4

Curie temperature Tc (K) 477.35

Reference temperature T0 (K) 298.15

Magnetization rate w0 0.2650

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Fig. 2. Schematic diagrams of (a) the experimental setup and (b) test section in the present study.

Fig. 3. Schematic diagrams of the cubic container and the heat generating object.

Table 2The material properties of the brass heat generating stick at T0=298.15 K.

Density r0 (kg/m3) 8.520�103

Specific heat Cp (J/kg K) 3.85�102

Thermal conductivity l (W/(m K) 1.110�10–1

H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 698–704700

approximated linear temperature distribution across the heatconductive blocks in Fig. 2, respectively; m0 is the magneticpermeability of vacuum.

In the present study, two heat generating sticks are tested andthey are

�

Case 1: longitude length=dc, side length: ds=dc/3 � Case 2: longitude length=dc, side length: ds=dc/2

Fig. 4. Distributions of the representative magnetic field.

The above two cases are investigated at a range of 0oGro160 based on the different magnetic Grashof numbers. Thetemperatures on the upper and bottom walls are fixed atTU=TL=T0=298.15 K.

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x

y

u = 0, Tu = 0

u = 0

∂xT = 0

u = 0 ∂yT = 0

u = 0 ∂yT = 0

u = 0

∂xT = 0

u = 0, Tb = 0

H = H0

z

g

u = 0Th = 1

u = 0∂yT = 0

Fig. 5. Sketch of the thermal magnetic natural convection flow condition in the

cubic cavity.

Fig. 6. Variations of Nun against Gr at four Grm of 0, 4.89�103, 1.10�104 and

1.96�104 for Case 1.

H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 698–704 701

3. Numerical methodology

In terms of the Lattice Boltzmann theory, the natural convec-tion of the TSMF in a cubic container with a heat generatingsquare cylinder stick inside and under a uniform magnetic fieldcan be solved by the following Lattice Boltzmann scheme byemploying three distribution functions for velocity, thermal andmagnetic field [9], respectively, as

faðrþxadt ; tþdtÞ � faðr; tÞ ¼ �faðr; tÞ � f eq

a ðr; tÞ

tf

þwaðtf � 0:5Þdt

tf c2s

FUðxa � uÞ ð1Þ

gaðrþxadt ; tþdtÞ � gaðr; tÞ ¼ �gaðr; tÞ � geq

a ðr; tÞ

tgþwaSdt ð2Þ

haðrþxadt ; tþdtÞ � haðr; tÞ ¼ �haðr; tÞ � heq

a ðr; tÞ

thð3Þ

with the equilibrium distribution functions defined as

f eqa ðr; tÞ ¼wa rþr0

xaUu

c2s

þ1

2c2s

ðxaUuÞ2

c2s

� u2

!" #( )ð4Þ

geqa ðr; tÞ ¼waT 1þ

xaUu

c2s

þ1

2c2s

ðxaUuÞ2

c2s

� u2

!( )ð5Þ

heqa ðr; tÞ ¼waj 1þ

xaUguT

c2s

þ1

2c2s

ðxaUguT Þ2

c2s

� ðguT Þ2

!( )ð6Þ

and the force and source term are

F¼ m0MrH � r0bðT � T0Þg ð7Þ

S¼ �m0TMU

DHDt

r0CP � m0HU@M@T

� �H

h i ð8Þ

where r=r(x,y,z) is the spatial vector; r is the density withconstant r0, u is the velocity, and T is the temperature; m0 is themagnetic permeability of vacuum, b is the expansion coefficientunder the Boussinesq approximation; H is the modulus of themagnetic intensity H and f is the scalar potential and rf=H forthe non-conductive magnetic fluid; M is the modulus of themagnetization M;Cp is the specific heats of respective fluid atconstant pressure; uT=�r(M/H)=w0rT/(Tc�T0) is defined as aneffective velocity with T0 and Tc being the reference and Curietemperatures, respectively, and w0 is the magnetization rate at T0;g is an adjustable preconditioning parameter introduced to ensurethat the solution of the LB scheme (3) remains close to a solutionof Maxwell equation of the magnetic field. The relaxationparameters in Eqs. (1–3) are given by

