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Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure
Advanced Powder Technology xxx (2016) xxx–xxx
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Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt
Original Research Paper
Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure A. Purusothaman a,b,⇑, N. Nithyadevi a, H.F. Oztop c, V. Divya a, K. Al-Salem d a
Department of Mathematics, Bharathiar University, Coimbatore, India Department of Mathematics, National Institute of Technology, Tiruchirappalli, India c Department of Mechanical Engineering, Technology Faculty, Firat University, Elazig, Turkey d Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia b
a r t i c l e
i n f o
Article history: Received 11 August 2015 Received in revised form 18 November 2015 Accepted 25 December 2015 Available online xxxx Keywords: Natural convection Equipment cooling Heater array Nanofluid
a b s t r a c t This study deals with the numerical analysis of 3D natural convection equipment cooling with a 3 3 array of isothermal heaters mounted on one vertical wall of the nanofluid filled enclosure. The enclosure is filled with water based nanofluid containing Copper (Cu) or Alumina (Al2O3) nanoparticles. The transport equations are solved by the finite volume method based on the SIMPLE algorithm with power-law scheme. The influence of pertinent parameters such as Rayleigh number (105 6 Ra 6 107 ), nanoparticle volume fraction (0% 6 / 6 10%) and enclosure side aspect ratio (1:0 6 AS 6 7:5) on the fluid flow and heat transfer characteristics are investigated together with different nanofluids and enclosure boundary conditions. It is observed that Cu–water nanofluid has the greatest effect on the equipment cooling performance compared with Al2O3–water nanofluid. Moreover, the row averaged Nusselt number increases monotonically with increase in both the Rayleigh number and the nanoparticle solid volume fraction. Ó 2015 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
1. Introduction The incredible developments in the electronic industry in the past few decades have resulted in a tremendous increase in microelectronic component density. The components generate heat and leave an adverse effect on the allowable operating temperature of the nearby components which in turn reduces the performance of the system. The motivation for the present work is the cooling of printed circuit boards in sealed enclosure. In this context, buoyancy induced natural convective flows play a significant role since passive cooling is the least expensive, noiseless, most reliable method of heating rejection and maintenance free. Natural convective heat transfer and fluid flow in enclosures with various shapes and wall conditions have been studied extensively by Ostrach [1] in the past. A comprehensive review of various cooling options which could be possibly used for electronic equipment cooling was discussed by Incropera [2]. Literature in this area is quite vast which includes theoretical as well as experimental results. Thermal control in electronic components using dielectric liquids has received increased attention due to inherently high heat ⇑ Corresponding author at: Department of Mathematics, Bharathiar University, Coimbatore, India. E-mail addresses: [email protected], [email protected] (A. Purusothaman).
removal capabilities of liquids compared to air. Wroblewski and Joshi [3] discussed the liquid immersion cooling of a substrate mounted protrusion in a three dimensional enclosure with opposite cold wall. The effect of the boundary conditions with the top cold wall and both top and opposite cold walls for the same physical configuration was further studied by Wroblewski and Joshi [4]. They found that when the ratio of substrate to fluid thermal conductivity is greater than 10, conduction plays a dominant role in cooling the chip through the substrate. Joshi et al. [5] carried out an experimental study of natural convection liquid immersion cooling with a 3 3 array of heaters embedded on one vertical wall of a 3D enclosure with top and bottom cold walls. They investigated the effect of the width of top and bottom surfaces and concluded that it was most significant at the lower power levels. The effect of the heater aspect ratio, Ahtr of 3 3 array of discrete heat sources flush mounted on one vertical wall of a rectangular enclosure with isothermally cooled opposite wall was studied numerically and experimentally by Heindel et al. [6,7]. For Ahtr 6 3 , average Nusselt number increased with decreasing Ahtr . However, for Ahtr P 3, the two and three-dimensional predictions differed within 5% of each other and results were approximately independent of Ahtr . These existing three dimensional numerical results were compared with the experimental predictions had archived with excellent agreement. The numerical
http://dx.doi.org/10.1016/j.apt.2015.12.012 0921-8831/Ó 2015 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
Please cite this article in press as: A. Purusothaman et al., Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2015.12.012
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A. Purusothaman et al. / Advanced Powder Technology xxx (2016) xxx–xxx
Nomenclature Ah AS b C cp D E g H k Lx ; Lz Nui Nui;k Nui;k p P Pr Ra t T u; v ; w U; V; W x; y; z
heater area (m2) enclosure side aspect ratio (¼ H=C) heater location (m) enclosure dimension in y direction (m) constant pressure specific heat (J kg1 K1) enclosure dimension in x direction (m) heater location (m) gravitational acceleration (m s2) enclosure dimension in z direction (m) thermal conductivity (W m1 K1) length and height of a heater (m) row averaged Nusselt number local Nusselt number of a heater average Nusselt number of a heater pressure (Pa) dimensionless pressure Prandtl number Rayleigh number dimensional time (s) temperature (K) dimensional velocity components (m s1) dimensionless velocity components dimensional cartesian coordinates (m)
X; Y; Z
dimensionless cartesian coordinates
Greek symbols thermal diffusivity (m2 s1) b thermal expansion coefficient (K1) l dynamic viscosity (Pa s) m kinematic viscosity (m2 s1) / nanoparticle volume fraction q density (kg m3) s dimensionless time h dimensionless temperature f general variable
a
Subscripts c cold f fluid h hot i row k column nf nanofluid p nanoparticle
analysis of natural convection liquid cooling of a 3 3 array of discrete heat sources flush-mounted on one vertical wall of a rectangular enclosure filled with various fluids (Pr ¼ 5; 9; 25 and 130) and cooled by the opposite wall was presented by Tou et al. [8]. The effects of modified Rayleigh number, Prandtl number and enclosure aspect ratio of heat transfer characteristics were investigated. They concluded that the row averaged Nusselt number increases with the Rayleigh number in the power of 0.27. Experimental investigation of an identical model at various inclinations was carried out by Tso et al. [9]. They found that the flow and temperature fields become complex and distorted when the enclosure was tilted.
The review of the literatures [3–9] indicates that the base fluids like water, oil and dielectric liquids have low thermal conductivity which provide some limitations upon achieving the maximum heat transfer performance. Hence, an innovative technique in which the colloidal suspension of nanometer sized particles in a base fluid was first introduced by Choi [10]. The novel approach of cooling by nanofluids improved the properties associated with heat transfer since then have substantially high thermal conductivities. Hence nanofluids are primarily used as coolant in heat transfer equipment such as heat exchangers, cooling of flush mounted electronic heaters and radiators. Studies on buoyancy driven convective heat transfer in an enclosure using nanofluids have drawn
z z g E Sx
Row 1 H
Row 2 Row 3 D
b H
Sz Lz Lx
x C
D/2
y
(a)
E
x
(b)
Fig. 1. Physical configuration and geometric parameters (a) full view of the enclosure (b) half view of sources mounted wall at Y = 0.
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A. Purusothaman et al. / Advanced Powder Technology xxx (2016) xxx–xxx Table 1 Properties of nanofluid material [13,23,24]. Property
Water
q (kg m3) 1
1
cp (J kg K ) K (W m1 K1) b 10
5
(1 K
1
)
a 107 (m2 s1)
Table 2 Benchmarking the present code with Tric et al. [29]. Nanoparticle
Rac
Nusselt number
Tric et al. [29]
Present
Error %
Cu
Al2 O3
4
10
997.1 4179 0.613 21
8933 385 400 1.67
3970 765 40 0.85
Nu Nump
2.0542 2.2505
2.0594 2.2558
0.25 0.23
105
Nu Nump
4.3370 4.6127
4.3497 4.6269
0.29 0.31
1.47
1163
131.7
106
Nu Nump
8.6407 8.8771
8.7096 8.9444
0.79 0.75
107
Nu Nump
16.3427 16.5477
16.7798 16.9813
2.60 2.55
attraction of many researchers in recent years. Khanafer et al. [11] numerically studied the natural convection heat transfer in a nanofluid filled enclosure with differentially heated vertical walls. They found that the existence of the nanoparticles in a base fluid enhanced the heat transfer rate for all the Grashof numbers considered. Ghasemi and Aminossadati [12] presented a numerical strategy of an inclined enclosure filled with CuO–water nanofluid. Their results showed that mixing nanoparticles with pure water improves the heat transfer performance and the inclination angle has a significant impact on the flow and temperature fields at high Rayleigh numbers. Buoyancy induced flow and heat transfer in a partially heated enclosure filled with nanofluid were receiving a great interest among the researchers due to their wide range of applications in the areas of cooling of buildings. Oztop and Abu-Nada [13] numerically examined the natural convection in a partially heated rectangular enclosure using nanofluids made with different types and concentrations of nanoparticles. They reported that the heat transfer enhancement using Cu–water nanofluids was more pronounced at lower aspect ratio than at a higher aspect ratio of the enclosure. A numerical investigation on a natural convection heat transfer in an inclined nanofluid filled enclosure with a sinusoidally heated and isothermally cooled walls was carried out by Oztop et al. [14]. It was found that the addition of nanoparticles with water affects the fluid flow and temperature distribution especially for higher Rayleigh numbers. Sheikhzadeh et al. [15] discussed the buoyancy driven fluid flow and heat transfer in a thermally active portion on Cu–water nanofluid filled rectangular enclosure with different positions. They concluded that the heat transfer rate was maximized either in the bottom–middle or middle–middle locations of the thermally active parts depending upon the Rayleigh numbers. Similar model along with the effects of inclination angle was studied by Hosseini et al. [16]. They found that an increment of the inclination angle from 0 to 40 provoked the fluid motion resulting in increased value of Nusselt number, while an opposite trend was observed when the inclination angle was assumed from 40 to 80 . An analysis of natural convection heat transfer, fluid flow and entropy generation in an odd shaped enclosure filled with nanofluid was numerically performed by Parvin and Chamkha [17]. It was found that the increasing Rayleigh number and solid volume fraction enhanced the heat transfer rate as well as entropy generation. Cho [18] measured the natural convection heat transfer and entropy generation in Al2O3 nanofluid filled enclosure with partially heated wavy surface. The results showed that the mean Nusselt number increased and the total entropy generation reduced as the nanoparticle volume fraction was increased. However the results were inconsistent when the amplitude and wavelength of the wavy surface were increased. Numerical studies of natural convection cooling of protruding heat source mounted in an enclosure filled with Cu–water nanofluid were carried out by Mahmoudi et al. [19]. Results indicated that the average Nusselt number increased linearly with the increase in the solid volume fraction of nanoparticles at a given Rayleigh number with definite heat source geometry.
