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Buoyancy induced flow in a nanofluid filled enclosure partially exposed to forced convection
Superlattices and Microstructures 51 (2012) 381–395
Contents lists available at SciVerse ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Buoyancy induced flow in a nanofluid filled enclosure partially exposed to forced convection Eiyad Abu-Nada a, Hakan F. Oztop b,⇑, Ioan Pop c a b c
Department of Mechanical Engineering, King Faisal University, Al-Ahsa 31982, Saudi Arabia Department of Mechanical Engineering, Fırat University, Elazig TR-23119, Turkey Faculty of Mathematics and Computer Science, Babesß-Bolyaiu University, 400082 Cluj-Napoca, Romania
a r t i c l e
i n f o
Article history: Received 5 October 2011 Received in revised form 22 December 2011 Accepted 3 January 2012 Available online 11 January 2012 Keywords: Nanofluid Natural convection Partially exposed enclosure
a b s t r a c t A numerical study was performed on natural convection for water– CuO nanofluid filled enclosure where the top surface was partially exposed to convection. The cavity has a square cross-section and differentially heated. Except exposed convection part on the top, all sides are adiabatic on horizontal walls. Effects of Rayleigh number (103 6 Ra 6 105), Biot number (0 6 Bi 6 1), length of partial convection (0.0 6 L 6 1.0) and volume fraction of nanoparticles (0.0 6 u 6 0.1) on heat and fluid flow were investigated. The results showed that for the case of high Biot number that heat transfer along the heated was enhanced by increasing the Rayleigh number mainly at the upper portion of the heated wall. When the top wall was totally exposed to convection, the results prevail that the heat transfer was more effective at high Biot number especially at the upper portion of the heated wall. For the case of high Biot number, the results prevailed that the heat transfer at the upper portion of the heated wall increases considerably at high exposed length to convection (L); however, for L 6 0.75 the effect of L was less pronounced. Contour maps for percentage of heat transfer enhancement were presented and it was shown that the location of maximum enhancement in heat transfer was sensitive to Ra, u and L. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Conventional heat transfer fluids, including oil, water, and ethylene glycol mixtures are poor heat transfer fluids due to the poor intrinsic thermal conductivity of these fluids. Numerous attempts have ⇑ Corresponding author. Tel.: +90 424 237 0000x4222; fax: +90 424 236 7064. E-mail address: [email protected] (H.F. Oztop). 0749-6036/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2012.01.002
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Nomenclature Bi cp g h H K ‘ L Nu Pr qw Ra T u,v U,V W x,y x0 ,y0
Biot number, Bi ¼ hk0 H nf specific heat at constant pressure (J kg1 K1) gravitational acceleration (m s2) local heat transfer coefficient (W m2 K1) distance between the bottom and top plates (m) hermal conductivity(W m1 K1) length of the exposed portion to convection (m) non-dimensional length of the exposed portion to convection, L ¼ ‘=H (see Fig. 1) Nusselt number, Nu = hH/kf Prandtl number, Pr = vf /af heat flux, (W m2) Rayleigh number, Ra = gb(TH TC)H3/vfaf dimensional temperature (°C) dimensional x and y components of velocity (m s1) dimensionless velocities, U = uH/af0 V = vH/af, width of the enclosure (m) dimensionless coordinates, x = x0 /H, y = y0 /H dimensional coordinates (m)
Greek symbols a thermal diffusivity, (m2 s1) b thermal expansion coefficient (K1) e numerical tolerance u nanoparticle volume fraction l dynamic viscosity, (N s m2) m kinematic viscosity (m2 s1) h dimensionless temperature, h = (T TC)/(TH TC) q density (kg m3) x dimensional vorticity (s1) X dimensionless vorticity, X = xH2/af w dimensional stream function (m2 s1) W dimensionless stream function, W ¼ w=af0 Subscripts avg average C cold f base fluid H hot nf nanofluid p particle w wall
been taken by various researchers during the recent years to improve the thermal conductivity of these fluids by suspending nano/micro particles in liquids [1]. Choi [2] is the first who used the term nanofluids to refer to such fluids having suspended nanoparticles. Choi et al. [3]showed that the addition of a small amount (less than 1% by volume) of nanoparticles can be effective in enhancing the heat transfer in liquids. There are numerous studies on forced convection applications using nanofluids and the reader is referred to the recent published review articles [4–7]. For natural convection, more publications are launched recently that study heat transfer enhancement using nanofluids. Putra et al. [8], Wen and Ding [9], Ho et al. [10], Agwa Nnanna [11] and Li and Peterson [12] studied experimentally the heat transfer enhancement due to the addition of nanoparticles. Besides, various computational
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Fig. 1. Physical model and coordinate system.
Table 1 Thermophysical properties of fluid and nanoparticles [16,18]. Physical Properties
Fluid phase (water)
CuO
cp (J/kgK) q (kg/m3) k (W/mK) a 107 (m2 s) b 105 (1/K)
4179 997.1 0.613 1.47 21
540 6500 18 131.7 0.85
studies focused on heat transfer enhancement due to the presence of nanoparticles. For example, Khanafer et al. [13], Oztop and Abu-Nada [14], Abu-Nada [15–17], Abu-Nada et al. [18], Abu-Nada and Chamkha [19], Aminossadati and Ghasemi [20], Ghasemi and Aminossadati [21], Kim et al. [22], Ög˘üt [23] studied numerically heat transfer enhancement due to the presence of nanoparticles in natural convection. Other theoretical studies that focused on natural convection are those of Hwang et al. [24] and Santra et al. [25] and Ho et al. [26].Besides, different effects on mechanisms of heat transfer in natural convection have been investigated recently by researchers to such as the role of Brownian motion and thermophoresis effects [27–29]. The main aim of this work is to present the effects of partial heating by convection The current study will evaluate the impact of nanoparticles on heat transfer enhancement in natural convection where the top wall of the enclosure is exposed to forced convection. The enhancement in heat transfer will be investigated under a wide range of volume fraction of nanoparticles, wide range of Rayleigh numbers, Biot number and the length of the partially exposed part to convection. 2. Model and governing equations Fig. 1 presents a schematic diagram of the differentially heated enclosure where the top wall is exposed partially to forced convection. The left wall was heated and maintained at a constant temperature (TH) higher than the right cold wall temperature (TC). The top wall is exposed to forced convection. The length of the exposed part to convection is denoted by ‘ where it changes 0 ‘ W. For the case of ‘ ¼ 0 the top wall is completely insulated and the case of ‘ ¼ W, the top wall is completely exposed to convection. The enclosure is considered square (i.e., W = H). The nanofluid is assumed incompressible and the flow is assumed to be laminar. It is assumed that the base fluid (i.e. water) and the nanoparticles are in thermal equilibrium and no slip occurs between
