reading international journal of applied...
TRANSCRIPT
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
International Journal of Applied MechanicsVol. 7, No. 4 (2015) 1550052 (16 pages)c© Imperial College PressDOI: 10.1142/S1758825115500520
Effect of Magnetic Field on Free Convection in InclinedCylindrical Annulus Containing Molten Potassium
Masoud Afrand∗,§, Nima Sina∗, Hamid Teimouri∗, Ali Mazaheri∗,Mohammad Reza Safaei†, Mohammad Hemmat Esfe∗,
Jamal Kamali∗ and Davood Toghraie‡∗Department of Mechanical Engineering
Faculty of Engineering, Najafabad BranchIslamic Azad University, Najafabad, Isfahan, Iran
†Young Researchers and Elite ClubMashhad Branch, Islamic Azad University
Mashhad, Iran‡Young Researchers and Elite Club
Khomeinishahr BranchIslamic Azad University, Isfahan, Iran
§[email protected]§masoud [email protected]
Received 8 July 2014Revised 28 April 2015Accepted 28 April 2015
Published 11 August 2015
Three-dimensional (3D) numerical simulation of natural convection of an electrically con-ducting fluid under the influence of a magnetic field in an inclined cylindrical annulushas been performed. The inner and outer cylinders are maintained at uniform temper-atures and other walls are thermally insulated. The governing equations of this fluidsystem are solved by a finite volume (FV) code based on SIMPLER solution scheme.Detailed numerical results of heat transfer rate, Lorentz force, temperature and electricfields have been presented for a wide range of Hartmann number (0 ≤ Ha ≤ 60) andinclination angle (0 ≤ γ ≤ 90). The results indicate that a magnetic field can control themagnetic convection of an electrically conducting fluid. Depending on the direction andstrength of the magnetic field, the suppression of convective motion was observed. Forvertical cylindrical annulus, increasing the strength of the magnetic field causes the losssymmetry, and as the consequence, isotherms lose their circular shape. With increasingthe Hartmann number the average Nusselt number approaches a constant value. Forvertical annulus, the effect of Hartmann number on the average Nusselt number is notprominent compared to the case of horizontal annulus.
Keywords: Inclined cylindrical annulus; local Nusselt number; magnetic field; 3D numer-ical investigation; electric potential.
§Corresponding author.
1550052-1
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
M. Afrand et al.
1. Introduction
Magnetic field plays a vital role in many industrial applications such as coolingof electronic equipment, metal solidification, solar technology, heat exchangers andthe like [Mahmoodi et al., 2015; Afrand et al., 2015]. In the casting process, freeconvection happens in a cavity filled with a liquid metal (electrically conductingfluid) due to a temperature difference between the solid walls and the liquid metal.Free convection flow in the melt causes the impurities to move. This phenomenonaffects the structure of the final product including metal slab, ingot and so forth.Therefore, the melt must have a uniform temperature throughout the cavity toreduce thermal stresses and free convection. In recent decades, this problem hasbeen untangled using magnetohydrodynamics (MHD), and free convection flow hasdecreased within the cavity by applying the magnetic field. In MHD, electricallyconducting fluid flow in the presence of a magnetic field is investigated.
Investigation of hydromagnetic flow of an electrically conducting, viscous andincompressible fluid the within the channel, duct and cavity is of much signifi-cance due to its varied and wide applications in science and engineering. Severalresearchers investigated such fluid flow problem considering different aspects of theproblem. Mention may be made of the research studies by Abbas et al. [2006], Hayatet al. [2007, 2008], Seth et al. [2010, 2012], Srinivas and Muthuraj [2010], Jha andApere [2011], Awasthi and Agrawal [2012], Ram and Kumar [2012], Cai et al. [2012]and Asadi et al. [2013]. In addition, hydromagnetic convective flow of an electricallyconducting fluid has been studied by many researchers including Pop et al. [2001],Ghosh et al. [2002, 2013], Guria and Jana [2007], Seth and Singh [2008], Seth andAnsari [2009] and Afrand et al. [2014].
When an electrically conducting fluid is exposed to a magnetic field, the Lorentzforce is also active and interacts with the buoyancy force in governing the flow andtemperature fields, and it can also control growing crystals. The phenomenon hasalso been successfully used in solidification processes to weaken the buoyancy drivenflows. In this regard, Gao et al. [2007] displayed that totally equiaxed grains couldbe obtained in pure Al under the external effect of the pulsed magnetic field; yet,only thin columnar grains are formed in high-melting steel even when treated bythe higher magnetic intensity.
