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Numerical Simulation on Buoyancy-driven Heat Transfer Enhancement of Nanofluids in an Inclined Triangular Enclosure
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 90 (2014) 517 – 523
10th International Conference on Mechanical Engineering, ICME 2013
Numerical simulation on buoyancy-driven heat transfer enhancement of nanofluids in an inclined triangular enclosure M. M. Billaha,b,* M. M. Rahmanc, Uddin M. Sharif b, M.N. Islamb a
Department of Arts and Sciences, Ahsanullah University of Science and Technology (AUST), Dhaka-1208, Bangladeh b Department of Mathematics, Jahangirnagar University (JU), Savar, Dhaka-1342, Bangladeh c Department of Mathematics, Bangladesh University of Engineering and Technology (BUET), Dhaka-1000, Bangladeh
Abstract Heat transfer enhancement in a two-dimensional inclined lid-driven triangular enclosure utilizing nanofluids is investigated numerically for various pertinent parameters. The present model is developed to analyze the heat transfer performance of nanofluids inside an enclosure taking into account the solid volume fraction δ, variable velocity and thermal boundary conditions. Fluid mechanics and conjugate heat transfer, described in terms of continuity, linear momentum and energy equations. The transport equations are solved numerically using the Galerkin finite element method. Results are obtained for a wide range of parameters such as the Grashof numbers. Copper-water nanofluids is used with Prandtl number, Pr = 6.2 and solid volume fraction δ is varied from 0% - 20%. The streamlines, isotherm plots and heat transfer correlation of the average Nusselt number at the hot surface as well as average fluid temperature in the enclosure is presented and discussed in detailed. It is found that heat transfer increased by 28% as volume fraction δ increases from 0% to 20% at Gr =105. Moreover, the variation of the average Nusselt number and average fluid temperature in the cavity is linear with the solid volume fraction. © 2014 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2014The TheAuthors. Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Engineering Engineering and Technology (BUET). and Technology (BUET)
Keywords: Finite element method; nanofluids; lid-driven triangular enclosure.
* Corresponding author. Tel.: +88-01715 400 423; fax: +88-2-9860564. E-mail address: [email protected]; [email protected] (Md. Masum Billah)
1877-7058 © 2014 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/3.0/). Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Engineering and Technology (BUET) doi:10.1016/j.proeng.2014.11.766
518
M.M. Billah et al. / Procedia Engineering 90 (2014) 517 – 523
1. Introduction Nanotechnology is measured to be one of the most important forces that drive the next major industrial revolution of this latest century. That represents the key relevant technological cutting edge currently being explored. It aims at manipulating the structure of the matter at the molecular level with the goal for innovation in almost every industry and public endeavor including biological sciences, physical sciences, electronics cooling, transportation, the environment and also national security. Khanafer et al. [1] made a numerical model to find out natural convection heat transfer in nanofluids. Jou and Tzeng [2] examined numerically the heat transfer enhancement utilizing nanofluids in a two-dimensional rectangular enclosure for different pertinent parameters. Tiwari and Das [3] presented a numerical model of heat transfer augmentation in a lid-driven square cavity filled with nanofluids. A computational model is performed to analyze the transport mechanism of mixed convection in a lid-driven enclosure packed with nanofluids by Muthtamilselvan et al. [4]. Ghasemi and Aminossadati [5] considered mixed convection heat transfer in a lid-driven triangular enclosure filled with a water–Al2O3 nanofluid. Mixed convection flow in a lid-driven inclined square enclosure filled with water-Al2O3 nanofluid was presented by Nada and Chamkha [6]. Mansour et al. [7] performed a numerical simulation on mixed convection flow in