Behzad Ghasemi and Saiied Mostafa Aminossadati
NUMERICAL SIMULATION OF MIXED CONVECTION IN A RECTANGULAR ENCLOSURE WITH DIFFERENT NUMBERS AND ARRANGEMENTS OF DISCRETE HEAT SOURCES Behzad Ghasemi Engineering Faculty, Shahrekord University Shahrekord, P.O.Box 115, IRAN
and Saiied Mostafa Aminossadati * School of Engineering, The University of Queensland CRC Mining, Brisbane, Australia
:اﻟﺨﻼﺻـﺔ اﻟﺤﻠﻮل اﻟﺤﺴﺎﺑﻴﺔ ﻷداء ﺗﺒﺮﻳﺪ اﻟﻘﻄﻊ واﻷﺟﺰاء اﻹﻟﻜﺘﺮوﻧﻴﺔ ﺁﺧﺬﻳﻦ- ﻓﻲ هﺬﻩ اﻟﺪراﺳﺔ- ﺳﻮف ﻧﺴﺘﻘﺼﻲ ، وﺳﻮف ﻧﺴﺘﺨﺪم ﻟﺬﻟﻚ ﺣﺎوﻳﺔ ﻣﺴﺘﻄﻴﻠﺔ ﺗﺤﺖ ﺗﺄﺛﻴﺮ اﻟﺤﻤﻞ اﻟﺤﺮاري اﻟﻄﺒﻴﻌﻲ واﻟﻘﺼﺮي.ﺑﺎﻻﻋﺘﺒﺎر ﻋﺪدهﺎ وﺗﺮﺗﻴﺒﻬﺎ وﻳُﻨﺘـَﺞ اﻟﻤﺼﺪر اﻟﺤﺮاري ﻓﻲ اﻟﺤﺎوﻳﺔ اﻟﺤﻤﻞ اﻟﺤﺮاري اﻟﻄﺒﻴﻌﻲ وﻣﺼﺪرًا ﺧﺎرﺟﻴًﺎ ﻟﺪﻓﻊ.ﻻ ﻣﻦ أرﻗﺎم راﻳﻠﻲ ً وﻣﺠﺎ وﻗﺪ أﺑﺎﻧﺖ اﻟﻨﺘﺎﺋﺞ أن أرﻗﺎم راﻳﻠﻲ اﻟﻜﺒﻴﺮة أدّت إﻟﻰ ﺗﺤﺴﻦ.اﻟﻬﻮاء ﺧﻼل اﻟﺤﺎوﻳﺔ ﻹﻧﺘﺎج اﻟﺤﻤﻞ اﻟﺤﺮاري اﻟﻘﺼﺮي ﺑﻴﻨﻤﺎ أدّت أرﻗﺎم راﻳﻠﻲ اﻟﺼﻐﻴﺮة اﻟﻤﺼﺤﻮﺑﺔ ﺑﺰﻳﺎدة ﻋﺪد اﻟﻤﺼﺎدر اﻟﺤﺮارﻳﺔ ﻓﻲ، آﺒﻴﺮ ﻓﻲ اﻻﻧﺘﻘﺎل اﻟﺤﺮاري ﺑﻴﻨﻤﺎ آﺎن ﻟﺘﺮﺗﻴﺐ وﻋﺪد اﻟﻤﺼﺎدر، وﺑﺎﻟﺘﺎﻟﻲ زﻳﺎدة درﺟﺔ اﻟﺤﺮارة اﻟﻘﺼﻮى، اﻟﺤﺎوﻳﺔ إﻟﻰ ﺗﻨﺎﻗﺺ اﻻﻧﺘﻘﺎل اﻟﺤﺮاري . ﺑﻴﻨﻤﺎ أدت أرﻗﺎم راﻳﻠﻲ اﻟﻜﺒﻴﺮة إﻟﻰ ﻧﺘﺎﺋﺞ ﻣﻌﺎآﺴﺔ، اﻟﺤﺮارﻳﺔ أﺛ ٌﺮ ﻋﻠﻰ أداء اﻟﺘﺒﺮﻳﺪ
* Address for correspondence: Dr. S. M. Aminossadati School of Engineering, The University of Queensland QLD 4072, Australia E-mail:
[email protected] Paper Received 12 April 2006; Revised 15 June 2007; Accepted 28 November 2007
April 2008
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
189
Behzad Ghasemi and Saiied Mostafa Aminossadati
ABSTRACT The objective of this paper is to numerically investigate the cooling performance of electronic devices with an emphasis on the effects of the arrangement and number of electronic components. The analysis uses a two dimensional rectangular enclosure under combined natural and forced convection flow conditions and considers a range of Rayleigh numbers. Heat sources in the enclosure generate the natural convection flow and an externally sourced air stream through the enclosure generates the forced convection flow. The results show that increasing the Rayleigh number significantly improves the enclosure heat transfer process. At low Rayleigh numbers, placing more heat sources within the enclosure reduces the heat transfer rate from the sources and consequently increases their overall maximum temperature. The arrangement and number of heat sources have a considerable contribution to the cooling performance. However, increasing the Rayleigh number reduces this contribution. Key words : heat transfer, mixed convection, rectangular enclosures.