Natural convection in a rectangular cavity has been investigated bymany researchers including Rudraiah et al. [1995], Piazza and Ciofalo [2006],Pirmohammadi and Ghassemi [2009], Sheikhzadeh et al. [2011] and Yu et al. [2013].The annular geometry has been extensively investigated by many researchers usingexperimental, analytical and numerical methods. Afrand et al. [2014a] studied asteady, laminar, natural-convection flow under different directions of uniform mag-netic field in a long horizontal annulus with isothermal walls. The fluid was gallium.The method of simulation is finite volume (FV) based on Patankar’s SIMPLERmethod. Their results show that with increasing Hartmann number (Ha) the con-vective heat transfer decreases. In addition, it was observed that at low Rayleigh
1550052-2
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
Magnetic Field Effects on Free Convection in Inclined Cylindrical Annulus
number, as the strength of the magnetic field increases, convection is damped andthe heat transfer in the enclosure is mostly due to the conduction mode. Wrobelet al. [2010] investigated convection in an annular enclosure with a round rod coreand a cylindrical outer wall containing paramagnetic fluid using an experimentaland numerical analysis. It was concluded that the magnetic field yielded heat trans-fer values four times as high as those of the thermal Rayleigh number; thus, itinfluences heat transfer more efficiently, in comparison with increasing the thermalRayleigh number. Steady fully developed laminar natural convective flow in openended vertical concentric annuli in the presence of a radial magnetic field was con-sidered analytically by Singh and Singh [2012]. They obtained temperature field forthe cases of isothermal and constant heat flux on the inner cylinder of concentricannuli. Also, it was found that Hartmann number and the gap between cylindersplay important roles in controlling the behavior of fluid flow. Natural convectionin concentric horizontal annuli containing magnetic fluid under nonuniform mag-netic fields was investigated experimentally by Sawada et al. [1993]. Two concen-tric cylinders made of copper were placed horizontally and maintained at constanttemperatures. Various kinds of experiments were performed to clarify the effects ofdirection and the strength of magnetic fields on the natural convection. Their resultsshowed that when a magnetic field gradient is applied in the same direction as thegravity, a wall-temperature distribution is observed as if an apparent gravity wereincreased. However, the clear influence of the magnetic field was not found. Also,when a magnetic field gradient is applied in the opposite direction of the gravity, thereverse natural convection occurs. Sankar et al. [2006] used implicit finite differencescheme to simulate the effect of axial or radial magnetic field on the natural convec-tion in a vertical cylindrical annulus at low Prandtl number. The inner and outercylinders were kept at constant temperatures and the horizontal top and bottomwalls were thermally insulated. It was found that the flow and heat transfer weresuppressed more effectively by an axial magnetic field in shallow cavity, whereas intall cavities, a radial magnetic field was more effective. Also, it was observed thatthe average Nusselt number is increased with radii ratio but decreased with theHartmann number. Furthermore, Venkatachalappa et al. [2011] reported the effectof axial or radial magnetic field on the double-diffusive natural convection in a ver-tical cylindrical annular cavity. From their results, it is found that the magneticfield suppresses the double diffusive convection only for small buoyancy ratios. But,for a larger buoyancy ratio, the magnetic field is effective in suppressing the ther-mal convective flow. Exact solutions for fully developed natural convection in anopen-ended vertical concentric annuli under a radial magnetic field were presentedby Singh et al. [1997]. It is observed that both velocity and temperature of the fluidare higher in case of isothermal condition compared to constant heat flux case whenthe gap between cylinders is less than or equal to the radius of inner cylinder whilereverse phenomena occur when the gap between cylinders is greater than the radiusof inner cylinder. The space between two concentric vertical cylinders containing a
1550052-3
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
M. Afrand et al.
viscous incompressible and electrically conducting fluid exposed to a radial mag-netic field was considered by Kumar and Singh [2013]. The effect of the Hartmannnumber and buoyancy force distribution parameter on the fluid velocity, inducedmagnetic field and induced current density have been analyzed using the graphs. Itis observed that the fluid velocity and induced magnetic field are rapidly decreas-ing with the increase in the value of Hartmann number. Kakarantzas et al. [2011]investigated laminar and turbulent regimes of the liquid metal flow in a verticalannulus under constant horizontal magnetic field numerically. They have consid-ered that flow is driven by using the volumetric heating and temperature differencebetween the two cylindrical walls. Their results illustrated that when the magneticfield increases, the flow becomes laminar. Also, they observed that the magneticfield causes the loss of axisymmetry of flow.
