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Effect of inside heated cylinder on the natural convection heat transfer of nanofluids in a wavy-wall enclosure
International Journal of Heat and Mass Transfer 103 (2016) 1053–1057
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
Effect of inside heated cylinder on the natural convection heat transfer of nanofluids in a wavy-wall enclosure M. Hatami ⇑, H. Safari Department of Mechanical Engineering, Esfarayen University of Technology, Esfarayen, North Khorasan, Iran
a r t i c l e
i n f o
Article history: Received 24 June 2016 Received in revised form 5 August 2016 Accepted 9 August 2016
Keywords: Wavy-wall cavity Heated cylinder Nusselt number FEM Nanofluid
a b s t r a c t In this study, natural convection heat transfer of nanofluids in a wavy-wall enclosure is studied while a heated cylinder is located inside the cavity. The governing equations for a simple cavity are studied based on the previous work and changed it to complicated two-phase nanofluid and solved by finite element method (FEM). Because the location on inside cylinder has a significant effect on the heat transfer and Nusselt number, it is important to find the best location for this cylinder. In our study the cylinder is movable in X and Y directions which affects the temperatures, stream lines and nanoparticles concentration contours. The results indicate that center location for the cylinder assists the heat transfer in both wavy side walls. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction One way to improve heat transfer rate in various industrial processes is to increase thermal conductivity of the fluid. With respect to the thermal properties of the solid nanoparticles the liquid thermal conductivity can be increased. Also, the heat transfer enhancement is achieved by utilizing of nanofluid without huge penalty in pumping power [1–3]. Sheremet et al. [4] studies the heat transfer in a different square porous cavity which filled with nanofluids, and two-temperature model has been used for heat transfer, and also the Tiwari and Das nanofluid model has been studied numerically. They found that to add the solid nanoparticles to the base fluid will suppresses convective flow. Tazraei and Riasi [5] discussed how adding nanoparticles to liquids can improve the flow dynamics as to shear stress profile and heat-transfer capabilities. Also, Tazraei [6] studied how the radiation mechanism can affect the heat transfer rate in porous media, along with other contributing factors, such as the strength of the Brownian motion, thermophoresis, and the viscous dissipation. Researches confirm that in most cases, nanofluid flows show the non-linear constitutive behavior which needs special treatment [7–9]. Domairry and Hatami [10] applied the differential transformation method (DTM) to investigate the Cu-water nanofluid treatment between parallel plates. Ahmadi et al. [11] studied the nanofluids heat transfer over an unsteady stretching flat plate. ⇑ Corresponding author. E-mail address: [email protected] (M. Hatami). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.029 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
Due to huge potential application of natural convection heat transfer of nanofluids, many studies are performed in this area, for instance, Hatami and Ganji [12] investigated convection of sodium alginate (SA) non-Newtonian nanofluid. Also, Kalaoka and Witayangkurn [13] investigated the natural convection in a partially cooled square enclosure filled with porous medium by numerical method and found that varied Darcy and Rayleigh numbers can lead to difference temperature flow and heat fields. Like the above studied related to natural convection, other works are also done for different cavity geometries such as prismatic [14], T-shaped [15], Triangular Containing Cylindrical Rods [16], Trapezoidal with Wavy Top [17] and wavy-wall [18] filled by pure fluid or nanofluilds. In present study, a novel and effective model has been built to investigate the natural convection heat transfer process of nanofluids in wavy cavities including heated cylinder using finite element method (FEM). In order to find the best location of heated cylinder inside the cavity, different geometries were designed and analyzed. Obtained results are discussed on heat transfer efficiency by Nusselt number, isotherm, streamlines and nanoparticle concentration in wavy-wall cavity as well as cylinder around. 2. Geometry definition and boundary conditions The enclosure is made of two horizontal flat walls and two vertical wavy walls with inside circular cylinder (Fig. 1). The space between the inside cylinder and the enclosure walls is filled with nanofluid. The flat horizontal walls are kept adiabatic while the
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M. Hatami, H. Safari / International Journal of Heat and Mass Transfer 103 (2016) 1053–1057
wavy walls and the inner cylinder are isothermal but kept at different temperatures (the temperature of inner cylinder is higher). The position of the inside cylinder can be moved along a vertical line X = 0 or Horizontal line Y = 0. The definition of the boundaries for the left wavy wall is:
1 p 2pY k þ k 1 sin ; þ 2 A 2
X¼
06Y 6A
ð1Þ
Here k denotes to the vertical waviness = a/W, and A is the aspect ratio = H/W = 1.5 = 4D, W is the average width of the enclosure. 3. Mathematical modeling The governing equations for the studied problem are presented based on the previous studies [8]. Since the condition of this study is natural convection of nanofluids in the cavity, density of the nanofluids is considered as [8]:
n
q ¼ /qp þ ð1 /Þqp ffi /qp þ ð1 /Þ qf 0 ð1 bðT T c ÞÞ
Isotherm lines
Streamlines
-0.5
Nanoparticles volume fraction
0 .5
-1.5 1 .5
0.05 2.
