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Analysis of MHD mixed convection in a flexible walled and nanofluids filled lid-driven cavity with volumetric heat generation
Author’s Accepted Manuscript Analysis of MHD mixed convection in a flexible walled and nanofluids filled lid-driven cavity with volumetric heat generation Fatih Selimefendigil, Hakan F. Öztop www.elsevier.com/locate/ijmecsci
PII: DOI: Reference:
S0020-7403(16)30217-X http://dx.doi.org/10.1016/j.ijmecsci.2016.09.011 MS3416
To appear in: International Journal of Mechanical Sciences Received date: 6 August 2015 Revised date: 5 September 2016 Accepted date: 6 September 2016 Cite this article as: Fatih Selimefendigil and Hakan F. Öztop, Analysis of MHD mixed convection in a flexible walled and nanofluids filled lid-driven cavity with volumetric heat generation, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2016.09.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analysis of MHD mixed convection in a flexible walled and nanofluids filled lid-driven cavity with volumetric heat generation b ¨ Fatih Selimefendigila,∗, Hakan F. Oztop a
Department of Mechanical Engineering, Celal Bayar University, 45140 Manisa, Turkey b Department of Mechanical Engineering, Technology Faculty, Fırat University, 23119 Elazı˘g, Turkey
Abstract In this study, MHD mixed convection in a CuO-nanofluid filled lid-driven cavity having an elastic side wall and volumetric heat generation is numerically investigated. The left vertical wall is moving with constant velocity in +y direction. The left vertical wall of the cavity is maintained at constant cold temperature while the right vertical wall is at hot temperature and the other walls of the cavity are insulated. The governing equations are solved with finite element formulation. The Arbitrary-Lagrangian-Eulerian method is used to describe the fluid motion with the elastic wall in the fluid-structure interaction model. The influence of Richardson number (between 0.01 and 100), internal Rayleigh number (between 103 and 106 ), Hartmann number (between 0 and 50), inclination angle of the magnetic field (between 0o and 90o ), Young’s modulus of flexible wall (between 5×102 N/m2 and 106 N/m2 ), and nanoparticle volume fraction (between 0 and 0.05) on the fluid flow and ∗
Corresponding author, Tel.: +90 236 241 21 44; Fax: +90 236 241 21 43. Email addresses: [email protected] (Fatih Selimefendigil), ¨ [email protected] (Hakan F. Oztop)
Preprint submitted to XXXXXXX
September 19, 2016
heat transfer were numerically investigated. The effect of Brownian motion on the effective thermal conductivity was taken into account. The averaged heat transfer decreases with increasing Richardson and Hartmann and internal Rayleigh numbers. Absolute value of the averaged heat transfer enhances by 14.34% and 8.83% at Richardon number of 1 and 100 and deteriorates by 6.51% at Richardson number of 0.01 for Young’s modulus of the elastic wall 500 when compared to configuration for Young’s modulus of 106 . The local and averaged heat transfer enhance as the value of solid volume fraction of the nanoparticle increases and this is more effective for higher values of Richardson number where heat transfer process is effective. Keywords: elastic wall, magnetic field, nanofluids, fluid-structure interaction, finite element method
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Nomenclature a
local acceleration
B0
magnetic field strength Greek Characters
E Young’s modulus f
α thermal diffusivity
body force per unit volume gβf (Th −Tc )H 3 νf2
Gr Grashof number,
local heat transfer coefficient σ Ha Hartmann number, B0 H ρnfnfνf h
k
thermal conductivity
H
length of the enclosure
n
unit normal vector local Nusselt number
Num
averaged Nusselt number
Q
θ
non-dimensional temperature
ν
kinematic viscosity
Vw H νf Gr Re2
temperature
u, v
x-y velocity components
x, y
Cartesian coordinates
stress tensor
Subscripts
volumetric heat generation rate
Ri Richardson number, T
magnetic inclination angle
φ solid volume fraction
νf αf
Re Reynolds number,
γ
ρ density of the fluid
pressure
Pr Prandtl number,
expansion coefficient
Γ electrical conductivity
σ
Nuy
p
β
3
c
cold
h
hot
m
average
nf
nanofluid
p
solid particle
st
static
1. Introduction Mixed convection in lid-driven cavity is important in many engineering applications such as cooling of electronic devices, food processing, solidification, float glass production, nuclear reactors, microelectronic devices, coating and solar power. The complicated interaction between the shear driven flow and natural convection has important effects on the enhancement in the flow mixing and heat transfer process. Volumetric internal heat generation is an important issue in engineering such as nuclear power plants, concrete or inside of earth. Convection in enclosures with heat generating fluids was extensively studied [1–3]. Acharya and Goldstein [1] numerically studied the free convection inside an inclined cavity having uniformly distributed internal energy source. They observed distinct flow pattern systems depending on the ratio of the external to internal Rayleigh number. Oztop and Bilgen [4] numerically investigated the natural convection of differentially heated, partitioned square cavity containing heat generating fluid. They observed that the heat transfer was generally reduced when the ratio of internal and external Rayleigh numbers was from 10 to 100. In another study by Oztop et al. [5], numerical study of natural convection in a wavy-wall cavity with volumetric heat source was performed. They observed that both the wavy wall and the ratio of internal Rayleigh number to external Rayleigh number affect the heat transfer and fluid flow significantly. In order to control the fluid flow and heat transfer characteristics within a cavity different strategies can be utilized. In one of these methods, magnetic field can be used to control of heat transfer [6–21]. The effects of magnetic field on the fluid flow and heat transfer were studied by many researchers due 4
to its importance in many engineering applications such as coolers of nuclear reactors, purification of molten metals etc. Rahman et al. [22] 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 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. Al-Salem et al. [23] studied the effects of moving lid direction on MHD mixed convection in a cavity with linearly heated bottom wall using finite volume method. They observed that direction of lid is more effective on heat transfer and fluid flow in the cavity and heat transfer is decreased with increasing of magnetic field parameter. Oztop et al. [24] studied 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. Magnetic field with nanofluids have received some attention due to higher thermal conductivity of the nanoparticles added to the base fluid [25–37]. Ghasemi et al. [28] studied the MHD natural convection in an enclosure filled with water - Al2 O3 nanofluid. They observed that an enhancement or deterioration of the heat transfer may be obtained with an increase of the nanoparticle volume fraction depending on the value of Hartmann and Rayleigh numbers. Mahmoudi et al. [34] numerically studied the MHD natural convection in a triangular enclosure filled with nanofluid. The impact of the Rayleigh number, Hartmann number and nanoparticle volume fraction on the heat transfer and fluid flow were numerically investigated.
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Deformable walls can be used to control the heat transfer and flow inside the cavity [38, 39] along with the corrugated walls [40–42]. Al-Amiri and Khanafer [39] numerically studied the laminar mixed convection in a lid driven cavity having a flexible bottom wall. They showed that elasticity of the bottom weall has significant effect on the heat transfer enhancement. In another study, Khanafer [38] numerically investigated the mixed convection in a lid driven cavity with a flexible bottom wall. He observed that compared to flat wall case significant heat transfer enhancement with flexible wall is achieved. Based on the above literature survey and to the best of authors’ knowledge, MHD mixed convection in a nanofluid filled lid- driven cavity under the influence of an inclined magnetic field having an elastic side wall has never been reported in the literature. Especially, the mixed convection configurations with deformable walls were rarely studied in the literature. In this study, MHD effect coupled with fluid-structure interaction model in a nanofluid filled lid driven cavity problem will be numerically investigated for the volumetric internal heat generation case for a range of flow parameters . The present numerical study aims at investigating the effects of Richardson number, Hartmann number, inclination angle of the magnetic field, Young’s modulus of the flexible wall and solid volume fraction of the nanoparticles on the fluid flow and heat transfer characteristics in a lid driven cavity having an elastic side wall.
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2. Mathematical formulation A schematic description of the physical problem and boundary conditions are depicted in Fig. 1 (a). The left vertical wall of the cavity is moving with constant velocity of uw in +y direction. The left vertical wall of the cavity is maintained at constant cold temperature of Tc while the right vertical wall is at hot temperature of Th and the other walls of the cavity are insulated. The gravity acts in the negative y-direction. The cavity is filled with CuO-water nanofluid under the influence of a vertical magnetic field. The cavity is filled ˙ A uniform with a uniform heat generating fluid with volumetric rate of Q.