Numerical studies of natural convection heat transfer due to periodic sinusoidal temperature distribution of heat source mounted at the center of the nanofluid filled enclosure were done by Sourtiji et al. [20]. Their augmentation of heat transfer by using Al2O3 nanoparticle reported that the percentage of heat transfer enhancement deviated over the wide range of Rayleigh numbers. The shape and orientation of several pairs of non-protruding heaters and coolers (HACs) inside the square enclosure subjected to natural convection filled with different nanofluids were modeled by Garoosi et al. [21]. They found that the vertically oriented rectangular HACs enhanced the rate of heat transfer between 20% and 27% compared to the horizontally oriented HACs. However it differs by 8–11% when compared to the square shape HACs for different Rayleigh numbers. Numerical investigation of an identical model dealt with mixed convection of nanofluids filled lid driven enclosure was examined by Garoosi et al. [22]. It was confirmed that the rate of heat transfer was increased by changing the orientation of the HAC from horizontal to vertical position for the entire range of Richardson number. Alsabery et al. [23] performed a numerical investigation of natural convection in a trapezoidal enclosure partly filled with nanofluid porous layer and partly with non-Newtonian fluid layer. It was found that convection plays a remarkable role with the addition of silver–water nanofluid and the heat transfer rate was affected by the inclination angle of the enclosure. Recently Kolsi et al. [24] computed the three dimensional natural convection and entropy generation nanofluid filled differentially heated cubical enclosure. The above studies confirmed that the performance of fluid flow and heat transfer rate
Present Study Tou et al. [8]
10
1
Ro
Nui
w
Ro
3
w
2
Ro
10
4
10
w
5
Ra
1
10
6
10
7
Fig. 2. Row-averaged Nusselt number against Ra.
Please cite this article in press as: A. Purusothaman et al., Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2015.12.012
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A. Purusothaman et al. / Advanced Powder Technology xxx (2016) xxx–xxx
Fig. 3. Case (i) for Cu–water nanofluid with / = 0.05, Ra ¼ 107 and AS ¼ 7:5 – (a) isothermal surfaces, isotherms at (b) Y = 0, (c) Y = 0.05, (d) Y = 0.9, (e) X = 0.3, 1.3, 2.3 & 3.3; velocity vector at (f) Y = 0.05, (g) Y = 0.9, and (h) X = 0.3, 1.3, 2.3, & 3.3.
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Fig. 4. Case (ii) for Cu–water nanofluid with / = 0.05, Ra ¼ 107 and AS ¼ 7:5 – (a) isothermal surfaces, isotherms at (b) Y = 0, (c) Y = 0.05, (d) Y = 0.9, (e) X = 0.3, 1.3, 2.3 & 3.3; Velocity vector at (f) Y = 0.05, (g) Y = 0.9, and (h) X = 0.3, 1.3, 2.3, & 3.3.
Please cite this article in press as: A. Purusothaman et al., Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2015.12.012
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Fig. 5. Case (iii) for Cu–water nanofluid with / = 0.05, Ra ¼ 107 and AS ¼ 7:5 – (a) isothermal surfaces, isotherms at (b) Y = 0, (c) Y = 0.05, (d) Y = 0.9, (e) X = 0.3, 1.3, 2.3 & 3.3; Velocity vector at (f) Y = 0.05, (g) Y = 0.9, (h) X = 0.3, 1.3, 2.3, & 3.3.
increased while increasing both the Rayleigh number and solid volume fraction of nanoparticles.
From the above literature review, it is very clear that the technique for improving heat transfer performance using nanofluids in
Please cite this article in press as: A. Purusothaman et al., Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2015.12.012
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Fig. 6. Isothermal surfaces for Cu–water nanofluid with AS ¼ 2:0, / = 0.05, Ra ¼ 107 – (a) Case (i), (b) Case (ii), (c) Case (iii); isotherms for Case (iii) at (d) Y = 7.5 & X = 3.75 for AS ¼ 1:0, (e) Y = 3.75 & X = 3.75 for AS ¼ 2:0, and (f) X = 0.0 & 3.75, Y = 1.0 for AS ¼ 7:5.
3D models is very limited besides the available work [24]. Hence, taking this opportunity, the aim of the present study is to evaluate the natural convection heat transfer performance with an array of non-protruding heat source mounted in a rectangular enclosure filled with nanofluids of different types and concentrations of nanoparticles. This configuration may give some additional knowledge in designing hermetically sealed electronic packages encountered in the microelectronics industry. 2. Mathematical formulation A schematic plan of the system considered in the present study and its fixed geometric parameters are displayed in Fig. 1. The
enclosure is three dimensional and rectangular and its height, width and length are denoted by H; C and D respectively. A 3 3 array of discrete heat sources is mounted on one vertical wall (y = 0) of a nanofluid filled enclosure. The remaining part of the wall in which the sources are mounted and the other walls are adiabatic unless otherwise specified. Three different symmetrical thermal boundary conditions are considered on the enclosure walls. Accordingly Case (i) opposite vertical wall, Case (ii) side vertical walls and Case (iii) top and bottom horizontal walls are maintained cold. The gravity acts perpendicular to the xy plane. The water based nanofluids containing solid spherical nanoparticles (Cu or Al2O3) and their thermophysical properties are listed in Table 1. The flow is assumed to be three dimensional,
Please cite this article in press as: A. Purusothaman et al., Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2015.12.012
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Fig. 7. Isotherms for water (—: / = 0.0) and Cu–water nanofluid (– – – –: / = 0.05) – (a) Case (i), (b) Case (ii), and (c) Case (iii) at Y = 0.01, Ra ¼ 107 and AS ¼ 7:5.
time-dependent, laminar, incompressible with negligible viscous dissipation and radiation effects. The fluid properties are assumed to be constant except the density in the buoyancy term that follows the Boussinesq approximation. Taking into account the above mentioned assumptions, the governing equations of the continuity, momentum and energy can be written in nondimensional form as follows:
r:~ V ¼ 0;
ð1Þ
qf bnf m @~ V Vþ þ ð~ V:rÞ~ V¼ rP þ nf Pr r2 ~ RaPrh~ k; @s qnf mf bf
ð2Þ
anf 2 @h þ ð~ V:rÞh ¼ r h; @s af
ð3Þ
where ~ V is the velocity vector with components U; V and W in the X; Y and Z directions, respectively and ~ k denotes the vector acting along Z-direction. Eqs. (1)–(3) were cast in non-dimensional form by using the following variables:
s ¼ taf =L2z ; P ¼ pL2z =qf a2f ; U ¼ uLz =af ; V ¼ v Lz =af ; W ¼ wLz =af ; h ¼ ðT T c Þ=ðT H T C Þ; Ra ¼ gbf ðT H T C ÞL3z =ðmf af Þ; Pr ¼ mf =af : ð4Þ
X ¼ x=Lz ;
Y ¼ y=Lz ;
Z ¼ z=Lz ;
Density qnf and thermal diffusivity anf of nanofluid are calculated as
qnf ¼ ð1 /Þqf þ /qp ;
ð5Þ
k anf ¼ nf : ðqcp Þnf
ð6Þ
The heat capacitance and thermal expansion coefficient of the nanofluid can be determined as follows:
ðqcp Þnf ¼ ð1 /Þðqcp Þf þ /ðqcp Þp ;
ð7Þ
ðqbÞnf ¼ ð1 /ÞðqbÞf þ /ðqbÞp :
ð8Þ
The effective thermal conductivity of the nanofluid knf is approximated by the Maxwell–Garnett model [25] as
knf kp þ 2kf 2/ðkf kp Þ ¼ : kf kp þ 2kf þ /ðkf kp Þ
ð9Þ
The effective dynamic viscosity of the nanofluid lnf can be approximated using the Brinkman [26] relation as
lnf ¼
lf ð1 /Þ2:5
:
ð10Þ
The dimensionless boundary conditions at the enclosure walls are:
D @h @h ¼ 0; ðiiÞ h ¼ 0; ðiiiÞ ¼ 0; : U ¼ V ¼ W ¼ 0; ðiÞ Lz @X @X ( h ¼ 1; source region; At Y ¼ 0 : U ¼ V ¼ W ¼ 0; @h ¼ 0; otherwise; @X
At X ¼ 0;
C @h ¼ 0; : U ¼ V ¼ W ¼ 0; ðiÞ h ¼ 0; ðiiÞ and ðiiiÞ Lz @Y H @h ¼ 0; ðiiiÞ h ¼ 0; ð11Þ : U ¼ V ¼ W ¼ 0; ðiÞ and ðiiÞ At Z ¼ 0; Lz @Z
At Y ¼
Heindel et al. [6,7], Tou et al. [8] discussed the plane of symmetry in their three dimensional natural convection problem. This assertion was experimentally confirmed by Ozoe et al. [27]. Following this, the plane at X ¼ D=2Lz is taken as a plane of symmetry and the computational domain is reduced by half. This is possible for all three cases due to symmetrically imposed boundary conditions at the enclosure walls and hence the plane of symmetry is defined as follows:
At X ¼
D @V @W @h ¼ ¼ 0; ðiÞ; ðiiÞ and ðiiiÞ ¼ 0: :U¼ 2Lz @X @X @X
ð12Þ
The heat transfer rate within the enclosure can be obtained through the Nusselt number. The local and average Nusselt numbers are calculated from the resulting temperature field at each heat source and are defined as follows:
The local Nusselt number : Nuik ðX; ZÞ ¼
knf @h ; kf @Y
The average Nusselt number : Nuik ðX; ZÞ ¼
1 Ah
Z
Ah
ð13Þ
Nui;k ðX; ZÞdAh ;
0
ð14Þ The row averaged Nusselt number : Nui ¼
3 1X Nui;k ðX; ZÞ; 3 i¼1
ð15Þ where Ah is the heater area and the subscript ik refers to the specified heater in row i and column k.