Review of these articles show that various parameters were studied in differentworks. From each work, different aspects of convection heat transfer in the presenceof magnetic field have been studied and different results have been obtained eachof which could be useful in its position. However, the effect of the magnetic field onan inclined annulus (by changing the enclosure from horizontal to vertical) is notpresented in the literature. Subsequently, it seems that such research can make agood contribution to the field. Temperature and electric fields inside the enclosureare calculated and the effects of the magnetic field strength on transport phenomenaare determined. The aim of the present study is to investigate free convection in aninclined annulus filled with molten potassium under horizontal magnetic field. Themodel used in this study is selected because of its applications in industry includingtubing industry, heat exchangers, power generation and the like. In the presentstudy, an FV scheme is used to solve nonlinear governing differential equations ofthe problem. In this study, the induced electric potential equation is solved and itseffect on the Lorentz force is given.
2. Mathematical Formulation
The modeling considered in the present study is an inclined cylindrical annulusformed by two concentric cylinders of inner and outer radii, ri and ro, respectivelyas shown in Fig. 1. The inner and outer cylinders are maintained at isothermalbut different temperatures, Th and Tc, respectively. The top and bottom walls areassumed to be adiabatic. The annulus is filled with a low Prandtl number fluid (Pr =0.072), which is electrically conducting. The three-dimensional (3D) cylindrical co-ordinates (r, θ, z) with their corresponding velocity components (u, v, w) are clarifiedin Fig. 1. It must be noted that in Fig. 1, g is the gravity acceleration, γ is the angleof incline and B0 is a uniform magnetic field, which is applied orthogonally tothe z- and y-direction. All walls are assumed to be electrically insulated. Also, weassume that the induced magnetic field produced by the motion of an electricallyconducting fluid is negligible compared to the applied magnetic field B0. By usingthe Boussinesq approximation, the equations governing steady, laminar, Newtonian,
1550052-4
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
Magnetic Field Effects on Free Convection in Inclined Cylindrical Annulus
Fig. 1. Geometry and co-ordinates of annulus configuration with constant horizontal magneticfield effect.
viscous, incompressible and electrically conducting fluid, after neglecting viscousand Ohmic dissipations in the 3D cylindrical co-ordinates (r, θ, z), are defined asfollows:
Continuity equation:
∇ ·V = 0. (1)
Momentum equation:
(V · ∇)V = −1ρ∇p + ν∇2V + g + F. (2)
Energy equation:
(V · ∇)T = α∇2T. (3)
Electric potential equation:
∇2φ = ∇ · (V × B0) = V · (∇ × B0) + B0 · (∇ × V). (4)
Since the magnetic field is constant (B0 = cte.), the first term of the right-handside of Eq. (4) is zero
J = σ(E + V × B0), E = −∇φ and F = J × B0. (5)
V is the vector of velocity, p is the pressure, µ is the dynamic viscosity, ρ is thefluid density, T is the temperature, α is the thermal diffusivity, β is the coefficient
1550052-5
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
M. Afrand et al.
of thermal expansion, σ is the electrical conductivity, E is the induced electric fieldvector, J is the electric current density vector, F is the Lorentz force vector and φ
is the electric potential. Also, ∇ and ∇2 are the gradient and Laplacian operators,respectively.
For convenience, the governing equations are nondimensionalized using the fol-lowing parameters:
U =uD
α, V =
vD
α, W =
wD
α, R =
r
D, Z =
z
L, A =
L
D,
P =pD2
ρα2, T ∗ =
T − Tc
Th − Tc, Φ =
φ
B0α, E∗ =
ED
B0α, λ =
ro
ri.
(6)
In Eq. (6), D and L are the height and gap of annulus, respectively.With some mathematical manipulations, the governing equations given by
Eqs. (1)–(4) can be expressed in dimensionless form as
Dimensionless continuity equation:
∂U
∂R+
1R
∂V
∂θ+
1A
∂W
∂Z= 0. (7)
Dimensionless momentum equation:
r-component:
U∂U
∂R+
V
R
∂U
∂θ− V 2
R+
1A
W∂U
∂Z
= −∂P
∂R+ Pr
[∂2U
∂R2+
1R
∂U
∂R+
1R2
∂2U
∂θ2− U
R2− 2
R2
∂V
∂θ+
1A2
∂2U
∂Z2
]
+ Ra · Pr ·T ∗ cos γ cos θ + Ha2 · Pr[− 1
A
∂Φ∂Z
cos θ − U(1 + cos2 θ) +V
2sin 2θ
].