= Bx i + By j is applied and it makes angle of γ with the magnetic field of B horizontal axis. The cavity is filled with CuO-water nanofluid (different solid volume fraction of φ) under the influence of the applied uniform magnetic field. Thermal equilibrium between the fluid phase and nanoparticles and no slip between them are assumed. Homogeneous mixture of CuO and water is considered. A homogeneous mixture can be obtained and the nanoparticles can be de-agglomerated by using an ultrasonic homogenizer [43, 44]. Thermo-physical properties of water and CuO at the reference temperature are presented in Table 1 [34]. The thermo-physical properties of the fluid are assumed to be constant except for the density variation which is modeled by the Boussinesq approximation in the buoyancy term.The effects of joule heating, displacement currents and induced magnetic field are assumed to be negligible. The flow inside the cavity is assumed to be laminar, steady and two dimensional. The Arbitrary Lagrangian-Eulerian method is utilized to describe the fluid motion with the flexible wall of the cavity in the fluid-structure interaction model. Navier-Stokes and energy equations with 7
effective thermophysical properties are used to describe the fluid domain. The conservation equation of mass, momentum and energy in a two dimensional Cartesian coordinate system for the fluid domain can be written as follows [38]: .u = 0
(1)
ρnf (u − ug ). u = .σnf + ρnf ffb
(2)
u T = αnf 2 T +
Q˙ ρcp
(3)
where ffb , σnf , ug denote the body force per unit volume, stress tensor and velocity of moving coordinate. The body force contains the Lorentz force due to the magnetic field in the x momentum equation such as: Γnf B02 v sin(γ) cos(γ) − u sin2 (γ)
(4)
and also the body force due to the buoyancy and Lorentz force due to the magnetic field in the y-momentum equation: βnf g(T − Tc ) + Γnf B02 u sin(γ) cos(γ) − v cos2 (γ)
(5)
The equation for the solid domain of the fluid-structure interaction model is given by [38]: ρs as = .σs + fs b
(6)
where as , fs b , σs represent the local acceleration, externally applied body force and solid stress tensor, respectively. 8
Nanofluid Thermophyscial Properties The effective density, specific heat, thermal expansion coefficient and electrical conductivity of nanofluid are given by the following formulas: ρnf = (1 − φ)ρf + φρp
(7)
(ρcp )nf = (1 − φ)(ρcp )f + φ(ρcp )p
(8)
(ρβ)nf = (1 − φ)(ρβ)f + φ(ρβ)p
(9)
Γnf = (1 − φ)Γf + φΓp
(10)
where the subscripts f, nf and p denote the base fluid, nanofluid and solid particle, respectively. The effective thermal conductivity of the nanofluid includes the effect of Brownian motion. In this model, the effects of particle size, particle volume fraction and temperature dependence are taken into account and it is given by the following formula [45]:
knf = kst + kBrownian where kst is the static thermal conductivity as given by [46] (kp + 2kf ) − 2φ(kf − kp ) kst = kf (kp + 2kf ) + φ(kf − kp )
(11)
(12)
The interaction between the nanoparticles and the effect of temperature are included in the models as
kBrownian = 5 × 104 φρf cp,f
κb T f (T, φ, dp ) ρp dp
where the function f for CuO-water nanofluid is given in [45]. 9
(13)
The effective viscosity of the nanofluid due to micro mixing in the suspension was given in [45] μnf = μst + μBrownian = μf (1 − φ)−0.25 +
kBrownian μf × kf P rf
(14)
where the first terms on the right hand side of equation is the viscosity of the nanofluid given in ref. ([47]). Thermo-physical properties of water-nanofluid are given in Table 1 [48]. Boundary Conditions The boundary conditions in the dimensional form can be defined as: • For the horizontal top and bottom walls: u = v = 0,
∂T ∂y
=0
• For the right vertical wall: u = v = 0, T = Th • For the left vertical wall: u = 0, v = Vw , T = Tc At the fluid structure interface displacement compatibility (df = ds ) and traction equilibrium (σf = σs ) must be satisfied [38]. Eqs.(1) - (4) can be converted to the dimensionless form and the relevant dimensionless numbers are: Gr =
gβf (Th −Tc )H 3 , νf2
Ha = B0 H
σf μf
Pr =
, RaI = 10
νf αf
, Re =
˙ 5 gβf QH , αf νf kf
Vw H , ν
Ri =
Gr Re2
(15)
Local Nusselt number along the left vertical wall is calculated as: knf ∂θ Nuy = − kf ∂X X=0
(16)
where θ represents the non-dimensional temperature and X denotes the nondimensional x- coordinate. Averaged Nusselt number is obtained after integrating the local Nusselt number along the hot wall of the cavity as
1 Nuy dY. (17) Num = 0
3. Solution method Galerkin weighted residual finite element formulation is used to solve the governing equations in Eqs. (1-4). The application of the method and the involved procedural steps were given in [49, 50]. The computational domain is discretized into triangular elements and Lagrange finite elements of different orders are used for each of the flow variables within the computational domain. Residuals for each of the conservation equation is obtained by substituting the approximations into the governing equations and boundary conditions. The resulting nonlinear and coupled algebraic equations from the discretization were solved by using segregated-solution method. The convergence of the solution is assumed when the relative error for each of the variables (Φ) satisfy the following convergence criteria: |
Φn+1 − Φn | ≤ 10−6 Φn+1
(18)
Numerical experiments with various grid sizes are performed in order to obtain an optimal grid distribution with accurate results and minimal 11
computational time. The obtained velocity field is divergence-free. The results of averaged Nusselt number for various grid sizes are shown in Table 2 (Ri=100, RaI =105 , Ha=20, γ = 45o , E=104 , φ = 0.05). G4 with 10916 nodes ensures grid independent solution and hence was used in the subsequent computations.The present code is validated against the existing results of Pirmohammadi and Ghassemi [51] and Sarris et al. [52]. Fig. 2 demonstrates the comparison results of streamlines and isotherms for (Ra = 7 × 103, Ha = 25) and (Ra = 7×105 , Ha = 100). The comparison results show good overall agreement. 4. Results and Discussion In this study, the effects of Richardson number (Ri =
Gr , Re2
0.01 ≤ Ri ≤
100), internal Rayleigh number (103 ≤ RaI ≤ 106 ), Hartmann number (0 ≤ Ha ≤ 50), inclination angle of the magnetic field (0o ≤ γ ≤ 90o ), Young’s modulus of flexible wall (5 × 102 N/m2 ≤ E ≤ 106 N/m2 ), and nanoparticle volume fraction (0 ≤ φ ≤ 0.05) on the flow and heat transfer were numerically investigated. The effects of different nanoparticle types and various shapes can also be investigated [53]. In this study, CuO nanoparticles with spherical shape were used. Higher conductivity of CuO-water nanofluids makes them attractive for various thermal engineering applications [54]. The cavity is filled with CuO-water nanofluid under the influence of an inclined magnetic field and volumetric heat generation. The effects of varying the Richardson number on the streamlines and isotherms are demonstrated in Fig. 3 for fixed values of (RaI =105 , Ha=20, γ = 45o , E=104 , φ = 0.02). At low Richardson number, the effect of forced convection due 12
to the mechanically-driven left wall is more pronounced compared to natural convection (Ri =
Gr ). Re2
The cavity is filled with a single main recirculation
zone and a small vortex is seen in the right bottom corner of the cavity. The flow due to the moving lid penetrates more into the cavity and the flexible wall has a concave shape at Ri=0.01. As the value of the Ri is increased to Ri=1, a secondary recirculation zone near the hot wall and a main vortex in the cavity are seen. At Ri=100, the effect of natural convection becomes important and the convection of the strength enhances. The shape of the flexible wall is convex due to the increasing effect of buoyancy. The effect of internal heating becomes more important when the Richardson number is low for fixed values of Reynolds number. The inner of the cavity is hotter than the right wall for this case especially in the lower part of the hot wall as shown in Fig 3 (d). As the Richardson number increases, the effect of external heating becomes important and steep temperature gradients are seen along the hot wall in the lower part of it. At Richardson number of 100, the isotherms become parallel to the horizontal wall indicates the dominance of convection inside the cavity. The local and averaged heat transfer plots are demonstrated in Fig. 4. At Ri=0.01, the local heat transfer is negative for the lower portion of the hot wall and increases toward the upper part of it. As the value of the Richardson number enhances, the local heat transfer increases in the lower part of the hot wall and deteriorates in the small portion of it. The averaged heat transfer enhances by 239.35% at Richardson number of 100 compared to case at Richardson number of 1. The effects of varying internal Rayleigh numbers on the flow and thermal patterns are demonstrated in Fig. 5 for fixed values of (Ri=1, Ha=25, γ =
13
45o , E=104 , φ = 0.02). The cavity is filled with two recirculating zones for internal Rayleigh numbers of RaI = 103 and RaI = 105 . The ratio of the internal heating compared to external heating is important for RaI = 106 and the number of vortices is three. The temperature gradients along the hot wall and the strength of the convection inside the cavity decreases as the value of the external Rayleigh number increases from RaI = 104 to RaI = 105 . At RaI = 106 , the effect of internal heating becomes important and interior of the cavity near the right wall is hotter than the hot wall. The local and the averaged Nusselt number plots are shown in Fig. 6 (a), (b) for various internal Rayleigh numbers. The local heat transfer decreases as the value of the RaI enhances and then negative values of the local heat transfer are obtained at higher values of internal Rayleigh numbers (RaI = 5 × 105 , RaI = 106 , in Fig. 6 (b)). This trend is also seen in the natural convection study in ref. [13] where different shaped obstacles are placed in the cavity to affect the heat transfer rate. The influence of varying Hartmann number on the flow patterns and isotherms is shown in Fig. 7 (a)-(f) for fixed values of (Ri=2.5, RaI =105 , γ = 45o , E=104 , φ = 0.02 ). In the absence of the magnetic field, the cavity is occupied with a main recirculation zone and another vortex adjacent to the left vertical wall is seen. The increment in the Hartmann number enhances the sterngth of the magnetic field which results in dampening of the flow motion. The recirculating zone adjacent to the left vertical wall increases in size whereas the other vortex diminishes in size and strength with increasing effect of Hartmann number. The tempretaure gradients are steep along the hot wall in the lower portions of it and isotherms become less clustered along