Please cite this article in press as: A. Purusothaman et al., Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2015.12.012
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40
Row 3 Row 2 Row 1
As=2.0
30 As=7.5
Nui
20
As=1.0
10
0.025
0.0
0.075
0.05
0.1
φ
(a) 40
35
Row 3 Row 2 Row 1
As=2.0
30
Row 3 Row 2 Row 1
25 As=2.0
Nui
Nui
20
15 As=7.5
As=7.5
As=1.0
As=1.0
10
0.025
0.0
0.05
0.075
0.1
5
0.0
φ
0.025
0.05
0.075
0.1
φ
(b)
(c)
Fig. 8. Average Nusselt number against / for Cu–water nanofluid (a) Case (i), (b) Case (ii), and (c) Case (iii) at Ra ¼ 107 .
3. Numerical procedure The governing nondimensional Eqs. (1)–(3) have been solved numerically based on finite volume technique described by Patankar [28]. A uniform staggered grid is considered in the computational domain. The scalar quantities (P and h) are stored in the nodal points whereas the components of the velocity (U; V and W) are stored at the cell face of the control volumes of the nodal points. A fully implicit time marching scheme is employed. The numerical procedure namely Semi-Implicit Method for Pressure Linked Equation (SIMPLE) algorithm is used to handle the pressure velocity coupling. The convective and diffusive terms in Eqs. (2) and (3) are handled by adopting the power-law scheme. Finally the discretized set of algebraic equations are solved by the lineby-line Tri-Diagonal Matrix Algorithm (TDMA). To ensure convergence of the numerical algorithm, the following criterion applied to all dependent variables over the solution domain
P
fm1 i;j;k 6 105 ; P m f i;j;k i;j;k
m i;j;k fi;j;k
ð16Þ
where f represents the dependent variables U; V; W and h, the subscripts i; j; k indicate the space coordinates and the superscript m represents the time step. Based on the above numerical procedure, the present problem is solved by the developed FORTRAN code. In order to examine the effect of the grid on the numerical study, the numerical code was examined for Ra ¼ 107 ; / ¼ 0:0 and AS ¼ 7:5 using five different computational grids, namely, 55 31 91, 61 31 91, 76 91 91, 91 91 91 and 109 91 121, for the Case (i). It is observed that a refinement of the grid from 91 91 91 to 109 91 121 does not have a significant effect on the results in terms of row averaged Nusselt number and the difference in Nusselt numbers between these two grids is found to be less than 0.1%. Hence, by considering both
Please cite this article in press as: A. Purusothaman et al., Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2015.12.012
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Fig. 9. Case (iii) for Cu–water nanofluid with AS ¼ 7:5 and / = 0.05 – isotherms at Y = 0.01 (a) Ra ¼ 105 , (b) Ra ¼ 107 ; W – velocity component at Y = 0.1, (c) Ra ¼ 105 , and (d) Ra ¼ 107 .
the accuracy and computational time, the present calculations are all performed with a 91 91 91 spaced grid system.
of the numerical results produced by the present code is satisfactory with those provided in the literature.
3.1. Validation of the study Before proceeding further, it is of interest to compare the numerical results of the three dimensional empty enclosure which is heated differentially at two vertical side walls with the available results in the literature. Hence the code was validated against the natural convection in a 3-D empty enclosure which was studied by Tric et al. [29] and the Nusselt numbers for Ra ¼ 104 107 are given in Table 2. The results show that the maximum percentage deviation in the average Nusselt number values is 0.8% up to Ra ¼ 106 . But when Ra ¼ 107 , they deviate to a maximum of 2.6%. Fig. 2 presents another comparative study with Tou et al. [8] in terms of the variation of row averaged Nusselt number with the Rayleigh number for a fixed aspect ratio A ¼ 7:5 and Prandtl number Pr ¼ 9. It is observed that the deviation in the Nusselt number lies within 5.6%, for 104 6 Ra 6 107 . Hence the agreement
4. Results and discussion Buoyancy driven natural convection cooling of a 3 3 array of discrete heaters mounted in a nanofluid filled three dimensional rectangular enclosure is numerically investigated with three different symmetric thermal boundary conditions. To ensure an effectual way of cooling performance on the enclosure, it is considered that two different nanofluids consisting of water and different volume concentrations of Cu or Al2O3 nanoparticles. Thus the Prandtl number (Pr) of the base fluid (water) is fixed to be 6.2 throughout this study. Moreover computations are performed for wide ranges of Rayleigh number (105 6 Ra 6 107 ), enclosure side aspect ratio (1:0 6 AS 6 7:5) and nanoparticles solid volume fraction (0:0 6 / 6 0:1).
Please cite this article in press as: A. Purusothaman et al., Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2015.12.012
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40
36
Row 3 Row 2
Cu-Water
Row 1
Case (i)
Al 2O3-Water
35
Case (i)
27
Case (ii) Case (iii)
Case (ii) Case (iii)
30
Nui
Nu3
18
25
20
9 15
0
10
5
10
6
10
7
Ra Fig. 10. Average Nusselt number against Ra for Cu–water nanofluid with AS ¼ 7:5 and / = 0.05.
4.1. 3D nature of temperature and flow field The nature of the isotherms and velocity vectors within Cu–water nanofluid filled enclosure at the selected plane sections parallel to the enclosure walls corresponding to Case (i) are shown in Fig. 3 (a)–(h) for the fixed value of Ra = 107, / = 0.05 and AS ¼ 7:5. It is found that the isotherms around the heater modules are more compact and the maximum fluid temperature within the enclosure is recorded at the top of each heater on account of receiving additional energy from the bottom heaters through the dominance of convection mechanism. It is very clear that the magnitude of the velocity vectors adjacent to the column heaters is maximum which are qualitatively expressed in Fig. 3(f)–(h). However the fluid near the adiabatic region on the side of the heater array and the adiabatic walls stays motionless. The stream trace in Fig. 3(f) visibly demonstrates that the raising plume like upward flow on the heaters hits the enclosure ceiling and turns naturally towards the opposite cooled wall. Adjacent to the opposite cold wall at Y = 0.9, the majority of the fluid motion is vertically downward and uniform eventually forming a simple closed flow pattern with a torpid core region. Moreover the flat isothermal surfaces in Fig. 3 (a) indicate that the originated heat from the heaters shifts evenly towards the opposite cold wall. As a result, the entire region of the enclosure is thermally stratified in the vertical direction (see Fig. 3 (e)). The consequences of the Case (ii) are presented in Fig. 4(a)–(h) in terms of isotherms and velocity vectors. It is noticed that the generated heat from the heater array is transferred towards the side cold walls with wavy isotherms. Moreover the fluid temperature above the core region increases significantly compared to Case (i). It can be reflected by the lack of heat removal effectiveness through the wavy temperature distribution which is equal to the reciprocal of the ideal efficiency in controlling the device operating temperature. However the vertical thermally stratified fluid core region is apparently noticed in this case also (see Fig. 4(a)–(e)). The velocity vectors for this case are depicted in Fig. 4(f)–(h). The rising fluid contiguous to the heater array moves upward and revolves toward the side cold walls with a bulk fluid motion which are assured by the stream traces in Fig. 4(f). However the fluid
10
0.0
0.025
0.05
φ
0.075
0.1
Fig. 11. Average Nusselt number against / for Cu–water and Al2O3–water nanofluids at AS ¼ 7:5 and Ra ¼ 107 .