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E. Abu-Nada et al. / Superlattices and Microstructures 51 (2012) 381–395 Table 2 Variation of average Nusselt number for Ra = 103 and L = 0.5 at different volume fraction and Biot number. Bi 0 1 10 1
u
Nu
0 0.1 0 0.1 0 0.1 0 0.1
1.126 1.418 1.165 1.466 1.277 1.609 1.363 1.715
Table 3 Variation of average Nusselt number for Ra = 104 and L = 0.5 at different volume fraction and Biot number. Bi 0 1 10 1
u
Nu
0 0.1 0 0.1 0 0.1 0 0.1
2.112 2.793 2.363 2.865 2.575 3.126 2.774 3.363
Table 4 Variation of average Nusselt number for Ra = 105 and L = 0.5 at different volume fraction and Biot number. Bi 0 1 10 1
u
Nu
0 0.1 0 0.1 0 0.1 0 0.1
4.90 5.951 4.968 6.037 5.291 6.436 5.729 6.953
Table 5 Variation of average Nusselt number for Ra = 105 and Bi = 1.0 at different volume fraction and Biot number. L
u
Nu
0.25
0 0.1 0 0.1 0 0.1
4.010 5.979 4.968 6.037 5.138 6.250
0.50 1.00
them. The thermo physical properties of the nanofluid are given in Table 1. Thermo-physical properties of the nanofluid are assumed to be constant except for the density variation, which is
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Table 6 Grid independency check for Bi = 0, u = 0.0, L = 0.5 and Ra = 105. Grid dimension
Average Nusselt number
31 31 41 41 51 51 61 61 71 71
3.89 4.30 4.90 4.91 4.93
approximated by the Boussinesq model. The governing equations for the laminar, two-dimensional, steady state natural convection in terms of the stream function-vorticity formulation are written as: 2.1. Vorticity
lnf @ x @ x @ @w @ @w ¼ þ g½ubp þ ð1 uÞbf x x þ @x0 @y0 @y0 @x0 qnf @x02 @y02
ð1Þ
2.2. Energy
! @ @w @ @w @2T @2T ¼ T T a þ nf @x0 @y0 @y0 @x0 @x02 @y02
ð2Þ
2.3. Kinematics
@2w @2w þ ¼ x @x02 @y02
ð3Þ
where lnf is the effective dynamic viscosity of the nanofluid, anf is the effective thermal diffusivity of the nanofluid and qnf is the effective density of the nanofluid, which are given by
lnf ¼
lf keff ; anf ¼ ; qnf ¼ ð1 uÞqf þ uqp 2:5 ð q cp Þnf ð1 uÞ
ð4Þ
(qcp) being the heat capacitance of the nanofluid and knf is the effective thermal conductivity of the nanofluid that are expressed as
ðqcp Þnf ¼ ð1 uÞðqcp Þf þ uðqcp Þp ;
knf kp þ 2kf 2uðkf kp Þ ¼ kf kp þ 2kf uðkf kp Þ
ð5Þ
The effective thermal conductivity of the nanofluid knf is approximated by the Maxwell–Garnetts model. This model is found to be appropriate for studying heat transfer enhancement using nanofluids as it is adopted by various researchers in literature. The effective viscosity of the nanofluid lnf is assumed to follow the Brinkman law as a viscosity of a base fluid lf containing dilute suspension of fine spherical particles. The horizontal and vertical velocities are given by the following relations,
u¼
@w ; @y0
m¼
@w ; @x0
The following dimensionless variables are introduced:
ð6Þ
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0
y ¼ yH ;
x ¼ xH ;
X ¼ a x=H2 ;
W ¼ awf ;
U ¼ a u=H ; f
f
h¼
V ¼ a m=H f
ð7Þ
TT C T H T c
where L = l/H. Thus, the governing equations are re-written in dimensionless form as: @ @x
X @@yW @y@ X @@xW ¼
@2 X þ 2 @y ð1uÞ @h þRa Pr½ð1 uÞ þ ubp =bf @h cos/ @y sin/ @x 0:25
Pr ð1uþuqp =qf
@2 X @x2
knf =kf @ @W @ @W ¼ h h @x @y @y @x ð1 u þ uðqcp Þp =ðqcp Þf
@2h @2h þ @x2 @y2
ð8Þ
!
@2W @2W þ 2 ¼ X @x2 @y
ð9Þ
ð10Þ
where Ra is the Rayleigh number and Pr is the Prandtl number, which are defined as
Ra ¼
gbðT H T L ÞH3
mf a f
;
Pr ¼
mf af
ð11Þ
The dimensionless horizontal and vertical velocities are converted to
U¼
@W ; @y
V ¼
@W ; @x
ð12Þ
The dimensionless boundary conditions can be written as:
on the left wall; i:e: x ¼ 0 :
W ¼ 0;
on the right wall i:e: x ¼ 1 : on the bottom walls : W ¼ 0;
2
X ¼ @@xW2 ;
W ¼ 0;
2
X ¼ @@xW2 ; 2
X ¼ @@yW2 ;
@h @y
h¼1 h¼0
ð13Þ
¼0
on the top wall (exposed length to convection):
q00conduction ¼ q00convection k dT ¼ hðT w T C Þ dy
ð14Þ
Using the non-dimensional quantities given in Eq. (7), from (14), we get
dh ¼ Bihw dy
ð15Þ
where, Bi ¼ hk0 H is the Biot number defined based on nanofluid properties, knf is the thermal conductivnf ity of the nanofluid and h0 is the local heat transfer coefficient between the fluid surrounding the enclosure and the top wall. Therefore, the boundary conditions on the exposed portion to forced convection at the top wall are
W ¼ 0;
X¼
@2W ; @y2
dh ¼ Bih dy
ð16Þ
For the non-exposed portion at the top wall, the boundary conditions are
W ¼ 0;
X¼
@2W ; @y2
dh ¼0 dy
ð17Þ
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3. Numerical implementation Eq. (8) through Eq. (10) and using the corresponding boundary conditions Eq. (13) are solved using a finite volume method [30–32]. The diffusion term in the vorticity and energy equations is approximated by a second-order central difference scheme which gives a stable solution. Furthermore, a second order upwind differencing scheme is adopted for the convective terms. The resulted algebraic equations are solved using successive over/under relaxation method. Successive under relaxation was used due to the non-linear nature of the governing equations especially for the vorticity equation at high Rayleigh numbers. The convergence criterion is defined by the following expression:
Pj¼M Pi¼N
nþ1 /n j i¼1 j/ < 106 Pj¼M Pi¼N nþ1 j/ j i¼1 j¼1
j¼1
e¼
ð18Þ
where u stands for W, X, and h and the symbol e is the tolerance; M and N are the number of grid points in the x and y directions, respectively. Grid independency check is also done for case of Bi = 0, u = 0.0, L = 0.5 and Ra = 105 in Table 6. As seen from the table, 51 51 grid dimension is enough for calculations. The vorticity at the enclosure walls is expressed as:
X¼
ð8Wi;j W2j Þ
ð19Þ
2ðDyÞ2
After solving for W, X, and h, more useful quantities for engineering applications are obtained. For example, the local Nusselt number can be expressed as:
Nuy ¼
hH kf
ð20Þ
The heat transfer coefficient is computed from:
h¼
qw TH TC
ð21Þ
The thermal conductivity of the nanofluid is expressed as:
knf ¼
qw @T=@x
ð22Þ
Fig. 2. Comparison of results with literature [13,32].