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the hot wall as the value of Hartmann number enhances. Isotherms become parallel to the vertical walls in the interior of the cavity with increasing strength of magnetic field which indicates dominance of conduction heat transfer. The local and averaged heat transfer deteriorates as the value of Ha increases due to the suppression of the convection with magnetic field. 146.28 % of averaged heat transfer enhancement is obtained when Hartmann number is reduced from Ha=50 to Ha=0. In the ref. [13], 68.1 % of average heat transfer enhancement is reported for the rigid wall and for natural convection configuration when the value of Hartamnn number is reduced from 50 to 0 ( RaE =106 , RaI =106 , φ = 0.06). Streamlines and isotherms for various inclination angles of the magnetic field are demonstrated in Fig. 9 for fixed values (Ri=1, RaI =105 , Ha=20, E=103 , φ = 0.02). The cavity is occupied with two recirculating zones which divide the cavity vertically in two halves for the inclination angles of γ = 0o , 45o and γ = 90o . The size and the extent of the secondary vortex near the right wall increases as the value of the magnetic inclination angle increases which is due to the Lorentz force acting in the momentum equations of flow dynamics which changes with inclination angle of the magnetic field. The isotherms are less clustered along the upper portion of the hot wall with increasing magnetic field angle and temperature gradients become slightly steep in the lower part of the hot wall as the value of γ enhances (Fig. 9 (d), (e), (f)). The local and averaged Nusselt number for various inclination angles of the magnetic field are shown in Fig.10. The inclination angle acts in a way to decrease the heat transfer in the upper part and to increase it in the lower part of the hot wall as the value of γ increases. The averaged heat
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transfer deteriorates by 11.53% and enhances by 5.75% at magnetic angles of γ = 30o and γ = 90o compared to inclination angle of γ = 0o . The effects of varying elastic modulus of the flexible wall on the streamlines and isotherms are demonstrated in Fig. 11 and Fig. 12 for three different Richardson numbers at fixed values of (RaI =105 , Ha=20, γ = 45o , φ = 0.02). As the value of the Young’s modulus decreases the deformation of the flexible wall enhances which provides more space for the hot rising fluid near the right wall. The deformation of the elastic wall is in +x direction and the cavity is occupied with a main recirculating zone whose center is near the left vertical wall. There is slightly change of the flow topology with elastic modulus of the flexible wall. At Richardson number of 1, the cavity is filled with two recirculating zones and the extent of the second vortex near the right wall decreases with increasing values of Young’s modulus due to the decreased space near the right wall. At Richardson number of 100, the deformation of the flexible wall is in -x direction the fluid motion increases near the left and right walls of the cavity. At low Richardson number, the effect of internal Rayleigh number become important since the value of external Rayleigh number decreases for fixed value of Reynolds number. The interior of the cavity is hotter than the hot wall of the cavity at Ri=0.01 (Fig. 12 (a)-(d)). The thermal patterns are similar within the cavity for different elastic modulus of the flexible wall, but the temperate gradients are different in the lower and upper portions of the right wall at Ri=0.01. As the value of the Richardson number increases, the external heating becomes more important and the isotherms become more clustered along the lower portion of the right wall at Ri=100. At this Richardson number, the isotherms are
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also parallel to the horizontal wall indicating the dominance of convection within the cavity. The temperate gradient increases in the lower portion of the hot wall with increasing value of the elastic modulus of the flexible wall at Ri=100 and thermal patterns remains similar for different elastic modulus in the interior of the cavity. The local and averaged heat transfer plots are demonstrated in Figs.13 and 14 for various values of Young’s modulus of the flexible wall and for different Richardson numbers. At Ri=0.01, the Nusselt numbers are negative due to the increasing effect of internal heating and increasing temperatures which are greater than that of the hot wall. At Ri=0.01 and Ri=1, the local Nusselt number achieves higher values in the upper and lower portion of the hot wall and at Ri=100, the local Nusselt number is higher in the interior parts of the hot wall as the value of the elastic modulus of the flexible wall increases. Absolute value of the Nusselt number decreases with Young’s modulus of the elastic wall at Ri=0.01 and enhances at Ri=1 and Ri=100. Absolute value of the averaged heat transfer enhances by 14.34% and 8.83% at Ri=1 and Ri=100 and deteriorates by 6.51% at Ri=0.01 for Young’s modulus of the elastic wall for E=500 when compared to case for E=106 . The local and averaged Nusselt numbers for various values of Richardson numbers are demonstrated in Fig. 15 and Fig. 16 for different nanoparticle volume fractions at fixed values of (RaI =2.5×104 , Ha=25, γ = 45o , E=5 × 104 ). As the solid volume fraction of the nanoparticles enhances, better thermal transport of the fluid within the enclosure is obtained due to the increment of effective thermal conductivity of the nanofluid. The increase in the local heat transfer with nanoparticle volume fraction is higher in the
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portions of the hot wall where heat transfer is locally high. The local and averaged heat transfer enhance as the value of φ increases and this is more pronounced for higher values of Richardson number. For higher values of Richardson number, anomalous heat transfer enhancement is seen which is similar in ref. [10] where the velocity slip significantly increases the heat transfer rate and it is effective when nanoparticles are added to the base fluid. Averaged heat transfer enhancements of 7.79%, 28.15% and 35.22% are achieved for solid volume fraction of φ = 0.05 compared to pure fluid (φ = 0). 5. Conclusions Numerical simulation of mixed convection under the influence of a magnetic effect in a nanofluid filled lid driven enclosure having a flexible side wall was performed. Following conclusions can be drawn from the numerical study as: • As the value of the Richardson number enhances, the local heat transfer increases in the lower part of the hot wall and deteriorates in the small portion of it. The averaged heat transfer enhances by 239.35% at Richardson number of 100 compared to case at Richardson number of 1. • The local heat transfer decreases as the value of the RaI and then negative values of the local heat transfer are obtained at higher values of internal Rayleigh numbers which is due to the fact that interior of the cavity is hotter than the hot wall. 18
• The local and averaged Nusselt number deteriorates as the strength of the magnetic field increases due to the dampening of the fluid motion and suppression of the convection with magnetic field. The averaged Nusselt number enhances by 146.28 % at Ha=50 compared to case at Ha=0. The inclination angle acts in a way to decrease the Nusselt number in the upper part and to increase it in the lower part of the hot wall as the value of the inclination angle increases. The averaged Nusselt number decreases by 11.53% and increases by 5.75% at inclination angles of 30o and 90o compared to inclination angle of 0o . • Absolute value of the Nusselt number decreases as the value of the elastic modulus of the flexible wall at Ri=0.01 and enhances at Ri=1 and Ri=100. Absolute value of the averaged Nusselt number increases by 14.34% and 8.83% at Ri=1 and Ri=100 and decreases by 6.51% at Ri=0.01 for Young’s modulus of the elastic wall for E=500 when compared to configuration for E=106 . • Better thermal transport of the fluid within the cavity is seen due to the increment of effective thermal conductivity of the nanofluid as the volume fraction of the solid nanoparticles increases. The local and averaged Nusselt number increase as the value of φ increases and this is more effective for higher values of Richardson number where heat transfer process is effective. Averaged Nusselt number enhancements of 7.79%, 28.15% and 35.22% are obtained for solid volume fraction of φ = 0.05 compared to base fluid (φ = 0). [1] Acharya, S., Goldstein, R.. Natural convection in an externally heated 19
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24
flow of a nanofluid past a vertical semi-infinite flat plate. Nonlinear Analysis: Real World Applications 2010;12:1338–1346. [34] Mahmoudi, A.H., Pop, I., Shahi, M.. Effect of magnetic field on natural convection in a triangular enclosure filled with nanofluid. International Journal of Thermal Sciences 2012;59:12–140. [35] Sarkar, S., Ganguly, S., Biswas, G.. Buoyancy driven convection of nanofluids in an ?nfinitely long channel under the effect of a magnetic field. International Journal of Heat and Mass Transfer 2014;71:328–340. [36] Mejri,
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filled open enclosure with a sinusoidal boundary condition. Chemical Engineering Research and Design 2015;98:1–16. [37] A. Malvandi, D.G.. Magnetic field and slip effects on free convection inside a vertical enclosure filled with alumina/water nanofluid. Chemical Engineering Research and Design 2015;94:355–364. [38] Khanafer, K.. Comparison of flow and heat transfer characteristics in a lid-driven cavity between flexible and modified geometry of a heated bottom wall. International Journal of Heat and Mass Transfer 2014;78:1032– 1041. [39] Al-Amiri, A., Khanafer, K.. Fluid-structure interaction analysis of mixed convection heat transfer in a lid-driven cavity with a flexible bottom wall. International Journal of Heat and Mass Transfer 54 (2011) 38263836 2011;54:3826–3836.