motion is practically stagnant near the opposite adiabatic wall at Y 0.9 plane owing to an adverse situation for fluid flow. The isothermal surface, temperature distribution and velocity vectors corresponding to Case (iii) are shown in Fig. 5(a)–(h). It is interesting to note from Fig. 5(a)–(e) that complex flow pattern evolution is observed which, to the author’s knowledge, has been reported previously by Tso et al. [9]. Moreover the raising fluid from the column heaters reaches the ceiling and loses its momentum due to the cooled top and falls down freely near the opposite and adjacent walls of the enclosure. But in the opposite wall, the nonuniform downward flow descends drastically in the region not facing the column heaters. This causes the existence of isothermal closed loops in the opposite wall which collapse the vertical thermal stratification above the core region of the enclosure (see Fig. 5(d)). This is mainly associated with nonuniform downward flows adjacent to the side and opposite walls which is contradictory to the cases (i) and (ii). However the bottom of the enclosure (Z 6 2) is thermally stratified. These observations turn out to be more obvious from the vector plots in Fig. 5(f)–(h). On comparing Figs. 3–5 for all three cases, it is observed that, though the constant temperature is maintained in all nine heaters, the fluid temperature in the upper region of each heater is maximum than in the other region. This is due to the fact that the fluid on top of the each heater receives additional heat energy from the bottom heaters through the buoyancy mechanism. However, the fluid temperature within the enclosure is recorded minimum in Case of (i) than in the Cases (ii) and (iii) for AS ¼ 7:5. Hence it is obviously revealed that the flat plane thermal distribution in Case (i) is more efficient transforming trend which successfully controls the device operating temperature within the limits. When AS ¼ 2:0, this flatness thermal distribution tendency is also retained for both Cases (i) and (ii) which stabilizes the system thermal efficiency and is shown in Fig. 6(a) and (b). In Case (iii), the isothermal surfaces and isothermal propagation quit the flatness and adopt more complex forms (see Figs. 5 and 6(c)–(f)). Moreover one can notice that the existence of the closed thermal loops is with local temperature maximum on adiabatic walls. This is one of the main objectives of electronic equipment cooling problems of finding the specific hot locations on the enclosure walls. This superfluous heat can be
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A. Purusothaman et al. / Advanced Powder Technology xxx (2016) xxx–xxx
absorbed by placing sinks, where it occurs, thus providing a pathway to improve the efficiency of equipment. In these simulations, the particle volume fraction is an important factor to be considered to obtain nanofluids with more thermal efficiency which are required for some applications relating to microelectronic manufacturing industries. For the sake of comparisons, the isotherms for base fluid and Cu–water nanofluid with 5% particle volume concentration are exhibited by solid line and dashed line respectively for three different cases shown in Fig. 7 (a)–(c) at Y = 0.01 plane section with a fixed value of AS ¼ 7:5
shown in Fig. 11. This indicates that Cu–water nanofluid has the greatest effect on the thermal performance compared with Al2O3–water nanofluid. Moreover the nanofluid effect and its performance are more prominent in Case (i) than in the Cases (ii) and (iii). The results obtained in this study are expected to lead to new guidelines for cooling the electronic equipment which will improve the reliability of the system performance.
5. Conclusion
7
and Ra ¼ 10 . The shape of the temperature distributions inside the enclosure shows how the dominant heat transfer mechanism changes in base fluid and nanofluid. It is shown that the thermal energy transfers faster as natural convection is intensified in nanofluid than in the base fluid. In order to know the effects of /; AS and boundary conditions in terms of the heat transfer rate, the graphs of row averaged Nusselt numbers, Nui (i = 1, 2, 3), are depicted in Fig. 8 for Ra ¼ 107 and Cu– water nanofluid. As a consequence, the solid volume fractions / are varied from 0% to 10% with an increment of 2.5% and the enclosure side aspect ratio is assumed the value between the ranges of 1 to 7.5. The results show that the heat transfer rate increases monotonically with increasing volume fraction in all the three cases. The trends shown by the nanofluid is due to the fact that the suspension of nanoparticles in basefluid enhances the thermal conductivity of the nanofluid leads to high heat transfer performance. Moreover on account of buoyancy mechanism, the row averaged Nusselt number for the top row heaters is less than the bottom row heaters irrespective of the value of AS , (that is, Nu1 < Nu2 < Nu3 ). This is consistent for all the three cases. When comparing Case (i), Case (ii) and Case (iii), the row average Nusselt numbers Nui for Case (i) attain a maximum value and the heat transfer rates Nui are extremely supreme for those corresponding to AS ¼ 2:0. Fig. 9 is presented to show the effect of Rayleigh numbers on the temperature distribution and flow field corresponding to the Case (iii) for Cu–water nanofluid with fixed AS ¼ 7:5 and / = 0.05. On the XZ-plane at Y = 0.01, the isotherms are illustrated in Fig. 9 (a) and (b) for the Rayleigh numbers Ra ¼ 105 and 107 respectively. Isotherms and local fluid temperature adjacent to the heater face region in Fig. 9(a) for Ra ¼ 105 indicates that heat transfer is more influenced by the conduction mode of heat transfer mechanism. With an increase in the Rayleigh number (Ra ¼ 107 ), the isotherms become more stratified adjacent to the heater face region because of induced buoyancy force and the convective heat transfer is more effective. The velocity vectors at Y ¼ 0:1 plane are shown in Fig. 9 (c) and (d) for the different Rayleigh numbers Ra ¼ 105 and 107 respectively. For discussing the magnitude of the fluid velocity, the W-velocity component is only taken as primary component which is more supreme in strength than the other two (U and V) components. It is found from Fig. 9(c) that flow is weakly strengthened for Ra ¼ 105 . When Ra ¼ 107 , the magnitude of the primary component increases as the buoyancy force begins to dominate the fluid motion. Moreover, it is noticed from Fig. 10, that the row averaged Nusselt number Nui (i = 1, 2, 3), increases with the increase of Ra due to the enhanced convection that carries heat energy with increased momentum. In addition, the present computations are carried out for two different nanofluids, namely, Cu–water and Al2O3–water. It is found that, with other parameters unaltered, maximum percentage deviation between these two nanofluids is only 2%. In order to know the results further in detail, the row average Nusselt number for row 3 (Nu3 ) is calculated for different solid volume fraction together with three boundary conditions and the variation is
A numerical study of natural convection heat transfer on a nanofluid filled three dimensional rectangular enclosure induced by a 3 3 array of wall mounted isothermal heaters is investigated in the context of cooling of electronic equipments. Computations are performed for wide ranges of Rayleigh number, enclosure side aspect ratio and nanoparticle volume fraction together with different nanofluids and enclosure boundary conditions. The study leads to the following significant conclusions: (a) The fluid temperature within the enclosure is recorded minimum in Case (i) than in the Cases (ii) and (iii). This is obviously achieved by the flat thermal distribution which has more efficient transforming trend which successfully controls the device operating temperature within the limits. (b) The enclosures of Cases (i) and (ii) clearly demonstrate a high degree of vertical thermal stratification with a stagnant core for higher values of Ra. But Case (iii) breaks the vertical thermal stratification associated with non-uniform downward flows from the cooled ceiling. (c) Moreover isothermal propagation in Case (iii) quits the flatness and adopts more complex forms of closed thermal loops with local temperature maximum on adiabatic or opposite walls and is highly sensitive for higher value of AS . (d) From the cooling performance and system geometry design point of view, AS ¼ 2:0 is an optimal enclosure side aspect ratio to dissipate the maximum heat from the isothermal heaters. (e) The suspension of nanoparticles in basefluid enhances the thermal conductivity of the mixture (nanofluid) leads to high heat transfer rate than the base fluid. Hence increasing the value of nanoparticle volume fraction (0–10%) yields better thermal performance than that of the base fluid for all three cases, irrespective of the values of Ra and AS . (f) The overall heat transfer within the enclosure for Ra ¼ 107 is enhanced in Case (i) than in the other two cases. This shows that maintaining the opposite wall to be cold is the optimum condition of cooling the dispersed heat from an electrical device. (g) Cu–water nanofluid has the greatest effect on the thermal performance compared with Al2O3–water nanofluid. Moreover the row averaged Nusselt number increases monotonically with increasing the Rayleigh number and the nanoparticle volume fraction.
References [1] S. Ostrach, Natural convection in enclosures, J. Heat Transfer 110 (4b) (1988) 1175–1190. [2] F.P. Incropera, Convection heat transfer in electronic equipment cooling, J. Heat Transfer 110 (4b) (1988) 1097–1111. [3] D.E. Wroblewski, Y. Joshi, Computations of liquid immersion cooling for a protruding heat source in a cubical enclosure, Int. J. Heat Mass Transfer 36 (5) (1993) 1201–1218. [4] D.E. Wroblewski, Y. Joshi, Liquid immersion cooling of a substrate-mounted protrusion in a three-dimensional enclosure: the effects of geometry and boundary conditions, J. Heat Transfer 116 (1) (1994) 112–119.
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A. Purusothaman et al. / Advanced Powder Technology xxx (2016) xxx–xxx [5] Y. Joshi, M.D. Kelleher, M. Powell, E.I. Torres, Natural convection heat transfer from an array of rectangular protrusions in an enclosure filled with dielectric liquid, J. Electron. Packag. 116 (2) (1994) 138–147. [6] T.J. Heindel, S. Ramadhyani, F.P. Incropera, Laminar natural convection in a discretely heated cavity: I—assessment of three dimensional effects, J. Heat Transfer 117 (4) (1995) 902–909. [7] T.J. Heindel, F.P. Incropera, S. Ramadhyani, Laminar natural convection in a discretely heated cavity: II—comparisons of experimental and theoretical results, J. Heat Transfer 117 (4) (1995) 910–917. [8] S.K.W. Tou, C.P. Tso, X. Zhang, 3-D numerical analysis of natural convective liquid cooling of a 3 3 heater array in rectangular enclosures, Int. J. Heat Mass Transfer 42 (17) (1999) 3231–3244. [9] C.P. Tso, L.F. Jin, S.K.W. Tou, X.F. Zhang, Flow pattern evolution in natural convection cooling from an array of discrete heat sources in a rectangular cavity at various orientations, Int. J. Heat Mass Transfer 47 (19–20) (2004) 4061–4073. [10] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, Dev. Appl. Non Newtonian Flows 66 (1995) 99–106. [11] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure, Int. J. Heat Mass Transfer 46 (19) (2003) 3639–3653. [12] B. Ghasemi, S.M. Aminossadati, Natural convection heat transfer in an inclined enclosure filled with a water–CuO nanofluid, Numer. Heat Transfer, Part A 55 (8) (2009) 807–823. [13] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (5) (2008) 1326–1336. [14] H.F. Oztop, E. Abu-Nada, Y. Varol, K. Al-Salem, Computational analysis of nonisothermal temperature distribution on natural convection in nanofluid filled enclosures, Superlattices Microstruct. 49 (4) (2011) 453–467. [15] G.A. Sheikhzadeh, A. Arefmanesh, M.H. Kheirkhah, R. Abdollahi, Natural convection of Cu–water nanofluid in a cavity with partially active side walls, Eur. J. Mech B/Fluids 30 (2) (2011) 166–176. [16] M. Hosseini, M.T. Mustafa, M. Jafaryar, E. Mohammadian, Nanofluid in tilted cavity with partially heated walls, J. Mol. Liq. 199 (2014) 545–551. [17] S. Parvin, A.J. Chamkha, An analysis on free convection flow, heat transfer and entropy generation in an odd-shaped cavity filled with nanofluid, Int. Commun. Heat Mass Transfer 54 (2014) 8–17.
