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Forced convection of ferrofluids in a vented cavity with a rotating cylinder
International Journal of Thermal Sciences 86 (2014) 258e275
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Forced convection of ferrofluids in a vented cavity with a rotating cylinder b € Fatih Selimefendigil a, *, Hakan F. Oztop a b
Department of Mechanical Engineering, Celal Bayar University, 45140 Manisa, Turkey , Turkey Department of Mechanical Engineering, Technology Faculty, Fırat University, 23119 Elazıg
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 October 2013 Received in revised form 15 July 2014 Accepted 15 July 2014 Available online
In this study, numerical investigation of the forced convection of ferrofluid in a square cavity with ventilation ports in the presence of an adiabatic rotating cylinder is carried out. The governing equations are solved with a finite element based solver. The effects of Reynolds number (20 Re 400), angular rotational speed of the cylinder (500 U 500), strength and location of the magnetic dipole (0 g 250), (0.2 a 0.8, 0.8 b 0.2) on the flow and thermal fields are numerically studied. It is observed that the length and size of the recirculation zones can be condtrolled with magnetic dipole strength and angular rotational speed of the cylinder. When the magnetic dipole is closer to the bottom wall of the cavity, flow is accelerated towards the bottom wall with larger influence area. The increasing values of the angular rotational speed of the cylinder in the clockwise direction enhance the heat transfer. © 2014 Elsevier Masson SAS. All rights reserved.
Keywords: Magnetic dipole Rotating cylinder Vented cavity Finite element method
1. Introduction Fluid flow around a rotating cylinder has a lot of practical applications such as rotating tube-heat exchangers, nuclear reactor fuel rods and drilling of oil wells. Several studies have been conducted to investigate the mixed or natural convection in enclosures with rotating or stationary cylinders [1e6]. Hussain and Hussein [7] have numerically investigated the mixed convection in an enclosure with a rotating cylinder using a finite volume method. The numerical experiment is carried out for a range of Reynolds number and Grashof numbers. Their results showed that rotating cylinder locations have an important effect in enhancing convection heat transfer in the square enclosure. Costa and Raimundo [8] have numerically studied the mixed convection in a differentially heated square enclosure with an active rotating circular cylinder. They observed that depending on the rotation, the free and forced convection can be combined or opposite. The effects of the radius, rotation velocity and thermal conductivity and thermal capacity of the cylinder on the mixed convection problem is studied.
* Corresponding author. Tel.: þ90 236 241 21 44. E-mail addresses: [email protected], € (F. Selimefendigil), [email protected] (H.F. Oztop). http://dx.doi.org/10.1016/j.ijthermalsci.2014.07.007 1290-0729/© 2014 Elsevier Masson SAS. All rights reserved.
[email protected]
Magnetic field effects on the fluid flow and heat transfer have received much attention during the recent years due to its importance in many technological applications such as coolers of nuclear reactors, micro-electronic devices and purification of molten metals. A review of heat transfer enhancement using ferrofluids is given in Ref. [9]. Due to the effect of the magnetic field, the fluid flow experiences a Lorentz force. Employing an external magnetic field can be used as a control method since magnetic field can suppress the convective flow field [10e13]. Finlayson [14] has studied the stability of ferromagnetic fluid for a fluid layer heated from below and subjected to a uniform vertical magnetic field. A temperature gradient was established across the fluid layer which causes a spatial variation in magnetization and hence convection. Strek and Jopek [15] have simulated the channel flow under the influence of magnetic dipole using a finite element code. They reported that inhomogeneous magnetic body force due to temperature gradient a convection similar to buoyancy force. Oztop et al. [16] have studied the mixed convection with a magnetic field in a top sided lid-driven cavity heated by a corner heater. They showed that heat transfer decreases with increasing the Hartmann number and magnetic field plays an important role to control heat transfer and fluid flow. Rahman et al. [17] have studied the conjugate effect of Joule heating and magnetic force, acting normal to the left vertical wall of an obstructed lid-driven cavity saturated with an
€ F. Selimefendigil, H.F. Oztop / International Journal of Thermal Sciences 86 (2014) 258e275
Nomenclature a, b B h H k L Mn n Nu p Pr Re T u, v x, y
location of the magnetic dipole magnetic induction local heat transfer coefficient, (W/m2 K) magnetic field thermal conductivity, (W/m K) length of the enclosure, (m) Magnetic number, m0 Hr2 =r0 n2r unit normal vector local Nusselt number, hL/k pressure, (Pa) Prandtl number, n/a Reynolds number, u0L/n temperature, (K) x-y velocity components, (m/s) Cartesian coordinates, (m)
259
Greek characters thermal diffusivity, (m2/s) strength of the dipole non-dimensional temperature, TTc/ThTc kinematic viscosity, (m2/s) density of the fluid, (kg/m3) viscous dissipation magnetic susceptibility nondimensional rotation velocity of cylinder, uL/2u0
a g q n r F c U
Subscripts c cold wall max maximum mean average h hot wall
electrically conducting fluid numerically using finite element method. They showed that the Joule heating parameter and the Hartmann number have notable effect on fluid flow and heat transfer. Jafari et al. [18] have studied the heat transfer and fluid flow characteristics for a kerosene based ferrofluid in two cylinders with different dimensions using computational fluid dynamics. They studied the effects of temperature gradients and uniform magnetic fields on the heat transfer and observed that magnetic field enhances the transport processes. They also showed that heat transfer increases when the magnetic field is perpendicular to the temperature gradient. Ishak et al. [19] have investigated the steady magnetohydrodynamic mixed convection flow adjacent to a vertical surface with prescribed heat flux. They found that magnetic parameter plays an important role in controlling the boundary layer separation.
In the present study, the effects of a rotating cylinder under the influence of magnetic dipole on the heat transfer enhancement and fluid flow characteristics are numerically studied in a vented cavity. To the best of the authors' knowledge, a numerical investigation for such a configuration has never been reported in the literature. The preset numerical study aims at investigating the effects of magnetic field parameters (strength and location of the magnetic dipole source) and angular rotational speed of the cylinder on the fluid
Fig. 1. Geometry and the boundary conditions for the ventilated cavity with an adiabatic rotating cylinder placed at the center of the cavity under the influence of magnetic dipole.