tf ¼Z

r0c2s dtþ0:5; ð9Þ

tg ¼D

ðDþ2Þ

l

r0CP � m0HU@M@T

� �H

h ic2

s dt

þ0:5; and ð10Þ

th ¼g

c2s dt

1þM

H

� �þ0:5; ð11Þ

respectively. Here Z is the dynamical viscosity, l are the coefficientsof thermal conductivity of the fluid. In numerical simulation, thevalue of the preconditioning parameter g is chosen by setting th=1.The sound speed cs, the weight coefficient wa and the discretevelocity xa used in the above LBM scheme can be referred to

the discrete velocity model of the D3Q19 for three-dimension (3D)[10]. The density, velocity, and temperature are calculated by

r¼P

afa

r0u¼Xa

faxaþ0:5dtF

T ¼Xa

ga

j¼Xa

ha ð12Þ

The boundary conditions of the macroscopic variables u, T aresketched in Fig. 5 in non-dimensional form. For the magnetic field,

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the magnetic potential boundary conditions are set by themagnetic intensity, which are described as follows:

@j@x x ¼ 0;L

¼ 0;@j@y y ¼ 0;L

¼ 0;@j@z z ¼ 0;L

¼H:

����������

����� ð13Þ

In our numerical simulation, the temperatures on the isolatedwalls and the scalar potentials on all walls are calculated based onthe second order extrapolation scheme. For the distributionfunctions of fa, ga and ha on the boundaries, the non-equilibriumbounce back boundary conditions [10] are employed when themacroscopic velocities, temperatures and scalar potentials on thewalls are known, and they are given as follows:

faðr; tÞ � f eqa ðr; tÞ ¼ faðr; tÞ � f eq

a ðr; tÞ; ð14Þ

gaðr; tÞ � geqa ðr; tÞ ¼ � gaðr; tÞ � geq

a ðr; tÞ� �

; ð15Þ

haðr; tÞ � heqa ðr; tÞ ¼ � haðr; tÞ � heq

a ðr; tÞ� �

; and ð16Þ

Fig. 7. Variations of Nun against Gr at four Grm of 0, 4.89�103, 1.10�104 and

1.96�104 for Case 2.

Fig. 8. The velocity vectors (color plotted against temperature) of Gr=75 insid

respectively. a in Eqs. (14–16) denotes directions of the unknowndistribution function, and a¼ � a. With the grid-independenceinvestigations of other two grids of 25�25�25 and 43�43�43in advance, all the simulations in the present study are carried outon the uniform grid of 37�37�37 in the range of experimentalGarshof numbers, and the results presented below are given in

the non-dimensional form by r/L, Uref ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigbLDTref

q �and T/DTref.

The calculated Nusselt number is given by

Nu¼

ZZ�@T

@Z

� �Z ¼ 0;L

dxdy: ð17Þ

4. Results and discussions

Figs. 6 and 7 show the variations of the effective Nusseltnumber Nu� against the Grashof number Gr at four magneticGrashof numbers Grm of 0, 4.89�103, 1.10�104 and 1.96�104

for Case 1 and Case 2, respectively. Experimental and numericalresults are all plotted in these two figures, and the symbols andlines represent the experimental data and LBM simulation results,respectively. As shown in Figs. 6 and 7, the calculated results are ingood agreement with the experimental data, all the effectiveNusselt numbers Nun measured at four magnetic Grashof numbersGrm for Cases 1 and 2 show an increasing tendency when theGrashof number Gr increases. This phenomenon is attributed tothe increase of the temperature differenceDT ¼ TH� ðTUþTLÞ=2.