a square lid-driven cavity partially heated from below using nanofluid. Talebi et al. [8] performed mixed convection flows in a square lid-driven cavity utilizing nanofluid. Rahman et al. [9] performed a numerical investigation on heat transfer enhancement of nanofluids in a lid-driven square enclosure. Billah et al. [10] presented a model to investigate the heat transfer enhancement of copper-water nanofluids in an inclined lid-driven triangular enclosure. Rahman et al. [11] have done a numerical investigation of heat transfer enhancement of nanofluids in an inclined liddriven triangular enclosure. Nomenclature cp g Gr H k L Nu p P Pr Re Ri T u U v V
Specific heat at constant pressure Y dimensionless vertical coordinate gravitational acceleration (ms-2) Grashof number Greek symbols enclosure height (m) α thermal diffusivity (m2s-1) -1 -1 thermal conductivity (Wm K ) β thermal expansion coefficient (K-1) length of the cavity (m) G solid volume fraction Nusselt number μ dynamic viscosity (kg m-1s-1) dimensional pressure (Nm-2) ν kinematic viscosity (m2s-1) dimensionless pressure θ non-dimensional temperature Prandtl number U density (kg m-3) Reynolds number ψ Streamfunction Richardson number J penalty parameter temperature (K) * general dependent variable horizontal velocity component (ms-1) θav Average temperature dimensionless horizontal velocity component Subscripts vertical velocity component (ms-1) av Average dimensionless vertical velocity component h Hot cavity volume (m3) c Cold V horizontal coordinate (m) f Fluid x X dimensionless horizontal coordinate nf Nanofluid y vertical coordínate (m) s Solid nanoparticle Owing to the rapid development of computer and computational techniques, finite element formulations have provided an alternative in dealing with the mixed convection heat transfer in enclosures using nanofluids. To the best knowledge of the authors, a little attention has been paid to investigate buoyancy-driven heat transfer characteristics of the nanofluids contained in an inclined lid-driven triangular enclosure with different solid volume fractions. This problem may be encountered in a number of electronic cooling and MEMS applications.
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M.M. Billah et al. / Procedia Engineering 90 (2014) 517 – 523
2. Mathematical Analysis The physical model under current study with the system of coordinates is shown in Fig. 1. The length of the base wall and height of the sliding left wall of the enclosure are denoted by L and H, respectively. In addition, the sliding
§ 5S · y ¸V0 . Moreover, © L ¹
wall of the cavity is kept adiabatic and allowed to move from bottom to top at a speed aˆ sin ¨ it is assumed that the temperature, T x,0
Th Th Tc x L 1 x L 2
2
of the bottom wall is higher than
the temperature (Tc) of the right inclined wall. Y § 5S · aˆ sin ¨ y ¸V0 © L ¹ adiabatic
g H
u = v = 0, T = Tc
I
L X
Fig. 1. Schematic of the problem with the domain and boundary conditions.
The free space in the enclosure is filled with copper water nanofluids. The nanofluid in the enclosure is Newtonian, incompressible and laminar. The nanoparticles are assumed to have uniform shape and size. It is assumed that thermal equilibrium exists between the base fluid and nanoparticles, and no slip occurs between the two media. The physical properties of the nanofluid are considered to be constant except the density variation in the body force term of the momentum equation which is satisfied by the Boussinesq’s approximation. Under the above assumptions, the system of equations governing the two-dimensional motion of a nanofluid is as follows: wU wV wX wY
w UV
w U2 wX
w UV wX
wY
w UT wX
(1)
0
w V2
wY w V T wY
§ Gr · UE nf 1 Uf 1 wP 2U ¨ T sin I ¸ wX Re Unf 1 G 2.5 © Re2 ¹ Unf E f
(2)
§ Gr · UE nf 1 wP 1 U f 2V ¨ T cosI ¸ 2.5 wY Re Unf 1 G © Re2 ¹ Unf E f
(3)
1 § D nf ¨ Re Pr ¨© D f
(4)
· 2 ¸ T ¸ ¹
where Reynolds number Re V0 L X f , Prandtl number Pr Q f D f and Grashof number Gr
E f g Th Tc L3 Xf2
The above equations (1)-(4) are non-dimensionalized by introducing the following dimensionless variables p U gy L2 , T T Tc x y u v ,Y ,U ,V ,P X L L V0 V0 Th Tc Unf V02
.