Fig. 2. (a)- Local and (b)- averaged Nusselt number distribution along the bottom wall of the cavity at (Re ¼ 400, U ¼ 500, g ¼ 250, (a,b) ¼ (0.5,0.25) for various grid densities.
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Fig. 4. Comparison of local Nusselt number distribution along the three walls of the cavity at Re ¼ 500 computed in Saeidi and Khodadadi [21] and present study. S denotes the dimensionless coordinate adopted for distance along the walls.
as two-dimensional. A magnetic dipole is located below the bottom wall of the enclosure and the fluid is electrically nonconducting (ferrofluid does not induce electromagnetic current). The momentum equation for an incompressible fluid and constant viscosity is modified by adding a term related to magnetic field as [14].
ru$Vu ¼ Vp þ V$ HB þ mV2 u
(1)
where H,B denote the magnetic field and magnetic induction. The energy equation for an incompressible fluid can be stated as [14,15]
rcu$VT ¼ kV2 T þ F m0 T Fig. 3. Mesh distribution for (a) full computational domain and (b) near the cylinder.
flow and heat transfer characteristics in a vented cavity which may be encountered in many engineering applications such as cooling of electronic devices, coating, solidification, float glass production, microelectronic devices, food processing and solar power. Convective heat transfer control by using a magnetic field in a cavity with a rotating cylinder can be utilized in nuclear reactor fuel rods, rotating tube-heat exchangers and drilling of oil wells. The results of this study can be utilized to find the appropriate flow parameters to achieve effective heat transfer enhancement in those systems. 2. Physical model and numerical study A schematic description of the physical problem is shown in Fig. 1. A vented square enclosure with a rotating adiabatic cylinder is considered. The height of the square cavity is L and a circular cylinder of radius R ¼ 0.1L is placed at the center of the cavity ((x0,y0) ¼ (0.5L,0.5L)). The inlet and outlet ports are placed at the left-top and right-bottom vertical cavity walls. The size of the ports is 0.1L. Top and bottom walls of the enclosure are kept at constant temperature Th. At the inlet port, a uniform velocity and temperature Tc < Th are imposed. The width of the square cavity is assumed to be long and the problem can be considered
vM vT
u$V H
(2)
where M denotes the magnetization. Maxwell's equation for an electrically non-conducting fluid can be written as [14,15]
V$B ¼ 0;
VH¼0
(3)
The constitutive relation between B,M and H can be stated as [14,15]
B ¼ m0 ðM þ HÞ
(4)
The magnetic field is induced with a magnetic dipole located below the bottom wall of the cavity. For the magnetostatic case, a magnetic scalar potential can be defined H ¼ VVm [15]
Vm ðxÞ ¼
g xa 2p ðx aÞ2 þ ðy bÞ2
(5)
where g, a and b denote the magnetic field strength, and position where the dipole is placed. The term in the momentum equation is the force per unit volume when the spatially non-uniform magnetic field is employed to the magnetic fluid. The relation between the magnetization vector M and magnetic field vector H can be written as [14,15]
M ¼ cm H where cm is the total magnetic susceptibility
(6)
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Fig. 5. Effect of magnetic dipole strength on the streamlines for fixed values of Re ¼ 200, U ¼ 250, (a,b) ¼ (0.5,0.25).
cm
At the exit port, pressure outlet boundary condition is used:
c0 ¼ 1 þ aT þ T0
(7)
Finally, using the constitutive equation in (4), the body force in the momentum equation can be stated as [14,15]
f ¼ m0 cm ð1 þ cm ÞðH$VÞH þ m0 cm HðH$VÞð1 þ cm Þ
(8)
In this study, natural convection effects are not taken into account but those effects may be of importance especially for the lowest considered values of dimensionless governing parameters. The boundary conditions for the considered problem in dimensional form can be expressed as: At the inlet port, velocity is unidirectional, temperature and velocity are uniform
u ¼ u0 ;
v ¼ 0;
T ¼ Tc :
p¼0 At the top and bottom wall, no-slip boundary condition with constant temperature walls are used
u ¼ 0;
v ¼ 0;
T ¼ Th
At the left and right walls of the cavity, no-slip boundary boundary condition with adiabatic walls are used
u ¼ 0;
v ¼ 0;
vT ¼0 vx
On the cylinder surface, specified velocity components with adiabatic wall boundary condition is used
u ¼ uðy y0 Þ;
v ¼ uðx x0 Þ;
vT ¼0 vn
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Fig. 6. Effect of magnetic dipole strength on the isotherms for fixed values of Re ¼ 200, U ¼ 250, (a,b) ¼ (0.5,0.25).
The relevant physical nondimensional numbers are Reynolds number (Re), Magnetic number (Mn) and angular rotational speed of the cylinder (U). Local Nusselt number is defined as.
Nux ¼
hx L vq ¼ : k vn wall
(9)
where hx, k, q and n represent the local heat transfer coefficient, thermal conductivity, nondimensional temperature and surface normal component, respectively. Spatial averaged Nusselt number is obtained after integrating the local Nusselt number along the bottom wall and top wall of the cavity as
1 Nu ¼ L
ZH Nux dx:
(10)
0
Eqs. (1)e(8) along with the boundary conditions are solved with COMSOL Multiphysics version 3.5 (a general purpose finite element solver [20]). P2 e P1 Lagrange finite elements are used to discretize velocity components and pressure, and Lagrangequadratic finite elements are chosen for temperature. The unstructured body-adapted mesh of appropriate size consists of
only triangular elements. In order to avoid the need for stabilizing convective terms in momentum equations, meshes are resolved fine enough. COMSOL solver adds artificial diffusion with the streamline upwind PetroveGalerkin method (SUPG) to handle local numerical instabilities. Segregated parametric solvers are used for fluid flow and heat transfer variables. Biconjugate gradient stabilized iterative method solver (BICGStab) is used for fluid flow and heat transfer modules of software. The computational domain is divided into 68216 triangular elements. The mesh is finer near the walls of cavity and cylinder to resolve the high gradients in the thermal and velocity boundary layer. Mesh independence study is also assured to obtain an appropriate grid distribution with accurate results and minimal computational time. Five different grid sizes are tested and the convergence in the length-averaged Nusselt number (along the bottom wall of the cavity) is checked. Local (a) and averaged (b) Nusselt number variations along the bottom wall of the cavity is shown in Fig. 2 for different grid sizes (Re ¼ 400, U ¼ 200, g ¼ 200, (a,b) ¼ (0.5,0.25). Grid size of G4 ¼ 68216 is decided to be fine enough to resolve the flow and thermal field for the given flow parameters. Distribution of the grid is depicted in Fig. 3. The code is validated against the result of Saeidi and Khodadadi [21]. The comparison is made for the case at Re ¼ 500 (based on the
Fig. 7. Variation of (a) local and (b) average Nusselt number distributions for different magnetic dipole strengths at Re ¼ 200, U ¼ 250, (a,b) ¼ (0.5,0.25).
