DOI 10.3233/JAE-150028 IOS Press
Natural convection of liquid metal in a
horizontal cylindrical annulus under radial magnetic field
Hamid Teimouri, Masoud Afrand∗, Nima Sina, Arash Karimipour and Amir Homayoon Meghdadi Isfahani
Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran
Abstract.In this article, the effect of a radial magnetic field on the steady, laminar and natural convection flow of molten potassium in a long horizontal annulus between two horizontal cylinders is investigated numerically. The inner and outer cylinders are atTHandTCtemperatures, respectively, withTH > TC. A finite volume code based on SIMPLER-algorithm is used to solve the governing equations of the model. Simulations are carried out for a wide range of the Rayleigh number (Ra), Hartmann number (Ha) and radii ratio (λ). It is shown that for a given radii ratio as the strength of magnetic field increases, the convection heat transfer reduces. For narrow annulus, the effect of Hartmann number on the maximum absolute value of the stream function is not eminent compared to the case of high radii ratio. Furthermore it is found that for a givenHaandRa, the average Nusselt number increases up to aboutλ=5 and then reduces gently asλenhances. Moreover, for all Hartmann numbers the effect of radii ratio on the average Nusselt number has almost the same trend.
Keywords: Liquid metal, radial magnetic field, free convection, horizontal annulus, numerical simulation
Nomenclature List of symbols
r radius TC cold wall temperature
ri inner radius T∗ dimensionless temperature
ro outer radius B0 magnitude of magnetic field
L annulus gap g acceleration due to gravity
Ri inner dimensionless radius Pr Prandtl number
Ro outer dimensionless radius Ha Hartmann number
vθ tangential velocity Ra Rayleigh number
vr radial velocity Nu Nusselt number
Vθ dimensionless tangential velocity J current density
Vr dimensionless radial velocity Rcond Conduction thermal resistance
p Pressure k thermal conductivity
P dimensionless pressure h heat transfer coefficient
TH hot wall temperature q heat transfer
∗Corresponding author: Masoud Afrand, Department of Mechanical Engineering, Najafabad Branch, Islamic Azad Univer- sity, Najafabad, Iran. E-mail: [email protected].
1383-5416/15/$35.00 c2015 – IOS Press and the authors. All rights reserved
Greek symbols
α thermal diffusivity ρ density
β volumetric expansion coefficient υ kinematic viscosity
σ electrical conductivity θ tangential component
1. Introduction
The free convection in an enclosure, in many engineering application, has been investigated. There are many notable examples which can be cited on this area such as cooling the electrical devices, nuclear reactors insulation, sun collectors and growing the grains in fluids and so on. Therefore, the practical ap- plication of this method is proliferated in a wide range of material industries. Extensive researches have been carried out on the rectangular and cylindrical enclosures due to their profound effect on the crystal grain grow and the products quality [1–6]. In order to demonstrate the impact of magnetic field on free convection in electrically conducting fluids on the cylindrical annulus, a wide variety of researches have been undertaken. Sankar et al. [7] investigated the influence of magnetic field on the free convection between the two coaxial vertical cylinders. This research was performed by the electrically conducting fluids, with the Prandtl Number, 0.054, influenced by axial and radial magnetic fields. The temperature of inner and outer cylinders is constant. Having used the numerical method, they managed to prove the effect of Rayleigh and Hartmann numbers on the temperature and stream lines. Moreover, Venkatacha- lappa et al. [8] by employing numerical methods studied the effect of radial and axial magnetic fields on the convection heat and mass transfer in a vertical hollow cylinder. They showed the influence of Hartmann number on the temperature and stream profiles where the magnetic field is radial and axial.