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M.M. Billah et al. / Procedia Engineering 90 (2014) 517 – 523
The effective density Unf of the nanofluid is defined by
1 G U f
Unf
GUs
(5)
where δ is the solid volume fraction of nanoparticles. In addition, the thermal diffusivity D nf of the nanofluid can be expressed as: knf
D nf
(6)
Uc p nf
The heat capacitance of nanofluids can be defined as: Uc p 1 G Uc p G Uc p
nf
f
s
(7)
Additionally, UE nf is the thermal expansion coefficient of the nanofluid and it can be determined by
UE nf 1 G UE f
G UE s
(8)
Furthermore, Pnf is the dynamic viscosity of the nanofluid introduced by Brikman [12] as Pf
Pnf
(9)
1 G 2.5
The effective thermal conductivity of nanofluid was intoduced by Kanafer et al.[1] as:
k s 2k f G k f k s
ks 2k f 2G k f ks
knf kf
(10)
where, ks is the thermal conductivity of the nanoparticles and kf is the thermal conductivity of base fluid. The appropriate non dimensional initial and boundary conditions for the governing equations are on the bottom wall: U V 0, T 1 X 2 1 X 2 on the left wall: U
0,V
wT 0 wN 0, T 0
aˆ sin 5S Y ,
on the right inclined wall: U V where N is the non-dimensional distances either X or Y direction acting normal to the surface. The average Nusselt number at the heated surface of the cavity may be expressed as Nuav
knf kf
L
wT
³ wY dX , and average fluid temperature in the enclosure may be defined as Tav ³T d V /V 0
The fluid motion is displayed using the stream function (\ ) obtained from velocity components U and V. The relationships between stream function and velocity components for two dimensional flows can be expressed as: U
w\ ,V wY
w\ wX
(11)
3. Solution Procedure The Galerkin weighted residual method of finite element formulation is employed as a computational scheme. The finite element method begins by the partition of the continuum area of interest into a number of simply shaped regions known as elements. These elements may be different shapes and sizes. Within each element, the dependent variables are approximated using interpolation functions. In the present study erratic grid size system is considered especially near the walls to capture the rapid changes in the dependent variables. The coupled governing equations (1)-(4) are transformed into sets of algebraic equations using finite element method to reduce the continuum domain into discrete triangular domains. The system of algebraic equations is solved by iteration technique. The solution
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M.M. Billah et al. / Procedia Engineering 90 (2014) 517 – 523
Gr = 104
Gr = 105
Gr = 106
process is iterated until the subsequent convergence condition is satisfied: *m1 *m d 106 where m is number of iteration and * is the general dependent variable.
δ=0
δ = 0.04
δ = 0.08
δ = 0.20
Fig. 2. Streamlines for different values of solid volume fraction δ and Grashof number Gr. 4. Results and Discussion The current study is carried out for copper-water nanofluids as working fluid with Prandtl number of 6.2. In this paper, our attention is taken into account to investigate the effects of controlling parameters namely the solid volume fraction (δ), and Grashof number (Gr). Here, the effect of the solid volume fractions is investigated over the range of 0% – 20% while the Re and tilt angle I are kept fixed at 100 and 60, respectively. It is worth to note that the value of Gr is varied from 104 to 106. In addition, the results of this study are presented in terms of streamlines and isotherms. Furthermore, the heat transfer effectiveness of the enclosure is displayed in terms of average Nusselt number (Nuav) and the average fluid temperature (θav) in the cavity. Fig. 2 shows the streamlines in an inclined lid-driven triangular enclosure for various values of the solid volume fractions δ and Gr. Visualization are given from δ = 0% to 20%. Based on literature on nanofluid which are cited in reference list, the maximum values of nanoparticle fraction was taken as 0.1. It can easily be seen from the bottom row of Fig. 2, the circulating cells are generated by the lid dragging the neighboring fluid. A counter clockwise rotating cell is observed near the lower part of the enclosure for all values of δ. The size and shape of the cell change with the increased of δ. It is also found from the streamlines that the flow strength of the anticlockwise rotating cell is decreasing very slowly when δ is increased. This is due to an increase as
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M.M. Billah et al. / Procedia Engineering 90 (2014) 517 – 523
Gr = 104
Gr = 105
Gr = 106
the volume fraction as a result of high-energy transport through the flow associated with the irregular motion of the ultrafine particles. The size of the anticlockwise rotating major cell increases while Gr increases from 104 to 105 and 106. The isotherm plots indicate the lines with equal intervals between unity (hot wall) and zero (cold wall). It is noticed from Fig. 3 that isotherm lines become very denser towards the hot surface for each value of δ. The isotherms distributions are nearly parallel to the horizontal wall of the cavity, indicating most of the heat transfer is carried out by conduction, which agrees with the result of Muthtamilselvan et al. [4]. On the other hand, for a fixed solid volume fraction δ, the isotherms become nonlinear with the rising Gr. For higher values of Gr, wavy variation is observed from isotherms and changing of nanoparticle volume fraction becomes effective.