Fig. 8. (a) x- velocity and (b) y- velocity distributions for different values of magnetic dipole strengths at Re ¼ 200, U ¼ 250, (a,b) ¼ (0.5,0.25).
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Fig. 9. Effect of angular rotational speed of the cylinder on the streamlines for fixed values of Re ¼ 200, g ¼ 150, (a,b) ¼ (0.5,0.25).
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Fig. 10. Effect of angular rotational speed of the cylinder on the isotherms for fixed values of Re ¼ 200, g ¼ 150, (a,b) ¼ (0.5,0.25).
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Fig. 11. Variation of (a) local and (b) average Nusselt number distributions for different angular rotational speed of the cylinder at Re ¼ 200, g ¼ 150, (a,b) ¼ (0.5,0.25).
Fig. 12. (a) x- velocity and (b) y- velocity distributions for different values of angular rotational speed of the cylinder at Re ¼ 200, g ¼ 150, (a,b) ¼ (0.5,0.25).
width of the inlet port) Fig. 4 shows the local Nusselt number distributions along the walls of the cavity for this flow condition computed with the COMSOL code (current settings) and computed in Saeidi and Khodadadi [21]. The comparison results show good overall agreement.
field within the cavity are affected. The streamlines on the right bottom of the cylinder elongate towards the left and their strengths increase whereas the flow patterns on the left top of the cylinder disappear as the strength of the dipole increases. Magnetic dipole strength is more effective under the object due to rotating motion of the object and captured flow at this region as seen from Fig. 5. The size of the recirculation zone on the right top corner of the cavity decreases and flow field is accelerated in clockwise direction. For g ¼ 250, Fig. 5(d), a recirculation bubble is formed on the bottom wall of the cavity. This effect is due to the spatial variation in the magnetization which is induced through temperature gradient. The inhomogeneous magnetic body force is responsible for this flow behavior [15,14]. The wavy appearance of the isotherms on the bottom wall of the cylinder and cavity is due to the convection currents of the magnetic dipole source. Local and length-averaged Nusselt number plots for both bottom wall of the cavity are shown in Fig. 7 (a, b) at Re ¼ 200, U ¼ 250 for varying magnetic dipole strengths. Increasing the magnetic dipole strength number, heat transfer is enhanced for all cases. There is a saturation type behavior in the averaged Nusselt number plot since the recirculation bubble appearing on the bottom wall increases in size and strength with increasing magnetic dipole strength. The external magnetic field acts in a way to decrease the local heat transfer in some locations and increase it in some others.
3. Results and discussion As stated earlier, the overall purpose of this study is to investigate the effects of rotating cylinder in the presence of a magnetic dipole located below the bottom wall of the cavity on the heat transfer and fluid flow characteristics. In this study, the influence of the Reynolds number (20 Re 400), angular rotational speed of the cylinder (500 U 500) [8], strength of the magnetic dipole (0 g 250) [15], horizontal and vertical location of the magnetic dipole (0.2 a 0.8, 0.8 b 0.2) are examined for convective heat transfer enhancement in a vented cavity. 3.1. Effects of magnetic dipole strength Figs. 5 and 6 demonstrate the effects of magnetic dipole strength on the structure of flow and thermal patterns for fixed values of Re ¼ 200, U ¼ 250, (a,b) ¼ (0.5,0.25). In this case, cylinder rotation is in the counterclockwise direction. It is seen that with increasing values of magnetic dipole strength, the flow
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Fig. 13. Effect of Reynolds number on the streamlines for fixed values of U ¼ 200, g ¼ 200, (a,b) ¼ (0.5,0.25).
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Fig. 14. Effect of Reynolds number on the isotherms for fixed values of U ¼ 200, g ¼ 200, (a,b) ¼ (0.5,0.25).
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with increasing angular rotational speed of the cylinder. The vortex shown up on the top right of the cavity wall in the motionless cylinder case, gets larger in size and strength. The isotherm plots in Fig. 10(c) show that when the cylinder does not rotate, it does not interact with the heat transfer. Depending on the rotation direction, it accelerates or decelerates the fluid flow from the inlet towards the outlet, thus aiding or opposing convection. Due to the formation of the vortices around the bottom part of the cylinder, the isotherms fluctuates more in that region at positive values of rotations angles (Fig. 10d, e). Local and length-averaged Nusselt number distribution along the bottom wall of the cavity at Re ¼ 200, g ¼ 150, (a,b) ¼ (0.5,0.25) for different angular rotational speed of the cylinder are shown in Fig. 11(a, b). The angular rotational speed of the cylinder acts to increase the local heat transfer in some locations (until x ¼ 0.8H and increase it in some others (from the value X/H ¼ 0.8 until the exit port) with increasing values. Averaged-Nusselt number values increases for negative values of angular rotational speed of the cylinder since in this case it has a positive impact on the convection from the inlet to the outlet port. Fig. 12(a, b) depicts the x- and y-component of the velocity at cavity section y ¼ 0.3 along the horizontal direction for different values of angular rotational speed of the cylinder. A negative values of the x-component of the velocity corresponding to a recirculation zone and it is seen that the length and size of the recirculation zone formed below the cylinder can be controlled with angular rotational speed of the cylinder. 3.3. Effects of Reynolds number Fig. 15. Variation of (a) local and (b) average Nusselt number distributions for various Reynolds numbers at U ¼ 200, g ¼ 200, (a,b) ¼ (0.5,0.25).