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[40] Hasan, M.N., Saha, S.C., Gu, Y.. Unsteady natural convection within a differentially heated enclosure of sinusoidal corrugated side walls. International Journal of Heat and Mass Transfer 2012;55:5696–5708. [41] Hussain, S.H., Hussein, A.K., Mohammed, R.N.. Studying the effects of a longitudinal magnetic field and discrete isoflux heat source size on natural convection inside a tilted sinusoidal corrugated enclosure. Computers and Mathematics with Applications 2012;64:476–488. [42] Selimefendigil,
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magnetic field interaction in a nanofluid filled lid-driven cavity with flexible side wall. European Journal of Mechanics - B/Fluids, In Press, Accepted Manuscript, 2016 (in press);. [43] Haddad, Z., Abid, C., Oztop, H.F., Mataoui, A.. A review on how the researchers prepare their nanofluids. International Journal of Thermal Sciences 2014;76:168–189. [44] Sridhara,
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[48] Sheikholeslami,
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Natural convection heat transfer in a cavity with sinusoidal wall filled with cuo-water nanofluid in presence of magnetic field. Journal of the Taiwan Institute of Chemical Engineers 2014;45:40–49. [49] Taylor, C., Hood, P.. A numerical solution of the navier-stokes equations using finite-element technique. Comput Fluids 1973;1:73–89. [50] Patnaik, B., Seetharamu, K., Narayana, P.. Simulation of laminar confined flow past a circular cylinder with integral wake splitter involving heat transfer. Int J Numer Meth Heat Fluid Flow 1996;6:65–81. [51] Pirmohammadi, M., Ghassemi, M.. Effect of magnetic field on convection heat transfer inside a tilted square enclosure. International Communications in Heat and Mass Transfer 2009;36:776–780. [52] Sarris, I., Zikos, G., Grecos, A., Vlachos, N.. On the limits of validity of the low magnetic reynolds number approximation in mhd naturalconvection heat transfer. Numer Heat Transfer Part B 2006;50:158–180. [53] Selimefendigil, F., Oztop, H.F., Abu-Hamdeh, N.. Mixed convection due to rotating cylinder in an internally heated and flexible walled cavity filled with sio2-water nanofluids: Effect of nanoparticle shape. International Communications in Heat and Mass Transfer 2016;71:9–19. [54] Manimaran, R., Palaniradja, K., Alagumurthi, N., Sendhilnathan, S., Hussain, J.. Preparation and characterization of copper oxide nanofluid for heat transfer applications. Appl Nanosci 2014;4:163–167.
27
(a)
(b)
Figure 1: Schematic decription of the physical model with boundary conditions (a) and grid distribution (b)
28
(a) results of Pirmohammadi and Ghassemi [51] Figure 2: Code verification with the results of Pirmohammadi and Ghassemi [51] and Sarris et al. [52]. Comparison of streamlines and isotherms
29
(a) Ri=0.01
(b) Ri=1
(c) Ri=100
Figure 3: Effects of varying Richardson number on the flow patterns and isotherms for fixed values of RaI =105 , Ha=20, γ = 45o , E=104, φ = 0.02
30
(d) Ri=0.01
35 15 30 25
Ri=0.01, 1, 10, 100 10
Num
Nuy
20 15 10
5 5 0 −5 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Y (a) local Nusselt number Figure 4: Local and averaged Nusselt number for various Richardson numbers (RaI =105 , Ha=20, γ = 45o , E=104, φ = 0.02)
31
0 −2 10
(a) RaI = 103
(b) RaI = 105
(c) RaI = 106
Figure 5: Streamlines and isotherms for various values of internal Rayleigh numbers for fixed values of Ri=1, Ha=25, γ = 45o , E=104 , φ = 0.02
32
(d) RaI = 103
2
3 2
1 1 0
0
Num
Nuy
−1 −2 −3 −4
−1 −2 −3
−5
−7 0
−4
RaI=104, 105, 106
−6 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Y (a) local Nusselt number Figure 6: Local and averaged Nusselt number for various internal Rayleigh numbers (Ri=1, Ha=25, γ = 45o , E=104, φ = 0.02)
33
−5 3 10
(a) Ha=0
(b) Ha=30
(c) Ha=50
Figure 7: Effects of varying Hartmann number on the flow patterns and isotherms for fixed values of Ri=2.5, RaI =105 , γ = 45o , E=104, φ = 0.02
34
(d) Ha=0
4
8 7
3.5 6 3
Nu
Nu
y
m
5 4
2.5
3 2 2
0 0
1.5
Ha=0, 20, 40, 50
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Y (a) local Nusselt number Figure 8: Local and averaged Nusselt number for various Hartmann number (Ri=2.5, RaI =105, γ = 45o , E=104, φ = 0.02)
35
1 0
5
(a) γ = 0o
(b) γ = 45o
(c) γ = 90o
Figure 9: Streamlines and isotherms for various inclination angles of the magnetic field for fixed values of Ri=1, RaI =105 , Ha=20, E=103, φ = 0.02
36
(d) γ = 0o
1.18
5 4.5
1.16
4
1.14
3.5
1.12 γ=0o, 30o, 45o, 60o, 90o
Num
Nuy
3 2.5
1.1 1.08
2
1.06
1.5
1.04
1
1.02
0.5
1
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Y (a) local Nusselt number Figure 10: Local and averaged Nusselt number for inclination angles of the magnetic field (Ri=1, RaI =105 , Ha=20, E=103 , φ = 0.02)
37
0.98 0
10
(a) E=500
(c) E=104
(b) E=5000
(d) E=106
Figure 11: Streamlines for various Young modulus of the flexible wall for three different Richardson number at fixed values of RaI =105 , Ha=20, γ = 45o , φ = 0.02
38
(e) E=500
(a) E=500
(c) E=104
(b) E=5000
(d) E=106
Figure 12: Isotherms for various Young modulus of the flexible wall for three different Richardson number at fixed values of RaI =105 , Ha=20, γ = 45o , φ = 0.02
39
(e) E=500
−30
4.5 E=5x 102, 5x103,104, 106
−40
4
−50
3.5 3
Nuy
Nu
y
−60 −70
2.5 2
2
3
4
6
E=5x 10 , 5x10 ,10 , 10
−80 1.5 −90
1
−100 −110 0
0.5 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0
0.1
0.2
Y
0.3
0.4
0.5
Y
(a) Ri=0.01
(b) Ri=1
Figure 13: Effects of varying Young modulus on the local Nusselt number distributions along the hot wall for three different Richardson number (RaI =105 , Ha=20, γ = 45o , φ = 0.02)
40
0.6
0.7
−38.5 1.14 −39
1.12 1.1
−39.5 −40
Num
Nu
m
1.08
−40.5
1.06 1.04 1.02
−41
1 −41.5 −42 2 10
0.98 3
10
4
5
10
10
6
10
0.96 2 10
3
10
4
10
E
E
(a) Ri=0.01
(b) Ri=1
Figure 14: Effects of varying Young modulus on the averaged Nusselt number distributions along the hot wall for three different Richardson number (RaI =105 , Ha=20, γ = 45o , φ = 0.02)
41
11
14
10 12
9 8
φ=0, 0.03, 0.05
10
y
6
Nu
Nu
y
7
5
8 6
4 3
4
φ=0, 0.03, 0.05
2
2
1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0
0.1
0.2
Y
0.3
0.4
0.5
0.6
0.7
Y
(a) Ri=0.01
(b) averaged Nusselt numb
Figure 15: Effect of varying nanoparticle voulme fractions on the local Nusselt number distributions along the hot wall for three different Richardson numbers (RaI =2.5×104, Ha=25, γ = 45o , E=5 × 104 )
42
25
Num
20
Ri=0.01 Ri=1 Ri=100
15
10
5
0 0
0.005
0.01
0.015
0.02
0.025
φ
0.03
0.035
0.04
0.045
0.05
Figure 16: Averaged Nusselt numbers for various nanoparticle volume fractions and Richardson numbers (RaI =2.5×104, Ha=25, γ = 45o , E=5 × 104 )
43
Table 1: Thermophysical properties of base fluid and nanoparticle [48]
Property
Water
CuO
ρ (kg/m3 )
997.1
6500
cp (J/kg K)
4179
540
k (W/mK)
0.61
18
Γ (S/m)
0.05
5 × 107
dp (mm)
–
29
44
Table 2: Grid independence test (Ri=100, RaI =105, Ha=20, γ = 45o, E=104 , φ = 0.05).