(8)
θ-component:
U∂V
∂R+
V
R
∂V
∂θ− UV
R+
1A
W∂V
∂Z
= − 1R
∂P
∂θ+ Pr
[∂2V
∂R2+
1R2
∂2V
∂θ2+
1R
∂V
∂R− V
R2− 2
R2
∂U
∂θ+
1A2
∂2V
∂Z2
]
−Ra · Pr ·T ∗ cos γ sin θ
+ Ha2 · Pr[
1A
∂Φ∂Z
sin θ − V (1 + sin2 θ) − W cos θ +U
2sin 2θ
]. (9)
1550052-6
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
Magnetic Field Effects on Free Convection in Inclined Cylindrical Annulus
z-component:
U∂W
∂R+
V
R
∂W
∂θ+
1A
W∂W
∂Z
= − 1A
∂P
∂Z+ Pr
[∂2W
∂R2+
1R2
∂2W
∂θ2+
1R
∂W
∂R+
1A2
∂2W
∂Z2
]
+ Ra · Pr ·T ∗ sin γ + Ha2 · Pr(
∂Φ∂R
cos θ − 1R
∂φ
∂θsin θ − W
). (10)
Dimensionless energy equation:
U∂T ∗
∂R+
V
R
∂T ∗
∂θ+
1A
W∂T ∗
∂Z=
∂2T ∗
∂R2+
1R2
∂2T ∗
∂θ2+
1R
∂T ∗
∂R+
1A2
∂2T ∗
∂Z2. (11)
Dimensionless electric potential equation:
∂2Φ∂R2
+1
R2
∂2Φ∂θ2
+1R
∂Φ∂R
+∂2Φ∂Z2
=(
1R
∂W
∂θ− 1
A
∂V
∂Z
)sin θ +
(1A
∂U
∂Z− ∂W
∂R
)cos θ.
(12)In the above equations, Pr is the Prandtl number and Ra is the Rayleigh numberdefined as follows:
Pr =υ
α, Ra =
gβ(TH − TC)D3
υα. (13)
The effect of the magnetic field is introduced into the momentum and potentialequations through the Hartmann number. The Hartmann number is defined as
Ha = B0D
√σ
µ. (14)
The local and average Nusselt numbers along the inner cylinder are defined asfollows:
Nu(θ, Z) =∂T ∗
∂R
∣∣∣∣R=Ri
, Nu =1
2πL
∫ 2π
0
∫ L
0
Nu(θ, Z)dZdθ. (15)
No-slip conditions are assumed on all walls. Constant temperatures are consideredat the inner (T ∗
i = 1) and outer (T ∗o = 0) cylindrical walls and adiabatic conditions
are considered at the bottom and top walls (∂T ∗/∂Z = 0). Also, all walls are assumedto be electrically insulated (∂Φ/∂n = 0).
3. Numerical Procedure
The governing equations along with the boundary conditions are solved numericallyusing the FV method. The diffusion terms in the governing equations are discretizedusing a second-order central difference scheme; meanwhile, for the hybrid scheme,which is a combination of the central difference scheme and the upwind scheme, it
1550052-7
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
M. Afrand et al.
is used to discretize the convection terms. A staggered grid system together withthe SIMPLER algorithm is adopted to solve the governing equations. The coupledsystems of discretized equations are solved iteratively using the TDMA method[Patankar, 1980]. To obtain the converged solutions, an under-relaxation scheme isemployed.
4. Grid Study and Validation
Different grid sizes are examined to ensure grid independence results. The testedgrids and the obtained average Nusselt numbers are demonstrated in Table 1. Theresults are obtained for inclined annulus with λ = 6.0, A = 3 and γ = 45◦ containingliquid metal with Pr = 0.072. Also, Hartmann and Rayleigh numbers are 30 and105, respectively. According to Table 1, a 60 × 60 × 60 grid is sufficiently fine in r,θ and z directions, respectively.