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Fig. 3. Effects of Biot number on streamlines (left column) and isotherms (right column) for Ra = 105, L = 0.25 and u = 0.1: (a) Bi = 0; (b) Bi = 10; (c) Bi = 1.
Substituting Eqs. (21) and (22) into Eq. (20), and using the dimensionless quantities, the local Nusselt number along the heated wall can be written as:
knf @h Nuy ¼ kf @x
ð23Þ
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where ðK nf =kf Þ is calculate using the MG model. Finally, the average Nusselt number is determined from:
Nuavg ¼
Z
1
Nuy ðyÞdy
ð24Þ
0
Fig. 4. Effects of partial convection length on streamlines (left column) and isotherms (right column) for Ra = 105, Bi = 1 and u = 0.1: (a) L = 0.5; (b) L = 0.75; (c) L = 1.0.
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Fig. 5. Effects of volume fraction on streamlines (left column) and isotherms (right column) for Ra = 105, Bi = 1 and L = 0.5: (a) u = 0.04; (b) u = 0.06; (c) u = 0.08.
To evaluate Eq. (24), a 1/3 Simpson’s rule of integration is implemented. For convenience, a normalized local and average Nusselt numbers are defined as the ratio of Nusselt number at any volume fraction of nanoparticles to that of pure water and are given, respectively as:
uÞ ¼ Nuð
NuðuÞ Nuavg aver ðuÞ ¼ ; Nu Nuðu ¼ 0Þ Nuavg ðu ¼ 0Þ
ð25Þ
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4. Results and discussion Natural convection heat transfer in a square enclosure exposed to partial convection heat transfer is studied numerically. The enclosure is filled with water/CuO nanofluid with different volume fractions of nanoparticles. The calculations were carried out for different Rayleigh number, Biot number, volume fraction of nanoparticles, and the exposed length to convection (L), see Fig. 1. Fig. 2 shows the comparison of results with literature. Experimental works are not available to compare studied data. But as seen from the figure, comparisonal data show good agreement with literature. Fig. 3 illustrates the streamlines (on the left) and isotherms (on the right) for different Biot numbers at Ra = 105, l = 0.25L and u ¼ 0:1. For Bi = 0, the problem illustrates the typical natural convection problem in a differentially heated enclosure where the top wall is completely insulated. It is demonstrated that the flow strength increases with increasing of Biot number. Near the part exposed to convection the isotherms are distorted due to increasing of convection (Fig. 3(b)). Length of main cell also becomes shorter by increasing Bi. For the case of Bi = 1, the isotherms at the top partial exposed part
Fig. 6. Effects of Rayleigh number on the variation of local Nusselt number along the heated wall for L = 0.5, Bi = 104 and u = 0.1.
Fig. 7. Variation of the local Nusselt number for different Biot numbers and volume fraction for Ra = 104 and L = 1.0.
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Fig. 8. Variation of the local Nusselt numbers for different L: Ra = 105, Bi = 1000 and u = 0.1.
Fig. 9. Variation of the local Nusselt numbers using different Biot numbers for Ra = 105, L = 0.5, u = 0.1 and Bi = 0.0.
are almost parallel to the wall, which indicates higher rates of heat transfer from the enclosure as portrayed in Fig. 3(c). For the case of Bi = 1, Fig. 4 portrays the effects of partial exposed convection length (L) on the flow and heat patterns. It is noticed that flow strength is directly related to L. Shape of main cell and its inclination from horizontal is affected by L. Also, it is evident how the thermal boundary layer at the heated wall is affected by L where the thickness of the thermal boundary layer approached zero at the upper end of the left heated wall for the case of L = 1.0, which make the heat transfer more effective at the upper portion of the heated wall. Fig. 5 demonstrates the effect of volume fraction of nanoparticles on the streamlines and isotherms. It is shown that more addition of nanoparticles causes the flow strength to increase since the heat transfer within the fluid is enhanced due to the presence of high thermal conductivity of nanoparticles which enhances thermo convection motion within the enclosure and accordingly the flow strength. The thermal boundary thickness at the heated and cold walls seems to be not influenced due to the increase percentage of nanoparticles, but the heat transfer will definitely be affected as will be shown later in this section when we discuss the Nusselt number.
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Fig. 10. Non-dimensional Nusselt numbers contour maps for Bi = 1: (a) Ra = 105; (b) Ra = 104; (c) Ra = 103.