Kakarantzas et al. [9], studied MHD free convection of a liquid metal in a vertical annulus with a sinu- soidal temperature distribution at the upper wall and the other surfaces being adiabatic. They investigated the effect of the Hartmann and Rayleigh numbers in laminar and turbulent streams. Their results show that the vertical magnetic field (in the zdirection) decreased the Nu number more than the horizontal one in ther,θplane. Furthermore, Mozayyeni and Rahimi [10] numerically studied mixed convection of a fluid in the fully developed region in a horizontal concentric cylindrical annulus with different uni- form wall temperatures. The forced flow is induced by the cold rotating outer cylinder at slow constant angular velocity. Considerable researches are performed for different combinations of non-dimensional group numbers; Reynolds number, Rayleigh number, Hartmann number, Eckert number and annulus gap width ratio. A finite volume scheme is applied to solve the governing equations, which are continuity, two-dimensional momentum and energy, by the SIMPLE algorithm. They concluded that in an external magnetic field flow and heat transfer are suppressed more effectively. Afrand et al. [11] studied a steady, laminar, natural-convection of gallium under different directions of uniform magnetic field in a long horizontal annulus with isothermal walls. The simulation was done using finite volume method (FVM) based on the SIMPLER algorithm. Their results reveal that increasing the value of Hartmann number (Ha) leads to decreasing the convection heat transfer. In addition, it is observed that at the low Rayleigh number, as the strength of the magnetic field is increased, convection is damped and the heat transfer in the enclosure is mostly due to the conduction mode. Sawada et al. [12] conducted experiments on natural convection in concentric horizontal annuli filled with magnetic fluid under non-uniform magnetic fields.
Two concentric cylinders were made of copper, put horizontally, and kept at constant temperatures. In order to specify how natural convection is influenced by the direction and the strength of magnetic fields, different types of experiments were carried out. Their experiments confirm that when a magnetic field gradient is applied in the same direction as the gravity, a wall-temperature distribution is observed, as
if an apparent gravity is increased; however, the clear influence of the magnetic field was not found.
Moreover, when a magnetic field gradient is applied in the reverse direction of the gravity, the natural convection occurs reversely.
For 2D annulus, it is reasonable to ignore the effects from the electric potential. However, for 3D annulus, electric potential should be considered when the magnetic field is not along the flow direction.
In recent years, several studies have been performed on the effect of magnetic fields on convective heat transfer in 3D cylindrical enclosures considering electric potential. In this regard, Kakarantzas et al. [13]
numerically investigated the effect of a horizontal magnetic field on unsteady natural convection in the vertical annulus filled with electrically conducting fluid. They reported that the horizontal magnetic field causes the loss of axisymmetry of flow. Moreover, Afrand et al. [14,15] performed 3D numerical studies on the effect of various magnetic fields and inclination angle of annulus on natural convection in the cylindrical annulus containing liquid potassium. They showed that the magnetic field can control natural convection by using various magnetic fields. Their results also revealed the effect of the magnetic field direction on Lorentz force, induced electric field and temperature distribution. Recently, laminar steady magneto-natural convection in a vertical cylindrical annulus containing liquid potassium under constant horizontal magnetic field was studied numerically by Afrand et al. [16]. Their results showed that by applying the horizontal magnetic field flow became asymmetric. They found that this fact is due to the growth of Roberts and Hartmann layers near the walls parallel and normal to the magnetic field, respectively. It was also found that for a given value of Hartman number, the average Nusselt number is greater in the case of solving the electric potential equation. The results show that there is a large difference in the Nusselt number.
Since one of the most important shapes of molds is horizontal annulus enclosures, the aim of the present study is to investigate numerically the problem of mixed convection of a fluid between two horizontally concentric cylinders with infinite lengths, in the presence of a radial magnetic field. The effects of radii ratio, Rayleigh number and Hartmann number on the fluid flow and heat transfer are investigated in the low Prandtl number fluid (Pr =0.072) and steady state conditions. The buoyancy effect is regarded through Boussinesq approximation.