In order to approve the numerical procedure, two test cases are examined usingthe proposed code. In the first test, the results are compared with the existingexperimental results in the literature. In the absence of magnetic field, the numericalresults are first validated with the experimental data obtained by Wrobel et al. [2010]as illustrated in left panel of Fig. 2. The maximum discrepancy is within 8%. Inthe second test, a comparison is made between the present results and Kumar andKalam’s [1991] numerical data as depicted in the right panel of Fig. 2. As shown,there is an agreement between the results.
Table 1. Grid independence test.
Grid 15 × 60 × 15 30 × 60 × 30 60 × 60 × 60 120 × 60 × 120Nu 4.285 4.923 5.245 5.321Error 12.5% 6.1% 1.4% —
Fig. 2. Comparison of average Nusselt number results with data in the literature. Experimentaldata (left) and numerical data (right).
1550052-8
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
Magnetic Field Effects on Free Convection in Inclined Cylindrical Annulus
5. Results and Discussion
Natural convection of a low Prandtl number electrically conducting fluid in aninclined cylindrical annulus with isothermally heated and cooled vertical walls andadiabatic top and bottom walls in the presence of uniform magnetic field is studiednumerically. According to the work done by Kakarantzas et al. [2011], this annuluswas considered with an aspect ratio A = 3, and radii ratio λ = 6, which is used inindustries including the tubing industry, heat exchangers, power generation and thelike. Simulations are performed for a wide range of Hartmann number (0 ≤ Ha ≤60) and inclination angle (0 ≤ γ ≤ 90). Also, it is assumed that Ra = 105 andPr = 0.072.
Figures 3 and 4 show the effect of the magnetic field (different Hartmann num-bers from 0 to 40) on the temperature distribution inside the cylindrical annulus.This figure shows the isotherms at different inclination angles from γ = 0◦ to γ = 90◦
Fig. 3. Isotherms for different inclination angles and Ha = 0.
Fig. 4. Isotherms for different inclination angles and Ha = 40.
1550052-9
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
M. Afrand et al.
in two cross sections perpendicular to x-direction and z-direction. At γ = 0◦, moredense isotherms can be observed under the hot wall, which is parallel to its direction;hence, higher heat transfer rate is achieved. Moreover, the flow is in symmetry alongthe z-direction due to absence of the buoyancy force. This symmetry is lost withincreasing inclination angle; as a result, more dense isotherms are seen near annulusbottom. Comparison of Figs. 3 and 4 show that with increasing Ha, the density ofisotherms is reduced and consequently, heat transfer is reduced for all inclinationangles. This is due to the fact that with increasing the Hartmann number, Lorentzforce is increased against the buoyancy force. Inspection of isotherms in differentinclination angles makes it clear while at γ = 90◦, increasing the Hartmann numbercauses the loss of symmetry in θ-direction. The isotherms lose their circular shape,if being stretched in the y-direction.
The induced electric field and the Lorentz force distribution at different sectionsunder the influence of magnetic field (Ha = 20) at different inclination angles arepresented in Figs. 5 and 6. As expressed in Eq. (5), the Lorentz force depends ontwo factors, E and (V × B0). It is found from Fig. 5 that the maximum valueof Lorentz force occurs at the magnetic field direction. More distance from thisdirection corresponds to lower values of Lorentz force. Also, Fig. 6 reveals that forthe induced electric field, the above phenomenon is reversed. Therefore, in eacharea where the induced electric field is stronger, the Lorentz force is weaker. FromEq. (5) and Fig. 4, it can be deduced that the induced electric field decreases theLorentz force and (V × B0) increases the Lorentz force in the present model.
Figure 7 shows the effect of inclination angle and Hartmann number on the aver-age Nusselt number for Ra = 105. As expected, we can observe that for a given valueof inclination angle average Nusselt number is a decreasing function of Hartmannnumber. This is due to the fact that with the increase in Hartmann number, theconvection is progressively reduced by the magnetic drag resulting in a lower heattransfer. Also, with increasing the Hartmann number, the average Nusselt number
Fig. 5. Numerical results for Lorentz force and Ha = 20.
1550052-10
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
Magnetic Field Effects on Free Convection in Inclined Cylindrical Annulus
Fig. 6. Numerical results for induced electric field and Ha = 20.