δ=0
δ = 0.04
δ = 0.08
δ = 0.20
Fig. 3. Isotherms for different values of solid volume fraction δ and Grashof number Gr. The average Nusselt number (Nuav) is calculated along the hot surface, which is a measure of the overall heat transfer rate, is plotted in the left column of the Fig. 4 for the abovementioned values of the solid volume fractions (δ). These figures show a linear variation of the average Nusselt number with the solid volume fraction. As clearly be seen that the heat transfer increases with increasing Gr. Besides, the values of Nu are always maximum for the highest value of δ (= 20%). The effects of the solid fractions (δ) on average fluid temperature (θav) in the enclosure are depicted in the right column of the Fig. 4 for Gr = 104, 105, and 106. It is observed that θav decreases very swiftly with the increasing of Gr for all values of δ. It is clearly being seen that θav is maximum for the highest value of solid volume fraction (δ = 20%). 5. Conclusion Mixed convective flow and heat transfer in a lid-driven inclined triangular enclosure filled with Cu-water nanofluids is formulated and solved using Galarkin finite element method. Variable velocity and variable thermal boundary conditions are applied here. In this paper, the flow and heat transport structures are described by
M.M. Billah et al. / Procedia Engineering 90 (2014) 517 – 523
streamlines and isotherms respectively. Computed results for various parametric situation namely solid volume fraction (δ), and the Grashof numbers (Gr) and discussed. From the above study, the following conclusions can be drawn: x The inclusion of nanoparticles into the base fluid has produced an augmentation of the heat transfer coefficient, which increases appreciably with an increase of nanoparticles volume concentration. x Both flow and thermal fields are influenced by the solid volume fraction, because raising the solid volume fraction leads to reduce both of action of fluid movement and temperature. However, it leads to increase the corresponding average Nusselt number at the heated surface. x The flow and thermal fields as well as the heat transfer rate inside the enclosure are strongly dependent on the Grashof number. x Nanofluids are capable to modify the flow pattern.
Fig. 4. Effects of average Nusselt (left) and average temperature (right) for different Gr.
