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Unsteady MHD Mixed Convection inside L-shaped Enclosure in the Presence of Ferrofluid (Fe3O4)
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 194 (2017) 494 – 501
10th International Conference on Marine Technology, MARTEC 2016
Unsteady MHD Mixed Convection inside L-shaped Enclosure in the Presence of Ferrofluid (Fe3O4) Nandita Chakrabarty Jhumur∗, Anuruddha Bhattacharjee Department of Mechanical Engineering, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh
Abstract Mixed convection has been one of the most interesting topics of research in the field of heat transfer for many years. In the present study, unsteady MHD mixed convection heat transfer inside a heated L-shaped enclosure filled with Fe3 O4 water has been investigated numerically. Lid-motion is introduced by the top wall to induce forced convection. For pure mixed convection (Ri = 1), Grashof number (Gr = 103 − 105 ) and time (τ) are varied, whereas the solid volume fraction of nanofluid (φ = 0.1) and Hartman number (Ha = 25) are kept constant. Results have been presented in terms of isotherm lines, streamlines, average Nusselt (Nu) number and average temperature (θay ). The obtained results show that MHD causes significant effects on heat transfer rate. Interestingly, natural convection is more effective in this case of mixed convection as higher values of Nu are obtained for higher values of Gr. © Published by Elsevier Ltd. This c 2017 2017The TheAuthors. Authors. Published by Elsevier Ltd.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 10th International Conference on Marine Technology. Peer-review under responsibility of the organizing committee of the 10th International Conference on Marine Technology. Keywords: Unsteady; MHD; Mixed Convection; Ferrofluid.
1. Introduction Mixed convection heat transfer inside enclosures has a wide variety of technological applications such as thermalhydraulics of nuclear reactors, flow include crystal growth, combustion of atomized liquid fuels industries etc. In recent years, mixed convection inside different shaped enclosures with nanofluids has attracted attention in variety of engineering applications. Ahmed et al. [1] studied mixed convection from a discrete heat source in enclosures with two adjacent moving walls and filled with micropolar nanofluids. The laminar mixed convection of water-Cu nanofluid in an inclined shallow driven cavity using the lattice Boltzmann method was numerically investigated by Karimipour et al. [2]. Khanafer et al. [3] investigated unsteady laminar mixed convection heat transfer in a lid driven cavity. Najam et al. [4] studied laminar unsteady mixed convection in a two dimensional horizontal channel containing heating blocks periodically mounted on its lower wall. The study of MHD flow has gained much attraction of researchers due to the effect of magnetic field on the flow control. Mukhopadhyay et al. [5] studied MHD mixed convection slip flow and heat transfer over a vertical porous plate. Sandeep et al. [6] investigated dual solutions for ∗
Corresponding author. Tel.: +880-168-156-3128. E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 10th International Conference on Marine Technology.
doi:10.1016/j.proeng.2017.08.176
Nandita Chakrabarty Jhumur and Anuruddha Bhattacharjee / Procedia Engineering 194 (2017) 494 – 501
unsteady mixed convection flow of MHD micro polar fluid. Arani et al. [7] investigated natural convection in an inclined L-shaped cavity filled with copper-water nano fluid. The fluid flow and heat transfer inside L-shaped enclosure filled with Cu-water nanofluid has been investigated numerically by Mliki et al. [8]. Aminfar et al. [9] studied the hydro-thermal characteristics of a Ferrofluid (water and 4vol% Fe3 O4 ) in a vertical rectangular duct which is exposed to a non-uniform transverse magnetic field. Selimefendigil et al. [10] numerically investigated natural convection of ferrofluid in a partially heated square cavity. This paper numerically investigates the flow characteristics of unsteady MHD mixed convection inside L-shaped enclosure in the presence of Ferro fluid Fe3 O4 -Water). Lid-motion is introduced by the top wall to induce forced convection. For pure mixed convection (Ri = 1), Grashof number (Gr) and time (τ) are varied, whereas the solid volume fraction of nanofluid (φ = 0.1) and Hartman number (Ha = 25) are kept constant. Nomenclature L Cp g Gr k Ha Nu p P Pr Re Ri B0 T u, v U, V x, y X, Y