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[18] C.C. Cho, Heat transfer and entropy generation of natural convection in nanofluid-filled square cavity with partially-heated wavy surface, Int. J. Heat Mass Transfer 77 (2014) 818–827. [19] A.H. Mahmoudi, M. Shahi, A.H. Raouf, A. Ghasemian, Numerical study of natural convection cooling of horizontal heat source mounted in a square cavity filled with nanofluid, Int. Commun. Heat Mass Transfer 37 (8) (2010) 1135–1141. [20] E. Sourtiji, D.D. Ganji, M. Gorji-Bandpy, S.M. Seyyedi, Numerical study of periodic natural convection in a nanofluid-filled enclosure due to transitional temperature of heat source, Powder Technol. 259 (2014) 65–73. [21] F. Garoosi, G. Bagheri, F. Talebi, Numerical simulation of natural convection of nanofluids in a square cavity with several pairs of heaters and coolers (HACs) inside, Int. J. Heat Mass Transfer 67 (2013) 362–376. [22] F. Garoosi, L. Jahanshaloo, S. Garoosi, Numerical simulation of mixed convection of the nanofluid in heat exchangers using a Buongiorno model, Powder Technol. 269 (2015) 296–311. [23] A.I. Alsabery, A.J. Chamkha, S.H. Hussain, H. Saleh, I. Hashim, Heatline visualization of natural convection in a trapezoidal cavity partly filled with nanofluid porous layer and partly with non-Newtonian fluid layer, Adv. Powder Technol. 26 (4) (2015) 1230–1244. [24] L. Kolsi, A.K. Hussein, M.N. Borjini, H.A. Mohammed, H.B. Aissia, Computational analysis of three-dimensional unsteady natural convection and entropy generation in a cubical enclosure filled with water–Al2O3 nanofluid, Arab J. Sci. Eng. 39 (11) (2014) 7483–7493. [25] J. Maxwell, A Treatise on Electricity and Magnetism, Oxford University Press, Cambridge, UK, 1904. [26] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571–581. [27] H. Ozoe, N. Sato, S.W. Churchill, Experimental confirmation of the three dimensional helical streaklines previously computed for natural convection in inclined rectangular enclosures, Int. Chem. Eng. 19 (1979) 454–462. [28] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing, Corporation, Taylor and Francis Group, New York, 1980. [29] E. Tric, G. Labrosse, M. Betrouni, A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions, Int. J. Heat Mass Transfer 43 (21) (2000) 4043–4056.
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Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt
Original Research Paper
Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure A. Purusothaman a,b,⇑, N. Nithyadevi a, H.F. Oztop c, V. Divya a, K. Al-Salem d a
Department of Mathematics, Bharathiar University, Coimbatore, India Department of Mathematics, National Institute of Technology, Tiruchirappalli, India c Department of Mechanical Engineering, Technology Faculty, Firat University, Elazig, Turkey d Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia b
a r t i c l e
i n f o
Article history: Received 11 August 2015 Received in revised form 18 November 2015 Accepted 25 December 2015 Available online xxxx Keywords: Natural convection Equipment cooling Heater array Nanofluid
a b s t r a c t This study deals with the numerical analysis of 3D natural convection equipment cooling with a 3 3 array of isothermal heaters mounted on one vertical wall of the nanofluid filled enclosure. The enclosure is filled with water based nanofluid containing Copper (Cu) or Alumina (Al2O3) nanoparticles. The transport equations are solved by the finite volume method based on the SIMPLE algorithm with power-law scheme. The influence of pertinent parameters such as Rayleigh number (105 6 Ra 6 107 ), nanoparticle volume fraction (0% 6 / 6 10%) and enclosure side aspect ratio (1:0 6 AS 6 7:5) on the fluid flow and heat transfer characteristics are investigated together with different nanofluids and enclosure boundary conditions. It is observed that Cu–water nanofluid has the greatest effect on the equipment cooling performance compared with Al2O3–water nanofluid. Moreover, the row averaged Nusselt number increases monotonically with increase in both the Rayleigh number and the nanoparticle solid volume fraction. Ó 2015 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
1. Introduction The incredible developments in the electronic industry in the past few decades have resulted in a tremendous increase in microelectronic component density. The components generate heat and leave an adverse effect on the allowable operating temperature of the nearby components which in turn reduces the performance of the system. The motivation for the present work is the cooling of printed circuit boards in sealed enclosure. In this context, buoyancy induced natural convective flows play a significant role since passive cooling is the least expensive, noiseless, most reliable method of heating rejection and maintenance free. Natural convective heat transfer and fluid flow in enclosures with various shapes and wall conditions have been studied extensively by Ostrach [1] in the past. A comprehensive review of various cooling options which could be possibly used for electronic equipment cooling was discussed by Incropera [2]. Literature in this area is quite vast which includes theoretical as well as experimental results. Thermal control in electronic components using dielectric liquids has received increased attention due to inherently high heat ⇑ Corresponding author at: Department of Mathematics, Bharathiar University, Coimbatore, India. E-mail addresses: [email protected], [email protected] (A. Purusothaman).
removal capabilities of liquids compared to air. Wroblewski and Joshi [3] discussed the liquid immersion cooling of a substrate mounted protrusion in a three dimensional enclosure with opposite cold wall. The effect of the boundary conditions with the top cold wall and both top and opposite cold walls for the same physical configuration was further studied by Wroblewski and Joshi [4]. They found that when the ratio of substrate to fluid thermal conductivity is greater than 10, conduction plays a dominant role in cooling the chip through the substrate. Joshi et al. [5] carried out an experimental study of natural convection liquid immersion cooling with a 3 3 array of heaters embedded on one vertical wall of a 3D enclosure with top and bottom cold walls. They investigated the effect of the width of top and bottom surfaces and concluded that it was most significant at the lower power levels. The effect of the heater aspect ratio, Ahtr of 3 3 array of discrete heat sources flush mounted on one vertical wall of a rectangular enclosure with isothermally cooled opposite wall was studied numerically and experimentally by Heindel et al. [6,7]. For Ahtr 6 3 , average Nusselt number increased with decreasing Ahtr . However, for Ahtr P 3, the two and three-dimensional predictions differed within 5% of each other and results were approximately independent of Ahtr . These existing three dimensional numerical results were compared with the experimental predictions had archived with excellent agreement. The numerical
http://dx.doi.org/10.1016/j.apt.2015.12.012 0921-8831/Ó 2015 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
Please cite this article in press as: A. Purusothaman et al., Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2015.12.012
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A. Purusothaman et al. / Advanced Powder Technology xxx (2016) xxx–xxx
Nomenclature Ah AS b C cp D E g H k Lx ; Lz Nui Nui;k Nui;k p P Pr Ra t T u; v ; w U; V; W x; y; z
heater area (m2) enclosure side aspect ratio (¼ H=C) heater location (m) enclosure dimension in y direction (m) constant pressure specific heat (J kg1 K1) enclosure dimension in x direction (m) heater location (m) gravitational acceleration (m s2) enclosure dimension in z direction (m) thermal conductivity (W m1 K1) length and height of a heater (m) row averaged Nusselt number local Nusselt number of a heater average Nusselt number of a heater pressure (Pa) dimensionless pressure Prandtl number Rayleigh number dimensional time (s) temperature (K) dimensional velocity components (m s1) dimensionless velocity components dimensional cartesian coordinates (m)
X; Y; Z
dimensionless cartesian coordinates
Greek symbols thermal diffusivity (m2 s1) b thermal expansion coefficient (K1) l dynamic viscosity (Pa s) m kinematic viscosity (m2 s1) / nanoparticle volume fraction q density (kg m3) s dimensionless time h dimensionless temperature f general variable
a
Subscripts c cold f fluid h hot i row k column nf nanofluid p nanoparticle
analysis of natural convection liquid cooling of a 3 3 array of discrete heat sources flush-mounted on one vertical wall of a rectangular enclosure filled with various fluids (Pr ¼ 5; 9; 25 and 130) and cooled by the opposite wall was presented by Tou et al. [8]. The effects of modified Rayleigh number, Prandtl number and enclosure aspect ratio of heat transfer characteristics were investigated. They concluded that the row averaged Nusselt number increases with the Rayleigh number in the power of 0.27. Experimental investigation of an identical model at various inclinations was carried out by Tso et al. [9]. They found that the flow and temperature fields become complex and distorted when the enclosure was tilted.
The review of the literatures [3–9] indicates that the base fluids like water, oil and dielectric liquids have low thermal conductivity which provide some limitations upon achieving the maximum heat transfer performance. Hence, an innovative technique in which the colloidal suspension of nanometer sized particles in a base fluid was first introduced by Choi [10]. The novel approach of cooling by nanofluids improved the properties associated with heat transfer since then have substantially high thermal conductivities. Hence nanofluids are primarily used as coolant in heat transfer equipment such as heat exchangers, cooling of flush mounted electronic heaters and radiators. Studies on buoyancy driven convective heat transfer in an enclosure using nanofluids have drawn
z z g E Sx
Row 1 H
Row 2 Row 3 D
b H
Sz Lz Lx
x C
D/2
y
(a)
E
x
(b)
Fig. 1. Physical configuration and geometric parameters (a) full view of the enclosure (b) half view of sources mounted wall at Y = 0.