190
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
April 2008
Behzad Ghasemi and Saiied Mostafa Aminossadati
NOMENCLATURE Re
Reynolds number, ui hi ν
Ric
Richardson number, Ric = Ra Pr Re 2
t
Time, Thickness of heat source
t1
Thickness of heat source cover
T
Temperature
1 Heat source at the bottom
ui
Inlet flow velocity
BB
2 Heat sources at the bottom
u ,v
x and y components of velocity
BBB
3 Heat sources at the bottom
U ,V
BBL
3 Heat sources (2 bottom, 1 left)
Dimensionless velocities, u ui , v ui
BBR
3 Heat sources (2 bottom, 1 right)
Ve
Volume of heat source per length
BLR
3 Heat sources (1 bottom, 1 left, 1 right)
x, y
Coordinates
B1
1st Heat source at the bottom
X ,Y
Dimensionless coordinates, x hi , y hi
B2
2nd Heat source at the bottom
Z1
Heat source diffusivity ratio, α e α
B3
3rd Heat source at the bottom
Z2
Heat source cover diffusivity ratio, α c α
d
Distance between the heat sources
Z3
Heat source conductivity ratio, ke k
D
Dimensionless parameter, Ve/(As hi)
α
Heat diffusivity coefficient
g
Gravity
β
Fluid volume expansion coefficient
h
Height of entry and exit sections
∆T
H
Height of enclosure
Ref. temperature difference, hi2q ke
Conduction coefficient
θ
k
Dimensionless temp., (T − Ti )/∆T
L
Length of enclosure
θ s,max
Maximum dimensionless temperature
n
Normal vector to the surface
µ
Fluid viscosity
Num
Average Nusselt number
ν
Kinematic fluid viscosity
Nu m, F
Average Nusselt number, forced convection
ρ
Density
p
Pressure
τ
Dimensionless time, uit hi
P
Dimensionless pressure, p / ρui 2
p
Modified pressure, p+ ρi gy
c
Heat source cover
Pr
Prandtl number, ν α
e
Heat source
q
Heat generation rate per volume
i
Inlet (entry section)
qs
Heat flux from the heat source, q Ve / A s
o
Outlet (exit section)
Ra
Rayleigh number, gβ h 3i ∆T ν α
s
Surface
a
Length of the heat source
a1
Length of the heat source cover
As
Heat transfer area per length, a+2t
As
Dimensionless form of As, As/hi
AR
Length to height ratio, L H
B
April 2008
Subscripts
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
191
Behzad Ghasemi and Saiied Mostafa Aminossadati
NUMERICAL SIMULATION OF MIXED CONVECTION IN A RECTANGULAR ENCLOSURE WITH DIFFERENT NUMBERS AND ARRANGEMENTS OF DISCRETE HEAT SOURCES 1. INTRODUCTION The continuing miniaturization of electronic devices has significantly contributed to rapid developments in electronic technology. Miniaturization is, however, characterized by high heat dissipation per unit area of electronic components. Therefore, an effective cooling strategy is required to avoid premature failure and to ensure enhanced reliability and life expectancy of electronic components [1]. Despite various cooling techniques and recent developments in the electronic industry [2–3], air cooling is still considered to be not only a cheap and readily available option but also a very effective technique. The performance of air cooling by natural convection is extensively influenced by the physical characteristics of the electronic components such as number, size, spacing, and position arrangements [4–10]. A combination of forced and natural convection, known as mixed convection, has been recommended for high heat dissipating electronic components, where natural convection is incapable of providing satisfactory cooling [11]. Heat generating components of electronic devices are generally considered as discrete heat sources in numerical analysis. The influence of the physical characteristics of heat sources on the mixed convection heat transfer performance has been examined by many researchers. Yilbas et al. [12] studied the influence of the aspect ratio of a protruding body on a square cavity heat transfer and entropy characteristics. They showed that the heat transfer increases at high aspect ratios, while the irreversibility, generated in the cavity, reduces. Papanicolaou and Jaluria [13] studied the effects the arrangement of the positions of the heat sources on mixed convection heat transfer in a two-dimensional square enclosure. Chen et al. [14] experimentally investigated the effects of different arrangements of obstacles on the cooling of electronic packages. They concluded that the conventional equi-spaced arrangement of obstacles is not the optimum option to achieve a good cooling performance. Mixed convection cooling of board mounted components was studied by Kehoe et al. [15] for a horizontally printed circuit board. They argued that the optimum location of critical components in mixed convection systems is of paramount importance. da Silva et al. [16] examined the optimal distribution of discrete heat sources on a plate with laminar forced convection. They stated that the heat sources should be placed nonuniformly, with the smallest distance between them near the tip of the plate. This is similar to the results reported by Young and Vafai [17]. At