4. 5 3. 5
0.4
0.15
75 0.
-1 .5
5
65
95
5 0.1 5
0.
5
0.3 05.0 0.1 0.55 55
5
5 0.5
0.0
0 .6
0.5 0. 65 15 50. 0. 2 .95
5
0.35
0.5 0.95 0.85 0 .2 5 5
0 .8
1.5
0.25
5
-3 .5 -2.5 -0 .5
0.45
5
0. 05
-3.5
0.2 5
0.
0. 7
2. 5
0.0
5
5
-2.5
0.35
0
0
0.05
05 0.35
0.05
0. 650.9 5
0.05
0 .8
0. 35
12
-12 4
0. 65
-8
4
-4
0.6
05
5
8
0.
-4 0
0. 8
0.2
-8
05
0 .8
0.
0
8
0.8
2
-4
0 .2
0.5
0.5
0.
0.5
-4 0
35 0. 65
0.
8
-8
4
0.
95
4
0
0.2 0.
5
350 .2
0 .5 0 .3 5 0 .2
0.3
0. 65
0.5
0
5
0. 5 0.35
5
8 -12 -8
12
0.55
-4
0.
5 0.
0.2 0.05
5 0.4 0 .2 5
4
0 .3
05
15
8
5
0
-4
25
0
0.3
0.45
-16
5 0.
0. 05
16
4
-8
0.1
5
5
0.
0.2
0.6 5
0.4 5
-16
-12
0 .3
0 0.
5
12
5
0 .0
-4 0
8 0.35
0.55
0.3
0.9 5
0
-4
65
-8
0.
-8
0.
0.45 0.25
4
5
5
4
0.8
0 .1
0.95
8
0.2
0.35
0.55 75 0. 5 0 .8
0.
8 0.
0.15
0
0. 05
5
05
0.2
0.0
0
0.7 5 0.6
0.
ð2Þ
where q, qp and qf are densities of nanofluid, nanoparticles and base fluid, respectively, Tc is the reference temperature, qf0 is the base fluid’s density at the reference temperature, and b is the volumetric coefficient of expansion.
Fig. 1. Enclosure geometry and boundary conditions.
0.15
o
0
Fig. 2. Effect of Yc (0.5, 0, 0.5) for inside cylinder on different countours.
2
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M. Hatami, H. Safari / International Journal of Heat and Mass Transfer 103 (2016) 1053–1057
Streamlines
0
Isotherm lines
0 .0
4
5
5
0.65
0. 05
0 .0
0 .6
5
0.5
25
4
0.
12
5
0
0 .3
5
0.0
8
0. 15 0 .2
Nanoparticles volume fraction
-4
0. 8 0 .3 5
16
4
0.2
8
12
0.35
0.1 5
8
85
12 0.
65
0. 8
4
0. 4 5
0.45 0 .5 5
0.8 5 0. 7 5 0.55
0.
15 0.0 5 0.
16 0
35 0.
5 0.3 0.2
0.75
0
5
0.2
0.9 5
5
0.45
0.9
0.
05
0.5
8
0.
4
0.05
0.2 05 5
25
0
0
0.05
0.
05
2
0.05
0. 650.9 5
0.05
0 .8
0. 35
12
-12 4
0. 65
-4 0
-8
0.
4
05
-4
0.6
5
8
0. 8
0.2
-8
05
0 .8
0.
0
8
0.8
0.
-4
0 .2
0.5
0.5
0.35 0.5
-4
8
-8
0
35 0. 65
0.
95
4
0.
4
0
0.2
5
350 .2
0 .5 0 .3 5 0.2
0.3
0. 65
0.5
0
0.0
5
0
-4
0 .0
-8
5
65
0.6
5
5
0.5 0.9
0.
5
0 .2
0
2 -1
55
0.0
5
0.
25
0. 2
0.0
0 .35 0.45
-4
0.15
0.
0
0
5
2
05
0.
8
0.
0.7 5 00..1 055
5 0.8
0
0.5
25
0.35
0.
-1 2
-8
0
-4
0
5 0.20.0
0.05
0.
0.3 5
0.8
-4
5
0.2
5 3 65 0 .
4
0.7 0.
0. 45
-12
-8 -4
5 0.9 5
0.55
-8
0
-16
45
0.8 5
0.1
0.35
0.