Grid name
Grid size
Averaged Nusselt number
G1
1865
9.900
G2
3842
9.855
G3
7460
9.838
G4
10916
9.777
G5
28417
9.774
45
PII: DOI: Reference:
S0020-7403(16)30217-X http://dx.doi.org/10.1016/j.ijmecsci.2016.09.011 MS3416
To appear in: International Journal of Mechanical Sciences Received date: 6 August 2015 Revised date: 5 September 2016 Accepted date: 6 September 2016 Cite this article as: Fatih Selimefendigil and Hakan F. Öztop, Analysis of MHD mixed convection in a flexible walled and nanofluids filled lid-driven cavity with volumetric heat generation, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2016.09.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analysis of MHD mixed convection in a flexible walled and nanofluids filled lid-driven cavity with volumetric heat generation b ¨ Fatih Selimefendigila,∗, Hakan F. Oztop a
Department of Mechanical Engineering, Celal Bayar University, 45140 Manisa, Turkey b Department of Mechanical Engineering, Technology Faculty, Fırat University, 23119 Elazı˘g, Turkey
Abstract In this study, MHD mixed convection in a CuO-nanofluid filled lid-driven cavity having an elastic side wall and volumetric heat generation is numerically investigated. The left vertical wall is moving with constant velocity in +y direction. The left vertical wall of the cavity is maintained at constant cold temperature while the right vertical wall is at hot temperature and the other walls of the cavity are insulated. The governing equations are solved with finite element formulation. The Arbitrary-Lagrangian-Eulerian method is used to describe the fluid motion with the elastic wall in the fluid-structure interaction model. The influence of Richardson number (between 0.01 and 100), internal Rayleigh number (between 103 and 106 ), Hartmann number (between 0 and 50), inclination angle of the magnetic field (between 0o and 90o ), Young’s modulus of flexible wall (between 5×102 N/m2 and 106 N/m2 ), and nanoparticle volume fraction (between 0 and 0.05) on the fluid flow and ∗
Corresponding author, Tel.: +90 236 241 21 44; Fax: +90 236 241 21 43. Email addresses: [email protected] (Fatih Selimefendigil), ¨ [email protected] (Hakan F. Oztop)
Preprint submitted to XXXXXXX
September 19, 2016
heat transfer were numerically investigated. The effect of Brownian motion on the effective thermal conductivity was taken into account. The averaged heat transfer decreases with increasing Richardson and Hartmann and internal Rayleigh numbers. Absolute value of the averaged heat transfer enhances by 14.34% and 8.83% at Richardon number of 1 and 100 and deteriorates by 6.51% at Richardson number of 0.01 for Young’s modulus of the elastic wall 500 when compared to configuration for Young’s modulus of 106 . The local and averaged heat transfer enhance as the value of solid volume fraction of the nanoparticle increases and this is more effective for higher values of Richardson number where heat transfer process is effective. Keywords: elastic wall, magnetic field, nanofluids, fluid-structure interaction, finite element method
2
Nomenclature a
local acceleration
B0
magnetic field strength Greek Characters
E Young’s modulus f
α thermal diffusivity
body force per unit volume gβf (Th −Tc )H 3 νf2
Gr Grashof number,
local heat transfer coefficient σ Ha Hartmann number, B0 H ρnfnfνf h
k
thermal conductivity
H
length of the enclosure
n
unit normal vector local Nusselt number
Num
averaged Nusselt number
Q
θ
non-dimensional temperature
ν
kinematic viscosity
Vw H νf Gr Re2
temperature
u, v
x-y velocity components
x, y
Cartesian coordinates
stress tensor
Subscripts
volumetric heat generation rate
Ri Richardson number, T
magnetic inclination angle
φ solid volume fraction
νf αf
Re Reynolds number,
γ
ρ density of the fluid
pressure
Pr Prandtl number,
expansion coefficient
Γ electrical conductivity
σ
Nuy
p
β
3
c
cold
h
hot
m
average
nf
nanofluid
p
solid particle
st
static
1. Introduction Mixed convection in lid-driven cavity is important in many engineering applications such as cooling of electronic devices, food processing, solidification, float glass production, nuclear reactors, microelectronic devices, coating and solar power. The complicated interaction between the shear driven flow and natural convection has important effects on the enhancement in the flow mixing and heat transfer process. Volumetric internal heat generation is an important issue in engineering such as nuclear power plants, concrete or inside of earth. Convection in enclosures with heat generating fluids was extensively studied [1–3]. Acharya and Goldstein [1] numerically studied the free convection inside an inclined cavity having uniformly distributed internal energy source. They observed distinct flow pattern systems depending on the ratio of the external to internal Rayleigh number. Oztop and Bilgen [4] numerically investigated the natural convection of differentially heated, partitioned square cavity containing heat generating fluid. They observed that the heat transfer was generally reduced when the ratio of internal and external Rayleigh numbers was from 10 to 100. In another study by Oztop et al. [5], numerical study of natural convection in a wavy-wall cavity with volumetric heat source was performed. They observed that both the wavy wall and the ratio of internal Rayleigh number to external Rayleigh number affect the heat transfer and fluid flow significantly. In order to control the fluid flow and heat transfer characteristics within a cavity different strategies can be utilized. In one of these methods, magnetic field can be used to control of heat transfer [6–21]. The effects of magnetic field on the fluid flow and heat transfer were studied by many researchers due 4
to its importance in many engineering applications such as coolers of nuclear reactors, purification of molten metals etc. Rahman et al. [22] 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 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. Al-Salem et al. [23] studied the effects of moving lid direction on MHD mixed convection in a cavity with linearly heated bottom wall using finite volume method. They observed that direction of lid is more effective on heat transfer and fluid flow in the cavity and heat transfer is decreased with increasing of magnetic field parameter. Oztop et al. [24] studied 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. Magnetic field with nanofluids have received some attention due to higher thermal conductivity of the nanoparticles added to the base fluid [25–37]. Ghasemi et al. [28] studied the MHD natural convection in an enclosure filled with water - Al2 O3 nanofluid. They observed that an enhancement or deterioration of the heat transfer may be obtained with an increase of the nanoparticle volume fraction depending on the value of Hartmann and Rayleigh numbers. Mahmoudi et al. [34] numerically studied the MHD natural convection in a triangular enclosure filled with nanofluid. The impact of the Rayleigh number, Hartmann number and nanoparticle volume fraction on the heat transfer and fluid flow were numerically investigated.
