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

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

Masoud Afrand

^a,⇑

, Sara Rostami

^a

, Mohammad Akbari

^a

, Somchai Wongwises

^b,⇑

, Mohammad Hemmat Esfe

^a

, Arash Karimipour

^a

aDepartment of Mechanical Engineering, Faculty of Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Isfahan, Iran

bFluid Mechanics, Thermal Engineering and Multiphase Flow Research Lab. (FUTURE), Department of Mechanical Engineering, Faculty of Engineering, King Mongkut’s University of Technology Thonburi, Bangmod, Bangkok 10140, Thailand

a r t i c l e i n f o

Article history:

Received 4 April 2015

Received in revised form 18 June 2015 Accepted 21 June 2015

Keywords:

Cylindrical annulus Magnetic field

Magneto-natural convection Electric field

CFD

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 the other 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 the horizontal 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 field results in a reduction in the Nusselt number in most of the regions of the annulus. This reduction is high in the Hartmann layers but low in the Roberts layers. Moreover, it was found that for a given value of Hartman 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.

Ó2015 Elsevier Ltd. All rights reserved.

1. Introduction

Magnetic field is widely used in many applications such as the military 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 causes buoyancy forces that result in a natural convection flow.

Buoyancy force causes the motions that can generate undesirable microscopic 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 the natural convection in liquid metals[5–8]. Many authors numeri- cally studied the convective heat transfer in a rectangular cavity exposed to a magnetic field and showed that with an increase in the 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 annulus with a rotating outer cylinder under a radial magnetic field was

investigated numerically by Mozayyeni and Rahimi [18]. They investigated the effect on heat transfer of various dimensionless numbers such as Reynolds number, Rayleigh number, Hartmann number, Eckert number, and radii ratio. Numerical results showed that the flow and heat transfer are significantly suppressed by applying an external magnetic field. In another study, Afrand et al.[19]studied numerically a steady natural convection under different directions of magnetic field in a 2D cylindrical annulus filled with liquid gallium. Their results showed that by increasing the Hartmann number, natural convection is decreased. They found that at low Rayleigh numbers with an increase in Hartmann number, natural convection in the annulus is frequently due 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, natural convective heat transfer in a horizontal cylindrical annulus con- taining magnetic fluid was experimentally investigated by Sawada et al.[20]. The cylindrical walls were maintained at a con- stant temperature, and a nonuniform magnetic field was applied to the annulus. Results revealed the influence of the magnetic field changes the heat transfer. Experimental and numerical analyses of a thermo-magnetic convective flow of paramagnetic fluid in the space between two vertical cylinders influenced by the

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.06.059 0017-9310/Ó2015 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).

Contents lists available atScienceDirect

International Journal of Heat and Mass Transfer

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j h m t

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

In the last decade, several studies have been performed on the effect 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 of magnetic fields on natural convection in the space between two vertical concentric cylinders. This study was carried out for electri- cally conducting fluid (Pr = 0.054) under a constant axial or radial magnetic field. The inner and outer cylinders were kept at constant temperatures. The results show that flow and heat transfer are suppressed more effectively by a radial magnetic field in tall enclo- sures, whereas in shallow enclosures, an axial magnetic field is more effective. Furthermore, the effect of a radial or axial magnetic field on double-diffusive natural convection in a vertical cylindrical annulus was presented by Venkatachalappa et al. [23]. They claimed 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 electrically conducting fluid in the presence of a radial magnetic field. The effect of the Hartmann number and buoyancy force distribution parameters on the fluid velocity, induced magnetic field, and induced current density was numerically analyzed. It was observed that the fluid velocity and induced magnetic field rapidly decrease with the increase in the value of Hartmann number. Kakarantzas et al.[25]used numerical methods to study the magneto-natural convection in a vertical cylindrical enclosure with sinusoidal upper wall temperature under a tilted magnetic field. The computational results demonstrated that the Nusselt number is decreased more effectively by an axial magnetic field compared to a horizontal one. In another research, Kakarantzas et al. [26] performed a numerical study on the effect of a horizontal magnetic field on unsteady natural convection in the vertical annulus containing liq- uid metal. They reported that by increasing the magnetic field, the flow 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 investigation of 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 electric field.