Height of cavity Constant pressure specific heat Gravitational acceleration Grashof number Thermal conductivity Hartmann number Nusselt number Pressure Dimensionless pressure Prandtl number Reynolds number Richardson number Magnetic induction Temperature Cartesian velocity components Dimensionless velocity components Cartesian co-ordinates Dimensionless co-ordinates
Greek symbols α Thermal diffusivity β Coefficient of volume expansion μ Dynamic viscosity ν Kinematic viscosity ρ Density Θ Dimensionless temperature ψ Stream function Ψ Dimensionless stream function σ Electrical conductivity Subscripts c Cold (lower value) f Fluid h Hot (higher value) f f Ferrofluid o Initial value
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2. Problem Formulation 2.1. Physical Modelling The present configuration consists of a two-dimensional L-shaped cavity as shown in Fig. 1. The top wall of the cavity is lid-driven and moving at a constant velocity u0 in positive x-direction while the bottom and the other side walls are kept stationary. The gravity is acting in the negative direction of the y-axis. Length of the bottom wall and height of the cavity are remained fixed at the same length, L. Length of the horizontal heated wall, “a” is such that a/L = 0.5.Also, the length of the vertical heated wall, “b”, is such that b/L = 0.5. For present study, the ratio a/L is kept 1. The upper moving wall and the bottom wall are maintained at low temperature, T = T c . The horizontal and the vertical heated walls are at higher temperature, T = T h . The side walls of the cavity are thermally insulated. The free space of the cavity is filled with a water-based nanofluid containing Fe3 O4 nanoparticles under MHD effects. Laminar flow is assumed. Also for simplification of the problem, radiation effect and viscous dissipation along with internal heat generation have been neglected.
Fig. 1. Schematic diagram of the L-shaped cavity
2.2. Mathematical Modeling Mixed convection fluid flow inside the cavity follows the mass conservation equation for the Newtonian fluid, laminar and unsteady state flow that reads, in its dimensionless form: The continuity equation: ∂U ∂V + =0 ∂X ∂Y
(1)
The momentum equations: ∂U ∂U ∂P ν f f 1 ∂2 U ∂2 U ∂U +V =− + +U ( + ) ∂τ ∂X ∂Y ∂X ν f Re ∂X 2 ∂Y 2
(2)
βf f ∂U ∂P ν f f 1 ∂2 U ∂2 U ∂U ∂U (Ha)2 +V =− + +U ( 2 + )+ RiΘ V 2 ∂τ ∂X ∂Y ∂X ν f Re ∂X βf Re ∂Y
(3)
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Nandita Chakrabarty Jhumur and Anuruddha Bhattacharjee / Procedia Engineering 194 (2017) 494 – 501
The energy conservation equation: ∂Θ α f f 1 ∂2 Θ ∂2 Θ ∂Θ ∂Θ +V = +U ( + ) ∂τ ∂X ∂Y α f RePr ∂X 2 ∂Y 2
(4)
The governing equations are transformed into a dimensionless form under the following non-dimensional variables: X=
Pr =
kf f T − Tc ψ y u x v p ,Y = ,U = ,V = ,Θ = ,Ψ = ,P = ,τ = t 2 L L u0 v0 Th − Tc αf f ρ f f n0 (ρC p ) f f L2 νf f ρ f f u0 L gβ f f (T h − T c )L3 Gr , Re = , Gr = , Ri = 2 , Ha = B0 L αf f μf f νf f 3 Re
(5)
αf f μf f
(6)
Boundary conditions for analysis of pure mixed convection in L-shaped cavity are presented in non-dimensional from in Table 1. The stream function is obtained by Table 1. Boundary conditions for this problem in non-dimensional form. Boundary Wall
Flow Field Condition
Thermal Field Condition
Top Wall Bottom Wall Side Walls Vertical Heated Wall Horizontal Heated Wall
U = 1, V = 1 U=V=0 U=V=0 U=V=0 U=V=0
θ=0 θ=0 ∂θ ∂X = 0 θ=1 θ=1
U=
∂Ψ ∂Ψ ,V = − ∂Y ∂X
(7)