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A. Purusothaman et al. / Advanced Powder Technology xxx (2016) xxx–xxx Table 1 Properties of nanofluid material [13,23,24]. Property
Water
q (kg m3) 1
1
cp (J kg K ) K (W m1 K1) b 10
5
(1 K
1
)
a 107 (m2 s1)
Table 2 Benchmarking the present code with Tric et al. [29]. Nanoparticle
Rac
Nusselt number
Tric et al. [29]
Present
Error %
Cu
Al2 O3
4
10
997.1 4179 0.613 21
8933 385 400 1.67
3970 765 40 0.85
Nu Nump
2.0542 2.2505
2.0594 2.2558
0.25 0.23
105
Nu Nump
4.3370 4.6127
4.3497 4.6269
0.29 0.31
1.47
1163
131.7
106
Nu Nump
8.6407 8.8771
8.7096 8.9444
0.79 0.75
107
Nu Nump
16.3427 16.5477
16.7798 16.9813
2.60 2.55
attraction of many researchers in recent years. Khanafer et al. [11] numerically studied the natural convection heat transfer in a nanofluid filled enclosure with differentially heated vertical walls. They found that the existence of the nanoparticles in a base fluid enhanced the heat transfer rate for all the Grashof numbers considered. Ghasemi and Aminossadati [12] presented a numerical strategy of an inclined enclosure filled with CuO–water nanofluid. Their results showed that mixing nanoparticles with pure water improves the heat transfer performance and the inclination angle has a significant impact on the flow and temperature fields at high Rayleigh numbers. Buoyancy induced flow and heat transfer in a partially heated enclosure filled with nanofluid were receiving a great interest among the researchers due to their wide range of applications in the areas of cooling of buildings. Oztop and Abu-Nada [13] numerically examined the natural convection in a partially heated rectangular enclosure using nanofluids made with different types and concentrations of nanoparticles. They reported that the heat transfer enhancement using Cu–water nanofluids was more pronounced at lower aspect ratio than at a higher aspect ratio of the enclosure. A numerical investigation on a natural convection heat transfer in an inclined nanofluid filled enclosure with a sinusoidally heated and isothermally cooled walls was carried out by Oztop et al. [14]. It was found that the addition of nanoparticles with water affects the fluid flow and temperature distribution especially for higher Rayleigh numbers. Sheikhzadeh et al. [15] discussed the buoyancy driven fluid flow and heat transfer in a thermally active portion on Cu–water nanofluid filled rectangular enclosure with different positions. They concluded that the heat transfer rate was maximized either in the bottom–middle or middle–middle locations of the thermally active parts depending upon the Rayleigh numbers. Similar model along with the effects of inclination angle was studied by Hosseini et al. [16]. They found that an increment of the inclination angle from 0 to 40 provoked the fluid motion resulting in increased value of Nusselt number, while an opposite trend was observed when the inclination angle was assumed from 40 to 80 . An analysis of natural convection heat transfer, fluid flow and entropy generation in an odd shaped enclosure filled with nanofluid was numerically performed by Parvin and Chamkha [17]. It was found that the increasing Rayleigh number and solid volume fraction enhanced the heat transfer rate as well as entropy generation. Cho [18] measured the natural convection heat transfer and entropy generation in Al2O3 nanofluid filled enclosure with partially heated wavy surface. The results showed that the mean Nusselt number increased and the total entropy generation reduced as the nanoparticle volume fraction was increased. However the results were inconsistent when the amplitude and wavelength of the wavy surface were increased. Numerical studies of natural convection cooling of protruding heat source mounted in an enclosure filled with Cu–water nanofluid were carried out by Mahmoudi et al. [19]. Results indicated that the average Nusselt number increased linearly with the increase in the solid volume fraction of nanoparticles at a given Rayleigh number with definite heat source geometry.
Numerical studies of natural convection heat transfer due to periodic sinusoidal temperature distribution of heat source mounted at the center of the nanofluid filled enclosure were done by Sourtiji et al. [20]. Their augmentation of heat transfer by using Al2O3 nanoparticle reported that the percentage of heat transfer enhancement deviated over the wide range of Rayleigh numbers. The shape and orientation of several pairs of non-protruding heaters and coolers (HACs) inside the square enclosure subjected to natural convection filled with different nanofluids were modeled by Garoosi et al. [21]. They found that the vertically oriented rectangular HACs enhanced the rate of heat transfer between 20% and 27% compared to the horizontally oriented HACs. However it differs by 8–11% when compared to the square shape HACs for different Rayleigh numbers. Numerical investigation of an identical model dealt with mixed convection of nanofluids filled lid driven enclosure was examined by Garoosi et al. [22]. It was confirmed that the rate of heat transfer was increased by changing the orientation of the HAC from horizontal to vertical position for the entire range of Richardson number. Alsabery et al. [23] performed a numerical investigation of natural convection in a trapezoidal enclosure partly filled with nanofluid porous layer and partly with non-Newtonian fluid layer. It was found that convection plays a remarkable role with the addition of silver–water nanofluid and the heat transfer rate was affected by the inclination angle of the enclosure. Recently Kolsi et al. [24] computed the three dimensional natural convection and entropy generation nanofluid filled differentially heated cubical enclosure. The above studies confirmed that the performance of fluid flow and heat transfer rate
Present Study Tou et al. [8]
10
1
Ro
Nui
w
Ro
3
w
2
Ro
10
4
10
w
5
Ra
1
10
6
10
7
Fig. 2. Row-averaged Nusselt number against Ra.
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Fig. 3. Case (i) for Cu–water nanofluid with / = 0.05, Ra ¼ 107 and AS ¼ 7:5 – (a) isothermal surfaces, isotherms at (b) Y = 0, (c) Y = 0.05, (d) Y = 0.9, (e) X = 0.3, 1.3, 2.3 & 3.3; velocity vector at (f) Y = 0.05, (g) Y = 0.9, and (h) X = 0.3, 1.3, 2.3, & 3.3.
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A. Purusothaman et al. / Advanced Powder Technology xxx (2016) xxx–xxx
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Fig. 4. Case (ii) for Cu–water nanofluid with / = 0.05, Ra ¼ 107 and AS ¼ 7:5 – (a) isothermal surfaces, isotherms at (b) Y = 0, (c) Y = 0.05, (d) Y = 0.9, (e) X = 0.3, 1.3, 2.3 & 3.3; Velocity vector at (f) Y = 0.05, (g) Y = 0.9, and (h) X = 0.3, 1.3, 2.3, & 3.3.
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A. Purusothaman et al. / Advanced Powder Technology xxx (2016) xxx–xxx
Fig. 5. Case (iii) for Cu–water nanofluid with / = 0.05, Ra ¼ 107 and AS ¼ 7:5 – (a) isothermal surfaces, isotherms at (b) Y = 0, (c) Y = 0.05, (d) Y = 0.9, (e) X = 0.3, 1.3, 2.3 & 3.3; Velocity vector at (f) Y = 0.05, (g) Y = 0.9, (h) X = 0.3, 1.3, 2.3, & 3.3.
increased while increasing both the Rayleigh number and solid volume fraction of nanoparticles.
From the above literature review, it is very clear that the technique for improving heat transfer performance using nanofluids in
Please cite this article in press as: A. Purusothaman et al., Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2015.12.012
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Fig. 6. Isothermal surfaces for Cu–water nanofluid with AS ¼ 2:0, / = 0.05, Ra ¼ 107 – (a) Case (i), (b) Case (ii), (c) Case (iii); isotherms for Case (iii) at (d) Y = 7.5 & X = 3.75 for AS ¼ 1:0, (e) Y = 3.75 & X = 3.75 for AS ¼ 2:0, and (f) X = 0.0 & 3.75, Y = 1.0 for AS ¼ 7:5.
3D models is very limited besides the available work [24]. Hence, taking this opportunity, the aim of the present study is to evaluate the natural convection heat transfer performance with an array of non-protruding heat source mounted in a rectangular enclosure filled with nanofluids of different types and concentrations of nanoparticles. This configuration may give some additional knowledge in designing hermetically sealed electronic packages encountered in the microelectronics industry. 2. Mathematical formulation A schematic plan of the system considered in the present study and its fixed geometric parameters are displayed in Fig. 1. The
enclosure is three dimensional and rectangular and its height, width and length are denoted by H; C and D respectively. A 3 3 array of discrete heat sources is mounted on one vertical wall (y = 0) of a nanofluid filled enclosure. The remaining part of the wall in which the sources are mounted and the other walls are adiabatic unless otherwise specified. Three different symmetrical thermal boundary conditions are considered on the enclosure walls. Accordingly Case (i) opposite vertical wall, Case (ii) side vertical walls and Case (iii) top and bottom horizontal walls are maintained cold. The gravity acts perpendicular to the xy plane. The water based nanofluids containing solid spherical nanoparticles (Cu or Al2O3) and their thermophysical properties are listed in Table 1. The flow is assumed to be three dimensional,
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Fig. 7. Isotherms for water (—: / = 0.0) and Cu–water nanofluid (– – – –: / = 0.05) – (a) Case (i), (b) Case (ii), and (c) Case (iii) at Y = 0.01, Ra ¼ 107 and AS ¼ 7:5.