References [1]
K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transfer 46 (2003) 3639-3653. [2] R.Y. Jou, S.C. Tzeng, Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures, Int. Commun. Heat Mass Transfer 33 (2006) 727-736. [3] R.K. Tiwari, M.K. Das, Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Transfer 50 (2007) 2002-2018. [4] M. Muthtamilselvan, P. Kandaswamy, J. Lee, Heat transfer enhancement of copper-water nanofluids in a lid-driven enclosure, Commun. Nonlinear Sci Numer Simulat, 15 (2010) 1501-1510. [5] B. Ghasemi, S.M. Aminossadati, Mixed convection in a lid-driven triangular enclosure filled with nanofluids, Int. Commun. Heat Mass Transfer 37 (2010) 1142–1148. [6] E.A. Nada, A.J. Chamkha, Mixed convection flow in a lid-driven inclined square enclosure filled with a nanofluid, European J. Mechanics B/Fluids 29 (2010) 472-482. [7] M.A. Mansour, R.A. Mohamed, M.M. Abd-Elaziz, S.E. Ahmed, Numerical simulation of mixed convection flows in a square lid-driven cavity partially heated from below using nanofluid, Int. Commun. Heat Mass Transfer 37 (2010) 1504–1512. [8] F. Talebi, A.H. Mahmoudi, M. Shahi, Numerical study of mixed convection flows in a square lid-driven cavity utilizing nanofluid, Int. Commun. Heat Mass Transfer 37 (2010) 79-90. [9] M.M. Rahman, M.M. Billah, M. Hasanuzzaman, R. Saidur and N.A. Rahim, Heat transfer enhancement of nanofluids in a lid-driven square enclosure, Numer. Heat Transfer, Part A, 62 (2012) 973-991. [10] M.M. Billah, M.M. Rahman, M. Shahabuddin and A.K. Azad, Heat Transfer Enhancement of Copper-Water Nanofluids in an Inclined LidDriven Triangular Enclosure, J. Sci. Res. 3 (2) (2011) 525-538. [11] M.M. Rahman, M.M. Billah, A.T.M.M. Rahman, M.A. Kalam and A. Ahsan, Numerical investigation of heat transfer enhancement of nanofluids in an inclined liddriven triangular enclosure, Int. Commun. Heat Mass Transfer 38 (2011) 1360-1367. [12] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571-581.
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ScienceDirect Procedia Engineering 90 (2014) 517 – 523
10th International Conference on Mechanical Engineering, ICME 2013
Numerical simulation on buoyancy-driven heat transfer enhancement of nanofluids in an inclined triangular enclosure M. M. Billaha,b,* M. M. Rahmanc, Uddin M. Sharif b, M.N. Islamb a
Department of Arts and Sciences, Ahsanullah University of Science and Technology (AUST), Dhaka-1208, Bangladeh b Department of Mathematics, Jahangirnagar University (JU), Savar, Dhaka-1342, Bangladeh c Department of Mathematics, Bangladesh University of Engineering and Technology (BUET), Dhaka-1000, Bangladeh
Abstract Heat transfer enhancement in a two-dimensional inclined lid-driven triangular enclosure utilizing nanofluids is investigated numerically for various pertinent parameters. The present model is developed to analyze the heat transfer performance of nanofluids inside an enclosure taking into account the solid volume fraction δ, variable velocity and thermal boundary conditions. Fluid mechanics and conjugate heat transfer, described in terms of continuity, linear momentum and energy equations. The transport equations are solved numerically using the Galerkin finite element method. Results are obtained for a wide range of parameters such as the Grashof numbers. Copper-water nanofluids is used with Prandtl number, Pr = 6.2 and solid volume fraction δ is varied from 0% - 20%. The streamlines, isotherm plots and heat transfer correlation of the average Nusselt number at the hot surface as well as average fluid temperature in the enclosure is presented and discussed in detailed. It is found that heat transfer increased by 28% as volume fraction δ increases from 0% to 20% at Gr =105. Moreover, the variation of the average Nusselt number and average fluid temperature in the cavity is linear with the solid volume fraction. © 2014 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2014The TheAuthors. Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Engineering Engineering and Technology (BUET). and Technology (BUET)
Keywords: Finite element method; nanofluids; lid-driven triangular enclosure.