5
Deformable walls can be used to control the heat transfer and flow inside the cavity [38, 39] along with the corrugated walls [40–42]. Al-Amiri and Khanafer [39] numerically studied the laminar mixed convection in a lid driven cavity having a flexible bottom wall. They showed that elasticity of the bottom weall has significant effect on the heat transfer enhancement. In another study, Khanafer [38] numerically investigated the mixed convection in a lid driven cavity with a flexible bottom wall. He observed that compared to flat wall case significant heat transfer enhancement with flexible wall is achieved. Based on the above literature survey and to the best of authors’ knowledge, MHD mixed convection in a nanofluid filled lid- driven cavity under the influence of an inclined magnetic field having an elastic side wall has never been reported in the literature. Especially, the mixed convection configurations with deformable walls were rarely studied in the literature. In this study, MHD effect coupled with fluid-structure interaction model in a nanofluid filled lid driven cavity problem will be numerically investigated for the volumetric internal heat generation case for a range of flow parameters . The present numerical study aims at investigating the effects of Richardson number, Hartmann number, inclination angle of the magnetic field, Young’s modulus of the flexible wall and solid volume fraction of the nanoparticles on the fluid flow and heat transfer characteristics in a lid driven cavity having an elastic side wall.
6
2. Mathematical formulation A schematic description of the physical problem and boundary conditions are depicted in Fig. 1 (a). The left vertical wall of the cavity is moving with constant velocity of uw in +y direction. The left vertical wall of the cavity is maintained at constant cold temperature of Tc while the right vertical wall is at hot temperature of Th and the other walls of the cavity are insulated. The gravity acts in the negative y-direction. The cavity is filled with CuO-water nanofluid under the influence of a vertical magnetic field. The cavity is filled ˙ A uniform with a uniform heat generating fluid with volumetric rate of Q.
= Bx i + By j is applied and it makes angle of γ with the magnetic field of B horizontal axis. The cavity is filled with CuO-water nanofluid (different solid volume fraction of φ) under the influence of the applied uniform magnetic field. Thermal equilibrium between the fluid phase and nanoparticles and no slip between them are assumed. Homogeneous mixture of CuO and water is considered. A homogeneous mixture can be obtained and the nanoparticles can be de-agglomerated by using an ultrasonic homogenizer [43, 44]. Thermo-physical properties of water and CuO at the reference temperature are presented in Table 1 [34]. The thermo-physical properties of the fluid are assumed to be constant except for the density variation which is modeled by the Boussinesq approximation in the buoyancy term.The effects of joule heating, displacement currents and induced magnetic field are assumed to be negligible. The flow inside the cavity is assumed to be laminar, steady and two dimensional. The Arbitrary Lagrangian-Eulerian method is utilized to describe the fluid motion with the flexible wall of the cavity in the fluid-structure interaction model. Navier-Stokes and energy equations with 7
effective thermophysical properties are used to describe the fluid domain. The conservation equation of mass, momentum and energy in a two dimensional Cartesian coordinate system for the fluid domain can be written as follows [38]: .u = 0
(1)
ρnf (u − ug ). u = .σnf + ρnf ffb
(2)
u T = αnf 2 T +
Q˙ ρcp
(3)
where ffb , σnf , ug denote the body force per unit volume, stress tensor and velocity of moving coordinate. The body force contains the Lorentz force due to the magnetic field in the x momentum equation such as: Γnf B02 v sin(γ) cos(γ) − u sin2 (γ)
(4)
and also the body force due to the buoyancy and Lorentz force due to the magnetic field in the y-momentum equation: βnf g(T − Tc ) + Γnf B02 u sin(γ) cos(γ) − v cos2 (γ)
(5)
The equation for the solid domain of the fluid-structure interaction model is given by [38]: ρs as = .σs + fs b
(6)
where as , fs b , σs represent the local acceleration, externally applied body force and solid stress tensor, respectively. 8
Nanofluid Thermophyscial Properties The effective density, specific heat, thermal expansion coefficient and electrical conductivity of nanofluid are given by the following formulas: ρnf = (1 − φ)ρf + φρp
(7)
(ρcp )nf = (1 − φ)(ρcp )f + φ(ρcp )p
(8)
(ρβ)nf = (1 − φ)(ρβ)f + φ(ρβ)p
(9)
Γnf = (1 − φ)Γf + φΓp
(10)
where the subscripts f, nf and p denote the base fluid, nanofluid and solid particle, respectively. The effective thermal conductivity of the nanofluid includes the effect of Brownian motion. In this model, the effects of particle size, particle volume fraction and temperature dependence are taken into account and it is given by the following formula [45]:
knf = kst + kBrownian where kst is the static thermal conductivity as given by [46] (kp + 2kf ) − 2φ(kf − kp ) kst = kf (kp + 2kf ) + φ(kf − kp )
(11)
(12)
The interaction between the nanoparticles and the effect of temperature are included in the models as
kBrownian = 5 × 104 φρf cp,f
κb T f (T, φ, dp ) ρp dp
where the function f for CuO-water nanofluid is given in [45]. 9
(13)
The effective viscosity of the nanofluid due to micro mixing in the suspension was given in [45] μnf = μst + μBrownian = μf (1 − φ)−0.25 +
kBrownian μf × kf P rf
(14)
where the first terms on the right hand side of equation is the viscosity of the nanofluid given in ref. ([47]). Thermo-physical properties of water-nanofluid are given in Table 1 [48]. Boundary Conditions The boundary conditions in the dimensional form can be defined as: • For the horizontal top and bottom walls: u = v = 0,
∂T ∂y
=0
• For the right vertical wall: u = v = 0, T = Th • For the left vertical wall: u = 0, v = Vw , T = Tc At the fluid structure interface displacement compatibility (df = ds ) and traction equilibrium (σf = σs ) must be satisfied [38]. Eqs.(1) - (4) can be converted to the dimensionless form and the relevant dimensionless numbers are: Gr =
gβf (Th −Tc )H 3 , νf2
Ha = B0 H
σf μf
Pr =
, RaI = 10
νf αf
, Re =
˙ 5 gβf QH , αf νf kf
Vw H , ν
Ri =
Gr Re2
(15)
Local Nusselt number along the left vertical wall is calculated as: knf ∂θ Nuy = − kf ∂X X=0
(16)
where θ represents the non-dimensional temperature and X denotes the nondimensional x- coordinate. Averaged Nusselt number is obtained after integrating the local Nusselt number along the hot wall of the cavity as
1 Nuy dY. (17) Num = 0
3. Solution method Galerkin weighted residual finite element formulation is used to solve the governing equations in Eqs. (1-4). The application of the method and the involved procedural steps were given in [49, 50]. The computational domain is discretized into triangular elements and Lagrange finite elements of different orders are used for each of the flow variables within the computational domain. Residuals for each of the conservation equation is obtained by substituting the approximations into the governing equations and boundary conditions. The resulting nonlinear and coupled algebraic equations from the discretization were solved by using segregated-solution method. The convergence of the solution is assumed when the relative error for each of the variables (Φ) satisfy the following convergence criteria: |
Φn+1 − Φn | ≤ 10−6 Φn+1
(18)
Numerical experiments with various grid sizes are performed in order to obtain an optimal grid distribution with accurate results and minimal 11
computational time. The obtained velocity field is divergence-free. The results of averaged Nusselt number for various grid sizes are shown in Table 2 (Ri=100, RaI =105 , Ha=20, γ = 45o , E=104 , φ = 0.05). G4 with 10916 nodes ensures grid independent solution and hence was used in the subsequent computations.The present code is validated against the existing results of Pirmohammadi and Ghassemi [51] and Sarris et al. [52]. Fig. 2 demonstrates the comparison results of streamlines and isotherms for (Ra = 7 × 103, Ha = 25) and (Ra = 7×105 , Ha = 100). The comparison results show good overall agreement. 4. Results and Discussion In this study, the effects of Richardson number (Ri =
Gr , Re2
0.01 ≤ Ri ≤