Fig. 8(a, b) shows the x- and y-component of the velocity at cavity section y ¼ 0.3 along the horizontal direction for different values of magnetic dipole strength. As it is shown in Fig. 8, negative values of the x-component of the velocity corresponding to a recirculation zone. The length and size of the recirculation zone can be controlled with magnetic dipole strength. 3.2. Effects of angular rotational speed of the cylinder Figs. 9 and 10 show the effect of varying angular rotational speed of the cylinder (U) on the streamlines and isotherms for fixed values of Re ¼ 200, g ¼ 150 and (a,b) ¼ (0.5,0.25). The case U ¼ 0 corresponds to a stationary cylinder which is shown in Fig. 9(c). In this case, a vortex is shown up on the right top corner of the cavity and also a vortex adjacent to the cylinder bottom surface is formed. A negative value of the rotation U indicates clockwise rotation of the cylinder. When the cylinder rotates in the clockwise direction, this has a positive impact on the convection from the inlet to the outlet port. In this case, the vortices appeared around the cylinder when it is motionless disappears and the size and strength of the vortex on the top right corner decreases slightly (Fig. 9a, b). When the cylinder rotates in the counter clockwise direction, the vortices formed near the bottom wall of the cylinder in the motionless state, gets bigger in size and strength and flow field in the vicinity of the cylinder accelerates
The effects of Reynolds number on the flow and thermal patterns are demonstrated in Figs. 13 and 14 for fixed values of U ¼ 200, g ¼ 200 and (a,b) ¼ (0.5,0.25). For Re ¼ 20, a complex flow structure in the cavity is seen where the recirculation zones are formed below the cylinder and in the vicinity of the exit port. With increasing the Reynolds number, the forced flow overcomes the convection effects of magnetic field. At Re ¼ 100, the vortice shown up at the exit port disappears. On the right top corner of the cavity a vortex is formed and increases in size and strength with increasing values of Reynolds numbers. Increase of the Reynolds number leads to increased temperature gradient on the bottom wall fluctuation of the isotherms due to the forced convection. Local and averaged Nusselt number plots are depicted in Fig. 15(a,b). Averaged and maximum Nusselt number values increase with increasing values of Reynolds number. 3.4. Effects of magnetic dipole location The magnetic dipole location also affects the flow and thermal patterns within the cavity. Figs. 16 and 17 demonstrate the influence of horizontal location of the magnetic dipole on the streamline and isotherms for fixed values of Re ¼ 100, g ¼ 50, U ¼ 200 and b ¼ 0.25. The recirculation bubbles shown up in the vicinity of the left wall of the cavity moves and gets distorted in the direction of the magnetic dipole movement. Isotherm plots show that temperature gradients move and intensified along the direction of the magnetic dipole movement. Fig. 18(a, b) demonstrate the effect of x-position of the magnetic dipole on the local and averaged Nusselt numbers. First local peak of the Nusselt number shifts towards positive x direction for increasing
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Fig. 16. Effect of horizontal location of the magnetic dipole on the streamlines for fixed values of Re ¼ 100, U ¼ 200, g ¼ 50, b ¼ 0.25.
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Fig. 17. Effect of horizontal location of the magnetic dipole on the isotherms for fixed values of Re ¼ 100, U ¼ 200, g ¼ 50, b ¼ 0.25.
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Fig. 18. Variation of (a) local and (b) average Nusselt number distributions for various horizontal locations of the magnetic dipole at Re ¼ 100, U ¼ 200, g ¼ 50, b ¼ 0.25.
horizontal distance values of the magnetic dipole. From the averaged Nusselt number plot, it is seen that maximum value is attained for the horizontal location of the magnetic dipole x ¼ H/2. Figs. 19 and 20 depict the influence of vertical location of magnetic dipole source on the flow and thermal patterns for fixed values at Re ¼ 100, g ¼ 50, U ¼ 200 and a ¼ 0.5. As it is shown in Fig. 19, when the magnetic dipole is closer to the bottom wall of the cavity, flow is accelerated towards the bottom wall with larger influence area. The isotherms show a wavy behavior in the vicinity of the bottom part of the cylinder due to the recirculation bubble extended in this region. Fig. 21 shows that maximum and averaged Nusselt number increase when the magnetic dipole gets closer towards the bottom wall of the cavity.
4. Conclusions In the present study, forced convection in a vented square cavity filled with ferrofluid in the presence of an adiabatic rotating cylinder is numerically studied for a range of Reynolds number, angular rotational speed of the cylinder, strength and location of the magnetic dipole source. Following results are obtained: The spatial variation in the magnetization is induced through temperature gradient which acts as an inhomogeneous magnetic body force and affects the recirculation bubbles formed in the vicinity of the cylinder and on the bottom wall of the cavity. The length and size of the recirculation zone can be controlled with magnetic dipole strength and angular rotational speed of the cylinder.
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Fig. 19. Effect of vertical location of the magnetic dipole on the streamlines for fixed values of Re ¼ 100, U ¼ 200, g ¼ 50, a ¼ 0.5.
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Fig. 20. Effect of vertical location of the magnetic dipole on the isotherms for fixed values of Re ¼ 100, U ¼ 200, g ¼ 50, a ¼ 0.5.
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References
Fig. 21. Variation of (a) local and (b) average Nusselt number distributions for various vertical locations of the magnetic dipole at Re ¼ 100, U ¼ 200, g ¼ 50, a ¼ 0.5.
When the magnetic dipole is closer to the bottom wall of the cavity, flow is accelerated towards the bottom wall with larger influence area. Averaged-Nusselt number values increase for negative values of angular rotational speed of the cylinder since in this case it has a positive impact on the convection from the inlet to the outlet port. The external magnetic field acts in a way to decrease the local heat transfer in some locations and increase it in some others. Averaged and maximum Nusselt number values increase with increasing values of Reynolds number. The range of parameters shows the results for theoretical works. In the real physical conditions of the practical interest, they can be extend for higher velocities.