Fig. 3. Effect of Xc (0.25, 0, 0.25) for inside cylinder on different countours.
q ffi /qp þ ð1 /Þfq0 ð1 bðT T c ÞÞg
ð3Þ
@u @ v þ ¼0 @x @y
ð4Þ
qf u
T ¼ T h ; / ¼ /h on the inner cylinder boundary
T ¼ T c ; / ¼ /c
! @u @u @p @2u @2u ¼ þl þ þv @x @y @x @x2 @y2
ð5Þ
! @v @v @p @2v @2v qf u þ v þ ¼ þl @y @x @y @x2 @y2 ð/ /c Þðqp qf 0 Þg þ ð1 /c Þqf 0 ðT T c Þg ! ðqcÞp @T @T @2T @2T @/ @T @/ @T þ 2 þ DB þv ¼a þ u 2 @x @y @x @y @x @x @y @y ðqcÞf ( 2 )# 2 @T @T þðDT =T c Þ þ @x @x ( ) ( ) @/ @/ @2/ @2/ DT @2T @2T þ þ þ þv ¼ DB u @x @y @x2 @y2 @x2 @y2 Tc
on the outer wavy boundaries
ð9Þ
@T=@n ¼ @/=@n ¼ 0 on two flat insulation boundaries w ¼ 0 all the solid boundaries The stream function and vorticity are defined as follows:
u¼
@w ; @y
v ¼
@w ; @x
w¼
@ v @u @x @y
ð10Þ
Non-dimensional variables employed in this study are shown in Eq. (11)
ð6Þ
x y wL2 w T Tc / /c X ¼ ; Y ¼ ; X¼ ; W¼ ; H¼ ; U¼ L L a a Th Tc /h /c
ð11Þ
By using these dimensionless parameters the equations become:
ð7Þ
ð8Þ
Continuity, momentum under Boussinesq approximation and energy equations for the laminar and steady state natural convection in a two-dimensional form are shown in Eqs. (4)–(8). Boundary conditions are as follows:
! @W @X @W @X @2X @2X þ ¼ Pr @Y @X @X @Y @X 2 @Y 2 @H @H þ PrRa Nr @X @X
@W @H @W @H @2H @2H @U @H @U @H þ þ Nb ¼ @Y @X @X @Y @X @X @Y @Y @X 2 @Y 2 2 2 ! @H @H þ þ Nt @X @Y
ð12Þ
ð13Þ
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M. Hatami, H. Safari / International Journal of Heat and Mass Transfer 103 (2016) 1053–1057
Xc=0,Yc=-0.5 Xc=0,Yc=0 Xc=0,Yc=0.5 Xc=0.25, Yc=0 Xc=-0.25, Yc=0
20
Xc=0,Yc=-0.5 Xc=0,Yc=0 Xc=0,Yc=0.5 Xc=0.25, Yc=0 Xc=-0.25,Yc=0
14 12
Local Nusselt
Local Nusselt
15
10
10 8 6 4
5
2
0
0
0.5
1
0
1.5
0
0.2
0.4
0.6
0.8
1
1.2
Length of Curve
Length of Curve
(a)
(b) Xc=0,Yc=-0.5 Xc=0,Yc=0 Xc=0,Yc=0.5 Xc=0.25, Yc=0 Xc=-0.25, Yc=0
20
Local Nusselt
15
10
5
0
0
0.5
1
1.5
Length of Curve
(c) Fig. 4. Local Nusselt number for (a) left wall, (b) inside cylinder wall and (c) right wall.
! @W @U @W @U 1 @2U @2U þ ¼ Le @X 2 @Y 2 @Y @X @X @Y Nt @2H @2H þ þ Nb Le @X 2 @Y 2 @2W @X 2
þ
@2W @Y 2
Nuloc ¼ ! ð14Þ
ð15Þ
where thermal Rayleigh number, the buoyancy ratio number, Prandtl number, the Brownian motion parameter, the thermophoretic parameter and Lewis number of nanofluid are defined as the same as [8]. Boundary conditions of those dimensionless equations are as follows (Fig. 1)
H ¼ 1; U ¼ 1 on the inner cylinder boundary H ¼ 0; U ¼ 0 on the outer wavy boundaries @ H=@n ¼ @ U=@n ¼ 0 on two flat insulation boundaries W ¼ 0 on all solid boundaries
ð16Þ
The local Nusselt number on the cold circular wall can be expressed as:
ð17Þ
where n is the direction normal to the outer cylinder surface. The average number on the cold circular wall is evaluated as:
Nuav e ¼ ¼ X
@H @n
1 L
Z
L
Nuloc ðnÞdn
ð18Þ
0
4. Finite element method (FEM) Finite element methods are numerical methods for approximating the solutions of mathematical problems that are usually formulated so as to precisely state an idea of some aspects of physical reality. A finite element method is characterized by a variational formulation, a discretization strategy, one or more solution algorithms and post-processing procedures. A typical work out of this method involves (1) dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem, followed by (2) systematically recombining all sets of element equations into a global system of equations for the final calculation. In the finite element
M. Hatami, H. Safari / International Journal of Heat and Mass Transfer 103 (2016) 1053–1057
1057
average Nusselt number for all these three boundaries. It is evident that approximately in all boundaries, center location of cylinder (Xc = Yc = 0) makes higher average Nusselt number.