The effect of the magnetic field can also be observed in thesetwo figures. Figs 6 and 7 display that the effective Nusselt numberincreases obviously with the increasing magnetic field, suggestingthat the TSMF has a good capability of the heat transfer under themagnetic field, and the stronger the magnetic field, the better theheat transfer characteristic. The size of the heat generating objecthas also remarkable effects on the heat transfer performance ofthe TSMF in the cubic container. This can be seen by comparingFig. 6 for Case 1 with Fig. 7 for Case 2. When the size of the heatgenerating object becomes larger, the effective Nusselt numberreduces, implying that the heat transfer of the TSMF in the cubiccavity becomes poor as the size of the inside heat generatingobject increases. This is because when the heat generating objectsize becomes bigger, the space where the fluids go throughbecomes narrower and flow is hugely obstructed.

The findings of the Nusselt numbers increasing with themagnetic Grashof number can be explained more clearly by thefollowing LBM results for Case 1, which depict details of the flowand magnetic fields. Fig. 8 shows the velocity vectors of Gr=75 for

e the cavity at Grm=1.10�104 (left) and 1.96�104 (right), respectively.

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Fig. 9. The temperature contours and velocity vectors of Gr=75 at four planes of the cavity at Grm=1.10�104 and 1.96�104, respectively.

Fig. 10. The coefficient CI as a function of Grm at different Gr for Case 1. Fig. 11. The coefficient CI as a function of Grm at different Gr for Case 2.

H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 698–704 703

Case 1 at Grm=1.10�104 (left) and 1.96�104 (right). As shown inFig. 8, four symmetric weak flow rolls occur near the isolate wallsinside cavity due to effects of the magnetic forces. Because the

temperature gradient is larger near the heat generating object, thevelocity is larger at that region. Moreover, the rolls becomestronger with the increase of the magnetic Grashof number. Fig. 9

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H. Yamaguchi et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 698–704704

(a) and (b) show the flow patterns and temperature fields ofGr=75 at four planes of the cavity for Case 1 at Grm=1.10�104

and 1.96�104, respectively. Corresponding to the velocity fieldshown in Fig. 8, the low-temperature fluid near the upper andbottom cooling walls is brought to the region near the heatgenerating object, and the high-temperature fluid near the heatgenerating object is brought to the side isolated walls, alongwhich it is further brought to the upper and bottom cooling walls.The heat transport rate is larger at high magnetic Grashof numberthan that at low magnetic Grashof number. As shown in Fig. 8 (a)and (b), it can also be observed that the heat is gradually transportfrom the hot side to the cool side along the heat generating object.

Figs. 10 and 11 show the coefficient CI of the heat transferincreasing ratio as a function of the magnetic Grashof numbersGrm at different Grashof numbers for Case 1 and Case 2,respectively. In the experiment, eight magnetic Grashofnumbers of 0, 1.22�103, 4.89�103, 7.64�103, 1.10�104,1.96�104, 3.06�104 and 4.40�104 are tested. As shown inthese two figures, the coefficient CI of the heat transfer increasingratio increases as Gr and Grm increases, and decreases as theshape size of the heat generating object increases. The results inFigs. 10 and 11 further confirm the observations and analysesfrom Figs. 6 and 7.

5. Conclusions

Experiment and numerical studies are conducted to investi-gate the heat transfer characteristics of a TSMF filled in a cubiccontainer with a heat generating square cylinder stick inside. Theexperimental results show that, regardless of the heat generatingobject sizes, the heat transfer characteristic of the TSMF isenhanced when the magnetic field is applied to the TSMF.However, the heat transfer of the TSMF becomes poor as the sizeof the inside heat generating object increases because the space

where the fluids go through becomes narrower and the flow isobstructed when the heat generating object size becomes bigger.Numerical simulation based on the Lattice Boltzmann methodconfirmed the experimental findings, and disclosed that, due tothe non-uniform distribution of the temperature caused by thenatural convection of the magnetic flow inside cavity, foursymmetric weak flow rolls occur near the isolate walls insidecavity due to effects of the magnetic force, and they bring the low-temperature fluid near the upper and bottom cooling walls to theregion near the heat generating object, and further stream thehigh-temperature fluid near the heat generating object to the sideisolated and the upper/bottom cooling walls.

Acknowledgement

This work was supported by a grant-in-aid for ScientificResearch (C) from the Ministry of Education, Culture, Sports,Science and Technology, Japan.

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