* Corresponding author. Tel.: +88-01715 400 423; fax: +88-2-9860564. E-mail address: [email protected]; [email protected] (Md. Masum Billah)
1877-7058 © 2014 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/3.0/). Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Engineering and Technology (BUET) doi:10.1016/j.proeng.2014.11.766
518
M.M. Billah et al. / Procedia Engineering 90 (2014) 517 – 523
1. Introduction Nanotechnology is measured to be one of the most important forces that drive the next major industrial revolution of this latest century. That represents the key relevant technological cutting edge currently being explored. It aims at manipulating the structure of the matter at the molecular level with the goal for innovation in almost every industry and public endeavor including biological sciences, physical sciences, electronics cooling, transportation, the environment and also national security. Khanafer et al. [1] made a numerical model to find out natural convection heat transfer in nanofluids. Jou and Tzeng [2] examined numerically the heat transfer enhancement utilizing nanofluids in a two-dimensional rectangular enclosure for different pertinent parameters. Tiwari and Das [3] presented a numerical model of heat transfer augmentation in a lid-driven square cavity filled with nanofluids. A computational model is performed to analyze the transport mechanism of mixed convection in a lid-driven enclosure packed with nanofluids by Muthtamilselvan et al. [4]. Ghasemi and Aminossadati [5] considered mixed convection heat transfer in a lid-driven triangular enclosure filled with a water–Al2O3 nanofluid. Mixed convection flow in a lid-driven inclined square enclosure filled with water-Al2O3 nanofluid was presented by Nada and Chamkha [6]. Mansour et al. [7] performed a numerical simulation on mixed convection flow in a square lid-driven cavity partially heated from below using nanofluid. Talebi et al. [8] performed mixed convection flows in a square lid-driven cavity utilizing nanofluid. Rahman et al. [9] performed a numerical investigation on heat transfer enhancement of nanofluids in a lid-driven square enclosure. Billah et al. [10] presented a model to investigate the heat transfer enhancement of copper-water nanofluids in an inclined lid-driven triangular enclosure. Rahman et al. [11] have done a numerical investigation of heat transfer enhancement of nanofluids in an inclined liddriven triangular enclosure. Nomenclature cp g Gr H k L Nu p P Pr Re Ri T u U v V
Specific heat at constant pressure Y dimensionless vertical coordinate gravitational acceleration (ms-2) Grashof number Greek symbols enclosure height (m) α thermal diffusivity (m2s-1) -1 -1 thermal conductivity (Wm K ) β thermal expansion coefficient (K-1) length of the cavity (m) G solid volume fraction Nusselt number μ dynamic viscosity (kg m-1s-1) dimensional pressure (Nm-2) ν kinematic viscosity (m2s-1) dimensionless pressure θ non-dimensional temperature Prandtl number U density (kg m-3) Reynolds number ψ Streamfunction Richardson number J penalty parameter temperature (K) * general dependent variable horizontal velocity component (ms-1) θav Average temperature dimensionless horizontal velocity component Subscripts vertical velocity component (ms-1) av Average dimensionless vertical velocity component h Hot cavity volume (m3) c Cold V horizontal coordinate (m) f Fluid x X dimensionless horizontal coordinate nf Nanofluid y vertical coordínate (m) s Solid nanoparticle Owing to the rapid development of computer and computational techniques, finite element formulations have provided an alternative in dealing with the mixed convection heat transfer in enclosures using nanofluids. To the best knowledge of the authors, a little attention has been paid to investigate buoyancy-driven heat transfer characteristics of the nanofluids contained in an inclined lid-driven triangular enclosure with different solid volume fractions. This problem may be encountered in a number of electronic cooling and MEMS applications.
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M.M. Billah et al. / Procedia Engineering 90 (2014) 517 – 523
2. Mathematical Analysis The physical model under current study with the system of coordinates is shown in Fig. 1. The length of the base wall and height of the sliding left wall of the enclosure are denoted by L and H, respectively. In addition, the sliding
§ 5S · y ¸V0 . Moreover, © L ¹
wall of the cavity is kept adiabatic and allowed to move from bottom to top at a speed aˆ sin ¨ it is assumed that the temperature, T x,0
Th Th Tc x L 1 x L 2
2
of the bottom wall is higher than
the temperature (Tc) of the right inclined wall. Y § 5S · aˆ sin ¨ y ¸V0 © L ¹ adiabatic
g H
u = v = 0, T = Tc
I
L X
Fig. 1. Schematic of the problem with the domain and boundary conditions.