high Reynolds number, the heat sources should be mounted flush against each other near the channel entrance. Mixed convection from four heat sources distributed in a three dimensional channel was numerically investigated by Cheng et al. [18]. Dogan et al. [19] experimentally investigated mixed convection in a horizontal channel with discrete heat sources at the top and at the bottom. They, in line with [16], suggested that when dealing with high power densities in an environment of mixed convection, electronic components with the greatest heat dissipation should be placed on the first and last two rows at the bottom of channel, whereas low heat dissipating components should always be placed around the middle section. A numerical investigation of laminar mixed convection in a rectangular cavity with differentially heated side walls was conducted by Singh and Sharif [20]. The objective of their work was to identify the optimum placement of the inlet and exit for best cooling effectiveness. Their results showed that for the configuration of the cavity with the inlet near the bottom and the exit near the top (S-flow), the forced and natural convection assist each other in the heat removal process. Rhee and Wong [21] similarly argued that for most of the chassis designs in telecommunication equipment with populated components in the front panel, the S-flow pattern provides a better cooling in comparison to the other patterns. In any electronic device, there is at least one or more highly sensitive or very high heat dissipating component. A proposed cooling strategy should not only take into account the overall heat dissipation from the electronic device, but it should also consider the allowable conditions for highly sensitive and very high heat dissipating components. While the overall heat dissipation rate is significant, close attention should also be paid to the temperature distribution in the whole flow region to avoid hot spots. As such, the objective of this research is to study a two-dimensional air pattern (S-flow) while taking into account the two factors of heat dissipation from the heat sources and temperature distribution in the enclosure. Moreover, in the following discussion, the effects of the number and arrangement of electronic components on the cooling performance are analyzed. This leads to an appropriate design for the arrangement of highly sensitive electronic components which, in turn, results in a superior cooling performance for the small-scale electronic device.
192
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
April 2008
Behzad Ghasemi and Saiied Mostafa Aminossadati
2. PROBLEM DEFINITION Figure 1 shows a schematic diagram of a two-dimensional rectangular enclosure used in the first stage of the analysis. The objective of this stage is to examine the effects of number of heat sources on the thermal behavior of the enclosure. This is achieved by performing a comparison between the flow and temperature fields and the heat transfer rates for the enclosure with one, two, and three heat sources. The heat sources simulate the components of electrical devices generating a natural convection flow in the enclosure. They are considered to be made of silicon covered with a layer of ceramics. In addition, an externally sourced uniform air stream with a specific temperature enters the enclosure from a section located at the bottom of the front panel (entry section) and exhausts from the upper rear section (exit section). This S-pattern airflow generates a forced convection flow in the enclosure. The entry and the exit sections have the same height, being one fourth of the height of the enclosure.
y x
Figure 1. A schematic diagram of the enclosure with one, two, and three heat sources located at the bottom of the enclosure
The second stage of the analysis investigates the mixed convection heat transfer, with an emphasis on the effects of the arrangement of three heat sources in the enclosure. The flow and temperature fields and the heat transfer rates at various arrangements of the heat sources are determined and compared. Similarly to the first stage, the heat sources generate a natural convection flow and the externally-sourced air stream generates a forced convection flow in the enclosure. Figure 2 shows four different arrangements of the three heat sources in the enclosure. BBB refers to the case where all three heat sources are located at the bottom of the enclosure. In the case of BBR, two heat sources are located at the bottom of the enclosure while one is placed on the right wall. BBL refers to the enclosure where two heat sources are located at the bottom and one source is located on the left wall of the enclosure. In the case of BLR, three heat sources are located at the bottom, on the left wall and on the right wall of the enclosure.
