International Journal of Heat and Mass Transfer 90 (2015) 418–426

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International Journal of Heat and Mass Transfer

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

Effect of induced electric field on magneto-natural convectionin a vertical cylindrical annulus filled with liquid potassium

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.06.0590017-9310/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding authors.E-mail addresses: masoud.afrand@pmc.iaun.ac.ir, masoud_afrand@yahoo.com

(M. Afrand), somchai.won@kmutt.ac.th (S. Wongwises).

Masoud Afrand a,⇑, Sara Rostami a, Mohammad Akbari a, Somchai Wongwises b,⇑,Mohammad Hemmat Esfe a, Arash Karimipour a

a Department of Mechanical Engineering, Faculty of Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Isfahan, Iranb Fluid Mechanics, Thermal Engineering and Multiphase Flow Research Lab. (FUTURE), Department of Mechanical Engineering, Faculty of Engineering, King Mongkut’s Universityof Technology Thonburi, Bangmod, Bangkok 10140, Thailand

a r t i c l e i n f o

Article history:Received 4 April 2015Received in revised form 18 June 2015Accepted 21 June 2015

Keywords:Cylindrical annulusMagnetic fieldMagneto-natural convectionElectric fieldCFD

a b s t r a c t

Laminar steady magneto-natural convection in a vertical cylindrical annulus formed by two coaxial cylin-ders filled with liquid potassium is studied numerically. The cylindrical walls are isothermal, and theother walls are assumed to be adiabatic. A constant horizontal magnetic field is also applied on the enclo-sure. The results show that flow is axisymmetric in the absence of the magnetic field; but by applying thehorizontal magnetic field, it becomes asymmetric. This is due to the growth of Roberts and Hartmann lay-ers near the walls parallel and normal to the magnetic field, respectively. The applied magnetic fieldresults in a reduction in the Nusselt number in most of the regions of the annulus. This reduction is highin the Hartmann layers but low in the Roberts layers. Moreover, it was found that for a given value ofHartman number, the average Nusselt number is greater in the case of solving the electric potential equa-tion. The results show that there is a large difference in the Nusselt number obtained by solving the elec-tric potential equation, compared with by neglecting the electric potential.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Magnetic field is widely used in many applications such as themilitary and aerospace industries [1], measuring [2], pumping [3],and electric generation [4]. In the crystal growth process, the tem-perature difference between the wall and the melt region causesbuoyancy forces that result in a natural convection flow.Buoyancy force causes the motions that can generate undesirablemicroscopic heterogeneities in production. Therefore, in such pro-cesses reduction in natural convection is taken it consideration.The use of magnetic field is an effective method for reducing thenatural convection in liquid metals [5–8]. Many authors numeri-cally studied the convective heat transfer in a rectangular cavityexposed to a magnetic field and showed that with an increase inthe magnetic field, natural convective heat transfer is reduced[9–17].

Some researchers have numerically studied the effect of mag-netic fields on convective heat transfer in 2D cylindrical enclosures.Steady and unsteady mixed convection in a 2D cylindrical annuluswith a rotating outer cylinder under a radial magnetic field was

investigated numerically by Mozayyeni and Rahimi [18]. Theyinvestigated the effect on heat transfer of various dimensionlessnumbers such as Reynolds number, Rayleigh number, Hartmannnumber, Eckert number, and radii ratio. Numerical results showedthat the flow and heat transfer are significantly suppressed byapplying an external magnetic field. In another study, Afrandet al. [19] studied numerically a steady natural convection underdifferent directions of magnetic field in a 2D cylindrical annulusfilled with liquid gallium. Their results showed that by increasingthe Hartmann number, natural convection is decreased. Theyfound that at low Rayleigh numbers with an increase inHartmann number, natural convection in the annulus is frequentlydue to the conduction mode.