11.5 11
Average Nusselt
10.5 10
6. Conclusion
9.5
In this paper, natural convection heat transfer of the nanofluids in a wavy-wall cavity is studied when a heated cylinder is located in it. To reach the maximum Nusselt number and consequently more heat transfer, cylinder location is assumed to be variable. By applying the FEM for solving and analyzing the results, it is concluded that the best location (among the tested cases) is when Xc = Yc = 0 which had maximum Nusselt number. Also, in this study the effect of cylinder location on stream/isotherm lines and nanoparticles concentration are investigated.
9 8.5 8 7.5 7 Yc - Inside Cylinder Yc-Right Wall Yc-Left Wall Xc-Inside Cylinder Xc-Right Wall Xc-Left Wall
6.5 6 5.5 5
-0.4
-0.2
0
0.2
References
0.4
Yc or Xc Fig. 5. Average Nusselt number for described wall and different cylinder position.
method, the solution region is considered as built up of many small, interconnected sub-regions called finite elements [8]. 5. Results and discussions As mentioned, a wavy-wall cavity including a heated cylinder is studied under the natural convection of two-phase nanofluids. In this function, Xc and Yc (cylinder center location) are considered as the variable parameters for this cavity as shown in Fig. 1. In this study, firstly the equations are derived for a two-phase nanofluid in cavity and then it is aimed to find the best location for these parameters to have the highest Nusselt number for both wavy walls and outside cylinder wall. By using the FEM code, the solution of governing equations can be obtained and local/average Nusselt number for all boundary walls were calculated when Pr = 7, Ra = 10,000, Nt = 0.5, Nb = 0.5, Le = 2 and Nr = 3. Center point location of cylinder is changed in two steps. Firstly it is moved along Y-direction and secondly moved in X-directions. The considered locations for Yc is 0.5, 0, 0.5 and Xc locations are 0.25, 0, 0.25. Figs. 2 and 3 are obtained from the numerical methods for the movements in Y and X directions, respectively. These figures show the isotherm, streamlines and nanoparticles concentrations obtained from numerical solution. As seen in these contours, when the cylinder is moved in X or Y directions, maximum temperatures and aggregation of nanoparticles move towards X or Y directions, too. To find the effect of these change on the heat transfer, local Nusselt number on both wavy side walls (right and left) is calculated as well as outer surface of heated cylinder and depicted in Fig. 4. As seen in this shape, local Nusselt number for right and left wavy walls are symmetry due to the symmetry shape of enclosure geometry and boundary conditions. To have and easier conclusion on the Xc and Yc effect on the heat transfer, Fig. 5 is depicted for
[1] Y. Xuan, Q. Li, Heat transfer enhancement of nanofluids, Int. J. Heat Fluid Flow 21 (1) (2000) 58–64. [2] J. Bayat, A.H. Nikseresht, Investigation of the different base fluid effects on the nanofluids heat transfer and pressure drop, Heat Mass Transfer 47 (9) (2011) 1089–1099. [3] M.H. Esfe, S. Saedodin, M. Mahmoodi, Experimental studies on the convective heat transfer performance and thermophysical properties of MgO–water nanofluid under turbulent flow, Exp. Therm. Fluid Sci. 52 (1) (2014) 68–78. [4] M.A. Sheremet, I. Pop, R. Nazar, Natural convection in a square cavity filled with a porous medium saturated with a nanofluid using the thermal nonequilibrium model with a Tiwari and Das nanofluid model, Int. J. Mech. Sci. 100 (2015) 312–321. [5] P. Tazraei, A. Riasi, Quasi-two-dimensional numerical analysis of fast transient flows considering non-Newtonian effects, J. Fluids Eng. 138 (1) (2016) 011203. [6] P. Tazraei, The extension of the QL method to solve the radiative heat transfer problem in a 3D square enclosure containing non-grey gas, Am. J. Mech. Eng. 4 (2) (2016) 71–81. [7] P. Tazraei, A. Riasi, B. Takabi, The influence of the non-Newtonian properties of blood on blood hammer through the posterior cerebral artery, Math. Biosci. 264 (2015) 119–127. [8] M. Hatami, D. Song, D. Jing, Optimization of a circular-wavy cavity filled by nanofluid under the natural convection heat transfer condition, Int. J. Heat Mass Transfer 98 (July 2016) 758–767. [9] M. Hatami, G.R.M. Ahangar, D.D. Ganji, Refrigeration efficiency analysis for fully wet semi-spherical porous fins, Energy Convers. Manage. 