The free space in the enclosure is filled with copper water nanofluids. The nanofluid in the enclosure is Newtonian, incompressible and laminar. The nanoparticles are assumed to have uniform shape and size. It is assumed that thermal equilibrium exists between the base fluid and nanoparticles, and no slip occurs between the two media. The physical properties of the nanofluid are considered to be constant except the density variation in the body force term of the momentum equation which is satisfied by the Boussinesq’s approximation. Under the above assumptions, the system of equations governing the two-dimensional motion of a nanofluid is as follows: wU wV wX wY
w UV
w U2 wX
w UV wX
wY
w UT wX
(1)
0
w V2
wY w V T wY
§ Gr · UE nf 1 Uf 1 wP 2U ¨ T sin I ¸ wX Re Unf 1 G 2.5 © Re2 ¹ Unf E f
(2)
§ Gr · UE nf 1 wP 1 U f 2V ¨ T cosI ¸ 2.5 wY Re Unf 1 G © Re2 ¹ Unf E f
(3)
1 § D nf ¨ Re Pr ¨© D f
(4)
· 2 ¸ T ¸ ¹
where Reynolds number Re V0 L X f , Prandtl number Pr Q f D f and Grashof number Gr
E f g Th Tc L3 Xf2
The above equations (1)-(4) are non-dimensionalized by introducing the following dimensionless variables p U gy L2 , T T Tc x y u v ,Y ,U ,V ,P X L L V0 V0 Th Tc Unf V02
.
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M.M. Billah et al. / Procedia Engineering 90 (2014) 517 – 523
The effective density Unf of the nanofluid is defined by
1 G U f
Unf
GUs
(5)
where δ is the solid volume fraction of nanoparticles. In addition, the thermal diffusivity D nf of the nanofluid can be expressed as: knf
D nf
(6)
Uc p nf
The heat capacitance of nanofluids can be defined as: Uc p 1 G Uc p G Uc p
nf
f
s
(7)
Additionally, UE nf is the thermal expansion coefficient of the nanofluid and it can be determined by
UE nf 1 G UE f
G UE s
(8)
Furthermore, Pnf is the dynamic viscosity of the nanofluid introduced by Brikman [12] as Pf
Pnf
(9)
1 G 2.5
The effective thermal conductivity of nanofluid was intoduced by Kanafer et al.[1] as:
k s 2k f G k f k s
ks 2k f 2G k f ks
knf kf
(10)
where, ks is the thermal conductivity of the nanoparticles and kf is the thermal conductivity of base fluid. The appropriate non dimensional initial and boundary conditions for the governing equations are on the bottom wall: U V 0, T 1 X 2 1 X 2 on the left wall: U
0,V
wT 0 wN 0, T 0
aˆ sin 5S Y ,
on the right inclined wall: U V where N is the non-dimensional distances either X or Y direction acting normal to the surface. The average Nusselt number at the heated surface of the cavity may be expressed as Nuav
knf kf
L
wT
³ wY dX , and average fluid temperature in the enclosure may be defined as Tav ³T d V /V 0
The fluid motion is displayed using the stream function (\ ) obtained from velocity components U and V. The relationships between stream function and velocity components for two dimensional flows can be expressed as: U
w\ ,V wY
w\ wX
(11)
3. Solution Procedure The Galerkin weighted residual method of finite element formulation is employed as a computational scheme. The finite element method begins by the partition of the continuum area of interest into a number of simply shaped regions known as elements. These elements may be different shapes and sizes. Within each element, the dependent variables are approximated using interpolation functions. In the present study erratic grid size system is considered especially near the walls to capture the rapid changes in the dependent variables. The coupled governing equations (1)-(4) are transformed into sets of algebraic equations using finite element method to reduce the continuum domain into discrete triangular domains. The system of algebraic equations is solved by iteration technique. The solution
521
M.M. Billah et al. / Procedia Engineering 90 (2014) 517 – 523
Gr = 104
Gr = 105
Gr = 106
process is iterated until the subsequent convergence condition is satisfied: *m1 *m d 106 where m is number of iteration and * is the general dependent variable.