Moreover, some researchers have to pay attention to experi-mental investigation of natural convective heat transfer in cylindri-cal annuluses containing magnetic fluid. For example, naturalconvective heat transfer in a horizontal cylindrical annulus con-taining magnetic fluid was experimentally investigated bySawada et al. [20]. The cylindrical walls were maintained at a con-stant temperature, and a nonuniform magnetic field was applied tothe annulus. Results revealed the influence of the magnetic fieldchanges the heat transfer. Experimental and numerical analysesof a thermo-magnetic convective flow of paramagnetic fluid inthe space between two vertical cylinders influenced by the

Nomenclature

A aspect ratioB0 magnitude of the external magnetic field (kg/s2 A)D annulus gap (D = ro�ri (m))E dimensional induced electric field (m kg/s3 A)E� dimensionless induced electric fieldF Lorentz force (N/m3)g acceleration due to gravity (m/s2)Ha Hartmann numberJ electric current density (A/m2)L height of the annulusNu Nusselt numberp pressure (N/m2)P dimensionless pressurePr Prandtl numberRa Rayleigh numberT dimensional temperature (K)T� dimensionless temperatureðr; zÞ radial and axial co-ordinatesðR; ZÞ dimensionless radial and axial co-ordinatesðri; roÞ radii of inner and outer cylinders (m)

ðu;v;wÞ dimensional velocity components in ðr; h; zÞ direction(m/s)

ðU;V ;WÞ dimensionless velocity components in ðr; h; zÞ directionðx; y; zÞ Cartesian co-ordinate components

Greek lettersa thermal diffusivity (m2/s)b fluid coefficient of thermal expansion (1/K)/ dimensional electrical potential ðm2kg=s3AÞU dimensionless electrical potentialc inclination anglek radii ratiol dynamic viscosity (kg/ms)h azimuthal angleq fluid density (kg/m3)r fluid electrical conductivity (s3 A2/m3 kg)

Subscriptsc condition at cold wallh condition at hot wall

M. Afrand et al. / International Journal of Heat and Mass Transfer 90 (2015) 418–426 419

magnetic field were carried out by Wrobel et al. [21]. They showedthat the effect of the Hartmann number on the heat transfer rate isfour times greater than the effect of Rayleigh number.

In the last decade, several studies have been performed on theeffect of magnetic fields on convective heat transfer in 3D cylindri-cal enclosures. In this regard, by focusing on the directions of mag-netic field, Sankar et al. [22] numerically studied the effect ofmagnetic fields on natural convection in the space between twovertical concentric cylinders. This study was carried out for electri-cally conducting fluid (Pr = 0.054) under a constant axial or radialmagnetic field. The inner and outer cylinders were kept at constanttemperatures. The results show that flow and heat transfer aresuppressed more effectively by a radial magnetic field in tall enclo-sures, whereas in shallow enclosures, an axial magnetic field ismore effective. Furthermore, the effect of a radial or axial magneticfield on double-diffusive natural convection in a vertical cylindricalannulus was presented by Venkatachalappa et al. [23]. Theyclaimed that for small buoyancy ratios, the magnetic field sup-presses double diffusive convection. Kumar and Singh [24] consid-ered a vertical cylindrical annulus filled with an electricallyconducting fluid in the presence of a radial magnetic field. Theeffect of the Hartmann number and buoyancy force distributionparameters on the fluid velocity, induced magnetic field, andinduced current density was numerically analyzed. It was observedthat the fluid velocity and induced magnetic field rapidly decreasewith the increase in the value of Hartmann number. Kakarantzaset al. [25] used numerical methods to study the magneto-naturalconvection in a vertical cylindrical enclosure with sinusoidal upperwall temperature under a tilted magnetic field. The computationalresults demonstrated that the Nusselt number is decreased moreeffectively by an axial magnetic field compared to a horizontalone. In another research, Kakarantzas et al. [26] performed anumerical study on the effect of a horizontal magnetic field onunsteady natural convection in the vertical annulus containing liq-uid metal. They reported that by increasing the magnetic field, theflow becomes laminar. Moreover, it can be observed that the hor-izontal magnetic field causes the loss of axisymmetry of flow.Recently, Afrand et al. [27] carried out a 3D numerical investigationof natural convection in a tilted cylindrical annulus filled with mol-ten potassium and controlled it by using various magnetic fields.Their results revealed the effect of the magnetic field direction

on temperature distribution, Lorentz force, and induced electricfield.