84 (84) (2014) 533–540. [10] G. Domairry, M. Hatami, Squeezing Cu–water nanofluid flow analysis between parallel plates by DTM-Padé Method, J. Mol. Liq. 193 (2014) 37–44. [11] A.R. Ahmadi, A. Zahmatkesh, M. Hatami, D.D. Ganji, A comprehensive analysis of the flow and heat transfer for a nanofluid over an unsteady stretching flat plate, Powder Technol. 258 (2014) 125–133. [12] M. Hatami, D.D. Ganji, Natural convection of sodium alginate (SA) nonNewtonian nanofluid flow between two vertical flat plates by analytical and numerical methods, Case Stud. Therm. Eng. 2 (2014) 14–22. [13] W. Kalaoka, S. Witayangkurn, Numerical simulation of natural convection in a partially cooled square enclosure filled with porous medium, Int. J. Pure Appl. Math. 96 (2014) 507–522. [14] S.J. Yaseen, Numerical study of steady natural convection flow in a prismatic enclosure with strip heater on bottom wall using flexpde, Diyala J. Eng. Sci. 7 (2014) 61–80. [15] M. Satyajit, S. Sourav, S. Sumon, M.A. Mamun, Combined effect of Reynolds and Grashof numbers on mixed convection in a lid-driven T-shaped cavity filled with water-Al2O3 nanofluid, J. Hydrodyn. 27 (2015) 782–794. [16] F.A. Abood, M.A. Ismael, Steady laminar natural convection heat transfer inside air filled horizontal triangular enclosure containing three cylindrical rods, Basrah J. Eng. Sci. (2011) 65–79. [17] P. Sompong, S. Witayangkurn, Natural convection in a trapezoidal enclosure with wavy top surface, J. Appl. Math. (2013) 840632. [18] M.A. Ismael, Natural convection in a wavy porous enclosure heated by an internal circular cylinder, Al-Qadisiya J. Eng. Sci. 4 (3) (2011).
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
Effect of inside heated cylinder on the natural convection heat transfer of nanofluids in a wavy-wall enclosure M. Hatami ⇑, H. Safari Department of Mechanical Engineering, Esfarayen University of Technology, Esfarayen, North Khorasan, Iran
a r t i c l e
i n f o
Article history: Received 24 June 2016 Received in revised form 5 August 2016 Accepted 9 August 2016
Keywords: Wavy-wall cavity Heated cylinder Nusselt number FEM Nanofluid
a b s t r a c t In this study, natural convection heat transfer of nanofluids in a wavy-wall enclosure is studied while a heated cylinder is located inside the cavity. The governing equations for a simple cavity are studied based on the previous work and changed it to complicated two-phase nanofluid and solved by finite element method (FEM). Because the location on inside cylinder has a significant effect on the heat transfer and Nusselt number, it is important to find the best location for this cylinder. In our study the cylinder is movable in X and Y directions which affects the temperatures, stream lines and nanoparticles concentration contours. The results indicate that center location for the cylinder assists the heat transfer in both wavy side walls. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction One way to improve heat transfer rate in various industrial processes is to increase thermal conductivity of the fluid. With respect to the thermal properties of the solid nanoparticles the liquid thermal conductivity can be increased. Also, the heat transfer enhancement is achieved by utilizing of nanofluid without huge penalty in pumping power [1–3]. Sheremet et al. [4] studies the heat transfer in a different square porous cavity which filled with nanofluids, and two-temperature model has been used for heat transfer, and also the Tiwari and Das nanofluid model has been studied numerically. They found that to add the solid nanoparticles to the base fluid will suppresses convective flow. Tazraei and Riasi [5] discussed how adding nanoparticles to liquids can improve the flow dynamics as to shear stress profile and heat-transfer capabilities. Also, Tazraei [6] studied how the radiation mechanism can affect the heat transfer rate in porous media, along with other contributing factors, such as the strength of the Brownian motion, thermophoresis, and the viscous dissipation. Researches confirm that in most cases, nanofluid flows show the non-linear constitutive behavior which needs special treatment [7–9]. Domairry and Hatami [10] applied the differential transformation method (DTM) to investigate the Cu-water nanofluid treatment between parallel plates. Ahmadi et al. [11] studied the nanofluids heat transfer over an unsteady stretching flat plate. ⇑ Corresponding author. E-mail address: [email protected] (M. Hatami). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.08.029 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