The average Nusselt (Nu) number in the vertical and horizontal heated wall is defined as k f f L 1 ∂Θ k f f L 1 ∂Θ Nuv = ( )dY, Nuh = ( )dX k f b b/L ∂X k f a a/L ∂Y Average temperature of the flow field can be calculated as, 1 Θa v = ΘdA A A where, the area of the cavity, A = (1 −
(8)
(9)
ab ) L2
2.3. Properties of Ferrofluid The thermo-physical properties of nanoparticle (Fe3 O4 ) and base fluid (water) have been summarized in Table 2. Theseproperties have been considered constant except for density variation according to Boussinesq approximation. Table 2. Thermo-physical properties of Fe3 O4 (nanoparticles) and water (base fluid). Property
(Fe3 O4 )
Water
Heat capacitance (JKg−1 K −1 ) Density (Kgm−3 ) Thermal conductivity (Wm−1 K −1 ) Thermal expansion coefficient (K −1 ) Dynamic viscosity (N sm−2 )
670 5200 6 1.18 × 10−5 -
4179 997.1 0.613 2.1 × 10−4 0.001003
498
Nandita Chakrabarty Jhumur and Anuruddha Bhattacharjee / Procedia Engineering 194 (2017) 494 – 501
3. Numerical Procedure This specific problem was solved using Galerkin weighted residual scheme of finite element method (FEM). Six nodded triangular mesh has been employed to discretize the entire cavity with finer mesh applied near the boundary walls. An iterative method has been used to find out the solution by the PARDISO solver with convergence criterion of 10−5 . 3.1. Grid Independency Test The grid independency test result is presented in Fig.2. From the figure it is evident that for the lower grid element numbers, Nu varies significantly. When the number of grid elements reaches to 21540, Nu becomes more or less constant. So the grid having 21540 elements has been taken for the entire numerical simulation of the specified problem.
Fig. 2. Variation of Nusselt (Nu) nuber with Grid Elements for Gr = 103 and τ = 5, at vertical heated wall.
4. Result and Discussions Results are generated for nanofluid (Fe3 O4 - Water) with φ = 0.1 at Ha = 25. Streamline and isotherm contours are presented for different dimensionless time (τ = 1, 5) and different Grashof (Gr) numbers. Also, average Nusselt (Nu) number at the horizontal and the vertical heated walls and average temperature (Θav ) for the entire cavity have been presented with varying time (τ) and Grashof (Gr) numbers. 4.1. Effect of Time (τ) and Grashof (Gr) Number The sensitivity of the streamline and isotherm patterns due to the variation in Grashof (Gr) number and dimensionless time (τ) is presented in Fig. 3. There are two major vortices inside the cavity. The moving lid induces a vortex along its direction of motion. On the other hand, the thermal buoyancy effect induces another vortex. For pure mixed convection (Ri = 1), both vortices are significant even if the Grashof (Gr) number is increased. As the Grashof (Gr) number increases, the intensity of convection intensifies within the cavity due to the increase in buoyancy effect. It is also seen that the strength of the streamline contours is decreased with the increment of time (τ). There is no significant effect of buoyancy near the horizontal heated wall and the primary mode of heat transfer in here is conduction. It is also evident from the isotherm contours, which become closely packed and almost parallel near the horizontal heated wall. The convection is mostly dependent on the intensity of the lid velocity and viscous effects are dominant here. However, the isotherms become distorted when the Grashof (Gr) number is increased keeping the Richardson (Ri) number constant at 1.
Nandita Chakrabarty Jhumur and Anuruddha Bhattacharjee / Procedia Engineering 194 (2017) 494 – 501
Fig. 3. Streamlines for (a) τ = 1.0; (b) τ = 5.0; Isotherms for (c) τ = 1.0; (d) τ = 5.0; at Gr = 103 , Ri = 1. Streamlines for (e) τ = 1.0; (f) τ = 5.0; Isotherms for (g) τ = 1.0; (h) τ = 5.0; at Gr = 105 , Ri = 1.
4.2. Variation of Average Nusselt (Nu) Number and Average Temperature (Θav )
Fig. 4. Variation of (a) Nu and (b) Θav with τ at Gr = 103 , Ri = 1.; Variation of (c) Nu and (d) Θav with τ at Gr = 104 , Ri = 1.; Variation of (e) Nu and (f) Θav with Gr at τ = 10, Ri = 1.