According to the above-mentioned works [1–27], it was foundthat some of these studies reported the effect of magnetic fieldon the Lorentz force and on heat transfer. Several other worksdemonstrated electric field distribution in the enclosure by solvingelectric potential equations. Scant attention has been given toinduced electric field and how it affects the amount of heat transferin a vertical cylindrical annulus in the presence of a horizontalmagnetic field. The aim of the present work is to examine the effectof induced electric field on the natural convection in a verticalcylindrical annulus filled with liquid potassium (Pr = 0.072). Sucha study helps in understanding the electric field induced by themotion of electrically conducting fluid exposed to a magnetic field,which has some relevance in material processes.

2. Mathematical formulation

2.1. Problem statement

In this work, laminar steady natural convection in a 3D verticalcylindrical annulus filled with liquid potassium (Pr = 0.072) isstudied. The annulus formed by two coaxial cylinders is displayedin Fig. 1. In this figure, ri and ro are the radii of the inner and outercylinders, respectively, and L is the length of the annulus. Thecylindrical walls are isothermal, and the other walls are assumedto be adiabatic. B is the horizontally applied uniform magnetic fieldwhile the induced magnetic field is assumed negligible.

2.2. Governing equations

The governing equations of steady laminar natural convectionflow of an incompressible electrically conducting fluid by usingBoussinesq approximation after neglecting viscous and ohmic dis-sipations in the 3D cylindrical coordinate are defined below:

Continuity equation:

r:v ¼ 0 ð1Þ

Momentum equation:

qðv � rÞv ¼ �rpþ lr2v þ qgþ F ð2Þ

Fig. 1. Schematic view of the vertical cylindrical annulus.

420 M. Afrand et al. / International Journal of Heat and Mass Transfer 90 (2015) 418–426

Energy equation:

ðv � rÞT ¼ ar2T ð3Þ

Electric potential equation:

r2/ ¼ r � ðv � BÞ ¼ v � ðr � BÞ þ B � ðr � vÞ ð4Þ

Electric current density, induced electric field, and Lorentz forceis obtained as follows:

J ¼ rðEþ v � BÞ;E ¼ �r/ and F ¼ J� B ð5Þ

here, p is the pressure, a is the thermal diffusivity, b is the coeffi-cient of volumetric expansion, q is the fluid density, l is thedynamic viscosity, r is the electrical conductivity, v is the vectorof velocity, T is the temperature, E is the induced electric field vec-tor, J is the electric current density vector, F is the Lorentz force vec-tor, and / is the electric potential.

For simplicity, the governing equations are nondimensionalizedusing the following parameters:

U ¼ uDa ; V ¼ vD

a ; W ¼ wDa ; R ¼ r

D ; Z ¼ zL ; A ¼ L

D ;

P ¼ pD2

qa2 ; T� ¼ T�TcTh�Tc

; U ¼ /Ba ; E� ¼ ED

Ba

ð6Þ

By inserting these parameters to Eqs. (1)–(4), the dimensionlessform of the governing equations can be expressed as follows:

Dimensionless continuity equation:

@U@Rþ 1

R@V@hþ 1

A@W@Z¼ 0 ð7Þ

Dimensionless momentum equation:r component:

U@U@Rþ V

R@U@h� V2

Rþ 1

AW@U@Z¼ � @P

@R

þ Pr@2U

@R2 þ1R@U@Rþ 1

R2

@2U

@h2 þ1

A2

@2U

@Z2 �U

R2 �2R2

@V@h

" #

þ Ha2:Pr �1A@U@Z

Cosh� UCos2hþ :V2

Sin2h

� �ð8Þ

h component:

U@V@Rþ V

R@V@hþ UV

Rþ 1

AW@V@Z¼ �1

R@P@h

þ Pr@2V

@R2 þ1R2

@2V

@h2 þ1R@V@Rþ 1

A2

@2V

@Z2 �V

R2 þ2R2

@U@h

" #

þHa2:Pr1A@U@ZðSinhÞ �WCoshþ :U

2Sin2h� VSin2h

� �ð9Þ

z component:

U@W@Rþ V

R@W@hþ 1

AW@W@Z¼ �1

A@P@Z

þ Pr@2W

@R2 þ1R2

@2W

@h2 þ1R@W@Rþ 1

A2

@2W

@Z2

" #þ Ra:Pr:T�

þHa2:Pr@U@R

Cosh� 1R@U@h

Sinh�WSin2h�WCos2h

� �ð10Þ

Dimensionless energy equation:

U@T�

@Rþ V

R@T�

@hþ 1

AW@T�

@Z¼ @

2T�

@R2 þ1R2

@2T�

@h2 þ1R@T�

@Rþ 1

A2

@2T�

@Z2 ð11Þ

Dimensionless electric potential equation:

@2U

@R2 þ1R2

@2U

@h2 þ1R@U@Rþ@

2U

@Z2 ¼1R@W@h�1

A@V@Z

� �Sinhþ 1

A@U@Z�@W@R

� �Cosh

ð12Þ

where, it is worth mentioning that in the case of not solving theelectric potential, there is no need for solving the Eq. (12), and com-ponents of rU can be neglected in Eqs. (8)–(11).

In the above equations, Ra and Pr are defined as follows:

Ra ¼ gbðTH � TCÞD3

ta; Pr ¼ t

að13Þ

The effect of the magnetic field is introduced into the momen-tum and potential equations through the Hartmann numberdefined as:

Ha ¼ B0Dffiffiffiffirl

rð14Þ

The local and average Nusselt numbers along the inner cylinderare defined as follows:

Nu ðh; ZÞ ¼ @T�

@R

����R¼Ri

; Nu ¼ 12p

Z 2p

0

Z 1

0Nu ðh; ZÞdZdh ð15Þ

2.3. Boundary conditions

No-slip conditions, thermal and electrical boundary conditionsare defined as below,

At R ¼ Ri; U ¼ V ¼W ¼ 0; T� ¼ 1 and@U@R¼ 0 ð16Þ

At R ¼ Ro; U ¼ V ¼W ¼ 0; T� ¼ 0 and@U@R¼ 0 ð17Þ

At Z ¼ 0; U ¼ V ¼W ¼ 0;@T�

@Z¼ 0 and

@U@Z¼ 0 ð18Þ

At Z ¼ 1; U ¼ V ¼W ¼ 0;@T�

@Z¼ 0 and

@U@Z¼ 0 ð19Þ

Table 1Grid independence test for Ra = 105 and Ha = 0.

Nr Nh Nz Nu Difference

11 16 16 5.4522 9.30%21 31 31 6.0115 3.83%41 61 61 6.2508 0.53%61 91 91 6.2838 0.34%81 121 121 6.3051 0.11% U

101 151 151 6.3122 –

M. Afrand et al. / International Journal of Heat and Mass Transfer 90 (2015) 418–426 421

3. Numerical details

3.1. Numerical procedure

The electric potential equation is coupled to the equations ofcontinuity, momentum, and energy, and is then solved using thefinite volume (FV) method. In this method, a mesh of points is firstfitted on the solution domain as shown in Fig. 2. Then, on thesepoints, the main control volumes and velocity control volumesare made. After defining the mesh dimensions, the momentumequation over each control volume is integrated. The staggeredmesh is used to solve the momentum equation, while the mainmesh is used to calculate the scalar quantities. The governing equa-tions are solved by using boundary conditions. A second-order cen-tral difference scheme is used to discretize the diffusion terms inthe governing equations. Moreover, a hybrid scheme is applied todiscretize the convection terms. A hybrid scheme is a combinationof the upwind scheme and the central difference scheme. The alge-braic equation system obtained by using the under-relaxation fac-tor can be very useful in avoiding the divergence in the iteratingsolution. In this work, the under-relaxation factors 0.5 and 0.7for momentum and energy are used for the numerical solution ofalgebraic equations. The coupled systems of discretized equationsare solved iteratively using the TDMA method [28].

3.2. Grid independence tests

Various structured grids are evaluated to ensure grid indepen-dence results. The tested grids and the obtained average Nusseltnumbers are shown in Table 1. Here,Nr ;Nh; and Nz correspond tothe number of grid points in the r; h; and z directions, respectively.As can be observed, the 81� 121� 121 grid is sufficiently fine inthe r, h, and z directions, respectively, and provides more accuratenumerical results.