Due to huge potential application of natural convection heat transfer of nanofluids, many studies are performed in this area, for instance, Hatami and Ganji [12] investigated convection of sodium alginate (SA) non-Newtonian nanofluid. Also, Kalaoka and Witayangkurn [13] investigated the natural convection in a partially cooled square enclosure filled with porous medium by numerical method and found that varied Darcy and Rayleigh numbers can lead to difference temperature flow and heat fields. Like the above studied related to natural convection, other works are also done for different cavity geometries such as prismatic [14], T-shaped [15], Triangular Containing Cylindrical Rods [16], Trapezoidal with Wavy Top [17] and wavy-wall [18] filled by pure fluid or nanofluilds. In present study, a novel and effective model has been built to investigate the natural convection heat transfer process of nanofluids in wavy cavities including heated cylinder using finite element method (FEM). In order to find the best location of heated cylinder inside the cavity, different geometries were designed and analyzed. Obtained results are discussed on heat transfer efficiency by Nusselt number, isotherm, streamlines and nanoparticle concentration in wavy-wall cavity as well as cylinder around. 2. Geometry definition and boundary conditions The enclosure is made of two horizontal flat walls and two vertical wavy walls with inside circular cylinder (Fig. 1). The space between the inside cylinder and the enclosure walls is filled with nanofluid. The flat horizontal walls are kept adiabatic while the
1054
M. Hatami, H. Safari / International Journal of Heat and Mass Transfer 103 (2016) 1053–1057
wavy walls and the inner cylinder are isothermal but kept at different temperatures (the temperature of inner cylinder is higher). The position of the inside cylinder can be moved along a vertical line X = 0 or Horizontal line Y = 0. The definition of the boundaries for the left wavy wall is:
1 p 2pY k þ k 1 sin ; þ 2 A 2
X¼
06Y 6A
ð1Þ
Here k denotes to the vertical waviness = a/W, and A is the aspect ratio = H/W = 1.5 = 4D, W is the average width of the enclosure. 3. Mathematical modeling The governing equations for the studied problem are presented based on the previous studies [8]. Since the condition of this study is natural convection of nanofluids in the cavity, density of the nanofluids is considered as [8]:
n
q ¼ /qp þ ð1 /Þqp ffi /qp þ ð1 /Þ qf 0 ð1 bðT T c ÞÞ
Isotherm lines
Streamlines
-0.5
Nanoparticles volume fraction
0 .5
-1.5 1 .5
0.05 2.
4. 5 3. 5
0.4
0.15
75 0.
-1 .5
5
65
95
5 0.1 5
0.
5
0.3 05.0 0.1 0.55 55
5
5 0.5
0.0
0 .6
0.5 0. 65 15 50. 0. 2 .95
5
0.35
0.5 0.95 0.85 0 .2 5 5
0 .8
1.5
0.25
5
-3 .5 -2.5 -0 .5
0.45
5
0. 05
-3.5
0.2 5
0.
0. 7
2. 5
0.0
5
5
-2.5
0.35
0
0
0.05
05 0.35
0.05
0. 650.9 5
0.05
0 .8
0. 35
12
-12 4
0. 65
-8
4
-4
0.6
05
5
8
0.
-4 0
0. 8
0.2
-8
05
0 .8
0.
0
8
0.8
2
-4
0 .2
0.5
0.5
0.
0.5
-4 0
35 0. 65
0.
8
-8
4
0.
95
4
0
0.2 0.
5
350 .2
0 .5 0 .3 5 0 .2
0.3
0. 65
0.5
0
5
0. 5 0.35
5
8 -12 -8
12
0.55
-4
0.
5 0.
0.2 0.05
5 0.4 0 .2 5
4
0 .3
05
15
8
5
0
-4
25
0
0.3
0.45
-16
5 0.
0. 05
16
4
-8
0.1
5
5
0.
0.2
0.6 5
0.4 5
-16
-12
0 .3
0 0.
5
12
5
0 .0
-4 0
8 0.35
0.55
0.3
0.9 5
0
-4
65
-8
0.
-8
0.
0.45 0.25
4
5
5
4
0.8
0 .1
0.95
8
0.2
0.35
0.55 75 0. 5 0 .8
0.
8 0.
0.15
0
0. 05
5
05
0.2
0.0
0
0.7 5 0.6
0.
ð2Þ
where q, qp and qf are densities of nanofluid, nanoparticles and base fluid, respectively, Tc is the reference temperature, qf0 is the base fluid’s density at the reference temperature, and b is the volumetric coefficient of expansion.
Fig. 1. Enclosure geometry and boundary conditions.
0.15
o
0
Fig. 2. Effect of Yc (0.5, 0, 0.5) for inside cylinder on different countours.
2
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M. Hatami, H. Safari / International Journal of Heat and Mass Transfer 103 (2016) 1053–1057
Streamlines
0
Isotherm lines
0 .0
4
5
5
0.65
0. 05
0 .0
0 .6
5
0.5
25
4
0.
12
5
0
0 .3
5
0.0
8
0. 15 0 .2
Nanoparticles volume fraction
-4
0. 8 0 .3 5
16
4
0.2
8
12
0.35
0.1 5
8
85
12 0.