Fig. 7. Effect of inclination angle and Hartmann number on the average Nusselt number.
approaches the constant value. This behavior indicates the conduction regime. Inaddition, it can be seen from this figure that for different inclination angles, theeffect of Hartmann number on the average Nusselt number has almost the sametrend. For higher inclination angles (vertical annulus), the effect of Hartmann num-ber on the average Nusselt number is not prominent compared to the case of lowinclination angles (horizontal annulus). At γ = 90◦, the minimum value of the aver-age the Nusselt number for different Hartmann numbers is observed. This meansthat with increasing the inclination angle, the convection mode of heat transfer issuppressed.
1550052-11
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
M. Afrand et al.
Fig. 8. Effect of Hartmann number and tilt on the local Nusselt number.
The numerical results for Ha = 0 and Ha = 30 at different inclination angles(γ = 0, γ = 45◦ and γ = 90◦) as a three-dimensional surface are plotted in Fig. 8.This figure illustrates the influence of inclination angle and Hartmann number onthe local Nusselt number at the inner wall. It is observed that with increasing theHartmann number, at all inclination angles, the local Nusselt number is decreased.This is due to the fact that by imposing the magnetic field, the fluid is retardeddue to the action of the Lorentz force and thus, convective heat transfer decreases.Figure 8(a) illustrates that the maximum Nusselt number occurs at 180◦. In thiscase, the Nusselt number along the length of the annulus is constant. Nusselt numbernear the annulus bottom is higher than its top while the annulus is consideredvertical. This event is because of the fluid movement in opposite direction of g (due
1550052-12
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
Magnetic Field Effects on Free Convection in Inclined Cylindrical Annulus
to the buoyancy force effect in free convection mechanism). Thickness of boundarylayer increases as it goes higher leading to decrease in the temperature gradient;as a result, the heat transfer rate (Nu) will decrease. Moreover, the symmetry canbe seen at the θ-direction of annulus in Fig. 8(e). The effects of magnetic field areshown in Figs. 8(b), 8(d) and 8(f). It should be noted that the Lorentz force affectsthe whole annulus volume in the same way while annulus is horizontal. However, theamount of Lorentz force would be higher at some places and lower at the other oneswith a change in annulus inclination until the least amounts of Nusselt numberoccur at the place of 90◦ and 270◦. These points are in a plane, parallel to themagnetic field direction. At these points, since there is no induced electric field,the maximum value for Lorenz force is obtained. More distance from these pointscorresponds to higher values of Nusselt number, and its greatest values would occurat 0◦ and 180◦; representing a plane perpendicular to the magnetic field direction.This phenomenon is obtained due to the higher values of electric field far awayfrom these points; consequently, the Lorentz force will be decreased. This is in totalagreement with the results obtained in Figs. 5 and 6.
6. Conclusion
Three-dimensional simulation of free convection in an inclined cylindrical annulusfilled with molten potassium in the presence of uniform magnetic field was per-formed. Simulations were carried out for a wide range of Hartmann number andinclination angle. Results have been presented in the form of isotherms, Nusseltnumber, Lorentz force and induced electric field distribution. In view of the resultsand discussion, the main contributions of the present paper are as follows:
• Magnetic convection of electrically conducting fluid could be controlled by a mag-netic field which depends on its direction and strength.
• For vertical cylindrical annulus, increasing the strength of the magnetic fieldcauses the loss of axisymmetry, and consequently isotherms lose their circularshape.
• The horizontal magnetic field induces the electric potential which, in turn, reducesthe Lorentz force against buoyancy force. Therefore, in each area, where theinduced electric field is stronger, the Lorentz force is weaker.
• With increasing the Hartmann number, the average Nusselt number approachesthe constant value, which indicates the conduction regime.
• For vertical annulus, the effect of Hartmann number on the average Nusselt num-ber is not prominent compared to the case of horizontal annulus.
• In horizontal annulus, the Nusselt number along the length of the annulus isconstant. Nusselt number near the annulus bottom is higher than its top while theannulus is considered vertical.
• The minimum value of Nusselt number occurs in x-direction, which is parallel tothe magnetic field direction.
1550052-13
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
M. Afrand et al.
References
Abbas, Z., Sajid, M. and Hayat, T. [2006] “MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel,” Theoretical and Computational FluidDynamics 20, 229–238.
Afrand, M., Farahat, S., Hossein Nezhad, A., Sheikhzadeh, G. A. and Sarhaddi, F. [2014a]“Numerical simulation of electrically conducting fluid flow and free convective heattransfer in an annulus on applying a magnetic field,” Heat Transfer Research 45,749–766.