Fig. 6 presents the effects of Rayleigh number on the local Nusselt number distribution along the left heated wall of the enclosure for the case of L = 0.5 and u ¼ 0:1, and Bi = 104. It is clear that by increasing Rayleigh number the local Nusselt number increase. However, for y approximately greater than 0.9 the heat transfer becomes less sensitive to Rayleigh number. Fig. 7 illustrates the effect of Bi and u on the local Nusselt number for L = 1.0 and Ra = 104. It is shown that the local Nusselt number increase dramatically for the case of high Bi for y > 0.9. Besides, it is clear that for the case of high Bi number the effect of nanoparticles are highly pronounced for y > 0.9 since the top wall will be totally exposed to convection and thermal boundary thickness the top portion of the left heated wall will be very small and the temperature gradient are very high. However, this behavior is not noticed for the case of low Bi number. Similar behavior is observed for the case of Ra = 105. Effect of L on the local Nusselt number distribution at the left heated wall is shown in Fig. 8. It is shown that the local Nusselt number distribution along the heated wall is almost the same for the cases of L 6 0.75; however, the case of L = 1.0 shows more enhancement in local Nusselt number at the top portion of the heated wall due to the increase in convection that will cause a very small thickness of the thermal boundary layer at the top portion of the heated wall. Fig. 9 shows the effect of the Biot number Bi on the local Nusselt number distribution at the heated wall where higher values of Bi shows higher rates of heat transfer at the heated wall. Finally, Fig. 10 gives contour maps for the variation of non-dimensional average Nusselt number for the whole range of Rayleigh numbers. It is shown that that location of maximum value of non-dimensional average
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Nusselt number is sensitive to Rayleigh number, L, and u. This figure is useful in identifying places where major enhancement of heat is taking place for a given value of L. Tables 2–4 illustrate the effects of Rayleigh number and L = 0.5 on average Nusselt number for different parameters for different Biot numbers and nano particle volume fraction. As seen from all tables, heat transfer increases with increasing of volume fraction and Biot numbers for low Rayleigh numbers. Convection heat transfer increases with increasing of Biot numbers even at low values of Rayleigh number. For Bi = 0 and u ¼ 0, conduction mode of heat transfer becomes dominant. Heat transfer is also increases with length of exposed convection part of enclosure. Table 5 illustrate the effects of heater length on heat transfer. As an expected result, heat transfer increases with increasing of heater length. 5. Conclusions A numerical study has been performed in this paper to present the thermal and flow field due to buoyancy force in a square enclosure exposed to partial convection filled with water based CuO–water nanofluid. The study was performed for pertinent parameters. The observed results showed that for the case of high Biot number the heat transfer along the heated is enhanced by increasing the Rayleigh number. However, the enhancement is less pronounced at the top portion of the heated wall (y > 0.9). When the top wall is totally exposed to convection, the results prevail that the heat transfer is more efficient at high Biot number especially at the top portion of the heated wall. For the case of high Biot number, the results show that the heat transfer at the upper portion of the heated wall increase considerably at high L and for L 6 0.75 the effect of the effect of L is less pronounced. Contour plots for the average non-dimensional Nusselt number (i.e., percentage of heat transfer enhancement) are presented and it shown that the location of maximum enhancement in heat transfer is sensitive to Ra, u and L. References [1] S. Kakac, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (2009) 3187–3196. [2] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: The Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, USA, ASME, FED 231/MD 66 (1995) 99–105. [3] S.U.S. Choi, Z.G. Zhang, W. Yu, F.E. Lockwood, E.A. Grulke, Anomalously thermal conductivity enhancement in nanotube suspensions, Appl. Phys. Lett. 79 (2001) 2252–2254. [4] W. Daungthongsuk, S. Wongwises, A critical review of convective heat transfer nanofluids, Renew. Sust. Energy Rev. 11 (2007) 797–817. [5] V. Trisaksri, S. Wongwises, Critical review of heat transfer characteristics of nanofluids, Renew. Sust. Energy Rev. 11 (2007) 512–523. [6] X.Q. Wang, A.S. Mujumdar, Heat transfer characteristics of nanofluids: a review, Int. J. Thermal Sci. 46 (2007) 1–19. [7] R. Saidur, K.Y. Leong, H.A. Mohammad, A review on applications and challenges of nanofluids, Renew. Sust. Energy Rev. 15 (2011) 1646–1668. [8] N. Putra, W. Roetzel, S.K. Das, Natural convection of nanofluids, Heat Mass Transfer 39 (2003) 775–784. [9] D. Wen, Y. Ding, Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions, Int. J. Heat Mass Transfer 47 (2004) 5181–5188. [10] C.J. Ho, W.K. Liu, Y.S. Chang, C.C. Lin, Natural conevction heat transfer of alumina–water nanofluid in vertical square enclosures: an experimental study, Int. J. Thermal Sci. 49 (2010) 1345–1353. [11] A.G. Agwa Nnana, Experimental model of temperature-driven nanofluid, ASME, J. Heat Transfer 129 (2007) 697–704. [12] C.H. Li, G.P. Peterson, Experimental studies of natural convection heat transfer of Al2O3/DI water nanoparticle suspensions (nanofluids), Adv. Mech. Eng. (2010) 742739. [13] 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. [14] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosure filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326–1336. [15] E. Abu-Nada, Effects of variable viscosity and thermal conductivity of Al2O3–water nanofluid on heat transfer enhancement in natural convection, Int. J. Heat Fluid Flow 30 (2009) 679–690. [16] E. Abu-Nada, Effects of variable viscosity and thermal conductivity of CuO–water nanofluid on heat transfer enhancement in natural convection: mathematical model and simulation, ASME J. Heat Transfer 132 (2010) 052401. [17] E. Abu-Nada, Rayleigh–Benard convection in nanofluids: effect of temperature dependent properties, Int. J. Thermal Sci. 50 (2011) 1720–1730. [18] E. Abu-Nada, Z. Masoud, H. Oztop, A. Campo, Effect of nanofluid variable properties on natural convection in enclosures, Int. J. Thermal Sci. 49 (2010) 479–491.
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395
[19] E. Abu-Nada, A. Chamkha, Effect of nanofluid variable properties on natural convection in enclosures filled with CuO–EG– water nanofluid, Int. J. Thermal Sci. 49 (2010) 2339–2352. [20] S.M. Aminossadati, B. Ghasemi, Natural convection cooling of a localised heat source at the bottom of a nanofluid-filled enclosure, Eur. J. Mech. B/Fluids 28 (2009) 630–640. [21] B. Ghasemi, S.M. Aminossadati, Brownian motion of nanoparticles in a triangular enclosure with natural convection, Int. J. Thermal Sci. 49 (2010) 931–940. [22] J. Kim, Y.T. Kang, C.K. Choi, Analysis of convective instability and heat characteristics of nanofluids, Phys. Fluids 16 (2004) 2395–2401. [23] E.B. Ög˘üt, Natural convection of water-based nanofluid in an inclined enclosure with a heat source, Int. J. Thermal Sci. 48 (2009) 2063–2073. [24] K.S. Hwang, J.H. Lee, S.P. Jang. Buoyancy-driven heat transfer of water-based Al2O3nanofluids in a rectangular cavity, Int. J. Heat Mass Transfer 50 (2007) 4003–4010. [25] A.K. Santra, S. Sen, N. Chakraborty, Study of heat transfer characteristics of copper–water nanofluid in a differentially heated square cavity with different viscosity models, J. Enhanced Heat Transfer 15 (2008) 273–287. [26] C.J. Ho, M.W. Chen, Z.W. Li, Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity, Int. J. Heat Mass Transfer 47 (2008) 4506–4516. [27] A.V. Kuznetsov, D.A. Nield, Natural convective boundary-layer flow of a nanofluid past a vertical plate, Int. J. Therm. Sci. 49 (2010) 243–247. [28] D.A. Nield, A.V. Kuznetsov, The onset of convection in a horizontal nanofluid layer of finite depth, Int. J. Mech. B/Fluids 29 (2010) 217–223. [29] N. Bachok, A. Ishak, I. Pop, Boundary-layer flow of nanofluids over a moving surface in a flowing fluid, Int. J. Thermal Sci. 49 (2010) 1663–1668. [30] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Taylor and Francis Group, New York, 1980. [31] H.K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamic: The Finite Volume Method, John Wiley, New York, 1995. [32] R.J. Krane, J. Jessee. Some detailed field measurements for a natural convection flow in a vertical square enclosure, in: 1st ASME–JSME Thermal Engineering Joint Conference 1 (1983) 323–329.