65
0. 8
4
0. 4 5
0.45 0 .5 5
0.8 5 0. 7 5 0.55
0.
15 0.0 5 0.
16 0
35 0.
5 0.3 0.2
0.75
0
5
0.2
0.9 5
5
0.45
0.9
0.
05
0.5
8
0.
4
0.05
0.2 05 5
25
0
0
0.05
0.
05
2
0.05
0. 650.9 5
0.05
0 .8
0. 35
12
-12 4
0. 65
-4 0
-8
0.
4
05
-4
0.6
5
8
0. 8
0.2
-8
05
0 .8
0.
0
8
0.8
0.
-4
0 .2
0.5
0.5
0.35 0.5
-4
8
-8
0
35 0. 65
0.
95
4
0.
4
0
0.2
5
350 .2
0 .5 0 .3 5 0.2
0.3
0. 65
0.5
0
0.0
5
0
-4
0 .0
-8
5
65
0.6
5
5
0.5 0.9
0.
5
0 .2
0
2 -1
55
0.0
5
0.
25
0. 2
0.0
0 .35 0.45
-4
0.15
0.
0
0
5
2
05
0.
8
0.
0.7 5 00..1 055
5 0.8
0
0.5
25
0.35
0.
-1 2
-8
0
-4
0
5 0.20.0
0.05
0.
0.3 5
0.8
-4
5
0.2
5 3 65 0 .
4
0.7 0.
0. 45
-12
-8 -4
5 0.9 5
0.55
-8
0
-16
45
0.8 5
0.1
0.35
0.
Fig. 3. Effect of Xc (0.25, 0, 0.25) for inside cylinder on different countours.
q ffi /qp þ ð1 /Þfq0 ð1 bðT T c ÞÞg
ð3Þ
@u @ v þ ¼0 @x @y
ð4Þ
qf u
T ¼ T h ; / ¼ /h on the inner cylinder boundary
T ¼ T c ; / ¼ /c
! @u @u @p @2u @2u ¼ þl þ þv @x @y @x @x2 @y2
ð5Þ
! @v @v @p @2v @2v qf u þ v þ ¼ þl @y @x @y @x2 @y2 ð/ /c Þðqp qf 0 Þg þ ð1 /c Þqf 0 ðT T c Þg ! ðqcÞp @T @T @2T @2T @/ @T @/ @T þ 2 þ DB þv ¼a þ u 2 @x @y @x @y @x @x @y @y ðqcÞf ( 2 )# 2 @T @T þðDT =T c Þ þ @x @x ( ) ( ) @/ @/ @2/ @2/ DT @2T @2T þ þ þ þv ¼ DB u @x @y @x2 @y2 @x2 @y2 Tc
on the outer wavy boundaries
ð9Þ
@T=@n ¼ @/=@n ¼ 0 on two flat insulation boundaries w ¼ 0 all the solid boundaries The stream function and vorticity are defined as follows:
u¼
@w ; @y
v ¼
@w ; @x
w¼
@ v @u @x @y
ð10Þ
Non-dimensional variables employed in this study are shown in Eq. (11)
ð6Þ
x y wL2 w T Tc / /c X ¼ ; Y ¼ ; X¼ ; W¼ ; H¼ ; U¼ L L a a Th Tc /h /c
ð11Þ
By using these dimensionless parameters the equations become:
ð7Þ
ð8Þ
Continuity, momentum under Boussinesq approximation and energy equations for the laminar and steady state natural convection in a two-dimensional form are shown in Eqs. (4)–(8). Boundary conditions are as follows:
! @W @X @W @X @2X @2X þ ¼ Pr @Y @X @X @Y @X 2 @Y 2 @H @H þ PrRa Nr @X @X
@W @H @W @H @2H @2H @U @H @U @H þ þ Nb ¼ @Y @X @X @Y @X @X @Y @Y @X 2 @Y 2 2 2 ! @H @H þ þ Nt @X @Y
ð12Þ
ð13Þ
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M. Hatami, H. Safari / International Journal of Heat and Mass Transfer 103 (2016) 1053–1057
Xc=0,Yc=-0.5 Xc=0,Yc=0 Xc=0,Yc=0.5 Xc=0.25, Yc=0 Xc=-0.25, Yc=0
20
Xc=0,Yc=-0.5 Xc=0,Yc=0 Xc=0,Yc=0.5 Xc=0.25, Yc=0 Xc=-0.25,Yc=0
14 12
Local Nusselt
Local Nusselt
15
10
10 8 6 4
5
2
0
0
0.5
1
0
1.5
0
0.2
0.4
0.6
0.8
1
1.2
Length of Curve
Length of Curve
(a)
(b) Xc=0,Yc=-0.5 Xc=0,Yc=0 Xc=0,Yc=0.5 Xc=0.25, Yc=0 Xc=-0.25, Yc=0
20
Local Nusselt
15
10
5
0
0
0.5
1
1.5
Length of Curve
(c) Fig. 4. Local Nusselt number for (a) left wall, (b) inside cylinder wall and (c) right wall.