time-dependent, laminar, incompressible with negligible viscous dissipation and radiation effects. The fluid properties are assumed to be constant except the density in the buoyancy term that follows the Boussinesq approximation. Taking into account the above mentioned assumptions, the governing equations of the continuity, momentum and energy can be written in nondimensional form as follows:
r:~ V ¼ 0;
ð1Þ
qf bnf m @~ V Vþ þ ð~ V:rÞ~ V¼ rP þ nf Pr r2 ~ RaPrh~ k; @s qnf mf bf
ð2Þ
anf 2 @h þ ð~ V:rÞh ¼ r h; @s af
ð3Þ
where ~ V is the velocity vector with components U; V and W in the X; Y and Z directions, respectively and ~ k denotes the vector acting along Z-direction. Eqs. (1)–(3) were cast in non-dimensional form by using the following variables:
s ¼ taf =L2z ; P ¼ pL2z =qf a2f ; U ¼ uLz =af ; V ¼ v Lz =af ; W ¼ wLz =af ; h ¼ ðT T c Þ=ðT H T C Þ; Ra ¼ gbf ðT H T C ÞL3z =ðmf af Þ; Pr ¼ mf =af : ð4Þ
X ¼ x=Lz ;
Y ¼ y=Lz ;
Z ¼ z=Lz ;
Density qnf and thermal diffusivity anf of nanofluid are calculated as
qnf ¼ ð1 /Þqf þ /qp ;
ð5Þ
k anf ¼ nf : ðqcp Þnf
ð6Þ
The heat capacitance and thermal expansion coefficient of the nanofluid can be determined as follows:
ðqcp Þnf ¼ ð1 /Þðqcp Þf þ /ðqcp Þp ;
ð7Þ
ðqbÞnf ¼ ð1 /ÞðqbÞf þ /ðqbÞp :
ð8Þ
The effective thermal conductivity of the nanofluid knf is approximated by the Maxwell–Garnett model [25] as
knf kp þ 2kf 2/ðkf kp Þ ¼ : kf kp þ 2kf þ /ðkf kp Þ
ð9Þ
The effective dynamic viscosity of the nanofluid lnf can be approximated using the Brinkman [26] relation as
lnf ¼
lf ð1 /Þ2:5
:
ð10Þ
The dimensionless boundary conditions at the enclosure walls are:
D @h @h ¼ 0; ðiiÞ h ¼ 0; ðiiiÞ ¼ 0; : U ¼ V ¼ W ¼ 0; ðiÞ Lz @X @X ( h ¼ 1; source region; At Y ¼ 0 : U ¼ V ¼ W ¼ 0; @h ¼ 0; otherwise; @X
At X ¼ 0;
C @h ¼ 0; : U ¼ V ¼ W ¼ 0; ðiÞ h ¼ 0; ðiiÞ and ðiiiÞ Lz @Y H @h ¼ 0; ðiiiÞ h ¼ 0; ð11Þ : U ¼ V ¼ W ¼ 0; ðiÞ and ðiiÞ At Z ¼ 0; Lz @Z
At Y ¼
Heindel et al. [6,7], Tou et al. [8] discussed the plane of symmetry in their three dimensional natural convection problem. This assertion was experimentally confirmed by Ozoe et al. [27]. Following this, the plane at X ¼ D=2Lz is taken as a plane of symmetry and the computational domain is reduced by half. This is possible for all three cases due to symmetrically imposed boundary conditions at the enclosure walls and hence the plane of symmetry is defined as follows:
At X ¼
D @V @W @h ¼ ¼ 0; ðiÞ; ðiiÞ and ðiiiÞ ¼ 0: :U¼ 2Lz @X @X @X
ð12Þ
The heat transfer rate within the enclosure can be obtained through the Nusselt number. The local and average Nusselt numbers are calculated from the resulting temperature field at each heat source and are defined as follows:
The local Nusselt number : Nuik ðX; ZÞ ¼
knf @h ; kf @Y
The average Nusselt number : Nuik ðX; ZÞ ¼
1 Ah
Z
Ah
ð13Þ
Nui;k ðX; ZÞdAh ;
0
ð14Þ The row averaged Nusselt number : Nui ¼
3 1X Nui;k ðX; ZÞ; 3 i¼1
ð15Þ where Ah is the heater area and the subscript ik refers to the specified heater in row i and column k.
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A. Purusothaman et al. / Advanced Powder Technology xxx (2016) xxx–xxx
40
Row 3 Row 2 Row 1
As=2.0
30 As=7.5
Nui
20
As=1.0
10
0.025
0.0
0.075
0.05
0.1
φ
(a) 40
35
Row 3 Row 2 Row 1
As=2.0
30
Row 3 Row 2 Row 1
25 As=2.0
Nui
Nui
20
15 As=7.5
As=7.5
As=1.0
As=1.0
10
0.025
0.0
0.05
0.075
0.1
5
0.0
φ
0.025
0.05
0.075
0.1
φ
(b)
(c)
Fig. 8. Average Nusselt number against / for Cu–water nanofluid (a) Case (i), (b) Case (ii), and (c) Case (iii) at Ra ¼ 107 .
3. Numerical procedure The governing nondimensional Eqs. (1)–(3) have been solved numerically based on finite volume technique described by Patankar [28]. A uniform staggered grid is considered in the computational domain. The scalar quantities (P and h) are stored in the nodal points whereas the components of the velocity (U; V and W) are stored at the cell face of the control volumes of the nodal points. A fully implicit time marching scheme is employed. The numerical procedure namely Semi-Implicit Method for Pressure Linked Equation (SIMPLE) algorithm is used to handle the pressure velocity coupling. The convective and diffusive terms in Eqs. (2) and (3) are handled by adopting the power-law scheme. Finally the discretized set of algebraic equations are solved by the lineby-line Tri-Diagonal Matrix Algorithm (TDMA). To ensure convergence of the numerical algorithm, the following criterion applied to all dependent variables over the solution domain
P
fm1 i;j;k 6 105 ; P m f i;j;k i;j;k
m i;j;k fi;j;k
ð16Þ
where f represents the dependent variables U; V; W and h, the subscripts i; j; k indicate the space coordinates and the superscript m represents the time step. Based on the above numerical procedure, the present problem is solved by the developed FORTRAN code. In order to examine the effect of the grid on the numerical study, the numerical code was examined for Ra ¼ 107 ; / ¼ 0:0 and AS ¼ 7:5 using five different computational grids, namely, 55 31 91, 61 31 91, 76 91 91, 91 91 91 and 109 91 121, for the Case (i). It is observed that a refinement of the grid from 91 91 91 to 109 91 121 does not have a significant effect on the results in terms of row averaged Nusselt number and the difference in Nusselt numbers between these two grids is found to be less than 0.1%. Hence, by considering both
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Fig. 9. Case (iii) for Cu–water nanofluid with AS ¼ 7:5 and / = 0.05 – isotherms at Y = 0.01 (a) Ra ¼ 105 , (b) Ra ¼ 107 ; W – velocity component at Y = 0.1, (c) Ra ¼ 105 , and (d) Ra ¼ 107 .
the accuracy and computational time, the present calculations are all performed with a 91 91 91 spaced grid system.
of the numerical results produced by the present code is satisfactory with those provided in the literature.
3.1. Validation of the study Before proceeding further, it is of interest to compare the numerical results of the three dimensional empty enclosure which is heated differentially at two vertical side walls with the available results in the literature. Hence the code was validated against the natural convection in a 3-D empty enclosure which was studied by Tric et al. [29] and the Nusselt numbers for Ra ¼ 104 107 are given in Table 2. The results show that the maximum percentage deviation in the average Nusselt number values is 0.8% up to Ra ¼ 106 . But when Ra ¼ 107 , they deviate to a maximum of 2.6%. Fig. 2 presents another comparative study with Tou et al. [8] in terms of the variation of row averaged Nusselt number with the Rayleigh number for a fixed aspect ratio A ¼ 7:5 and Prandtl number Pr ¼ 9. It is observed that the deviation in the Nusselt number lies within 5.6%, for 104 6 Ra 6 107 . Hence the agreement
4. Results and discussion Buoyancy driven natural convection cooling of a 3 3 array of discrete heaters mounted in a nanofluid filled three dimensional rectangular enclosure is numerically investigated with three different symmetric thermal boundary conditions. To ensure an effectual way of cooling performance on the enclosure, it is considered that two different nanofluids consisting of water and different volume concentrations of Cu or Al2O3 nanoparticles. Thus the Prandtl number (Pr) of the base fluid (water) is fixed to be 6.2 throughout this study. Moreover computations are performed for wide ranges of Rayleigh number (105 6 Ra 6 107 ), enclosure side aspect ratio (1:0 6 AS 6 7:5) and nanoparticles solid volume fraction (0:0 6 / 6 0:1).
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40
36
Row 3 Row 2
Cu-Water
Row 1
Case (i)
Al 2O3-Water
35
Case (i)
27
Case (ii) Case (iii)
Case (ii) Case (iii)
30
Nui
Nu3
18
25
20
9 15
0
10
5
10
6
10
7
Ra Fig. 10. Average Nusselt number against Ra for Cu–water nanofluid with AS ¼ 7:5 and / = 0.05.