In Fig. 4, variation of Nu and Θav are analyzed. In Fig. 4 (a), (b), (c), (d), τ has been varied in the range of 0 < τ < 12. With the increase of τ, both Nu and Θav decrease. This is because initially the flow was disturbed and
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unsettled in the initial stages of flow development. As the flow becomes steady, the mixing becomes more ordered and hence heat transfer also decreases. In Fig. 4 (e) and (f), as the Grashof number increases, the convection activities within the cavity intensify and, as a result, the average temperature (Θav ) within the cavity and the average Nusselt (Nu) number along both the vertical and the horizontal heated wall increase. 4.3. Effect of Hartmann (Ha) number on Average Nusselt (Nu) Number and Average Temperature (Θav )
Fig. 5. Variation of (a) Nu and (b) Θav with Ha at Gr = 105 , Ri = 1.
In Fig. 5, it is seen that with the increase of Hartmann (Ha) number, both Nu and Θav decrease very slowly. Applying external magnetic field to a flow creates Lorentz force, which acts in the opposite direction to its source and slows down the flow. 5. Conclusion In this paper a thorough discussion has been made on the problem specification, its background literature and numerical solution procedures. From the result analyses, the following conclusions can be drawn: • The buoyancy effect becomes prominent with higher Grashof (Gr) number. On the other hand, strength of the streamline contours decreases with the increase in time. • Effect of Grashof (Gr) number is insignificant for the horizontal heated wall because of conduction being the dominating mood of heat transfer. • Distortion of isotherm contours is very clear with the increment of Grashof (Gr) number. • Nusselt (Nu) number and average temperature (Θav ) both decrease with time, though both are with higher initial values. • With the increment of Grashof (Gr) number, Nusselt (Nu) number increases while average temperature (Θav ) decreases and then increases again proportionally. • Increment of Hartmann (Ha) number, decreases both the Nusselt (Nu) number and the average temperature (Θav ) proportionally. References [1] Sameh E. Ahmed, M.A. Mansour, Ahmed Kadhim Hussein, S. Sivasankaran, Mixed convection from a discrete heat source in enclosures with two adjacent moving walls and filled with micropolar nanofluids, Eng. Sci. Technol. Int. J. 19 (2016) 364376. [2] Arash Karimipour, Mohammad Hemmat Esfe, Mohammad Reza Safaei, Davood Toghraie Semiromic, S.N.Kazi, Mixed convection of copperwater nanofluid in a shallow inclined lid driven cavity using the lattice Boltzmann method, Physica A 402 (2014) 150168. [3] Khalil M. Khanafer, Abdalla M. Al-Amiri, Ioan Pop, Numerical simulation of unsteady mixed convection in a driven cavity using an externally excited sliding lid, European Journal of Mechanics B/Fluids 26 (2007) 669687. [4] M. Najam, A. Amahmid, M. Hasnaoui, M. El Alami, Unsteady mixed convection in a horizontal channel with rectangular blocks periodically distributed on its lower wall, Int. J. Heat and Fluid Flow 24 (2003) 726735.
Nandita Chakrabarty Jhumur and Anuruddha Bhattacharjee / Procedia Engineering 194 (2017) 494 – 501 [5] S. Mukhopadhyay, I.C. Mandal, Magnetohydrodynamic (MHD) mixed convection slip flow and heat transfer over a vertical porous plate, Eng. Sci.Technol. Int. J. 18 (2015) 98105. [6] N. Sandeep, C. Sulochana, Dual solutions for unsteady mixed convection flow of MHD micropolar fluid over a stretching/shrinking sheet with non-uniform heat source/sink, Eng. Sci. Technol. Int. J. 18 (2015) 738745. [7] A.A. Abbasian Arani, A.Z. Maghsoudi, A.H. Niroumand and S.M.E. Derakhshani, Study of Nanofluid Natural Convection in an Inclined L-Shaped Cavity, Scientia Iranica F 20 (2013) 2297-2305. [8] Bouchmel Mliki, Mohamed Ammar Abbassi, Kamel Guedri, Ahmed Omri, Lattice Boltzmann simulation of natural convection in an L-shaped enclosure in the presence of nanofluid, Eng. Sci.Technol. Int. J.18 (2015) 503511. [9] H. Aminfar, M. Mohammadpourfard, S. Ahangar Zonouzi, Numerical study of the ferrofluid flow and heat transfer through a rectangular duct in the presence of a non-uniform transverse magnetic field, J Magn. Magn. Mater 327 (2013) 3142. [10] Fatih Selimefendigil, Hakan F. ztop, Khaled Al-Salem, Natural convection of ferrofluids in partially heated square enclosures, J Magn. Magn. Mater 372 (2014) 122-133.