! @W @U @W @U 1 @2U @2U þ ¼ Le @X 2 @Y 2 @Y @X @X @Y Nt @2H @2H þ þ Nb Le @X 2 @Y 2 @2W @X 2
þ
@2W @Y 2
Nuloc ¼ ! ð14Þ
ð15Þ
where thermal Rayleigh number, the buoyancy ratio number, Prandtl number, the Brownian motion parameter, the thermophoretic parameter and Lewis number of nanofluid are defined as the same as [8]. Boundary conditions of those dimensionless equations are as follows (Fig. 1)
H ¼ 1; U ¼ 1 on the inner cylinder boundary H ¼ 0; U ¼ 0 on the outer wavy boundaries @ H=@n ¼ @ U=@n ¼ 0 on two flat insulation boundaries W ¼ 0 on all solid boundaries
ð16Þ
The local Nusselt number on the cold circular wall can be expressed as:
ð17Þ
where n is the direction normal to the outer cylinder surface. The average number on the cold circular wall is evaluated as:
Nuav e ¼ ¼ X
@H @n
1 L
Z
L
Nuloc ðnÞdn
ð18Þ
0
4. Finite element method (FEM) Finite element methods are numerical methods for approximating the solutions of mathematical problems that are usually formulated so as to precisely state an idea of some aspects of physical reality. A finite element method is characterized by a variational formulation, a discretization strategy, one or more solution algorithms and post-processing procedures. A typical work out of this method involves (1) dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem, followed by (2) systematically recombining all sets of element equations into a global system of equations for the final calculation. In the finite element
M. Hatami, H. Safari / International Journal of Heat and Mass Transfer 103 (2016) 1053–1057
1057
average Nusselt number for all these three boundaries. It is evident that approximately in all boundaries, center location of cylinder (Xc = Yc = 0) makes higher average Nusselt number.
11.5 11
Average Nusselt
10.5 10
6. Conclusion
9.5
In this paper, natural convection heat transfer of the nanofluids in a wavy-wall cavity is studied when a heated cylinder is located in it. To reach the maximum Nusselt number and consequently more heat transfer, cylinder location is assumed to be variable. By applying the FEM for solving and analyzing the results, it is concluded that the best location (among the tested cases) is when Xc = Yc = 0 which had maximum Nusselt number. Also, in this study the effect of cylinder location on stream/isotherm lines and nanoparticles concentration are investigated.
9 8.5 8 7.5 7 Yc - Inside Cylinder Yc-Right Wall Yc-Left Wall Xc-Inside Cylinder Xc-Right Wall Xc-Left Wall
6.5 6 5.5 5
-0.4
-0.2
0
0.2
References
0.4
Yc or Xc Fig. 5. Average Nusselt number for described wall and different cylinder position.
method, the solution region is considered as built up of many small, interconnected sub-regions called finite elements [8]. 5. Results and discussions As mentioned, a wavy-wall cavity including a heated cylinder is studied under the natural convection of two-phase nanofluids. In this function, Xc and Yc (cylinder center location) are considered as the variable parameters for this cavity as shown in Fig. 1. In this study, firstly the equations are derived for a two-phase nanofluid in cavity and then it is aimed to find the best location for these parameters to have the highest Nusselt number for both wavy walls and outside cylinder wall. By using the FEM code, the solution of governing equations can be obtained and local/average Nusselt number for all boundary walls were calculated when Pr = 7, Ra = 10,000, Nt = 0.5, Nb = 0.5, Le = 2 and Nr = 3. Center point location of cylinder is changed in two steps. Firstly it is moved along Y-direction and secondly moved in X-directions. The considered locations for Yc is 0.5, 0, 0.5 and Xc locations are 0.25, 0, 0.25. Figs. 2 and 3 are obtained from the numerical methods for the movements in Y and X directions, respectively. These figures show the isotherm, streamlines and nanoparticles concentrations obtained from numerical solution. As seen in these contours, when the cylinder is moved in X or Y directions, maximum temperatures and aggregation of nanoparticles move towards X or Y directions, too. To find the effect of these change on the heat transfer, local Nusselt number on both wavy side walls (right and left) is calculated as well as outer surface of heated cylinder and depicted in Fig. 4. As seen in this shape, local Nusselt number for right and left wavy walls are symmetry due to the symmetry shape of enclosure geometry and boundary conditions. To have and easier conclusion on the Xc and Yc effect on the heat transfer, Fig. 5 is depicted for
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