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Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Forced convection of ferrofluids in a vented cavity with a rotating cylinder b € Fatih Selimefendigil a, *, Hakan F. Oztop a b
Department of Mechanical Engineering, Celal Bayar University, 45140 Manisa, Turkey , Turkey Department of Mechanical Engineering, Technology Faculty, Fırat University, 23119 Elazıg
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 October 2013 Received in revised form 15 July 2014 Accepted 15 July 2014 Available online
In this study, numerical investigation of the forced convection of ferrofluid in a square cavity with ventilation ports in the presence of an adiabatic rotating cylinder is carried out. The governing equations are solved with a finite element based solver. The effects of Reynolds number (20 Re 400), angular rotational speed of the cylinder (500 U 500), strength and location of the magnetic dipole (0 g 250), (0.2 a 0.8, 0.8 b 0.2) on the flow and thermal fields are numerically studied. It is observed that the length and size of the recirculation zones can be condtrolled with magnetic dipole strength and angular rotational speed of the cylinder. When the magnetic dipole is closer to the bottom wall of the cavity, flow is accelerated towards the bottom wall with larger influence area. The increasing values of the angular rotational speed of the cylinder in the clockwise direction enhance the heat transfer. © 2014 Elsevier Masson SAS. All rights reserved.
Keywords: Magnetic dipole Rotating cylinder Vented cavity Finite element method
1. Introduction Fluid flow around a rotating cylinder has a lot of practical applications such as rotating tube-heat exchangers, nuclear reactor fuel rods and drilling of oil wells. Several studies have been conducted to investigate the mixed or natural convection in enclosures with rotating or stationary cylinders [1e6]. Hussain and Hussein [7] have numerically investigated the mixed convection in an enclosure with a rotating cylinder using a finite volume method. The numerical experiment is carried out for a range of Reynolds number and Grashof numbers. Their results showed that rotating cylinder locations have an important effect in enhancing convection heat transfer in the square enclosure. Costa and Raimundo [8] have numerically studied the mixed convection in a differentially heated square enclosure with an active rotating circular cylinder. They observed that depending on the rotation, the free and forced convection can be combined or opposite. The effects of the radius, rotation velocity and thermal conductivity and thermal capacity of the cylinder on the mixed convection problem is studied.
* Corresponding author. Tel.: þ90 236 241 21 44. E-mail addresses: [email protected], € (F. Selimefendigil), [email protected] (H.F. Oztop). http://dx.doi.org/10.1016/j.ijthermalsci.2014.07.007 1290-0729/© 2014 Elsevier Masson SAS. All rights reserved.
[email protected]
Magnetic field effects on the fluid flow and heat transfer have received much attention during the recent years due to its importance in many technological applications such as coolers of nuclear reactors, micro-electronic devices and purification of molten metals. A review of heat transfer enhancement using ferrofluids is given in Ref. [9]. Due to the effect of the magnetic field, the fluid flow experiences a Lorentz force. Employing an external magnetic field can be used as a control method since magnetic field can suppress the convective flow field [10e13]. Finlayson [14] has studied the stability of ferromagnetic fluid for a fluid layer heated from below and subjected to a uniform vertical magnetic field. A temperature gradient was established across the fluid layer which causes a spatial variation in magnetization and hence convection. Strek and Jopek [15] have simulated the channel flow under the influence of magnetic dipole using a finite element code. They reported that inhomogeneous magnetic body force due to temperature gradient a convection similar to buoyancy force. Oztop et al. [16] have studied the mixed convection with a magnetic field in a top sided lid-driven cavity heated by a corner heater. They showed that heat transfer decreases with increasing the Hartmann number and magnetic field plays an important role to control heat transfer and fluid flow. Rahman et al. [17] have studied the conjugate effect of Joule heating and magnetic force, acting normal to the left vertical wall of an obstructed lid-driven cavity saturated with an
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Nomenclature a, b B h H k L Mn n Nu p Pr Re T u, v x, y
location of the magnetic dipole magnetic induction local heat transfer coefficient, (W/m2 K) magnetic field thermal conductivity, (W/m K) length of the enclosure, (m) Magnetic number, m0 Hr2 =r0 n2r unit normal vector local Nusselt number, hL/k pressure, (Pa) Prandtl number, n/a Reynolds number, u0L/n temperature, (K) x-y velocity components, (m/s) Cartesian coordinates, (m)
259
Greek characters thermal diffusivity, (m2/s) strength of the dipole non-dimensional temperature, TTc/ThTc kinematic viscosity, (m2/s) density of the fluid, (kg/m3) viscous dissipation magnetic susceptibility nondimensional rotation velocity of cylinder, uL/2u0
a g q n r F c U
Subscripts c cold wall max maximum mean average h hot wall
electrically conducting fluid numerically using finite element method. They showed that the Joule heating parameter and the Hartmann number have notable effect on fluid flow and heat transfer. Jafari et al. [18] have studied the heat transfer and fluid flow characteristics for a kerosene based ferrofluid in two cylinders with different dimensions using computational fluid dynamics. They studied the effects of temperature gradients and uniform magnetic fields on the heat transfer and observed that magnetic field enhances the transport processes. They also showed that heat transfer increases when the magnetic field is perpendicular to the temperature gradient. Ishak et al. [19] have investigated the steady magnetohydrodynamic mixed convection flow adjacent to a vertical surface with prescribed heat flux. They found that magnetic parameter plays an important role in controlling the boundary layer separation.
In the present study, the effects of a rotating cylinder under the influence of magnetic dipole on the heat transfer enhancement and fluid flow characteristics are numerically studied in a vented cavity. To the best of the authors' knowledge, a numerical investigation for such a configuration has never been reported in the literature. The preset numerical study aims at investigating the effects of magnetic field parameters (strength and location of the magnetic dipole source) and angular rotational speed of the cylinder on the fluid
Fig. 1. Geometry and the boundary conditions for the ventilated cavity with an adiabatic rotating cylinder placed at the center of the cavity under the influence of magnetic dipole.