According to the above-mentioned works[1–27], it was found that some of these studies reported the effect of magnetic field on the Lorentz force and on heat transfer. Several other works demonstrated electric field distribution in the enclosure by solving electric potential equations. Scant attention has been given to induced electric field and how it affects the amount of heat transfer in a vertical cylindrical annulus in the presence of a horizontal magnetic field. The aim of the present work is to examine the effect of induced electric field on the natural convection in a vertical cylindrical annulus filled with liquid potassium (Pr= 0.072). Such a study helps in understanding the electric field induced by the motion 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 vertical cylindrical annulus filled with liquid potassium (Pr= 0.072) is studied. The annulus formed by two coaxial cylinders is displayed inFig. 1. In this figure,riandroare the radii of the inner and outer cylinders, respectively, and L is the length of the annulus. The cylindrical walls are isothermal, and the other walls are assumed to be adiabatic. B is the horizontally applied uniform magnetic field while the induced magnetic field is assumed negligible.

2.2. Governing equations

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

Continuity equation:

r:

v

¼0 ð1Þ

Momentum equation:

q

ð

v

rÞ

v

¼ rpþ

l

r²

v

þ

q

gþF ð2Þ Nomenclature

A aspect ratio

B0 magnitude of the external magnetic field (kg/s²A) D annulus gap (D=rori(m))

E dimensional induced electric field (m kg/s³A) E dimensionless induced electric field

F Lorentz force (N/m³)

g acceleration due to gravity (m/s²) Ha Hartmann number

J electric current density (A/m²) L height of the annulus Nu Nusselt number p pressure (N/m²) P dimensionless pressure Pr Prandtl number Ra Rayleigh number

T dimensional temperature (K) T dimensionless temperature ðr;zÞ radial and axial co-ordinates

ðR;ZÞ dimensionless radial and axial co-ordinates ðri;roÞ radii of inner and outer cylinders (m)

ðu;

v

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

ðU;V;WÞdimensionless velocity components inðr;h;zÞdirection ðx;y;zÞ Cartesian co-ordinate components

Greek letters

a

thermal diffusivity (m²/s)

b fluid coefficient of thermal expansion (1/K) / dimensional electrical potentialðm²kg=s³AÞ U dimensionless electrical potential

c

inclination angle k radii ratio

l

dynamic viscosity (kg/ms) h azimuthal angle

q

fluid density (kg/m³)

r

fluid electrical conductivity (s³A²/m³kg)

Subscripts

c condition at cold wall h condition at hot wall

Energy equation:

ð

v

^rÞT¼

^a

^{r2T ð3Þ}

Electric potential equation:

r²/¼r ð

v

BÞ ¼

v

ðrBÞ þB ðr

v

Þ ð4Þ Electric current density, induced electric field, and Lorentz force is obtained as follows:

J¼

r

ðEþ

v

^BÞ;E¼ r/andF¼JB ð5Þ here,pis the pressure,

a

is the thermal diffusivity,bis the coeffi- cient of volumetric expansion,

q

is the fluid density,

l

is the dynamic viscosity,

r

is the electrical conductivity,

v

is the vector of velocity,Tis the temperature,Eis the induced electric field vec- tor,Jis the electric current density vector,Fis the Lorentz force vec- tor, and/is the electric potential.

For simplicity, the governing equations are nondimensionalized using the following parameters:

U¼^uD_a; V¼^v_a^D; W¼^wD_a ; R¼_D^r; Z¼^z_L; A¼_D^L;

P¼^pD_qa²2; T¼_T^TTc

hTc; U¼_B^/_a; E¼^ED_Ba

ð6Þ

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

Dimensionless continuity equation:

@U

@Rþ1 R

@V

@hþ1 A

@W

@Z ¼0 ð7Þ

Dimensionless momentum equation:

rcomponent:

U@U

@RþV R

@U

@hV² R þ1

AW@U

@Z¼ @P

@R þPr @²U

@R²þ1 R

@U

@Rþ1 R²

@²U

@h²þ1 A²

@²U

@Z²U R²2

R²

@V

@h

" #

þHa²:Pr 1 A

@U

@ZCoshUCos²hþ:V 2Sin2h

ð8Þ

hcomponent:

U@V

@RþV R

@V

@hþUV R þ1

AW@V

@Z¼ 1 R

@P

@h þPr @²V

@R²þ1 R²

@²V

@h²þ1 R

@V

@Rþ 1 A²

@²V

@Z²V R²þ2

R²

@U

@h

" #

þHa²:Pr 1 A

@U

@ZðSinhÞ WCoshþ:U

2Sin2hVSin²h

ð9Þ zcomponent:

U@W

@R þV R

@W

@h þ1 AW@W

@Z ¼ 1 A

@P

@Z þPr @²W

@R² þ1 R²

@²W

@h² þ1 R

@W

@R þ1 A²

@²W

@Z²

" #

þRa:Pr:T

þHa²:Pr @U

@RCosh1 R

@U

@h SinhWSin²hWCos²h

ð10Þ Dimensionless energy equation:

U@T

@R þV R

@T

@h þ1 AW@T

@Z ¼@²T

@R²þ 1 R²

@²T

@h² þ1 R

@T

@Rþ 1 A²

@²T

@Z² ð11Þ Dimensionless electric potential equation:

@²U

@R²þ1 R²

@²U

@h²þ1 R

@U

@Rþ@²U

@Z²¼ 1 R

@W

@h1 A

@V

@Z

Sinhþ 1 A

@U

@Z@W

@R

Cosh ð12Þ where, it is worth mentioning that in the case of not solving the electric potential, there is no need for solving the Eq.(12), and com- ponents ofrUcan be neglected in Eqs.(8)–(11).

In the above equations,Raand Pr are defined as follows:

Ra¼gbðTHTCÞD³

ta

^{; Pr¼}

t

a

^ð13Þ

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

Ha¼B0D ffiffiffiffi

r l

r

ð14Þ

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

Nuðh;ZÞ ¼@T

@R _R¼R

i

; Nu¼ 1 2

p

Z 2p 0

Z 1 0

Nuðh;ZÞdZdh ð15Þ

2.3. Boundary conditions

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

At R¼Ri; U¼V¼W¼0; T¼1 and @U

@R¼0 ð16Þ

AtR¼Ro; U¼V¼W¼0; T¼0 and @U

@R¼0 ð17Þ

At Z¼0; U¼V¼W¼0; @T

@Z ¼0 and @U

@Z¼0 ð18Þ

At Z¼1; U¼V¼W¼0; @T

@Z ¼0 and @U

@Z¼0 ð19Þ

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

3. Numerical details 3.1. Numerical procedure

The electric potential equation is coupled to the equations of continuity, momentum, and energy, and is then solved using the finite volume (FV) method. In this method, a mesh of points is first fitted on the solution domain as shown inFig. 2. Then, on these points, the main control volumes and velocity control volumes are made. After defining the mesh dimensions, the momentum equation over each control volume is integrated. The staggered mesh is used to solve the momentum equation, while the main mesh 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 in the governing equations. Moreover, a hybrid scheme is applied to discretize the convection terms. A hybrid scheme is a combination of 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 iterating solution. In this work, the under-relaxation factors 0.5 and 0.7 for momentum and energy are used for the numerical solution of algebraic equations. The coupled systems of discretized equations are 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 Nusselt numbers are shown inTable 1. Here,Nr;Nh;andNzcorrespond to the number of grid points in ther;h;andzdirections, respectively.

As can be observed, the 81121121 grid is sufficiently fine in ther,h, andzdirections, respectively, and provides more accurate numerical results.

3.3. Model validation

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

The experimental data are illustrated inFig. 3and the maximum discrepancy is within 4.73%. Also, the accuracy of the numerical procedure was checked by comparing the results in a vertical cylin- drical annulus with the numerical results obtained by Sankar et al.

[22], as seen inFig. 4. These figures showed that the results of the code agree with the experimental and numerical data.

4. Results and discussion

The natural convection of the liquid potassium in a vertical cylindrical annulus with an aspect ratio of A¼3 and radii ratio ofk¼6 in the presence of a constant horizontal magnetic field is numerically studied. In this model, the inner and outer cylinders of annulus are heated and cooled isothermally, and top and bottom walls are adiabatic. Simulations are carried out for a wide range of Hartmann number. Furthermore, it is assumed that all walls are electrically insulated and thatRa¼10⁵. The influence of a horizon- tal magnetic field on average Nusselt number, distribution of Lorentz force, and electric field is also investigated.

To illustrate the importance of solving the electric potential equation, the variations of the average Nusselt number versus Hartmann number are presented inTable 2andFig. 5for two dif- ferent cases: with and without the electric potential equation (U).

As expected, the average Nusselt number decreases by increasing Hartmann number in both cases. This indicates that the Lorentz force is enhanced by increasing the Hartmann number, which leads to the lower average Nusselt number. Furthermore, it can be observed that there is a maximum difference of 32.71% in the aver- age Nusselt number among both cases of with and withoutU. For the case of withU, the average Nusselt number is greater than in the other case.

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

Table 1

Grid independence test for Ra = 10⁵andHa= 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 –

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

In order to evaluate the difference between both average Nusselt numbers (seeFig. 5, withUand withoutU), the local values ofJ,E,VBalong the centerline of the annulus are plotted inFig. 6.