100), internal Rayleigh number (103 ≤ RaI ≤ 106 ), Hartmann number (0 ≤ Ha ≤ 50), inclination angle of the magnetic field (0o ≤ γ ≤ 90o ), Young’s modulus of flexible wall (5 × 102 N/m2 ≤ E ≤ 106 N/m2 ), and nanoparticle volume fraction (0 ≤ φ ≤ 0.05) on the flow and heat transfer were numerically investigated. The effects of different nanoparticle types and various shapes can also be investigated [53]. In this study, CuO nanoparticles with spherical shape were used. Higher conductivity of CuO-water nanofluids makes them attractive for various thermal engineering applications [54]. The cavity is filled with CuO-water nanofluid under the influence of an inclined magnetic field and volumetric heat generation. The effects of varying the Richardson number on the streamlines and isotherms are demonstrated in Fig. 3 for fixed values of (RaI =105 , Ha=20, γ = 45o , E=104 , φ = 0.02). At low Richardson number, the effect of forced convection due 12
to the mechanically-driven left wall is more pronounced compared to natural convection (Ri =
Gr ). Re2
The cavity is filled with a single main recirculation
zone and a small vortex is seen in the right bottom corner of the cavity. The flow due to the moving lid penetrates more into the cavity and the flexible wall has a concave shape at Ri=0.01. As the value of the Ri is increased to Ri=1, a secondary recirculation zone near the hot wall and a main vortex in the cavity are seen. At Ri=100, the effect of natural convection becomes important and the convection of the strength enhances. The shape of the flexible wall is convex due to the increasing effect of buoyancy. The effect of internal heating becomes more important when the Richardson number is low for fixed values of Reynolds number. The inner of the cavity is hotter than the right wall for this case especially in the lower part of the hot wall as shown in Fig 3 (d). As the Richardson number increases, the effect of external heating becomes important and steep temperature gradients are seen along the hot wall in the lower part of it. At Richardson number of 100, the isotherms become parallel to the horizontal wall indicates the dominance of convection inside the cavity. The local and averaged heat transfer plots are demonstrated in Fig. 4. At Ri=0.01, the local heat transfer is negative for the lower portion of the hot wall and increases toward the upper part of it. As the value of the Richardson number enhances, the local heat transfer increases in the lower part of the hot wall and deteriorates in the small portion of it. The averaged heat transfer enhances by 239.35% at Richardson number of 100 compared to case at Richardson number of 1. The effects of varying internal Rayleigh numbers on the flow and thermal patterns are demonstrated in Fig. 5 for fixed values of (Ri=1, Ha=25, γ =
13
45o , E=104 , φ = 0.02). The cavity is filled with two recirculating zones for internal Rayleigh numbers of RaI = 103 and RaI = 105 . The ratio of the internal heating compared to external heating is important for RaI = 106 and the number of vortices is three. The temperature gradients along the hot wall and the strength of the convection inside the cavity decreases as the value of the external Rayleigh number increases from RaI = 104 to RaI = 105 . At RaI = 106 , the effect of internal heating becomes important and interior of the cavity near the right wall is hotter than the hot wall. The local and the averaged Nusselt number plots are shown in Fig. 6 (a), (b) for various internal Rayleigh numbers. The local heat transfer decreases as the value of the RaI enhances and then negative values of the local heat transfer are obtained at higher values of internal Rayleigh numbers (RaI = 5 × 105 , RaI = 106 , in Fig. 6 (b)). This trend is also seen in the natural convection study in ref. [13] where different shaped obstacles are placed in the cavity to affect the heat transfer rate. The influence of varying Hartmann number on the flow patterns and isotherms is shown in Fig. 7 (a)-(f) for fixed values of (Ri=2.5, RaI =105 , γ = 45o , E=104 , φ = 0.02 ). In the absence of the magnetic field, the cavity is occupied with a main recirculation zone and another vortex adjacent to the left vertical wall is seen. The increment in the Hartmann number enhances the sterngth of the magnetic field which results in dampening of the flow motion. The recirculating zone adjacent to the left vertical wall increases in size whereas the other vortex diminishes in size and strength with increasing effect of Hartmann number. The tempretaure gradients are steep along the hot wall in the lower portions of it and isotherms become less clustered along
14
the hot wall as the value of Hartmann number enhances. Isotherms become parallel to the vertical walls in the interior of the cavity with increasing strength of magnetic field which indicates dominance of conduction heat transfer. The local and averaged heat transfer deteriorates as the value of Ha increases due to the suppression of the convection with magnetic field. 146.28 % of averaged heat transfer enhancement is obtained when Hartmann number is reduced from Ha=50 to Ha=0. In the ref. [13], 68.1 % of average heat transfer enhancement is reported for the rigid wall and for natural convection configuration when the value of Hartamnn number is reduced from 50 to 0 ( RaE =106 , RaI =106 , φ = 0.06). Streamlines and isotherms for various inclination angles of the magnetic field are demonstrated in Fig. 9 for fixed values (Ri=1, RaI =105 , Ha=20, E=103 , φ = 0.02). The cavity is occupied with two recirculating zones which divide the cavity vertically in two halves for the inclination angles of γ = 0o , 45o and γ = 90o . The size and the extent of the secondary vortex near the right wall increases as the value of the magnetic inclination angle increases which is due to the Lorentz force acting in the momentum equations of flow dynamics which changes with inclination angle of the magnetic field. The isotherms are less clustered along the upper portion of the hot wall with increasing magnetic field angle and temperature gradients become slightly steep in the lower part of the hot wall as the value of γ enhances (Fig. 9 (d), (e), (f)). The local and averaged Nusselt number for various inclination angles of the magnetic field are shown in Fig.10. The inclination angle acts in a way to decrease the heat transfer in the upper part and to increase it in the lower part of the hot wall as the value of γ increases. The averaged heat
15
transfer deteriorates by 11.53% and enhances by 5.75% at magnetic angles of γ = 30o and γ = 90o compared to inclination angle of γ = 0o . The effects of varying elastic modulus of the flexible wall on the streamlines and isotherms are demonstrated in Fig. 11 and Fig. 12 for three different Richardson numbers at fixed values of (RaI =105 , Ha=20, γ = 45o , φ = 0.02). As the value of the Young’s modulus decreases the deformation of the flexible wall enhances which provides more space for the hot rising fluid near the right wall. The deformation of the elastic wall is in +x direction and the cavity is occupied with a main recirculating zone whose center is near the left vertical wall. There is slightly change of the flow topology with elastic modulus of the flexible wall. At Richardson number of 1, the cavity is filled with two recirculating zones and the extent of the second vortex near the right wall decreases with increasing values of Young’s modulus due to the decreased space near the right wall. At Richardson number of 100, the deformation of the flexible wall is in -x direction the fluid motion increases near the left and right walls of the cavity. At low Richardson number, the effect of internal Rayleigh number become important since the value of external Rayleigh number decreases for fixed value of Reynolds number. The interior of the cavity is hotter than the hot wall of the cavity at Ri=0.01 (Fig. 12 (a)-(d)). The thermal patterns are similar within the cavity for different elastic modulus of the flexible wall, but the temperate gradients are different in the lower and upper portions of the right wall at Ri=0.01. As the value of the Richardson number increases, the external heating becomes more important and the isotherms become more clustered along the lower portion of the right wall at Ri=100. At this Richardson number, the isotherms are
16