3.3. Model validation

The code validation was first confirmed by comparing of thecomputational results with those obtained by Wrobel et al. [21].

Fig. 2. An adapted structured mesh on the annulus.

The experimental data are illustrated in Fig. 3 and the maximumdiscrepancy is within 4.73%. Also, the accuracy of the numericalprocedure was checked by comparing the results in a vertical cylin-drical annulus with the numerical results obtained by Sankar et al.[22], as seen in Fig. 4. These figures showed that the results of thecode agree with the experimental and numerical data.

4. Results and discussion

The natural convection of the liquid potassium in a verticalcylindrical annulus with an aspect ratio of A ¼ 3 and radii ratioof k ¼ 6 in the presence of a constant horizontal magnetic field isnumerically studied. In this model, the inner and outer cylindersof annulus are heated and cooled isothermally, and top and bottomwalls are adiabatic. Simulations are carried out for a wide range ofHartmann number. Furthermore, it is assumed that all walls areelectrically insulated and that Ra ¼ 105. The influence of a horizon-tal magnetic field on average Nusselt number, distribution ofLorentz force, and electric field is also investigated.

To illustrate the importance of solving the electric potentialequation, the variations of the average Nusselt number versusHartmann number are presented in Table 2 and Fig. 5 for two dif-ferent cases: with and without the electric potential equation (U).As expected, the average Nusselt number decreases by increasingHartmann number in both cases. This indicates that the Lorentzforce is enhanced by increasing the Hartmann number, which leadsto the lower average Nusselt number. Furthermore, it can beobserved that there is a maximum difference of 32.71% in the aver-age Nusselt number among both cases of with and without U. Forthe case of with U, the average Nusselt number is greater than inthe other case.

Fig. 3. Comparison of average Nusselt number results with experimental data inthe literature.

Fig. 4. Comparison of isotherms results between present work (a and b) and Sankar et al. [22] work (c and d) for Ha = 0 (a and c) and Ha = 40 (b and d).

Table 2Variations of the average Nusselt number versus Hartmann number.

Ha Nu Difference (%)

With considering / Without considering /

0 6.31 6.31 0.0010 6.18 5.92 4.2120 5.82 4.95 14.9530 5.28 4.01 24.0540 4.88 3.43 29.7150 4.55 3.14 30.9960 4.31 2.90 32.71

Fig. 5. Variations of the average Nusselt number versus Hartmann number.

422 M. Afrand et al. / International Journal of Heat and Mass Transfer 90 (2015) 418–426

In order to evaluate the difference between both averageNusselt numbers (see Fig. 5, with U and without U), the local valuesof J, E,V � B along the centerline of the annulus are plotted in Fig. 6.According to Eq. (5), it is found that the electric current density andLorentz force are composed of two factors: V � B and E. As it is evi-dent in Fig. 6, the effects of these parameters on the electric currentdensity are exactly opposite. As a result, the electric potential is areducing factor to the Lorentz force and should be considered forthe analysis of these problems.

Fig. 7 shows the axial velocity and temperature profiles on the xand y axes in three sections for Ha = 0, which implies the flow is

Fig. 6. Local values of J, E, V � B and Lorentz force on the center line of the annulus(Z = 0.5).

Fig. 7. The dimensionless axial velocity (left) and the dimensionless temperature (right) along the x and y axis for Ha = 0.

Fig. 8. The dimensionless axial velocity along the (a) X axis (b) Y axis, and the dimensionless temperature along the (c) X axis (d) Y axis.

M. Afrand et al. / International Journal of Heat and Mass Transfer 90 (2015) 418–426 423

424 M. Afrand et al. / International Journal of Heat and Mass Transfer 90 (2015) 418–426

axisymmetric. This figure shows that the flow adjacent to the hotwall moves upward, while it would go downward along the coldwall. The maximum amount of velocity profile occurs at the vicin-ity of the hot wall. However, its value is two times as much as themaximum value the velocity profile near the cold wall. This is dueto the fact that the higher radius corresponds to lower velocityaccording to the continuity equation. Moreover, inspection ofvelocity profiles in Fig. 7 makes it clear that velocity value in the

Fig. 9. 3D distribution of Lorentz

Fig. 10. 3D distribution of induced electr

area near the bottom wall is low (z = 0.5), but due to the growthof the boundary layer, it increases far from the bottom (z = 2.5).