Fig. 2. (a)- Local and (b)- averaged Nusselt number distribution along the bottom wall of the cavity at (Re ¼ 400, U ¼ 500, g ¼ 250, (a,b) ¼ (0.5,0.25) for various grid densities.
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Fig. 4. Comparison of local Nusselt number distribution along the three walls of the cavity at Re ¼ 500 computed in Saeidi and Khodadadi [21] and present study. S denotes the dimensionless coordinate adopted for distance along the walls.
as two-dimensional. A magnetic dipole is located below the bottom wall of the enclosure and the fluid is electrically nonconducting (ferrofluid does not induce electromagnetic current). The momentum equation for an incompressible fluid and constant viscosity is modified by adding a term related to magnetic field as [14].
ru$Vu ¼ Vp þ V$ HB þ mV2 u
(1)
where H,B denote the magnetic field and magnetic induction. The energy equation for an incompressible fluid can be stated as [14,15]
rcu$VT ¼ kV2 T þ F m0 T Fig. 3. Mesh distribution for (a) full computational domain and (b) near the cylinder.
flow and heat transfer characteristics in a vented cavity which may be encountered in many engineering applications such as cooling of electronic devices, coating, solidification, float glass production, microelectronic devices, food processing and solar power. Convective heat transfer control by using a magnetic field in a cavity with a rotating cylinder can be utilized in nuclear reactor fuel rods, rotating tube-heat exchangers and drilling of oil wells. The results of this study can be utilized to find the appropriate flow parameters to achieve effective heat transfer enhancement in those systems. 2. Physical model and numerical study A schematic description of the physical problem is shown in Fig. 1. A vented square enclosure with a rotating adiabatic cylinder is considered. The height of the square cavity is L and a circular cylinder of radius R ¼ 0.1L is placed at the center of the cavity ((x0,y0) ¼ (0.5L,0.5L)). The inlet and outlet ports are placed at the left-top and right-bottom vertical cavity walls. The size of the ports is 0.1L. Top and bottom walls of the enclosure are kept at constant temperature Th. At the inlet port, a uniform velocity and temperature Tc < Th are imposed. The width of the square cavity is assumed to be long and the problem can be considered
vM vT
u$V H
(2)
where M denotes the magnetization. Maxwell's equation for an electrically non-conducting fluid can be written as [14,15]
V$B ¼ 0;
VH¼0
(3)
The constitutive relation between B,M and H can be stated as [14,15]
B ¼ m0 ðM þ HÞ
(4)
The magnetic field is induced with a magnetic dipole located below the bottom wall of the cavity. For the magnetostatic case, a magnetic scalar potential can be defined H ¼ VVm [15]
Vm ðxÞ ¼
g xa 2p ðx aÞ2 þ ðy bÞ2
(5)
where g, a and b denote the magnetic field strength, and position where the dipole is placed. The term in the momentum equation is the force per unit volume when the spatially non-uniform magnetic field is employed to the magnetic fluid. The relation between the magnetization vector M and magnetic field vector H can be written as [14,15]
M ¼ cm H where cm is the total magnetic susceptibility
(6)
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Fig. 5. Effect of magnetic dipole strength on the streamlines for fixed values of Re ¼ 200, U ¼ 250, (a,b) ¼ (0.5,0.25).
cm
At the exit port, pressure outlet boundary condition is used:
c0 ¼ 1 þ aT þ T0
(7)
Finally, using the constitutive equation in (4), the body force in the momentum equation can be stated as [14,15]
f ¼ m0 cm ð1 þ cm ÞðH$VÞH þ m0 cm HðH$VÞð1 þ cm Þ
(8)
In this study, natural convection effects are not taken into account but those effects may be of importance especially for the lowest considered values of dimensionless governing parameters. The boundary conditions for the considered problem in dimensional form can be expressed as: At the inlet port, velocity is unidirectional, temperature and velocity are uniform
u ¼ u0 ;
v ¼ 0;
T ¼ Tc :
p¼0 At the top and bottom wall, no-slip boundary condition with constant temperature walls are used
u ¼ 0;
v ¼ 0;
T ¼ Th
At the left and right walls of the cavity, no-slip boundary boundary condition with adiabatic walls are used
u ¼ 0;
v ¼ 0;
vT ¼0 vx
On the cylinder surface, specified velocity components with adiabatic wall boundary condition is used
u ¼ uðy y0 Þ;
v ¼ uðx x0 Þ;
vT ¼0 vn
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Fig. 6. Effect of magnetic dipole strength on the isotherms for fixed values of Re ¼ 200, U ¼ 250, (a,b) ¼ (0.5,0.25).
The relevant physical nondimensional numbers are Reynolds number (Re), Magnetic number (Mn) and angular rotational speed of the cylinder (U). Local Nusselt number is defined as.
Nux ¼
hx L vq ¼ : k vn wall
(9)
where hx, k, q and n represent the local heat transfer coefficient, thermal conductivity, nondimensional temperature and surface normal component, respectively. Spatial averaged Nusselt number is obtained after integrating the local Nusselt number along the bottom wall and top wall of the cavity as
1 Nu ¼ L
ZH Nux dx:
(10)
0
Eqs. (1)e(8) along with the boundary conditions are solved with COMSOL Multiphysics version 3.5 (a general purpose finite element solver [20]). P2 e P1 Lagrange finite elements are used to discretize velocity components and pressure, and Lagrangequadratic finite elements are chosen for temperature. The unstructured body-adapted mesh of appropriate size consists of
only triangular elements. In order to avoid the need for stabilizing convective terms in momentum equations, meshes are resolved fine enough. COMSOL solver adds artificial diffusion with the streamline upwind PetroveGalerkin method (SUPG) to handle local numerical instabilities. Segregated parametric solvers are used for fluid flow and heat transfer variables. Biconjugate gradient stabilized iterative method solver (BICGStab) is used for fluid flow and heat transfer modules of software. The computational domain is divided into 68216 triangular elements. The mesh is finer near the walls of cavity and cylinder to resolve the high gradients in the thermal and velocity boundary layer. Mesh independence study is also assured to obtain an appropriate grid distribution with accurate results and minimal computational time. Five different grid sizes are tested and the convergence in the length-averaged Nusselt number (along the bottom wall of the cavity) is checked. Local (a) and averaged (b) Nusselt number variations along the bottom wall of the cavity is shown in Fig. 2 for different grid sizes (Re ¼ 400, U ¼ 200, g ¼ 200, (a,b) ¼ (0.5,0.25). Grid size of G4 ¼ 68216 is decided to be fine enough to resolve the flow and thermal field for the given flow parameters. Distribution of the grid is depicted in Fig. 3. The code is validated against the result of Saeidi and Khodadadi [21]. The comparison is made for the case at Re ¼ 500 (based on the
Fig. 7. Variation of (a) local and (b) average Nusselt number distributions for different magnetic dipole strengths at Re ¼ 200, U ¼ 250, (a,b) ¼ (0.5,0.25).