2. Mathematical formulation 2.1. Problem statement
Figure 1 shows the geometry of the present problem. Inner cylinder with radius ofri and the outer cylinder with radius of ro are kept at constant temperature Ti and To respectively, where Ti > To. The annulus has infinite length. The annulus is filled with molten potassium. The physical properties of liquid potassium are presented in Table 1. A constant radial magnetic field is applied to the cylinders.
For mathematical modeling of the problem, assumptions considered are as follows:
– Since annulus length is long, the temperature and fluid variations along the annulus length are neglectable and their pattern is assumed the 2D model.
– Because of the low temperature difference between the annulus and liquid potassium, flow is con- sidered laminar. Therefore, the Boussinesq approximation can be used to estimate density.
– Because of low temperature difference, all other thermo-physical characteristics of the fluid are constant.
– Since the model has been considered as 2D, electric potential is not considered.
Table 1
Physical properties of liquid potassium
Parameter Value
Density (ρ) 856 (kg/m3)
Thermal conductivity (k) 52 (W/m·K) Specific heats (Cp) 753 (J/kg·K)
Viscosity (μ) 4.965×10−3(kg/m·s) Thermal expansion coefficient (β) 8.3×10−5(1/K) Electrical conductivity (σ) 7.0×106(1/ohm·m)
Fig. 1. A horizontal annulus with radial magnetic field.
2.2. Governing equations and boundary conditions
The non-dimensional parameters used in this article are:
R= r
L, Vθ = vθr
α , Vr = vrr
α , P = pr2
ρα2, T∗ = T−TC
TH −TC (1)
WhereVθ andVrare the velocity components in theθandr directions, respectively.P, T, αandρare the pressure, temperature, heat diffusion coefficient and fluid density respectively. The gap length,Lis equal withro−ri. With the assumption of Newtonian fluids, steady state and incompressible flow, the non-dimensional governing equations in polar system coordinates are as follows:
The continuity equation:
∂Vr
∂R + 1 R
∂Vθ
∂θ +Vr
R = 0 (2)
The momentum in the radial direction,r:
Vθ R
∂Vr
∂θ +Vr∂Vr
∂R − Vθ2
R =−∂P
∂R+R·Pr
1
R2
∂2Vr
∂θ2 + 1 R
∂Vr
∂R +∂2Vr
∂R2 − Vr R2 − 2
R2
∂Vθ
∂θ
+Ra PrT∗Cosθ (3)
The momentum in the tangential direction,θ:
Vθ R
∂Vθ
∂θ +Vr∂Vθ
∂R +VθVr
R =− 1 R
∂P
∂θ +R·Pr
1
R2
∂2Vθ
∂θ2 + 1 R
∂Vθ
∂R +∂2Vθ
∂R2 − Vθ R2 − 2
R2
∂Vr
∂θ
−Ra Pr RT∗Sinθ−Ha2PrVθ (4)
The energy equation:
Vθ R
∂T∗
∂θ +Vr∂T∗
∂R =R
∂2T∗
∂R2 + 1 R
∂T∗
∂R + 1 R2
∂2T∗
∂θ2
(5)
whichPr, the Prandtl number, andRa, the Rayleigh number, are:
Pr= ν
α, RaL= gβ(TH −TC)L3
αν (6)
In this equation, g is the gravity acceleration,υ is the kinematic acceleration and β is the thermal expansion coefficient.
The effect of imposing magnetic field is illustrated as the Hartmann number,Ha, in the Eqs (2) and (3) which is defined as:
Ha=BoL
σ
ρν (7)
WhereB0is the amount of constant magnetic field andσis the fluid electrical conductivity. Equations (3) and (4) are derived by using Hartman number and disregarding Lorentz force as it is applied in the following equation:
F=J×B, J=σ(V×B) (8)
HereJ is the density of electrical flow.