The axial velocity and temperature profiles along the x and yaxes in three sections for Ha = 60 are illustrated in Fig. 8. By com-paring Fig. 7 with Fig. 8, it becomes clear that applying horizontalmagnetic field makes the flow asymmetric. This phenomenon isdue to the formation of the Hartmann and Roberts layers nearthe walls normal and parallel to the magnetic field, respectively.

force for Ha = 30 and Ha = 60.

ic field and for Ha = 30 and Ha = 60.

M. Afrand et al. / International Journal of Heat and Mass Transfer 90 (2015) 418–426 425

Generally, the values of axial velocity decrease with an increase inHartmann number; however this reduction is more significant invelocity profiles along the x axis. Moreover, the velocity profilenear the top wall is most affected by the horizontal magnetic field,which means more Lorentz force is generated near the top wall.The Lorentz force suppresses the motion of fluid. It can also be seenthat Lorentz force value is more at the vicinity of upper wall, whichhas a higher velocity. In addition, the Ln (natural logarithm) tem-perature profile along the x axis in Fig. 8 shows that the dominantmode of heat transfer is conduction, which is in agreement withthe velocity profile.

Fig. 11. Distribution of local Nusselt number for Ha = 0, Ha = 30 and Ha = 60.

Figs. 9 and 10 illustrate the 3D distribution of Lorentz forceand induced electric field for two different Hartmann numbers.These figures confirm the result mentioned in Fig. 8 which wasrevealed before. As mentioned above, this phenomenon occursbecause of developing the Hartmann and Roberts layers nearthe walls perpendicular and parallel to the magnetic field [29–30]. The thickness of the Hartmann and Roberts layers is respec-tively proportional to Ha�1 and Ha�2 [31–32]; also the thicknessof Hartmann and Roberts layers decreases by increasingHartmann number from 30 to 60. The Hartmann layer developsnear the x axis, while the Roberts layer grows close to the y axis.The Lorentz force amount in the Hartmann layer is high, whilethe maximum value of induced electric field and the minimumamount of Lorentz force occur in the central region of the annulus(z = 1.5).

For a more detailed study of the effect of horizontal magneticfield on the natural convection in the annulus, the local Nusseltnumber on the inner cylinder of the annulus is presented inFig. 11. In the absence of the magnetic field, the local Nusselt num-ber is constant along the h (which is confirmed by Fig. 7). It can alsobe seen that the minimum and maximum values of the localNusselt number occur near the top and bottom sides of the annu-lus, respectively. The boundary layer develops from near the bot-tom wall. Hence, the minimum thickness of the boundary layeris achieved in this area, which has the most temperature gradientand convective heat transfer. Fig. 11 implies that the horizontalmagnetic field eliminates the flow axisymmetric, and two peaksappear in the local Nusselt number graph. The applying magneticfield results in a reduction of Nusselt number in most spaces ofthe annulus. This reduction is more significant at h = 90� andh = 270� (the place of Hartmann layers) than that at h = 0� andh = 180� (the place of Roberts layers). As mentioned before, theHartmann layer thickness is more than the thickness of theRoberts layer for a given Hartmann number. As a result, the tem-perature gradient in the thinner layer (Roberts layer) would belarge, which makes more heat transfer rate.

5. Conclusions

The effect of a horizontal magnetic field on the natural convec-tive heat transfer of liquid potassium in a vertical cylindrical annu-lus with an aspect ratio of 3 and radii ratio of 6 was studied. 3Dnumerical simulations were performed for various Hartmannnumbers, Ha = 0 to 60. The following results were obtained:

� In the absence of the magnetic field, the flow is axisymmetric.But if the horizontal magnetic field is applied, it becomesasymmetric.� The applying magnetic field resulted in a reduction of Nusselt

number in most spaces of the annulus. This reduction is moresignificant in the x-direction (the place of Hartmann layers)than that in the y-direction (the place of Roberts layers).� For a given value of Hartman number, the average Nusselt num-

ber is greater in the case of solving the electric potential equa-tion. The results show that there is a large difference in theNusselt number.� The electric potential is a reducing factor to the Lorentz force

and should be considered for the analysis of such problems.� The Lorentz force amount in the Hartmann layer is high, while

the great value of induced electric field and the little amount ofLorentz force occur in the Roberts layer.� The Hartmann layer thickness is more than the thickness of the

Roberts layer for a given Hartmann number. As a result, thetemperature gradient in the thinner layer (Roberts layer) wouldbe large, which makes more heat transfer rate.