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ScienceDirect Procedia Engineering 194 (2017) 494 – 501
10th International Conference on Marine Technology, MARTEC 2016
Unsteady MHD Mixed Convection inside L-shaped Enclosure in the Presence of Ferrofluid (Fe3O4) Nandita Chakrabarty Jhumur∗, Anuruddha Bhattacharjee Department of Mechanical Engineering, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh
Abstract Mixed convection has been one of the most interesting topics of research in the field of heat transfer for many years. In the present study, unsteady MHD mixed convection heat transfer inside a heated L-shaped enclosure filled with Fe3 O4 water has been investigated numerically. Lid-motion is introduced by the top wall to induce forced convection. For pure mixed convection (Ri = 1), Grashof number (Gr = 103 − 105 ) and time (τ) are varied, whereas the solid volume fraction of nanofluid (φ = 0.1) and Hartman number (Ha = 25) are kept constant. Results have been presented in terms of isotherm lines, streamlines, average Nusselt (Nu) number and average temperature (θay ). The obtained results show that MHD causes significant effects on heat transfer rate. Interestingly, natural convection is more effective in this case of mixed convection as higher values of Nu are obtained for higher values of Gr. © Published by Elsevier Ltd. This c 2017 2017The TheAuthors. Authors. Published by Elsevier Ltd.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 10th International Conference on Marine Technology. Peer-review under responsibility of the organizing committee of the 10th International Conference on Marine Technology. Keywords: Unsteady; MHD; Mixed Convection; Ferrofluid.
1. Introduction Mixed convection heat transfer inside enclosures has a wide variety of technological applications such as thermalhydraulics of nuclear reactors, flow include crystal growth, combustion of atomized liquid fuels industries etc. In recent years, mixed convection inside different shaped enclosures with nanofluids has attracted attention in variety of engineering applications. Ahmed et al. [1] studied mixed convection from a discrete heat source in enclosures with two adjacent moving walls and filled with micropolar nanofluids. The laminar mixed convection of water-Cu nanofluid in an inclined shallow driven cavity using the lattice Boltzmann method was numerically investigated by Karimipour et al. [2]. Khanafer et al. [3] investigated unsteady laminar mixed convection heat transfer in a lid driven cavity. Najam et al. [4] studied laminar unsteady mixed convection in a two dimensional horizontal channel containing heating blocks periodically mounted on its lower wall. The study of MHD flow has gained much attraction of researchers due to the effect of magnetic field on the flow control. Mukhopadhyay et al. [5] studied MHD mixed convection slip flow and heat transfer over a vertical porous plate. Sandeep et al. [6] investigated dual solutions for ∗
Corresponding author. Tel.: +880-168-156-3128. E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 10th International Conference on Marine Technology.