δ=0
δ = 0.04
δ = 0.08
δ = 0.20
Fig. 2. Streamlines for different values of solid volume fraction δ and Grashof number Gr. 4. Results and Discussion The current study is carried out for copper-water nanofluids as working fluid with Prandtl number of 6.2. In this paper, our attention is taken into account to investigate the effects of controlling parameters namely the solid volume fraction (δ), and Grashof number (Gr). Here, the effect of the solid volume fractions is investigated over the range of 0% – 20% while the Re and tilt angle I are kept fixed at 100 and 60, respectively. It is worth to note that the value of Gr is varied from 104 to 106. In addition, the results of this study are presented in terms of streamlines and isotherms. Furthermore, the heat transfer effectiveness of the enclosure is displayed in terms of average Nusselt number (Nuav) and the average fluid temperature (θav) in the cavity. Fig. 2 shows the streamlines in an inclined lid-driven triangular enclosure for various values of the solid volume fractions δ and Gr. Visualization are given from δ = 0% to 20%. Based on literature on nanofluid which are cited in reference list, the maximum values of nanoparticle fraction was taken as 0.1. It can easily be seen from the bottom row of Fig. 2, the circulating cells are generated by the lid dragging the neighboring fluid. A counter clockwise rotating cell is observed near the lower part of the enclosure for all values of δ. The size and shape of the cell change with the increased of δ. It is also found from the streamlines that the flow strength of the anticlockwise rotating cell is decreasing very slowly when δ is increased. This is due to an increase as
522
M.M. Billah et al. / Procedia Engineering 90 (2014) 517 – 523
Gr = 104
Gr = 105
Gr = 106
the volume fraction as a result of high-energy transport through the flow associated with the irregular motion of the ultrafine particles. The size of the anticlockwise rotating major cell increases while Gr increases from 104 to 105 and 106. The isotherm plots indicate the lines with equal intervals between unity (hot wall) and zero (cold wall). It is noticed from Fig. 3 that isotherm lines become very denser towards the hot surface for each value of δ. The isotherms distributions are nearly parallel to the horizontal wall of the cavity, indicating most of the heat transfer is carried out by conduction, which agrees with the result of Muthtamilselvan et al. [4]. On the other hand, for a fixed solid volume fraction δ, the isotherms become nonlinear with the rising Gr. For higher values of Gr, wavy variation is observed from isotherms and changing of nanoparticle volume fraction becomes effective.
δ=0
δ = 0.04
δ = 0.08
δ = 0.20
Fig. 3. Isotherms for different values of solid volume fraction δ and Grashof number Gr. The average Nusselt number (Nuav) is calculated along the hot surface, which is a measure of the overall heat transfer rate, is plotted in the left column of the Fig. 4 for the abovementioned values of the solid volume fractions (δ). These figures show a linear variation of the average Nusselt number with the solid volume fraction. As clearly be seen that the heat transfer increases with increasing Gr. Besides, the values of Nu are always maximum for the highest value of δ (= 20%). The effects of the solid fractions (δ) on average fluid temperature (θav) in the enclosure are depicted in the right column of the Fig. 4 for Gr = 104, 105, and 106. It is observed that θav decreases very swiftly with the increasing of Gr for all values of δ. It is clearly being seen that θav is maximum for the highest value of solid volume fraction (δ = 20%). 5. Conclusion Mixed convective flow and heat transfer in a lid-driven inclined triangular enclosure filled with Cu-water nanofluids is formulated and solved using Galarkin finite element method. Variable velocity and variable thermal boundary conditions are applied here. In this paper, the flow and heat transport structures are described by
M.M. Billah et al. / Procedia Engineering 90 (2014) 517 – 523
streamlines and isotherms respectively. Computed results for various parametric situation namely solid volume fraction (δ), and the Grashof numbers (Gr) and discussed. From the above study, the following conclusions can be drawn: x The inclusion of nanoparticles into the base fluid has produced an augmentation of the heat transfer coefficient, which increases appreciably with an increase of nanoparticles volume concentration. x Both flow and thermal fields are influenced by the solid volume fraction, because raising the solid volume fraction leads to reduce both of action of fluid movement and temperature. However, it leads to increase the corresponding average Nusselt number at the heated surface. x The flow and thermal fields as well as the heat transfer rate inside the enclosure are strongly dependent on the Grashof number. x Nanofluids are capable to modify the flow pattern.
Fig. 4. Effects of average Nusselt (left) and average temperature (right) for different Gr.
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