Fig. 8. (a) x- velocity and (b) y- velocity distributions for different values of magnetic dipole strengths at Re ¼ 200, U ¼ 250, (a,b) ¼ (0.5,0.25).
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Fig. 9. Effect of angular rotational speed of the cylinder on the streamlines for fixed values of Re ¼ 200, g ¼ 150, (a,b) ¼ (0.5,0.25).
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Fig. 10. Effect of angular rotational speed of the cylinder on the isotherms for fixed values of Re ¼ 200, g ¼ 150, (a,b) ¼ (0.5,0.25).
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Fig. 11. Variation of (a) local and (b) average Nusselt number distributions for different angular rotational speed of the cylinder at Re ¼ 200, g ¼ 150, (a,b) ¼ (0.5,0.25).
Fig. 12. (a) x- velocity and (b) y- velocity distributions for different values of angular rotational speed of the cylinder at Re ¼ 200, g ¼ 150, (a,b) ¼ (0.5,0.25).
width of the inlet port) Fig. 4 shows the local Nusselt number distributions along the walls of the cavity for this flow condition computed with the COMSOL code (current settings) and computed in Saeidi and Khodadadi [21]. The comparison results show good overall agreement.
field within the cavity are affected. The streamlines on the right bottom of the cylinder elongate towards the left and their strengths increase whereas the flow patterns on the left top of the cylinder disappear as the strength of the dipole increases. Magnetic dipole strength is more effective under the object due to rotating motion of the object and captured flow at this region as seen from Fig. 5. The size of the recirculation zone on the right top corner of the cavity decreases and flow field is accelerated in clockwise direction. For g ¼ 250, Fig. 5(d), a recirculation bubble is formed on the bottom wall of the cavity. This effect is due to the spatial variation in the magnetization which is induced through temperature gradient. The inhomogeneous magnetic body force is responsible for this flow behavior [15,14]. The wavy appearance of the isotherms on the bottom wall of the cylinder and cavity is due to the convection currents of the magnetic dipole source. Local and length-averaged Nusselt number plots for both bottom wall of the cavity are shown in Fig. 7 (a, b) at Re ¼ 200, U ¼ 250 for varying magnetic dipole strengths. Increasing the magnetic dipole strength number, heat transfer is enhanced for all cases. There is a saturation type behavior in the averaged Nusselt number plot since the recirculation bubble appearing on the bottom wall increases in size and strength with increasing magnetic dipole strength. The external magnetic field acts in a way to decrease the local heat transfer in some locations and increase it in some others.
3. Results and discussion As stated earlier, the overall purpose of this study is to investigate the effects of rotating cylinder in the presence of a magnetic dipole located below the bottom wall of the cavity on the heat transfer and fluid flow characteristics. In this study, the influence of the Reynolds number (20 Re 400), angular rotational speed of the cylinder (500 U 500) [8], strength of the magnetic dipole (0 g 250) [15], horizontal and vertical location of the magnetic dipole (0.2 a 0.8, 0.8 b 0.2) are examined for convective heat transfer enhancement in a vented cavity. 3.1. Effects of magnetic dipole strength Figs. 5 and 6 demonstrate the effects of magnetic dipole strength on the structure of flow and thermal patterns for fixed values of Re ¼ 200, U ¼ 250, (a,b) ¼ (0.5,0.25). In this case, cylinder rotation is in the counterclockwise direction. It is seen that with increasing values of magnetic dipole strength, the flow
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Fig. 13. Effect of Reynolds number on the streamlines for fixed values of U ¼ 200, g ¼ 200, (a,b) ¼ (0.5,0.25).
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Fig. 14. Effect of Reynolds number on the isotherms for fixed values of U ¼ 200, g ¼ 200, (a,b) ¼ (0.5,0.25).
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with increasing angular rotational speed of the cylinder. The vortex shown up on the top right of the cavity wall in the motionless cylinder case, gets larger in size and strength. The isotherm plots in Fig. 10(c) show that when the cylinder does not rotate, it does not interact with the heat transfer. Depending on the rotation direction, it accelerates or decelerates the fluid flow from the inlet towards the outlet, thus aiding or opposing convection. Due to the formation of the vortices around the bottom part of the cylinder, the isotherms fluctuates more in that region at positive values of rotations angles (Fig. 10d, e). Local and length-averaged Nusselt number distribution along the bottom wall of the cavity at Re ¼ 200, g ¼ 150, (a,b) ¼ (0.5,0.25) for different angular rotational speed of the cylinder are shown in Fig. 11(a, b). The angular rotational speed of the cylinder acts to increase the local heat transfer in some locations (until x ¼ 0.8H and increase it in some others (from the value X/H ¼ 0.8 until the exit port) with increasing values. Averaged-Nusselt number values increases for negative values of angular rotational speed of the cylinder since in this case it has a positive impact on the convection from the inlet to the outlet port. Fig. 12(a, b) depicts the x- and y-component of the velocity at cavity section y ¼ 0.3 along the horizontal direction for different values of angular rotational speed of the cylinder. A negative values of the x-component of the velocity corresponding to a recirculation zone and it is seen that the length and size of the recirculation zone formed below the cylinder can be controlled with angular rotational speed of the cylinder. 3.3. Effects of Reynolds number Fig. 15. Variation of (a) local and (b) average Nusselt number distributions for various Reynolds numbers at U ¼ 200, g ¼ 200, (a,b) ¼ (0.5,0.25).