doi:10.1016/j.proeng.2017.08.176
Nandita Chakrabarty Jhumur and Anuruddha Bhattacharjee / Procedia Engineering 194 (2017) 494 – 501
unsteady mixed convection flow of MHD micro polar fluid. Arani et al. [7] investigated natural convection in an inclined L-shaped cavity filled with copper-water nano fluid. The fluid flow and heat transfer inside L-shaped enclosure filled with Cu-water nanofluid has been investigated numerically by Mliki et al. [8]. Aminfar et al. [9] studied the hydro-thermal characteristics of a Ferrofluid (water and 4vol% Fe3 O4 ) in a vertical rectangular duct which is exposed to a non-uniform transverse magnetic field. Selimefendigil et al. [10] numerically investigated natural convection of ferrofluid in a partially heated square cavity. This paper numerically investigates the flow characteristics of unsteady MHD mixed convection inside L-shaped enclosure in the presence of Ferro fluid Fe3 O4 -Water). Lid-motion is introduced by the top wall to induce forced convection. For pure mixed convection (Ri = 1), Grashof number (Gr) and time (τ) are varied, whereas the solid volume fraction of nanofluid (φ = 0.1) and Hartman number (Ha = 25) are kept constant. Nomenclature L Cp g Gr k Ha Nu p P Pr Re Ri B0 T u, v U, V x, y X, Y
Height of cavity Constant pressure specific heat Gravitational acceleration Grashof number Thermal conductivity Hartmann number Nusselt number Pressure Dimensionless pressure Prandtl number Reynolds number Richardson number Magnetic induction Temperature Cartesian velocity components Dimensionless velocity components Cartesian co-ordinates Dimensionless co-ordinates
Greek symbols α Thermal diffusivity β Coefficient of volume expansion μ Dynamic viscosity ν Kinematic viscosity ρ Density Θ Dimensionless temperature ψ Stream function Ψ Dimensionless stream function σ Electrical conductivity Subscripts c Cold (lower value) f Fluid h Hot (higher value) f f Ferrofluid o Initial value
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Nandita Chakrabarty Jhumur and Anuruddha Bhattacharjee / Procedia Engineering 194 (2017) 494 – 501
2. Problem Formulation 2.1. Physical Modelling The present configuration consists of a two-dimensional L-shaped cavity as shown in Fig. 1. The top wall of the cavity is lid-driven and moving at a constant velocity u0 in positive x-direction while the bottom and the other side walls are kept stationary. The gravity is acting in the negative direction of the y-axis. Length of the bottom wall and height of the cavity are remained fixed at the same length, L. Length of the horizontal heated wall, “a” is such that a/L = 0.5.Also, the length of the vertical heated wall, “b”, is such that b/L = 0.5. For present study, the ratio a/L is kept 1. The upper moving wall and the bottom wall are maintained at low temperature, T = T c . The horizontal and the vertical heated walls are at higher temperature, T = T h . The side walls of the cavity are thermally insulated. The free space of the cavity is filled with a water-based nanofluid containing Fe3 O4 nanoparticles under MHD effects. Laminar flow is assumed. Also for simplification of the problem, radiation effect and viscous dissipation along with internal heat generation have been neglected.
Fig. 1. Schematic diagram of the L-shaped cavity
2.2. Mathematical Modeling Mixed convection fluid flow inside the cavity follows the mass conservation equation for the Newtonian fluid, laminar and unsteady state flow that reads, in its dimensionless form: The continuity equation: ∂U ∂V + =0 ∂X ∂Y
(1)
The momentum equations: ∂U ∂U ∂P ν f f 1 ∂2 U ∂2 U ∂U +V =− + +U ( + ) ∂τ ∂X ∂Y ∂X ν f Re ∂X 2 ∂Y 2
(2)
βf f ∂U ∂P ν f f 1 ∂2 U ∂2 U ∂U ∂U (Ha)2 +V =− + +U ( 2 + )+ RiΘ V 2 ∂τ ∂X ∂Y ∂X ν f Re ∂X βf Re ∂Y
(3)
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Nandita Chakrabarty Jhumur and Anuruddha Bhattacharjee / Procedia Engineering 194 (2017) 494 – 501
The energy conservation equation: ∂Θ α f f 1 ∂2 Θ ∂2 Θ ∂Θ ∂Θ +V = +U ( + ) ∂τ ∂X ∂Y α f RePr ∂X 2 ∂Y 2
(4)
The governing equations are transformed into a dimensionless form under the following non-dimensional variables: X=
Pr =