Contents lists available at SciVerse ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Buoyancy induced flow in a nanofluid filled enclosure partially exposed to forced convection Eiyad Abu-Nada a, Hakan F. Oztop b,⇑, Ioan Pop c a b c
Department of Mechanical Engineering, King Faisal University, Al-Ahsa 31982, Saudi Arabia Department of Mechanical Engineering, Fırat University, Elazig TR-23119, Turkey Faculty of Mathematics and Computer Science, Babesß-Bolyaiu University, 400082 Cluj-Napoca, Romania
a r t i c l e
i n f o
Article history: Received 5 October 2011 Received in revised form 22 December 2011 Accepted 3 January 2012 Available online 11 January 2012 Keywords: Nanofluid Natural convection Partially exposed enclosure
a b s t r a c t A numerical study was performed on natural convection for water– CuO nanofluid filled enclosure where the top surface was partially exposed to convection. The cavity has a square cross-section and differentially heated. Except exposed convection part on the top, all sides are adiabatic on horizontal walls. Effects of Rayleigh number (103 6 Ra 6 105), Biot number (0 6 Bi 6 1), length of partial convection (0.0 6 L 6 1.0) and volume fraction of nanoparticles (0.0 6 u 6 0.1) on heat and fluid flow were investigated. The results showed that for the case of high Biot number that heat transfer along the heated was enhanced by increasing the Rayleigh number mainly at the upper portion of the heated wall. When the top wall was totally exposed to convection, the results prevail that the heat transfer was more effective at high Biot number especially at the upper portion of the heated wall. For the case of high Biot number, the results prevailed that the heat transfer at the upper portion of the heated wall increases considerably at high exposed length to convection (L); however, for L 6 0.75 the effect of L was less pronounced. Contour maps for percentage of heat transfer enhancement were presented and it was shown that the location of maximum enhancement in heat transfer was sensitive to Ra, u and L. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Conventional heat transfer fluids, including oil, water, and ethylene glycol mixtures are poor heat transfer fluids due to the poor intrinsic thermal conductivity of these fluids. Numerous attempts have ⇑ Corresponding author. Tel.: +90 424 237 0000x4222; fax: +90 424 236 7064. E-mail address: [email protected] (H.F. Oztop). 0749-6036/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2012.01.002
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Nomenclature Bi cp g h H K ‘ L Nu Pr qw Ra T u,v U,V W x,y x0 ,y0
Biot number, Bi ¼ hk0 H nf specific heat at constant pressure (J kg1 K1) gravitational acceleration (m s2) local heat transfer coefficient (W m2 K1) distance between the bottom and top plates (m) hermal conductivity(W m1 K1) length of the exposed portion to convection (m) non-dimensional length of the exposed portion to convection, L ¼ ‘=H (see Fig. 1) Nusselt number, Nu = hH/kf Prandtl number, Pr = vf /af heat flux, (W m2) Rayleigh number, Ra = gb(TH TC)H3/vfaf dimensional temperature (°C) dimensional x and y components of velocity (m s1) dimensionless velocities, U = uH/af0 V = vH/af, width of the enclosure (m) dimensionless coordinates, x = x0 /H, y = y0 /H dimensional coordinates (m)
Greek symbols a thermal diffusivity, (m2 s1) b thermal expansion coefficient (K1) e numerical tolerance u nanoparticle volume fraction l dynamic viscosity, (N s m2) m kinematic viscosity (m2 s1) h dimensionless temperature, h = (T TC)/(TH TC) q density (kg m3) x dimensional vorticity (s1) X dimensionless vorticity, X = xH2/af w dimensional stream function (m2 s1) W dimensionless stream function, W ¼ w=af0 Subscripts avg average C cold f base fluid H hot nf nanofluid p particle w wall
been taken by various researchers during the recent years to improve the thermal conductivity of these fluids by suspending nano/micro particles in liquids [1]. Choi [2] is the first who used the term nanofluids to refer to such fluids having suspended nanoparticles. Choi et al. [3]showed that the addition of a small amount (less than 1% by volume) of nanoparticles can be effective in enhancing the heat transfer in liquids. There are numerous studies on forced convection applications using nanofluids and the reader is referred to the recent published review articles [4–7]. For natural convection, more publications are launched recently that study heat transfer enhancement using nanofluids. Putra et al. [8], Wen and Ding [9], Ho et al. [10], Agwa Nnanna [11] and Li and Peterson [12] studied experimentally the heat transfer enhancement due to the addition of nanoparticles. Besides, various computational
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Fig. 1. Physical model and coordinate system.
Table 1 Thermophysical properties of fluid and nanoparticles [16,18]. Physical Properties
Fluid phase (water)
CuO
cp (J/kgK) q (kg/m3) k (W/mK) a 107 (m2 s) b 105 (1/K)
4179 997.1 0.613 1.47 21
540 6500 18 131.7 0.85
studies focused on heat transfer enhancement due to the presence of nanoparticles. For example, Khanafer et al. [13], Oztop and Abu-Nada [14], Abu-Nada [15–17], Abu-Nada et al. [18], Abu-Nada and Chamkha [19], Aminossadati and Ghasemi [20], Ghasemi and Aminossadati [21], Kim et al. [22], Ög˘üt [23] studied numerically heat transfer enhancement due to the presence of nanoparticles in natural convection. Other theoretical studies that focused on natural convection are those of Hwang et al. [24] and Santra et al. [25] and Ho et al. [26].Besides, different effects on mechanisms of heat transfer in natural convection have been investigated recently by researchers to such as the role of Brownian motion and thermophoresis effects [27–29]. The main aim of this work is to present the effects of partial heating by convection The current study will evaluate the impact of nanoparticles on heat transfer enhancement in natural convection where the top wall of the enclosure is exposed to forced convection. The enhancement in heat transfer will be investigated under a wide range of volume fraction of nanoparticles, wide range of Rayleigh numbers, Biot number and the length of the partially exposed part to convection. 2. Model and governing equations Fig. 1 presents a schematic diagram of the differentially heated enclosure where the top wall is exposed partially to forced convection. The left wall was heated and maintained at a constant temperature (TH) higher than the right cold wall temperature (TC). The top wall is exposed to forced convection. The length of the exposed part to convection is denoted by ‘ where it changes 0 ‘ W. For the case of ‘ ¼ 0 the top wall is completely insulated and the case of ‘ ¼ W, the top wall is completely exposed to convection. The enclosure is considered square (i.e., W = H). The nanofluid is assumed incompressible and the flow is assumed to be laminar. It is assumed that the base fluid (i.e. water) and the nanoparticles are in thermal equilibrium and no slip occurs between