426 M. Afrand et al. / International Journal of Heat and Mass Transfer 90 (2015) 418–426

Conflict of interest

We would like to say that we and our institution don’t have anyconflict of interest and don’t have any financial or other relation-ship with other people or organizations that may inappropriatelyinfluence the author’s work.

Acknowledgement

The fourth author would like to thank the ‘‘Research ChairGrant’’ National Science and Technology Development Agency(NSTDA), the Thailand Research Fund (IRG5780005) and theNational Research University Project (NRU) for the support.

References

[1] P.A. Davidson, An Introduction to Magnetohydrodynamics, First published,Cambridge University Press, New York, 2001.

[2] B. Lu, L. Xu, X. Zhang, Three-dimensional MHD simulations of theelectromagnetic flowmeter for laminar and turbulent flows, Flow Meas.Instrum. 33 (2013) 239–243.

[3] H.R. Kim, The theoretical approach for viscous and end effect ts on an MHDpump for sodium circulation, Ann. Nucl. Energy 62 (2013) 103–108.

[4] S.P. Cicconardi, A.A. Perna, Performance analysis of integrated systems basedon MHD generators, Energy Procedia 45 (2014) 1305–1314.

[5] V. Vives, C. Perry, Effects of magnetically damped convection during thecontrolled solidification of metals and alloys, Int. J. Heat Mass Transfer 30(1987) 479–496.

[6] M. Afrand, N. Sina, H. Teimouri, A. Mazaheri, M.R. Safaei, M. Hemmat Esfe, J.Kamali, D. Toghraie, Effect of magnetic field on free convection in inclinedcylindrical annulus containing molten potassium, Int. J. Appl. Mech. (in press).

[7] P.X. Yu, J.X. Qiu, Q. Qin, Z.F. Tian, Numerical investigation of natural convectionin a rectangular cavity under different directions of uniform magnetic field, Int.J. Heat Mass Transfer 67 (2013) 1131–1144.

[8] M. Afrand, S. Farahat, A. Hossein Nezhad, G.A. Sheikhzadeh, F. Sarhaddi, S.Wongwises, Multi-objective optimization of natural convection in a cylindricalannulus mold under magnetic field using particle swarm algorithm, Int.Commun. Heat Mass Transfer 60 (2015) 13–20.

[9] A.A. Mohamad, R. Viskanta, Modeling of turbulent buoyant flow and heattransfer in liquid metals, Int. J. Heat Mass Transfer 36 (1993) 2815–2826.

[10] I.D. Piazza, M. Ciofalo, MHD free convection in a liquid-metal filled cubicenclosure. II. Internal heating, Int. J. Heat Mass Transfer 49 (2006) 4525–4535.

[11] P. Kandaswamy, S. Malliga Sundari, N. Nithyadevi, Magnetoconvection in anenclosure with partially active vertical walls, Int. J. Heat Mass Transfer 51(2008) 1946–1954.

[12] C. Revnic, T. Grosan, I. Pop, D.B. Ingham, Magnetic field effect on the unsteadyfree convection flow in a square cavity filled with a porous medium with aconstant heat generation, Int. J. Heat Mass Transfer 54 (2011) 1734–1742.

[13] G.A. Sheikhzadeh, A. Fattahi, M.A. Mehrabian, Numerical study of steadymagneto-convection around an adiabatic body inside a square enclosure inlow Prandtl numbers, Heat Mass Transfer 47 (2011) 27–34.

[14] M.A. Teamah, A.F. Elsafty, E.Z. Massoud, Numerical simulation of double-diffusive natural convective flow in an inclined rectangular enclosure in thepresence of magnetic field and heat source, Int. J. Therm. Sci. 52 (2012) 161–175.