kf f T − Tc ψ y u x v p ,Y = ,U = ,V = ,Θ = ,Ψ = ,P = ,τ = t 2 L L u0 v0 Th − Tc αf f ρ f f n0 (ρC p ) f f L2 νf f ρ f f u0 L gβ f f (T h − T c )L3 Gr , Re = , Gr = , Ri = 2 , Ha = B0 L αf f μf f νf f 3 Re
(5)
αf f μf f
(6)
Boundary conditions for analysis of pure mixed convection in L-shaped cavity are presented in non-dimensional from in Table 1. The stream function is obtained by Table 1. Boundary conditions for this problem in non-dimensional form. Boundary Wall
Flow Field Condition
Thermal Field Condition
Top Wall Bottom Wall Side Walls Vertical Heated Wall Horizontal Heated Wall
U = 1, V = 1 U=V=0 U=V=0 U=V=0 U=V=0
θ=0 θ=0 ∂θ ∂X = 0 θ=1 θ=1
U=
∂Ψ ∂Ψ ,V = − ∂Y ∂X
(7)
The average Nusselt (Nu) number in the vertical and horizontal heated wall is defined as k f f L 1 ∂Θ k f f L 1 ∂Θ Nuv = ( )dY, Nuh = ( )dX k f b b/L ∂X k f a a/L ∂Y Average temperature of the flow field can be calculated as, 1 Θa v = ΘdA A A where, the area of the cavity, A = (1 −
(8)
(9)
ab ) L2
2.3. Properties of Ferrofluid The thermo-physical properties of nanoparticle (Fe3 O4 ) and base fluid (water) have been summarized in Table 2. Theseproperties have been considered constant except for density variation according to Boussinesq approximation. Table 2. Thermo-physical properties of Fe3 O4 (nanoparticles) and water (base fluid). Property
(Fe3 O4 )
Water
Heat capacitance (JKg−1 K −1 ) Density (Kgm−3 ) Thermal conductivity (Wm−1 K −1 ) Thermal expansion coefficient (K −1 ) Dynamic viscosity (N sm−2 )
670 5200 6 1.18 × 10−5 -
4179 997.1 0.613 2.1 × 10−4 0.001003
498
Nandita Chakrabarty Jhumur and Anuruddha Bhattacharjee / Procedia Engineering 194 (2017) 494 – 501
3. Numerical Procedure This specific problem was solved using Galerkin weighted residual scheme of finite element method (FEM). Six nodded triangular mesh has been employed to discretize the entire cavity with finer mesh applied near the boundary walls. An iterative method has been used to find out the solution by the PARDISO solver with convergence criterion of 10−5 . 3.1. Grid Independency Test The grid independency test result is presented in Fig.2. From the figure it is evident that for the lower grid element numbers, Nu varies significantly. When the number of grid elements reaches to 21540, Nu becomes more or less constant. So the grid having 21540 elements has been taken for the entire numerical simulation of the specified problem.
Fig. 2. Variation of Nusselt (Nu) nuber with Grid Elements for Gr = 103 and τ = 5, at vertical heated wall.
4. Result and Discussions Results are generated for nanofluid (Fe3 O4 - Water) with φ = 0.1 at Ha = 25. Streamline and isotherm contours are presented for different dimensionless time (τ = 1, 5) and different Grashof (Gr) numbers. Also, average Nusselt (Nu) number at the horizontal and the vertical heated walls and average temperature (Θav ) for the entire cavity have been presented with varying time (τ) and Grashof (Gr) numbers. 4.1. Effect of Time (τ) and Grashof (Gr) Number The sensitivity of the streamline and isotherm patterns due to the variation in Grashof (Gr) number and dimensionless time (τ) is presented in Fig. 3. There are two major vortices inside the cavity. The moving lid induces a vortex along its direction of motion. On the other hand, the thermal buoyancy effect induces another vortex. For pure mixed convection (Ri = 1), both vortices are significant even if the Grashof (Gr) number is increased. As the Grashof (Gr) number increases, the intensity of convection intensifies within the cavity due to the increase in buoyancy effect. It is also seen that the strength of the streamline contours is decreased with the increment of time (τ). There is no significant effect of buoyancy near the horizontal heated wall and the primary mode of heat transfer in here is conduction. It is also evident from the isotherm contours, which become closely packed and almost parallel near the horizontal heated wall. The convection is mostly dependent on the intensity of the lid velocity and viscous effects are dominant here. However, the isotherms become distorted when the Grashof (Gr) number is increased keeping the Richardson (Ri) number constant at 1.