The local and average Nusselt number on the inner wall:
Nui =− ri Ti−Toln
ro ri
∂T
∂r
r=ri
; Nui = 1 2π
2π
0 Nui(θ)dθ (9)
and the boundary conditions:
VrandVθ = 0at all walls (r =ri, r=ro) (10)
T∗(Ri, θ) = 1, T∗(Ro, θ) = 0 (11) 2.3. Numerical procedure
The governing equations with the particular boundary conditions are discretized using the finite vol- ume method. A non-uniform grid is created. The governing equations are discretized on the grid points.
The diffusion terms are discretized using a central difference scheme, while a hybrid scheme, which is a combination of central difference and upwind schemes, is used to discretize the convective terms.
The pressure and velocity fields are coupled according to the SIMPLER algorithm based on a staggered grid. The set of algebraic equations is solved iteratively using the TDMA method. To acquire converged solutions, an under-relaxation scheme is employed [17].
2.4. Grid independent study and validation
In this section, the fitted mesh in the solution domain is revised. A sample of a fitted mesh in the domain is shown in Fig. 2. In this grid,N andM are the number of grids in the radial and tangential directions. In what follows, the number of mesh elements isM×N. To ensure that the numerical results are independent of the meshing, which is fitted into the solution domain, at the first mesh of optimum points is obtained by comparing the average Nusselt number for different meshes. In this case, 31×11,
Table 2
The average Nusselt number for various N;RaL=105,Pr=0.072,ro/ri=2.6, Ha=0
M N Nu Error %
31 11 3.257 9.88
31 21 2.965 3.67
31 41 2.859 1.37
31 81 2.821 –
Table 3
The average Nusselt number for various M;RaL =105,Pr =0.072,ro/ri = 2.6,Ha=0
M N Nu Error %
31 41 2.859 4.04
61 41 2.986 0.72
121 41 3.009 –
Fig. 2. A sample of mesh fitted on the solution domain.
Fig. 3. Comparison of local Nusselt number results with ex- perimental data of Kuehn and Goldstein [18].
Fig. 4. Comparison of local Nusselt number results with nu- merical data of Kuehn and Goldstein [18].
31×21, 31 ×41 and 31×81 were used to find the appropriate meshing in the tangential direction.
It can be seen in the Table 2 that, the 31×41 mesh, is the appropriate meshing. There is no important difference in the average Nusselt number for the smaller meshes; hence, the 41 elements in the radial direction are appropriate.
Table 3 illustrates different average Nusselt numbers for three types of meshes. As a result, 61×41 is the appropriate mesh in this study; moreover, there is no significant variation in the Nusselt number of the smaller meshes.
Eventually, we drew a comparison between the obtained results and experimental results to confirm the accuracy of numerical simulations. The wall temperature is kept constant and free convection of air (Pr = 0.7) between two horizontal coaxial cylinders for Ra =5 × 104 and ro/ri =2.6 is studied.
As indicated in Fig. 3, local Nusselt numbers on the inner and outer cylinders are compared with ex- perimental results of Kuehn and Goldstein [18]. As shown in Fig. 4, the tangential velocity profile in the radial direction at an angle of 0◦ and 90◦ are compared with the numerical results of Kuehn and Goldstein. In conclusion, these figures reveal that there is an appropriate analogy between theoretical and experimental results.
Fig. 5. Effect of the Hartmann number and Radii ratio on the (a) isotherms and (b) flow patterns.
Fig. 6. Maximum stream function versusλandHa forRa=105.
3. Results and discussions
Natural convection of a low Prandtl number fluid in a long cylindrical annulus with isothermally heated and cooled walls in the presence of radial magnetic field is investigated numerically. Simulations are performed for a wide range of Hartmann number (0Ha 70), Rayleigh number (103 Ra 106) and radii ratio (1.25λ20). Also, it is assumedPr=0.072.