also parallel to the horizontal wall indicating the dominance of convection within the cavity. The temperate gradient increases in the lower portion of the hot wall with increasing value of the elastic modulus of the flexible wall at Ri=100 and thermal patterns remains similar for different elastic modulus in the interior of the cavity. The local and averaged heat transfer plots are demonstrated in Figs.13 and 14 for various values of Young’s modulus of the flexible wall and for different Richardson numbers. At Ri=0.01, the Nusselt numbers are negative due to the increasing effect of internal heating and increasing temperatures which are greater than that of the hot wall. At Ri=0.01 and Ri=1, the local Nusselt number achieves higher values in the upper and lower portion of the hot wall and at Ri=100, the local Nusselt number is higher in the interior parts of the hot wall as the value of the elastic modulus of the flexible wall increases. Absolute value of the Nusselt number decreases with Young’s modulus of the elastic wall at Ri=0.01 and enhances at Ri=1 and Ri=100. Absolute value of the averaged heat transfer enhances by 14.34% and 8.83% at Ri=1 and Ri=100 and deteriorates by 6.51% at Ri=0.01 for Young’s modulus of the elastic wall for E=500 when compared to case for E=106 . The local and averaged Nusselt numbers for various values of Richardson numbers are demonstrated in Fig. 15 and Fig. 16 for different nanoparticle volume fractions at fixed values of (RaI =2.5×104 , Ha=25, γ = 45o , E=5 × 104 ). As the solid volume fraction of the nanoparticles enhances, better thermal transport of the fluid within the enclosure is obtained due to the increment of effective thermal conductivity of the nanofluid. The increase in the local heat transfer with nanoparticle volume fraction is higher in the
17
portions of the hot wall where heat transfer is locally high. The local and averaged heat transfer enhance as the value of φ increases and this is more pronounced for higher values of Richardson number. For higher values of Richardson number, anomalous heat transfer enhancement is seen which is similar in ref. [10] where the velocity slip significantly increases the heat transfer rate and it is effective when nanoparticles are added to the base fluid. Averaged heat transfer enhancements of 7.79%, 28.15% and 35.22% are achieved for solid volume fraction of φ = 0.05 compared to pure fluid (φ = 0). 5. Conclusions Numerical simulation of mixed convection under the influence of a magnetic effect in a nanofluid filled lid driven enclosure having a flexible side wall was performed. Following conclusions can be drawn from the numerical study as: • As the value of the Richardson number enhances, the local heat transfer increases in the lower part of the hot wall and deteriorates in the small portion of it. The averaged heat transfer enhances by 239.35% at Richardson number of 100 compared to case at Richardson number of 1. • The local heat transfer decreases as the value of the RaI and then negative values of the local heat transfer are obtained at higher values of internal Rayleigh numbers which is due to the fact that interior of the cavity is hotter than the hot wall. 18
• The local and averaged Nusselt number deteriorates as the strength of the magnetic field increases due to the dampening of the fluid motion and suppression of the convection with magnetic field. The averaged Nusselt number enhances by 146.28 % at Ha=50 compared to case at Ha=0. The inclination angle acts in a way to decrease the Nusselt number in the upper part and to increase it in the lower part of the hot wall as the value of the inclination angle increases. The averaged Nusselt number decreases by 11.53% and increases by 5.75% at inclination angles of 30o and 90o compared to inclination angle of 0o . • Absolute value of the Nusselt number decreases as the value of the elastic modulus of the flexible wall at Ri=0.01 and enhances at Ri=1 and Ri=100. Absolute value of the averaged Nusselt number increases by 14.34% and 8.83% at Ri=1 and Ri=100 and decreases by 6.51% at Ri=0.01 for Young’s modulus of the elastic wall for E=500 when compared to configuration for E=106 . • Better thermal transport of the fluid within the cavity is seen due to the increment of effective thermal conductivity of the nanofluid as the volume fraction of the solid nanoparticles increases. The local and averaged Nusselt number increase as the value of φ increases and this is more effective for higher values of Richardson number where heat transfer process is effective. Averaged Nusselt number enhancements of 7.79%, 28.15% and 35.22% are obtained for solid volume fraction of φ = 0.05 compared to base fluid (φ = 0). [1] Acharya, S., Goldstein, R.. Natural convection in an externally heated 19
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27
(a)
(b)
Figure 1: Schematic decription of the physical model with boundary conditions (a) and grid distribution (b)
28
(a) results of Pirmohammadi and Ghassemi [51] Figure 2: Code verification with the results of Pirmohammadi and Ghassemi [51] and Sarris et al. [52]. Comparison of streamlines and isotherms
29
(a) Ri=0.01
(b) Ri=1
(c) Ri=100
Figure 3: Effects of varying Richardson number on the flow patterns and isotherms for fixed values of RaI =105 , Ha=20, γ = 45o , E=104, φ = 0.02
30
(d) Ri=0.01
35 15 30 25
Ri=0.01, 1, 10, 100 10
Num
Nuy
20 15 10
5 5 0 −5 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Y (a) local Nusselt number Figure 4: Local and averaged Nusselt number for various Richardson numbers (RaI =105 , Ha=20, γ = 45o , E=104, φ = 0.02)
31
0 −2 10
(a) RaI = 103
(b) RaI = 105
(c) RaI = 106
Figure 5: Streamlines and isotherms for various values of internal Rayleigh numbers for fixed values of Ri=1, Ha=25, γ = 45o , E=104 , φ = 0.02
32
(d) RaI = 103
2
3 2
1 1 0
0
Num
Nuy
−1 −2 −3 −4
−1 −2 −3
−5
−7 0
−4
RaI=104, 105, 106
−6 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Y (a) local Nusselt number Figure 6: Local and averaged Nusselt number for various internal Rayleigh numbers (Ri=1, Ha=25, γ = 45o , E=104, φ = 0.02)
33
−5 3 10
(a) Ha=0
(b) Ha=30
(c) Ha=50
Figure 7: Effects of varying Hartmann number on the flow patterns and isotherms for fixed values of Ri=2.5, RaI =105 , γ = 45o , E=104, φ = 0.02
34
(d) Ha=0
4
8 7
3.5 6 3
Nu
Nu
y
m
5 4
2.5
3 2 2
0 0
1.5
Ha=0, 20, 40, 50
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Y (a) local Nusselt number Figure 8: Local and averaged Nusselt number for various Hartmann number (Ri=2.5, RaI =105, γ = 45o , E=104, φ = 0.02)
35
1 0
5
(a) γ = 0o
(b) γ = 45o
(c) γ = 90o
Figure 9: Streamlines and isotherms for various inclination angles of the magnetic field for fixed values of Ri=1, RaI =105 , Ha=20, E=103, φ = 0.02
36
(d) γ = 0o
1.18
5 4.5
1.16
4
1.14
3.5
1.12 γ=0o, 30o, 45o, 60o, 90o
Num
Nuy
3 2.5
1.1 1.08
2
1.06
1.5
1.04
1
1.02
0.5
1
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Y (a) local Nusselt number Figure 10: Local and averaged Nusselt number for inclination angles of the magnetic field (Ri=1, RaI =105 , Ha=20, E=103 , φ = 0.02)
37
0.98 0
10
(a) E=500
(c) E=104
(b) E=5000
(d) E=106
Figure 11: Streamlines for various Young modulus of the flexible wall for three different Richardson number at fixed values of RaI =105 , Ha=20, γ = 45o , φ = 0.02
38
(e) E=500
(a) E=500
(c) E=104
(b) E=5000
(d) E=106
Figure 12: Isotherms for various Young modulus of the flexible wall for three different Richardson number at fixed values of RaI =105 , Ha=20, γ = 45o , φ = 0.02
39
(e) E=500
−30
4.5 E=5x 102, 5x103,104, 106
−40
4
−50
3.5 3
Nuy
Nu
y
−60 −70
2.5 2
2
3
4
6
E=5x 10 , 5x10 ,10 , 10
−80 1.5 −90
1
−100 −110 0
0.5 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0
0.1
0.2
Y
0.3
0.4
0.5
Y
(a) Ri=0.01
(b) Ri=1
Figure 13: Effects of varying Young modulus on the local Nusselt number distributions along the hot wall for three different Richardson number (RaI =105 , Ha=20, γ = 45o , φ = 0.02)
40
0.6
0.7
−38.5 1.14 −39
1.12 1.1
−39.5 −40
Num
Nu
m
1.08
−40.5
1.06 1.04 1.02
−41
1 −41.5 −42 2 10
0.98 3
10
4
5
10
10
6
10
0.96 2 10
3
10
4
10
E
E
(a) Ri=0.01
(b) Ri=1
Figure 14: Effects of varying Young modulus on the averaged Nusselt number distributions along the hot wall for three different Richardson number (RaI =105 , Ha=20, γ = 45o , φ = 0.02)
41
11
14
10 12
9 8
φ=0, 0.03, 0.05
10
y
6
Nu
Nu
y
7
5
8 6
4 3
4
φ=0, 0.03, 0.05
2
2
1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0
0.1
0.2
Y
0.3
0.4
0.5
0.6
0.7
Y
(a) Ri=0.01
(b) averaged Nusselt numb
Figure 15: Effect of varying nanoparticle voulme fractions on the local Nusselt number distributions along the hot wall for three different Richardson numbers (RaI =2.5×104, Ha=25, γ = 45o , E=5 × 104 )
42
25
Num
20
Ri=0.01 Ri=1 Ri=100
15
10
5
0 0
0.005
0.01
0.015
0.02
0.025
φ
0.03
0.035
0.04
0.045
0.05
Figure 16: Averaged Nusselt numbers for various nanoparticle volume fractions and Richardson numbers (RaI =2.5×104, Ha=25, γ = 45o , E=5 × 104 )
43
Table 1: Thermophysical properties of base fluid and nanoparticle [48]
Property
Water
CuO
ρ (kg/m3 )
997.1
6500
cp (J/kg K)
4179
540
k (W/mK)
0.61
18
Γ (S/m)
0.05
5 × 107
dp (mm)
–
29
44
Table 2: Grid independence test (Ri=100, RaI =105, Ha=20, γ = 45o, E=104 , φ = 0.05).
Grid name
Grid size
Averaged Nusselt number
G1
1865
9.900
G2
3842
9.855
G3
7460
9.838
G4
10916
9.777
G5
28417
9.774
45