Afrand, M., Farahat, S., Hossein Nezhad, A., Sheikhzadeh, G. A. and Sarhaddi, F.[2014b] “3-D numerical investigation of natural convection in a tilted cylindrical annu-lus containing molten potassium and controlling it using various magnetic fields,”International Journal of Applied Electromagnetics and Mechanics 46, 809–821.
Afrand, M., Farahat, S., Hossein Nezhad, A., Sheikhzadeh, G. A., Sarhaddi, F. and Wong-wises, S. [2015a] “Multi-objective optimization of natural convection in a cylindri-cal annulus mold under magnetic field using particle swarm algorithm,” InternationalCommunications in Heat and Mass Transfer 60, 13–20.
Afrand, M., Rostami, S., Akbari, M., Wongwises, S., Hemmat Esfe, M., and Karimipour,A. [2015b] “Effect of induced electric field on magneto-natural convection in a verticalcylindrical annulus filled with liquid potassium,” International Journal of Heat andMass Transfer 90, 418–426.
Asadi, H., Javaherdeh, K. and Ramezani, S. [2013] “Finite element simulation of micropolarfluid flow in the lid-driven square cavity,” International Journal of Applied Mechanics5, 1350045.
Awasthi, M. K. and Agrawal, G. S. [2012] “Viscous contributions to the pressure forthe potential flow analysis of magnetohydrodynamic Kelvin–Helmholtz instability,”International Journal of Applied Mechanics 4, 1250001.
Cai, L., Zhou, J., Zhou, F., Xie, W. and Nie, Y. [2012] “Numerical simulation of 2D liquidsloshing,” International Journal of Applied Mechanics 4, 1250014.
Gao, Y. L., Li, Q. S., Gong, Y. Y. and Zhai, Q. J. [2007] “Comparative study on structuraltransformation of low-melting pure Al and high-melting stainless steel under externalpulsed magnetic field,” Material Letters 61, 4011–4014.
Ghosh, S. K., Anwar Beg, O. and Narhari, M. [2013] “A study of unsteady rotating hydro-magnetic free and forced convection in a channel subject to forced oscillation under anoblique magnetic field,” Journal of Applied Fluid Mechanics 6, 213–227.
Ghosh, S. K., Pop, I. and Nandi, D. K. [2002] “MHD fully developed mixed convectionflow with asymmetric heating of the wall,” International Journal of Applied Mechanicsand Engineering 7, 1211–1228.
Guria, M. and Jana, R. N. [2007] “Hall effects on the hydromagnetic convective flowthrough a rotating channel under general wall conditions,” Magnetohydrodynamics 43,287–300.
Hayat, T., Ahmed, N. and Sajid, M. [2008] “Analytical solution for MHD flow of a thirdorder fluid in a porous channel,” Journal of Computational and Applied Mathematics214, 572–582.
Hayat, T., Ahmed, N., Sajid, M. and Asghar, S. [2007] “On the MHD flow of a secondgrade fluid in a porous channel,” Computers and Mathematics with Applications 54,407–414.
Jha, B. K. and Apere, C. A. [2011] “Magnetohydrodynamic free convective couette flowwith suction and injection,” Journal of Heat Transfer 133, 092501.
1550052-14
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
Magnetic Field Effects on Free Convection in Inclined Cylindrical Annulus
Kakarantzas, S. C., Sarris, I. E. and Vlachos, N. S. [2011] “Natural convection of liquidmetal in a vertical annulus with lateral and volumetric heating in the presence ofa horizontal magnetic field,” International Journal of Heat and Mass Transfer 54,3347–3356.
Kumar, A. and Singh, A. K. [2013] “Effect of induced magnetic field on natural convectionin vertical concentric annuli heated/cooled asymmetrically,” Journal of Applied FluidMechanics 6, 15–26.
Kumar, R. and Kalam, M. A. [1991] “Laminar thermal convection between vertical coaxialisothermal cylinders,” International Journal of Heat and Mass Transfer 34, 513–524.
Mahmoodi, M., Esfe, M. H., Akbari, M., Karimipour, A. and Afrand, M. [2015] “Magneto-natural convection in square cavities with a source–sink pair on different walls,” Inter-national Journal of Applied Electromagnetics and Mechanics 47, 21–32.
Patankar, S. V. [1980] Numerical Heat Transfer and Fluid Flow (Taylor and Francis, NewYork).