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E. Abu-Nada et al. / Superlattices and Microstructures 51 (2012) 381–395 Table 2 Variation of average Nusselt number for Ra = 103 and L = 0.5 at different volume fraction and Biot number. Bi 0 1 10 1
u
Nu
0 0.1 0 0.1 0 0.1 0 0.1
1.126 1.418 1.165 1.466 1.277 1.609 1.363 1.715
Table 3 Variation of average Nusselt number for Ra = 104 and L = 0.5 at different volume fraction and Biot number. Bi 0 1 10 1
u
Nu
0 0.1 0 0.1 0 0.1 0 0.1
2.112 2.793 2.363 2.865 2.575 3.126 2.774 3.363
Table 4 Variation of average Nusselt number for Ra = 105 and L = 0.5 at different volume fraction and Biot number. Bi 0 1 10 1
u
Nu
0 0.1 0 0.1 0 0.1 0 0.1
4.90 5.951 4.968 6.037 5.291 6.436 5.729 6.953
Table 5 Variation of average Nusselt number for Ra = 105 and Bi = 1.0 at different volume fraction and Biot number. L
u
Nu
0.25
0 0.1 0 0.1 0 0.1
4.010 5.979 4.968 6.037 5.138 6.250
0.50 1.00
them. The thermo physical properties of the nanofluid are given in Table 1. Thermo-physical properties of the nanofluid are assumed to be constant except for the density variation, which is
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Table 6 Grid independency check for Bi = 0, u = 0.0, L = 0.5 and Ra = 105. Grid dimension
Average Nusselt number
31 31 41 41 51 51 61 61 71 71
3.89 4.30 4.90 4.91 4.93
approximated by the Boussinesq model. The governing equations for the laminar, two-dimensional, steady state natural convection in terms of the stream function-vorticity formulation are written as: 2.1. Vorticity
lnf @ x @ x @ @w @ @w ¼ þ g½ubp þ ð1 uÞbf x x þ @x0 @y0 @y0 @x0 qnf @x02 @y02
ð1Þ
2.2. Energy
! @ @w @ @w @2T @2T ¼ T T a þ nf @x0 @y0 @y0 @x0 @x02 @y02
ð2Þ
2.3. Kinematics
@2w @2w þ ¼ x @x02 @y02
ð3Þ
where lnf is the effective dynamic viscosity of the nanofluid, anf is the effective thermal diffusivity of the nanofluid and qnf is the effective density of the nanofluid, which are given by
lnf ¼
lf keff ; anf ¼ ; qnf ¼ ð1 uÞqf þ uqp 2:5 ð q cp Þnf ð1 uÞ
ð4Þ
(qcp) being the heat capacitance of the nanofluid and knf is the effective thermal conductivity of the nanofluid that are expressed as
ðqcp Þnf ¼ ð1 uÞðqcp Þf þ uðqcp Þp ;
knf kp þ 2kf 2uðkf kp Þ ¼ kf kp þ 2kf uðkf kp Þ
ð5Þ
The effective thermal conductivity of the nanofluid knf is approximated by the Maxwell–Garnetts model. This model is found to be appropriate for studying heat transfer enhancement using nanofluids as it is adopted by various researchers in literature. The effective viscosity of the nanofluid lnf is assumed to follow the Brinkman law as a viscosity of a base fluid lf containing dilute suspension of fine spherical particles. The horizontal and vertical velocities are given by the following relations,
u¼
@w ; @y0
m¼
@w ; @x0
The following dimensionless variables are introduced:
ð6Þ
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0
y ¼ yH ;
x ¼ xH ;
X ¼ a x=H2 ;
W ¼ awf ;
U ¼ a u=H ; f
f
h¼
V ¼ a m=H f
ð7Þ
TT C T H T c
where L = l/H. Thus, the governing equations are re-written in dimensionless form as: @ @x
X @@yW @y@ X @@xW ¼
@2 X þ 2 @y ð1uÞ @h þRa Pr½ð1 uÞ þ ubp =bf @h cos/ @y sin/ @x 0:25
Pr ð1uþuqp =qf
@2 X @x2
knf =kf @ @W @ @W ¼ h h @x @y @y @x ð1 u þ uðqcp Þp =ðqcp Þf
@2h @2h þ @x2 @y2
ð8Þ
!
@2W @2W þ 2 ¼ X @x2 @y
ð9Þ
ð10Þ
where Ra is the Rayleigh number and Pr is the Prandtl number, which are defined as
Ra ¼
gbðT H T L ÞH3
mf a f
;
Pr ¼
mf af
ð11Þ
The dimensionless horizontal and vertical velocities are converted to
U¼
@W ; @y
V ¼
@W ; @x
ð12Þ
The dimensionless boundary conditions can be written as:
on the left wall; i:e: x ¼ 0 :
W ¼ 0;
on the right wall i:e: x ¼ 1 : on the bottom walls : W ¼ 0;
2
X ¼ @@xW2 ;
W ¼ 0;
2
X ¼ @@xW2 ; 2
X ¼ @@yW2 ;
@h @y
h¼1 h¼0
ð13Þ
¼0
on the top wall (exposed length to convection):
q00conduction ¼ q00convection k dT ¼ hðT w T C Þ dy
ð14Þ
Using the non-dimensional quantities given in Eq. (7), from (14), we get
dh ¼ Bihw dy
ð15Þ
where, Bi ¼ hk0 H is the Biot number defined based on nanofluid properties, knf is the thermal conductivnf ity of the nanofluid and h0 is the local heat transfer coefficient between the fluid surrounding the enclosure and the top wall. Therefore, the boundary conditions on the exposed portion to forced convection at the top wall are
W ¼ 0;
X¼
@2W ; @y2
dh ¼ Bih dy
ð16Þ
For the non-exposed portion at the top wall, the boundary conditions are
W ¼ 0;
X¼
@2W ; @y2
dh ¼0 dy
ð17Þ
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3. Numerical implementation Eq. (8) through Eq. (10) and using the corresponding boundary conditions Eq. (13) are solved using a finite volume method [30–32]. The diffusion term in the vorticity and energy equations is approximated by a second-order central difference scheme which gives a stable solution. Furthermore, a second order upwind differencing scheme is adopted for the convective terms. The resulted algebraic equations are solved using successive over/under relaxation method. Successive under relaxation was used due to the non-linear nature of the governing equations especially for the vorticity equation at high Rayleigh numbers. The convergence criterion is defined by the following expression:
Pj¼M Pi¼N
nþ1 /n j i¼1 j/ < 106 Pj¼M Pi¼N nþ1 j/ j i¼1 j¼1
j¼1
e¼
ð18Þ
where u stands for W, X, and h and the symbol e is the tolerance; M and N are the number of grid points in the x and y directions, respectively. Grid independency check is also done for case of Bi = 0, u = 0.0, L = 0.5 and Ra = 105 in Table 6. As seen from the table, 51 51 grid dimension is enough for calculations. The vorticity at the enclosure walls is expressed as:
X¼
ð8Wi;j W2j Þ
ð19Þ
2ðDyÞ2
After solving for W, X, and h, more useful quantities for engineering applications are obtained. For example, the local Nusselt number can be expressed as:
Nuy ¼
hH kf
ð20Þ
The heat transfer coefficient is computed from:
h¼
qw TH TC
ð21Þ
The thermal conductivity of the nanofluid is expressed as:
knf ¼
qw @T=@x
ð22Þ
Fig. 2. Comparison of results with literature [13,32].