According to Eq.(5), it is found that the electric current density and Lorentz force are composed of two factors:VBandE. As it is evi- dent inFig. 6, the effects of these parameters on the electric current density are exactly opposite. As a result, the electric potential is a reducing factor to the Lorentz force and should be considered for the analysis of these problems.

Fig. 7shows the axial velocity and temperature profiles on thex andyaxes in three sections forHa= 0, which implies the flow is Fig. 4.Comparison of isotherms results between present work (a and b) and Sankar et al.[22]work (c and d) forHa= 0 (a and c) andHa= 40 (b and d).

Table 2

Variations of the average Nusselt number versus Hartmann number.

Ha Nu Difference (%)

With considering/ Without considering/

0 6.31 6.31 0.00

10 6.18 5.92 4.21

20 5.82 4.95 14.95

30 5.28 4.01 24.05

40 4.88 3.43 29.71

50 4.55 3.14 30.99

60 4.31 2.90 32.71

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

Fig. 6.Local values ofJ,E,VBand 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 thexandyaxis forHa= 0.

Fig. 8.The dimensionless axial velocity along the (a)Xaxis (b)Yaxis, and the dimensionless temperature along the (c)Xaxis (d)Yaxis.

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

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

The axial velocity and temperature profiles along thexandy axes in three sections forHa= 60 are illustrated inFig. 8. By com- paringFig. 7withFig. 8, it becomes clear that applying horizontal magnetic field makes the flow asymmetric. This phenomenon is due to the formation of the Hartmann and Roberts layers near the walls normal and parallel to the magnetic field, respectively.

Fig. 9.3D distribution of Lorentz force forHa= 30 andHa= 60.

Fig. 10.3D distribution of induced electric field and forHa= 30 andHa= 60.

Generally, the values of axial velocity decrease with an increase in Hartmann number; however this reduction is more significant in velocity profiles along thex axis. Moreover, the velocity profile near 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 seen that Lorentz force value is more at the vicinity of upper wall, which has a higher velocity. In addition, the Ln (natural logarithm) tem- perature profile along thexaxis inFig. 8shows that the dominant mode of heat transfer is conduction, which is in agreement with the velocity profile.

Figs. 9 and 10illustrate the 3D distribution of Lorentz force and induced electric field for two different Hartmann numbers.

These figures confirm the result mentioned in Fig. 8which was revealed before. As mentioned above, this phenomenon occurs because of developing the Hartmann and Roberts layers near the walls perpendicular and parallel to the magnetic field [29–

30]. The thickness of the Hartmann and Roberts layers is respec- tively proportional toHa¹andHa²[31–32]; also the thickness of Hartmann and Roberts layers decreases by increasing Hartmann number from 30 to 60. The Hartmann layer develops near thexaxis, while the Roberts layer grows close to theyaxis.

The Lorentz force amount in the Hartmann layer is high, while the maximum value of induced electric field and the minimum amount of Lorentz force occur in the central region of the annulus (z= 1.5).

For a more detailed study of the effect of horizontal magnetic field on the natural convection in the annulus, the local Nusselt number on the inner cylinder of the annulus is presented in Fig. 11. In the absence of the magnetic field, the local Nusselt num- ber is constant along theh(which is confirmed byFig. 7). It can also be seen that the minimum and maximum values of the local Nusselt 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 layer is achieved in this area, which has the most temperature gradient and convective heat transfer.Fig. 11 implies that the horizontal magnetic field eliminates the flow axisymmetric, and two peaks appear in the local Nusselt number graph. The applying magnetic field results in a reduction of Nusselt number in most spaces of the annulus. This reduction is more significant at h= 90° and h= 270° (the place of Hartmann layers) than that at h= 0° and h= 180° (the place of Roberts layers). As mentioned before, the Hartmann layer thickness is more than the thickness of the Roberts layer for a given Hartmann number. As a result, the tem- perature gradient in the thinner layer (Roberts layer) would be large, 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. 3D numerical simulations were performed for various Hartmann numbers,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 becomes asymmetric.

The applying magnetic field resulted in a reduction of Nusselt number in most spaces of the annulus. This reduction is more significant in the x-direction (the place of Hartmann layers) than that in they-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 the Nusselt 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 of Lorentz 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, the temperature gradient in the thinner layer (Roberts layer) would be large, which makes more heat transfer rate.

Fig. 11.Distribution of local Nusselt number forHa= 0,Ha= 30 andHa= 60.

Conflict of interest

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

Acknowledgement

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

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