Figure 5 shows the effect of the Hartmann number and radii ratio on the flow patterns and isotherms inside the enclosure. As shown by Fig. 5(a) for all radii ratios (λ), the temperature stratification in the core diminishes with an increase in the value of Ha. Specifically, for lower radii ratio (λ = 1.25, 2), by increasing the Ha, the isotherms are almost parallel and are nearly conduction like. However, the influence of magnetic field on isothermal lines with increasing radii ratio (λ =5, 10) is reduced. As the radii ratio increases to 10, the flow becomes stronger and the temperature stratification exists in the annulus even at higher Hartmann number. This is due to the free convection becomes stronger with increasing radii ratio and stronger magnetic field is required to reduce convection. It can be seen from Fig. 5(b) that the flow reveals a circulating pattern going up along the hot cylinder and falling down along the cold cylinder of the annulus. Also, it is observed that, with increasing Hartmann number, the center of the circulation shifts to the upper part of the annulus. For narrow annulus, by increasing the radii ratio the two circulations on the top part of the annulus are disappeared.
As shown by Fig. 6 for a given value of radii ratio it can be observed that the maximum absolute value of the stream function is a decreasing function of Hartmann number. Furthermore stream function decreases when radii ratio is increased which means natural convection is suppressed. For lower radii ratio (λ=1.25), the effect of Hartmann number on the maximum absolute value of the stream function is not eminent compared to the case of high radii ratio (λ2). Moreover, at higher Hartmann number (Ha >40), by increasing the radii ratio the variation of average Nusselt number is not tangible. This behavior is due to the fact that forHa>40, conduction is the dominant mode of heat transfer.
The effects of the Hartmann number on the average Nusselt number for variousRa andλis shown in Fig. 7. It is obvious from this figure that for a fixedRaandλ, by increasing the Hartmann number, the average Nusselt number decreases. Also, it is observed that for a specifiedHaandRa, the average Nusselt number increases up to about λ = 5 and then reduces gently as λ enhances. Inspection of
Fig. 7. Average Nusselt number versusλandHafor (a)Ra=105and (b)Ra=106.
Fig. 7(a) makes it clear that for lower Hartmann numbers (Ha < 40), the effect of radii ratio on the average Nusselt number has almost the same trend. However, at higher Hartmann numbers (Ha >40), by increasing the radii ratio the variation of average Nusselt number is not significant. This behavior is due to the fact that forHa>40 atRa=105, conduction is the dominant mode of heat transfer. As seen in Fig. 7(b), for all Hartmann numbers the effect of radii ratio on the average Nusselt number has almost the same trend. Moreover, for lower Hartmann numbers the effect of radii ratio on the average Nusselt number is more tangible.
Comparison of Figs 7(a) and (b) shows that whenRanumber is increased; the average Nusselt number is increased, as verified by heat transfer principles [19]. This means that, at higher Rayleigh numbers, a stronger magnetic field should be imposed to suppress the convective heat transfer.
4. Conclusion
In this article, a horizontal annulus at a constant wall temperature under radial magnetic field is studied.
The effect of the Hartmann number, the Rayleigh number, and the radii ratio on the streamlines, the temperature distribution, and the average Nusselt number were investigated and the following results have been achieved:
– The natural convection inside the annulus depends strongly on the radii ratio, strength of the mag- netic field, and the Rayleigh number.
– The natural convection heat transfer decreases with increase in the Hartmann number. With increas- ing radii ratio the influence of magnetic field on natural convection is reduced.
– The variations of average Nusselt number with the strength of magnetic field at the different Rayleigh numbers are similar to each other, but with enhance in the Rayleigh number, the aver- age Nusselt number increases.
– For lower radii ratio (λ=1.25), the effect of Hartmann number on the maximum absolute value of the stream function is not eminent compared to the case of high radii ratio (λ2).
– It is found that for a givenHaandRa, the average Nusselt number increases up to aboutλ=5 and then reduces gently asλenhances. Moreover, for all Hartmann numbers the effect of radii ratio on the average Nusselt number has almost the same trend.
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