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Fig. 3. Effects of Biot number on streamlines (left column) and isotherms (right column) for Ra = 105, L = 0.25 and u = 0.1: (a) Bi = 0; (b) Bi = 10; (c) Bi = 1.
Substituting Eqs. (21) and (22) into Eq. (20), and using the dimensionless quantities, the local Nusselt number along the heated wall can be written as:
knf @h Nuy ¼ kf @x
ð23Þ
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where ðK nf =kf Þ is calculate using the MG model. Finally, the average Nusselt number is determined from:
Nuavg ¼
Z
1
Nuy ðyÞdy
ð24Þ
0
Fig. 4. Effects of partial convection length on streamlines (left column) and isotherms (right column) for Ra = 105, Bi = 1 and u = 0.1: (a) L = 0.5; (b) L = 0.75; (c) L = 1.0.
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Fig. 5. Effects of volume fraction on streamlines (left column) and isotherms (right column) for Ra = 105, Bi = 1 and L = 0.5: (a) u = 0.04; (b) u = 0.06; (c) u = 0.08.
To evaluate Eq. (24), a 1/3 Simpson’s rule of integration is implemented. For convenience, a normalized local and average Nusselt numbers are defined as the ratio of Nusselt number at any volume fraction of nanoparticles to that of pure water and are given, respectively as:
uÞ ¼ Nuð
NuðuÞ Nuavg aver ðuÞ ¼ ; Nu Nuðu ¼ 0Þ Nuavg ðu ¼ 0Þ
ð25Þ
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4. Results and discussion Natural convection heat transfer in a square enclosure exposed to partial convection heat transfer is studied numerically. The enclosure is filled with water/CuO nanofluid with different volume fractions of nanoparticles. The calculations were carried out for different Rayleigh number, Biot number, volume fraction of nanoparticles, and the exposed length to convection (L), see Fig. 1. Fig. 2 shows the comparison of results with literature. Experimental works are not available to compare studied data. But as seen from the figure, comparisonal data show good agreement with literature. Fig. 3 illustrates the streamlines (on the left) and isotherms (on the right) for different Biot numbers at Ra = 105, l = 0.25L and u ¼ 0:1. For Bi = 0, the problem illustrates the typical natural convection problem in a differentially heated enclosure where the top wall is completely insulated. It is demonstrated that the flow strength increases with increasing of Biot number. Near the part exposed to convection the isotherms are distorted due to increasing of convection (Fig. 3(b)). Length of main cell also becomes shorter by increasing Bi. For the case of Bi = 1, the isotherms at the top partial exposed part
Fig. 6. Effects of Rayleigh number on the variation of local Nusselt number along the heated wall for L = 0.5, Bi = 104 and u = 0.1.
Fig. 7. Variation of the local Nusselt number for different Biot numbers and volume fraction for Ra = 104 and L = 1.0.
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Fig. 8. Variation of the local Nusselt numbers for different L: Ra = 105, Bi = 1000 and u = 0.1.
Fig. 9. Variation of the local Nusselt numbers using different Biot numbers for Ra = 105, L = 0.5, u = 0.1 and Bi = 0.0.
are almost parallel to the wall, which indicates higher rates of heat transfer from the enclosure as portrayed in Fig. 3(c). For the case of Bi = 1, Fig. 4 portrays the effects of partial exposed convection length (L) on the flow and heat patterns. It is noticed that flow strength is directly related to L. Shape of main cell and its inclination from horizontal is affected by L. Also, it is evident how the thermal boundary layer at the heated wall is affected by L where the thickness of the thermal boundary layer approached zero at the upper end of the left heated wall for the case of L = 1.0, which make the heat transfer more effective at the upper portion of the heated wall. Fig. 5 demonstrates the effect of volume fraction of nanoparticles on the streamlines and isotherms. It is shown that more addition of nanoparticles causes the flow strength to increase since the heat transfer within the fluid is enhanced due to the presence of high thermal conductivity of nanoparticles which enhances thermo convection motion within the enclosure and accordingly the flow strength. The thermal boundary thickness at the heated and cold walls seems to be not influenced due to the increase percentage of nanoparticles, but the heat transfer will definitely be affected as will be shown later in this section when we discuss the Nusselt number.
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Fig. 10. Non-dimensional Nusselt numbers contour maps for Bi = 1: (a) Ra = 105; (b) Ra = 104; (c) Ra = 103.