[15] S.K. Jena, V.K.R. Yettella, C.P.R. Sandeep, S.K. Mahapatra, A.J. Chamkha, Three-dimensional Rayleigh–Bénard convection of molten gallium in a rotatingcuboid under the influence of a vertical magnetic field, Int. J. Heat MassTransfer 78 (2014) 341–353.

[16] G.C. Bourantas, V.C. Loukopoulos, MHD natural-convection flow in an inclinedsquare enclosure filled with a micropolar-nanofluid, Int. J. Heat Mass Transfer79 (2014) 930–944.

[17] M. Mahmoodi, M. Hemmat Esfe, M. Akbari, A. Karimipour, M. Afrand,Magneto-natural convection in square cavities with a source-sink pair ondifferent walls, Int. J. Appl. Electromagn. Mech. 47 (2015) 21–32.

[18] H.R. Mozayyeni, A.B. Rahimi, Mixed convection in cylindrical annulus withrotating outer cylinder and constant magnetic field with an effect in the radialdirection, Sci. Iran. Trans. B. 19 (2012) 91–105.

[19] M. Afrand, S. Farahat, A. Hossein Nezhad, G.A. Sheikhzadeh, F. Sarhaddi,Numerical simulation of electricity conductor fluid flow and free convectionheat transfer in annulus using magnetic field, Heat Transfer Res. 45 (2014)749–766.

[20] T. Sawada, H. Kikura, A. Saito, T. Tanahashi, Natural convection of a magneticfluid in concentric horizontal annuli under nonuniform magnetic fields, Exp.Therm. Fluid. Sci. 7 (1993) 212–220.

[21] W. Wrobel, E. Fornalik-Wajs, J.S. Szmyd, Experimental and numerical analysisof thermo-magnetic convection in a vertical annular enclosure, Int. J. HeatFluid Flow 31 (2010) 1019–1031.

[22] M. Sankar, M. Venkatachalappa, I.S. Shivakumara, Effect of magnetic field onnatural convection in a vertical cylindrical annulus, Int. J. Eng. Sci. 44 (2006)1556–1570.

[23] M. Venkatachalappa, Y. Do, M. Sankar, Effect of magnetic field on the heat andmass transfer in a vertical annulus, Int. J. Eng. Sci. 49 (2011) 262–278.

[24] A. Kumar, A.K. Singh, Effect of induced magnetic field on natural convection invertical concentric annuli heated/cooled asymmetrically, J. Appl. Fluid. Mech. 6(2013) 15–26.

[25] S.C. Kakarantzas, I.E. Sarris, A.P. Grecos, N.S. Vlachos, Magnetohydrodynamicnatural convection in a vertical cylindrical cavity with sinusoidal upper walltemperature, Int. J. Heat Mass Transfer 52 (2009) 250–259.

[26] S.C. Kakarantzas, I.E. Sarris, N.S. Vlachos, Natural convection of liquid metal ina vertical annulus with lateral and volumetric heating in the presence of ahorizontal magnetic field, Int. J. Heat Mass Transfer 54 (2011) 3347–3356.

[27] M. Afrand, S. Farahat, A. Hossein Nezhad, G.A. Sheikhzadeh, F. Sarhaddi, 3-Dnumerical investigation of natural convection in a tilted cylindrical annuluscontaining molten potassium and controlling it using various magnetic fields,Int. J. Appl. Electromagn. Mech. 46 (2014) 809–821.

[28] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere PublishingCorp, Taylor & Francis, New York, 1980.

[29] L. Todd, Hartmann flow in an annular channel, J. Fluid Mech. 28 (1967) 371–384.

[30] S.C. Kakarantzas, A.P. Grecos, N.S. Vlachos, I.E. Sarris, B. Knaepen, D. Carati,Direct numerical simulation of a heat removal configuration for fusionblankets, Energy Convers. Manag. 48 (2007) 2775–2783.

[31] P.H. Roberts, Singularities of Hartmann layers, Proc. Roy. Soc. London 300(1967) 94–107.

[32] U. Müller, L. Bühler, Magnetofluiddynamics in Channels and Containers,Springer, 2001.