Nandita Chakrabarty Jhumur and Anuruddha Bhattacharjee / Procedia Engineering 194 (2017) 494 – 501
Fig. 3. Streamlines for (a) τ = 1.0; (b) τ = 5.0; Isotherms for (c) τ = 1.0; (d) τ = 5.0; at Gr = 103 , Ri = 1. Streamlines for (e) τ = 1.0; (f) τ = 5.0; Isotherms for (g) τ = 1.0; (h) τ = 5.0; at Gr = 105 , Ri = 1.
4.2. Variation of Average Nusselt (Nu) Number and Average Temperature (Θav )
Fig. 4. Variation of (a) Nu and (b) Θav with τ at Gr = 103 , Ri = 1.; Variation of (c) Nu and (d) Θav with τ at Gr = 104 , Ri = 1.; Variation of (e) Nu and (f) Θav with Gr at τ = 10, Ri = 1.
In Fig. 4, variation of Nu and Θav are analyzed. In Fig. 4 (a), (b), (c), (d), τ has been varied in the range of 0 < τ < 12. With the increase of τ, both Nu and Θav decrease. This is because initially the flow was disturbed and
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Nandita Chakrabarty Jhumur and Anuruddha Bhattacharjee / Procedia Engineering 194 (2017) 494 – 501
unsettled in the initial stages of flow development. As the flow becomes steady, the mixing becomes more ordered and hence heat transfer also decreases. In Fig. 4 (e) and (f), as the Grashof number increases, the convection activities within the cavity intensify and, as a result, the average temperature (Θav ) within the cavity and the average Nusselt (Nu) number along both the vertical and the horizontal heated wall increase. 4.3. Effect of Hartmann (Ha) number on Average Nusselt (Nu) Number and Average Temperature (Θav )
Fig. 5. Variation of (a) Nu and (b) Θav with Ha at Gr = 105 , Ri = 1.
In Fig. 5, it is seen that with the increase of Hartmann (Ha) number, both Nu and Θav decrease very slowly. Applying external magnetic field to a flow creates Lorentz force, which acts in the opposite direction to its source and slows down the flow. 5. Conclusion In this paper a thorough discussion has been made on the problem specification, its background literature and numerical solution procedures. From the result analyses, the following conclusions can be drawn: • The buoyancy effect becomes prominent with higher Grashof (Gr) number. On the other hand, strength of the streamline contours decreases with the increase in time. • Effect of Grashof (Gr) number is insignificant for the horizontal heated wall because of conduction being the dominating mood of heat transfer. • Distortion of isotherm contours is very clear with the increment of Grashof (Gr) number. • Nusselt (Nu) number and average temperature (Θav ) both decrease with time, though both are with higher initial values. • With the increment of Grashof (Gr) number, Nusselt (Nu) number increases while average temperature (Θav ) decreases and then increases again proportionally. • Increment of Hartmann (Ha) number, decreases both the Nusselt (Nu) number and the average temperature (Θav ) proportionally. References [1] Sameh E. Ahmed, M.A. Mansour, Ahmed Kadhim Hussein, S. Sivasankaran, Mixed convection from a discrete heat source in enclosures with two adjacent moving walls and filled with micropolar nanofluids, Eng. Sci. Technol. Int. J. 19 (2016) 364376. [2] Arash Karimipour, Mohammad Hemmat Esfe, Mohammad Reza Safaei, Davood Toghraie Semiromic, S.N.Kazi, Mixed convection of copperwater nanofluid in a shallow inclined lid driven cavity using the lattice Boltzmann method, Physica A 402 (2014) 150168. [3] Khalil M. Khanafer, Abdalla M. Al-Amiri, Ioan Pop, Numerical simulation of unsteady mixed convection in a driven cavity using an externally excited sliding lid, European Journal of Mechanics B/Fluids 26 (2007) 669687. [4] M. Najam, A. Amahmid, M. Hasnaoui, M. El Alami, Unsteady mixed convection in a horizontal channel with rectangular blocks periodically distributed on its lower wall, Int. J. Heat and Fluid Flow 24 (2003) 726735.
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