Fig. 8(a, b) shows the x- and y-component of the velocity at cavity section y ¼ 0.3 along the horizontal direction for different values of magnetic dipole strength. As it is shown in Fig. 8, negative values of the x-component of the velocity corresponding to a recirculation zone. The length and size of the recirculation zone can be controlled with magnetic dipole strength. 3.2. Effects of angular rotational speed of the cylinder Figs. 9 and 10 show the effect of varying angular rotational speed of the cylinder (U) on the streamlines and isotherms for fixed values of Re ¼ 200, g ¼ 150 and (a,b) ¼ (0.5,0.25). The case U ¼ 0 corresponds to a stationary cylinder which is shown in Fig. 9(c). In this case, a vortex is shown up on the right top corner of the cavity and also a vortex adjacent to the cylinder bottom surface is formed. A negative value of the rotation U indicates clockwise rotation of the cylinder. When the cylinder rotates in the clockwise direction, this has a positive impact on the convection from the inlet to the outlet port. In this case, the vortices appeared around the cylinder when it is motionless disappears and the size and strength of the vortex on the top right corner decreases slightly (Fig. 9a, b). When the cylinder rotates in the counter clockwise direction, the vortices formed near the bottom wall of the cylinder in the motionless state, gets bigger in size and strength and flow field in the vicinity of the cylinder accelerates
The effects of Reynolds number on the flow and thermal patterns are demonstrated in Figs. 13 and 14 for fixed values of U ¼ 200, g ¼ 200 and (a,b) ¼ (0.5,0.25). For Re ¼ 20, a complex flow structure in the cavity is seen where the recirculation zones are formed below the cylinder and in the vicinity of the exit port. With increasing the Reynolds number, the forced flow overcomes the convection effects of magnetic field. At Re ¼ 100, the vortice shown up at the exit port disappears. On the right top corner of the cavity a vortex is formed and increases in size and strength with increasing values of Reynolds numbers. Increase of the Reynolds number leads to increased temperature gradient on the bottom wall fluctuation of the isotherms due to the forced convection. Local and averaged Nusselt number plots are depicted in Fig. 15(a,b). Averaged and maximum Nusselt number values increase with increasing values of Reynolds number. 3.4. Effects of magnetic dipole location The magnetic dipole location also affects the flow and thermal patterns within the cavity. Figs. 16 and 17 demonstrate the influence of horizontal location of the magnetic dipole on the streamline and isotherms for fixed values of Re ¼ 100, g ¼ 50, U ¼ 200 and b ¼ 0.25. The recirculation bubbles shown up in the vicinity of the left wall of the cavity moves and gets distorted in the direction of the magnetic dipole movement. Isotherm plots show that temperature gradients move and intensified along the direction of the magnetic dipole movement. Fig. 18(a, b) demonstrate the effect of x-position of the magnetic dipole on the local and averaged Nusselt numbers. First local peak of the Nusselt number shifts towards positive x direction for increasing
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Fig. 16. Effect of horizontal location of the magnetic dipole on the streamlines for fixed values of Re ¼ 100, U ¼ 200, g ¼ 50, b ¼ 0.25.
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Fig. 17. Effect of horizontal location of the magnetic dipole on the isotherms for fixed values of Re ¼ 100, U ¼ 200, g ¼ 50, b ¼ 0.25.
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Fig. 18. Variation of (a) local and (b) average Nusselt number distributions for various horizontal locations of the magnetic dipole at Re ¼ 100, U ¼ 200, g ¼ 50, b ¼ 0.25.
horizontal distance values of the magnetic dipole. From the averaged Nusselt number plot, it is seen that maximum value is attained for the horizontal location of the magnetic dipole x ¼ H/2. Figs. 19 and 20 depict the influence of vertical location of magnetic dipole source on the flow and thermal patterns for fixed values at Re ¼ 100, g ¼ 50, U ¼ 200 and a ¼ 0.5. As it is shown in Fig. 19, when the magnetic dipole is closer to the bottom wall of the cavity, flow is accelerated towards the bottom wall with larger influence area. The isotherms show a wavy behavior in the vicinity of the bottom part of the cylinder due to the recirculation bubble extended in this region. Fig. 21 shows that maximum and averaged Nusselt number increase when the magnetic dipole gets closer towards the bottom wall of the cavity.
4. Conclusions In the present study, forced convection in a vented square cavity filled with ferrofluid in the presence of an adiabatic rotating cylinder is numerically studied for a range of Reynolds number, angular rotational speed of the cylinder, strength and location of the magnetic dipole source. Following results are obtained: The spatial variation in the magnetization is induced through temperature gradient which acts as an inhomogeneous magnetic body force and affects the recirculation bubbles formed in the vicinity of the cylinder and on the bottom wall of the cavity. The length and size of the recirculation zone can be controlled with magnetic dipole strength and angular rotational speed of the cylinder.
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Fig. 19. Effect of vertical location of the magnetic dipole on the streamlines for fixed values of Re ¼ 100, U ¼ 200, g ¼ 50, a ¼ 0.5.
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Fig. 20. Effect of vertical location of the magnetic dipole on the isotherms for fixed values of Re ¼ 100, U ¼ 200, g ¼ 50, a ¼ 0.5.
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References
Fig. 21. Variation of (a) local and (b) average Nusselt number distributions for various vertical locations of the magnetic dipole at Re ¼ 100, U ¼ 200, g ¼ 50, a ¼ 0.5.
When the magnetic dipole is closer to the bottom wall of the cavity, flow is accelerated towards the bottom wall with larger influence area. Averaged-Nusselt number values increase for negative values of angular rotational speed of the cylinder since in this case it has a positive impact on the convection from the inlet to the outlet port. The external magnetic field acts in a way to decrease the local heat transfer in some locations and increase it in some others. Averaged and maximum Nusselt number values increase with increasing values of Reynolds number. The range of parameters shows the results for theoretical works. In the real physical conditions of the practical interest, they can be extend for higher velocities.
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