Fig. 6 presents the effects of Rayleigh number on the local Nusselt number distribution along the left heated wall of the enclosure for the case of L = 0.5 and u ¼ 0:1, and Bi = 104. It is clear that by increasing Rayleigh number the local Nusselt number increase. However, for y approximately greater than 0.9 the heat transfer becomes less sensitive to Rayleigh number. Fig. 7 illustrates the effect of Bi and u on the local Nusselt number for L = 1.0 and Ra = 104. It is shown that the local Nusselt number increase dramatically for the case of high Bi for y > 0.9. Besides, it is clear that for the case of high Bi number the effect of nanoparticles are highly pronounced for y > 0.9 since the top wall will be totally exposed to convection and thermal boundary thickness the top portion of the left heated wall will be very small and the temperature gradient are very high. However, this behavior is not noticed for the case of low Bi number. Similar behavior is observed for the case of Ra = 105. Effect of L on the local Nusselt number distribution at the left heated wall is shown in Fig. 8. It is shown that the local Nusselt number distribution along the heated wall is almost the same for the cases of L 6 0.75; however, the case of L = 1.0 shows more enhancement in local Nusselt number at the top portion of the heated wall due to the increase in convection that will cause a very small thickness of the thermal boundary layer at the top portion of the heated wall. Fig. 9 shows the effect of the Biot number Bi on the local Nusselt number distribution at the heated wall where higher values of Bi shows higher rates of heat transfer at the heated wall. Finally, Fig. 10 gives contour maps for the variation of non-dimensional average Nusselt number for the whole range of Rayleigh numbers. It is shown that that location of maximum value of non-dimensional average
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Nusselt number is sensitive to Rayleigh number, L, and u. This figure is useful in identifying places where major enhancement of heat is taking place for a given value of L. Tables 2–4 illustrate the effects of Rayleigh number and L = 0.5 on average Nusselt number for different parameters for different Biot numbers and nano particle volume fraction. As seen from all tables, heat transfer increases with increasing of volume fraction and Biot numbers for low Rayleigh numbers. Convection heat transfer increases with increasing of Biot numbers even at low values of Rayleigh number. For Bi = 0 and u ¼ 0, conduction mode of heat transfer becomes dominant. Heat transfer is also increases with length of exposed convection part of enclosure. Table 5 illustrate the effects of heater length on heat transfer. As an expected result, heat transfer increases with increasing of heater length. 5. Conclusions A numerical study has been performed in this paper to present the thermal and flow field due to buoyancy force in a square enclosure exposed to partial convection filled with water based CuO–water nanofluid. The study was performed for pertinent parameters. The observed results showed that for the case of high Biot number the heat transfer along the heated is enhanced by increasing the Rayleigh number. However, the enhancement is less pronounced at the top portion of the heated wall (y > 0.9). When the top wall is totally exposed to convection, the results prevail that the heat transfer is more efficient at high Biot number especially at the top portion of the heated wall. For the case of high Biot number, the results show that the heat transfer at the upper portion of the heated wall increase considerably at high L and for L 6 0.75 the effect of the effect of L is less pronounced. Contour plots for the average non-dimensional Nusselt number (i.e., percentage of heat transfer enhancement) are presented and it shown that the location of maximum enhancement in heat transfer is sensitive to Ra, u and L. References [1] S. Kakac, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (2009) 3187–3196. [2] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: The Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, USA, ASME, FED 231/MD 66 (1995) 99–105. [3] S.U.S. Choi, Z.G. Zhang, W. Yu, F.E. Lockwood, E.A. Grulke, Anomalously thermal conductivity enhancement in nanotube suspensions, Appl. Phys. Lett. 79 (2001) 2252–2254. [4] W. Daungthongsuk, S. Wongwises, A critical review of convective heat transfer nanofluids, Renew. Sust. Energy Rev. 11 (2007) 797–817. [5] V. Trisaksri, S. Wongwises, Critical review of heat transfer characteristics of nanofluids, Renew. Sust. Energy Rev. 11 (2007) 512–523. [6] X.Q. Wang, A.S. Mujumdar, Heat transfer characteristics of nanofluids: a review, Int. J. Thermal Sci. 46 (2007) 1–19. [7] R. Saidur, K.Y. Leong, H.A. Mohammad, A review on applications and challenges of nanofluids, Renew. Sust. Energy Rev. 15 (2011) 1646–1668. [8] N. Putra, W. Roetzel, S.K. Das, Natural convection of nanofluids, Heat Mass Transfer 39 (2003) 775–784. [9] D. Wen, Y. Ding, Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions, Int. J. Heat Mass Transfer 47 (2004) 5181–5188. [10] C.J. Ho, W.K. Liu, Y.S. Chang, C.C. Lin, Natural conevction heat transfer of alumina–water nanofluid in vertical square enclosures: an experimental study, Int. J. Thermal Sci. 49 (2010) 1345–1353. [11] A.G. Agwa Nnana, Experimental model of temperature-driven nanofluid, ASME, J. Heat Transfer 129 (2007) 697–704. [12] C.H. Li, G.P. Peterson, Experimental studies of natural convection heat transfer of Al2O3/DI water nanoparticle suspensions (nanofluids), Adv. Mech. Eng. (2010) 742739. [13] 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. [14] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosure filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326–1336. [15] E. Abu-Nada, Effects of variable viscosity and thermal conductivity of Al2O3–water nanofluid on heat transfer enhancement in natural convection, Int. J. Heat Fluid Flow 30 (2009) 679–690. [16] E. Abu-Nada, Effects of variable viscosity and thermal conductivity of CuO–water nanofluid on heat transfer enhancement in natural convection: mathematical model and simulation, ASME J. Heat Transfer 132 (2010) 052401. [17] E. Abu-Nada, Rayleigh–Benard convection in nanofluids: effect of temperature dependent properties, Int. J. Thermal Sci. 50 (2011) 1720–1730. [18] E. Abu-Nada, Z. Masoud, H. Oztop, A. Campo, Effect of nanofluid variable properties on natural convection in enclosures, Int. J. Thermal Sci. 49 (2010) 479–491.
E. Abu-Nada et al. / Superlattices and Microstructures 51 (2012) 381–395
395
[19] E. Abu-Nada, A. Chamkha, Effect of nanofluid variable properties on natural convection in enclosures filled with CuO–EG– water nanofluid, Int. J. Thermal Sci. 49 (2010) 2339–2352. [20] S.M. Aminossadati, B. Ghasemi, Natural convection cooling of a localised heat source at the bottom of a nanofluid-filled enclosure, Eur. J. Mech. B/Fluids 28 (2009) 630–640. [21] B. Ghasemi, S.M. Aminossadati, Brownian motion of nanoparticles in a triangular enclosure with natural convection, Int. J. Thermal Sci. 49 (2010) 931–940. [22] J. Kim, Y.T. Kang, C.K. Choi, Analysis of convective instability and heat characteristics of nanofluids, Phys. Fluids 16 (2004) 2395–2401. [23] E.B. Ög˘üt, Natural convection of water-based nanofluid in an inclined enclosure with a heat source, Int. J. Thermal Sci. 48 (2009) 2063–2073. [24] K.S. Hwang, J.H. Lee, S.P. Jang. Buoyancy-driven heat transfer of water-based Al2O3nanofluids in a rectangular cavity, Int. J. Heat Mass Transfer 50 (2007) 4003–4010. [25] A.K. Santra, S. Sen, N. Chakraborty, Study of heat transfer characteristics of copper–water nanofluid in a differentially heated square cavity with different viscosity models, J. Enhanced Heat Transfer 15 (2008) 273–287. [26] C.J. Ho, M.W. Chen, Z.W. Li, Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity, Int. J. Heat Mass Transfer 47 (2008) 4506–4516. [27] A.V. Kuznetsov, D.A. Nield, Natural convective boundary-layer flow of a nanofluid past a vertical plate, Int. J. Therm. Sci. 49 (2010) 243–247. [28] D.A. Nield, A.V. Kuznetsov, The onset of convection in a horizontal nanofluid layer of finite depth, Int. J. Mech. B/Fluids 29 (2010) 217–223. [29] N. Bachok, A. Ishak, I. Pop, Boundary-layer flow of nanofluids over a moving surface in a flowing fluid, Int. J. Thermal Sci. 49 (2010) 1663–1668. [30] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Taylor and Francis Group, New York, 1980. [31] H.K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamic: The Finite Volume Method, John Wiley, New York, 1995. [32] R.J. Krane, J. Jessee. Some detailed field measurements for a natural convection flow in a vertical square enclosure, in: 1st ASME–JSME Thermal Engineering Joint Conference 1 (1983) 323–329.