4.1. 3D nature of temperature and flow field The nature of the isotherms and velocity vectors within Cu–water nanofluid filled enclosure at the selected plane sections parallel to the enclosure walls corresponding to Case (i) are shown in Fig. 3 (a)–(h) for the fixed value of Ra = 107, / = 0.05 and AS ¼ 7:5. It is found that the isotherms around the heater modules are more compact and the maximum fluid temperature within the enclosure is recorded at the top of each heater on account of receiving additional energy from the bottom heaters through the dominance of convection mechanism. It is very clear that the magnitude of the velocity vectors adjacent to the column heaters is maximum which are qualitatively expressed in Fig. 3(f)–(h). However the fluid near the adiabatic region on the side of the heater array and the adiabatic walls stays motionless. The stream trace in Fig. 3(f) visibly demonstrates that the raising plume like upward flow on the heaters hits the enclosure ceiling and turns naturally towards the opposite cooled wall. Adjacent to the opposite cold wall at Y = 0.9, the majority of the fluid motion is vertically downward and uniform eventually forming a simple closed flow pattern with a torpid core region. Moreover the flat isothermal surfaces in Fig. 3 (a) indicate that the originated heat from the heaters shifts evenly towards the opposite cold wall. As a result, the entire region of the enclosure is thermally stratified in the vertical direction (see Fig. 3 (e)). The consequences of the Case (ii) are presented in Fig. 4(a)–(h) in terms of isotherms and velocity vectors. It is noticed that the generated heat from the heater array is transferred towards the side cold walls with wavy isotherms. Moreover the fluid temperature above the core region increases significantly compared to Case (i). It can be reflected by the lack of heat removal effectiveness through the wavy temperature distribution which is equal to the reciprocal of the ideal efficiency in controlling the device operating temperature. However the vertical thermally stratified fluid core region is apparently noticed in this case also (see Fig. 4(a)–(e)). The velocity vectors for this case are depicted in Fig. 4(f)–(h). The rising fluid contiguous to the heater array moves upward and revolves toward the side cold walls with a bulk fluid motion which are assured by the stream traces in Fig. 4(f). However the fluid
10
0.0
0.025
0.05
φ
0.075
0.1
Fig. 11. Average Nusselt number against / for Cu–water and Al2O3–water nanofluids at AS ¼ 7:5 and Ra ¼ 107 .
motion is practically stagnant near the opposite adiabatic wall at Y 0.9 plane owing to an adverse situation for fluid flow. The isothermal surface, temperature distribution and velocity vectors corresponding to Case (iii) are shown in Fig. 5(a)–(h). It is interesting to note from Fig. 5(a)–(e) that complex flow pattern evolution is observed which, to the author’s knowledge, has been reported previously by Tso et al. [9]. Moreover the raising fluid from the column heaters reaches the ceiling and loses its momentum due to the cooled top and falls down freely near the opposite and adjacent walls of the enclosure. But in the opposite wall, the nonuniform downward flow descends drastically in the region not facing the column heaters. This causes the existence of isothermal closed loops in the opposite wall which collapse the vertical thermal stratification above the core region of the enclosure (see Fig. 5(d)). This is mainly associated with nonuniform downward flows adjacent to the side and opposite walls which is contradictory to the cases (i) and (ii). However the bottom of the enclosure (Z 6 2) is thermally stratified. These observations turn out to be more obvious from the vector plots in Fig. 5(f)–(h). On comparing Figs. 3–5 for all three cases, it is observed that, though the constant temperature is maintained in all nine heaters, the fluid temperature in the upper region of each heater is maximum than in the other region. This is due to the fact that the fluid on top of the each heater receives additional heat energy from the bottom heaters through the buoyancy mechanism. However, the fluid temperature within the enclosure is recorded minimum in Case of (i) than in the Cases (ii) and (iii) for AS ¼ 7:5. Hence it is obviously revealed that the flat plane thermal distribution in Case (i) is more efficient transforming trend which successfully controls the device operating temperature within the limits. When AS ¼ 2:0, this flatness thermal distribution tendency is also retained for both Cases (i) and (ii) which stabilizes the system thermal efficiency and is shown in Fig. 6(a) and (b). In Case (iii), the isothermal surfaces and isothermal propagation quit the flatness and adopt more complex forms (see Figs. 5 and 6(c)–(f)). Moreover one can notice that the existence of the closed thermal loops is with local temperature maximum on adiabatic walls. This is one of the main objectives of electronic equipment cooling problems of finding the specific hot locations on the enclosure walls. This superfluous heat can be
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absorbed by placing sinks, where it occurs, thus providing a pathway to improve the efficiency of equipment. In these simulations, the particle volume fraction is an important factor to be considered to obtain nanofluids with more thermal efficiency which are required for some applications relating to microelectronic manufacturing industries. For the sake of comparisons, the isotherms for base fluid and Cu–water nanofluid with 5% particle volume concentration are exhibited by solid line and dashed line respectively for three different cases shown in Fig. 7 (a)–(c) at Y = 0.01 plane section with a fixed value of AS ¼ 7:5
shown in Fig. 11. This indicates that Cu–water nanofluid has the greatest effect on the thermal performance compared with Al2O3–water nanofluid. Moreover the nanofluid effect and its performance are more prominent in Case (i) than in the Cases (ii) and (iii). The results obtained in this study are expected to lead to new guidelines for cooling the electronic equipment which will improve the reliability of the system performance.
5. Conclusion
7
and Ra ¼ 10 . The shape of the temperature distributions inside the enclosure shows how the dominant heat transfer mechanism changes in base fluid and nanofluid. It is shown that the thermal energy transfers faster as natural convection is intensified in nanofluid than in the base fluid. In order to know the effects of /; AS and boundary conditions in terms of the heat transfer rate, the graphs of row averaged Nusselt numbers, Nui (i = 1, 2, 3), are depicted in Fig. 8 for Ra ¼ 107 and Cu– water nanofluid. As a consequence, the solid volume fractions / are varied from 0% to 10% with an increment of 2.5% and the enclosure side aspect ratio is assumed the value between the ranges of 1 to 7.5. The results show that the heat transfer rate increases monotonically with increasing volume fraction in all the three cases. The trends shown by the nanofluid is due to the fact that the suspension of nanoparticles in basefluid enhances the thermal conductivity of the nanofluid leads to high heat transfer performance. Moreover on account of buoyancy mechanism, the row averaged Nusselt number for the top row heaters is less than the bottom row heaters irrespective of the value of AS , (that is, Nu1 < Nu2 < Nu3 ). This is consistent for all the three cases. When comparing Case (i), Case (ii) and Case (iii), the row average Nusselt numbers Nui for Case (i) attain a maximum value and the heat transfer rates Nui are extremely supreme for those corresponding to AS ¼ 2:0. Fig. 9 is presented to show the effect of Rayleigh numbers on the temperature distribution and flow field corresponding to the Case (iii) for Cu–water nanofluid with fixed AS ¼ 7:5 and / = 0.05. On the XZ-plane at Y = 0.01, the isotherms are illustrated in Fig. 9 (a) and (b) for the Rayleigh numbers Ra ¼ 105 and 107 respectively. Isotherms and local fluid temperature adjacent to the heater face region in Fig. 9(a) for Ra ¼ 105 indicates that heat transfer is more influenced by the conduction mode of heat transfer mechanism. With an increase in the Rayleigh number (Ra ¼ 107 ), the isotherms become more stratified adjacent to the heater face region because of induced buoyancy force and the convective heat transfer is more effective. The velocity vectors at Y ¼ 0:1 plane are shown in Fig. 9 (c) and (d) for the different Rayleigh numbers Ra ¼ 105 and 107 respectively. For discussing the magnitude of the fluid velocity, the W-velocity component is only taken as primary component which is more supreme in strength than the other two (U and V) components. It is found from Fig. 9(c) that flow is weakly strengthened for Ra ¼ 105 . When Ra ¼ 107 , the magnitude of the primary component increases as the buoyancy force begins to dominate the fluid motion. Moreover, it is noticed from Fig. 10, that the row averaged Nusselt number Nui (i = 1, 2, 3), increases with the increase of Ra due to the enhanced convection that carries heat energy with increased momentum. In addition, the present computations are carried out for two different nanofluids, namely, Cu–water and Al2O3–water. It is found that, with other parameters unaltered, maximum percentage deviation between these two nanofluids is only 2%. In order to know the results further in detail, the row average Nusselt number for row 3 (Nu3 ) is calculated for different solid volume fraction together with three boundary conditions and the variation is
A numerical study of natural convection heat transfer on a nanofluid filled three dimensional rectangular enclosure induced by a 3 3 array of wall mounted isothermal heaters is investigated in the context of cooling of electronic equipments. Computations are performed for wide ranges of Rayleigh number, enclosure side aspect ratio and nanoparticle volume fraction together with different nanofluids and enclosure boundary conditions. The study leads to the following significant conclusions: (a) The fluid temperature within the enclosure is recorded minimum in Case (i) than in the Cases (ii) and (iii). This is obviously achieved by the flat thermal distribution which has more efficient transforming trend which successfully controls the device operating temperature within the limits. (b) The enclosures of Cases (i) and (ii) clearly demonstrate a high degree of vertical thermal stratification with a stagnant core for higher values of Ra. But Case (iii) breaks the vertical thermal stratification associated with non-uniform downward flows from the cooled ceiling. (c) Moreover isothermal propagation in Case (iii) quits the flatness and adopts more complex forms of closed thermal loops with local temperature maximum on adiabatic or opposite walls and is highly sensitive for higher value of AS . (d) From the cooling performance and system geometry design point of view, AS ¼ 2:0 is an optimal enclosure side aspect ratio to dissipate the maximum heat from the isothermal heaters. (e) The suspension of nanoparticles in basefluid enhances the thermal conductivity of the mixture (nanofluid) leads to high heat transfer rate than the base fluid. Hence increasing the value of nanoparticle volume fraction (0–10%) yields better thermal performance than that of the base fluid for all three cases, irrespective of the values of Ra and AS . (f) The overall heat transfer within the enclosure for Ra ¼ 107 is enhanced in Case (i) than in the other two cases. This shows that maintaining the opposite wall to be cold is the optimum condition of cooling the dispersed heat from an electrical device. (g) Cu–water nanofluid has the greatest effect on the thermal performance compared with Al2O3–water nanofluid. Moreover the row averaged Nusselt number increases monotonically with increasing the Rayleigh number and the nanoparticle volume fraction.
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Please cite this article in press as: A. Purusothaman et al., Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2015.12.012