Piazza, I. D. and Ciofalo, M. [2006] “MHD free convection in a liquid-metal filled cubicenclosure. II. Internal heating,” International Journal of Heat and Mass Transfer 49,4525–4535.
Pirmohammadi, M. and Ghassemi, M. [2009] “Effect of magnetic field on convection heattransfer inside a tilted square enclosure,” International Communications in Heat andMass Transfer 36, 776–780.
Pop, I., Ghosh, S. K. and Nandi, D. K. [2001] “Effects of the hall current on free andforced convection flows in a rotating channel in the presence of an inclined magneticfield,” Magnetohydrodynamics 37, 348–359.
Ram, P. and Kumar, V. [2012] “Ferrofluid flow with magnetic field-dependent viscositydue to rotating disk in porous medium,” International Journal of Applied Mechanics4, 1250041.
Rudraiah, N., Barron, R. M., Venkatachalappa, M. and Subbaraya, C. K. [1995] “Effect ofa magnetic field on free convection in a rectangular enclosure,” International Journalof Engineering Science 33, 1075–1084.
Sankar, M., Venkatachalappa, M. and Shivakumara, I. S. [2006] “Effect of magnetic fieldon natural convection in a vertical cylindrical annulus,” International Journal of Engi-neering Science 44, 1556–1570.
Sawada, T., Kikura, H., Saito, A. and Tanahashi, T. [1993] “Natural convection of a mag-netic fluid in concentric horizontal annuli under nonuniform magnetic fields,” Experi-mental Thermal and Fluid Science 7, 212–220.
Seth, G. S., Ansari, M. S. and Nandkeolyar, R. [2010] “Unsteady hydromagnetic Couetteflow induced due to accelerated movement of one of the porous plates of the channelin a rotating system,” International Journal of Applied Mathematics and Mechanics6, 24–42.
Seth, G. S. and Ansari, S. [2009] “Magnetohydrodynamic convective flow in a rotatingchannel with Hall effects,” International Journal of Theoretical and Applied Mechanics4, 205–222.
Seth, G. S. and Singh, M. K. [2008] “Combined free and forced convection MHD flow in arotating channel with perfectly conducting walls,” International Journal of TheoreticalPhysics 56, 203–222.
Seth, G. S., Singh, J. K. and Mahto, G. K. [2012] “Effects of Hall current and rotation onunsteady hydromagnetic Couette flow within a porous channel,” International Journalof Applied Mechanics 4, 1250015.
Sheikhzadeh, G. A., Fattahi, A. and Mehrabian, M. A. [2011] “Numerical study of steadymagneto-convection around an adiabatic body inside a square enclosure in low Prandtlnumbers,” Heat and Mass Transfer 47, 27–34.
1550052-15
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.
2nd Reading
August 6, 2015 16:8 WSPC-255-IJAM S1758-8251 1550052
M. Afrand et al.
Singh, R. K. and Singh, A. K. [2012] “Effect of induced magnetic field on natural convectionin vertical concentric annuli,” Acta Mechanica Sinica 28(2), 315–323.
Singh, S. K., Jha, B. K. and Singh, A. K. [1997] “Natural convection in vertical concentricannuli under a radial magnetic field,” Heat and Mass Transfer 32, 399–401.
Srinivas, S. and Muthuraj, R. [2010] “Peristaltic transport of a Jeffrey fluid under the effectof slip in an inclined asymmetric channel,” International Journal of Applied Mechanics2, 437–455.
Venkatachalappa, M., Do, Y. and Sankar, M. [2011] “Effect of magnetic field on the heatand mass transfer in a vertical annulus,” International Journal of Engineering Science49, 262–278.
Wrobel, W., Fornalik-Wajs, E. and Szmyd, J. S. [2010] “Experimental and numericalanalysis of thermo-magnetic convectionin a vertical annular enclosure,” InternationalJournal of Heat and Fluid Flow 31, 1019–1031.
Yu, P. X., Qiu, J. X., Qin, Q. and Tian, Z. F. [2013] “Numerical investigation of naturalconvection in a rectangular cavity under different directions of uniform magnetic field,”International Journal of Heat and Mass Transfer 67, 1131–1144.
1550052-16
Int.
J. A
ppl.
Mec
hani
cs D
ownl
oade
d fr
om w
ww
.wor
ldsc
ient
ific
.com
by N
AN
YA
NG
TE
CH
NO
LO
GIC
AL
UN
IVE
RSI
TY
on
08/1
9/15
. For
per
sona
l use
onl
y.