April 2008
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
193
Behzad Ghasemi and Saiied Mostafa Aminossadati
Figure 2. A schematic diagram of the enclosure with different configurations of three heat sources
3. GOVERNING EQUATIONS The governing equations for a domain within a two-dimensional enclosure, which includes the fluid and the heat sources, are presented. A two-dimensional, laminar, Newtonian, and incompressible flow is assumed having an unsteady laminar convection with Boussinesq approximation. The continuity, momentum, and energy equations for the fluid are given as: The continuity equation: ∂u ∂v + =0 ∂x ∂y
(1)
The x-momentum equation: ρ
⎛ ∂ 2 u ∂ 2u ⎞ ∂u ∂u ∂u ∂P ⎟ + ρu + ρv =− + µ⎜ + ⎜ ∂x 2 ∂y 2 ⎟ ∂t ∂x ∂y ∂x ⎝ ⎠
(2)
ρ
⎛ ∂ 2v ∂ 2v ⎞ ∂v ∂v ∂v ∂P ⎟ − ρg + ρu + ρv =− + µ⎜ + ⎜ ∂x 2 ∂y 2 ⎟ ∂t ∂x ∂y ∂y ⎠ ⎝
(3)
The y-momentum equation:
The energy equation: ⎛ ∂ 2T ∂ 2T ∂T ∂T ∂T + +u +v =α ⎜ ⎜ ∂x 2 ∂y 2 ∂t ∂x ∂y ⎝
⎞ ⎟ ⎟ ⎠
(4)
Based on the Boussinesq approximation, density is assumed to be constant in all the terms except in the buoyancy term. The modified pressure is defined as P = P + ρi gy . Using the fluid thermal expansion coefficient yields: −
It is also known that
194
∂P ∂P − ρg = − + ρ i g β (T − Ti ) ∂y ∂y
(5)
∂P ∂P . = ∂x ∂x
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
April 2008
Behzad Ghasemi and Saiied Mostafa Aminossadati
It is assumed that the heat source units consist of two components with constant physical properties. One is the actual silicon heat source component, which generates heat, and the other one is the ceramic cover with no energy generation. The energy equations for the two components are two-dimensional conduction equations. The heat source: 1 ∂T ⎛⎜ ∂ 2T ∂ 2T + = α e ∂t ⎜⎝ ∂x 2 ∂y 2
⎞ q ⎟+ ⎟ ke ⎠
(6)
⎞ ⎟ ⎟ ⎠
(7)
where, q is the heat generation rate per volume. The cover: 1 ∂T ⎛⎜ ∂ 2T ∂ 2T = + α c ∂t ⎜⎝ ∂x 2 ∂y 2
The governing equations are presented in non-dimensional forms as ∂U ∂V + =0 ∂X ∂Y
(8)
2 ⎛ 2 ∂U 1 ⎜∂ U ∂ U ∂U ∂U ∂P +U +V =− + + ∂τ ∂X ∂Y ∂ X Re ⎜⎜ ∂ X 2 ∂Y2 ⎝ 2 ⎛ 2 ∂V ∂V ∂P ∂ V 1 ⎜∂ V ∂V +U +V = − + + ∂X ∂Y ∂ Y Re ⎜⎜ ∂ X 2 ∂ Y 2 ∂τ ⎝ 2 ⎛ 2 1 ⎜∂ θ ∂θ ∂θ ∂θ ∂ θ +U +V = + Pr Re ⎜⎜ ∂ X 2 ∂τ ∂X ∂Y ∂Y2 ⎝
⎞ ⎟ ⎟ ⎟ ⎠
⎞ ⎟ Ra θ ⎟+ 2 ⎟ Pr Re ⎠
⎞ ⎟ ⎟ ⎟ ⎠
(9)
(10)
(11)
2 ⎛ 2 Z1 ⎜ ∂ θ ∂ θ ∂θ = + Pr Re ⎜⎜ ∂ X 2 ∂τ ∂Y2 ⎝
⎞ Z1 ⎟ ⎟ + Pr Re ⎟ ⎠
(12)
2 ⎛ 2 Z2 ⎜ ∂ θ ∂θ ∂ θ = + Pr Re ⎜⎜ ∂ X 2 ∂τ ∂Y2 ⎝
⎞ ⎟ ⎟ ⎟ ⎠
(13)
In the above non-dimensional equations, τ is the time in a dimensionless form ( ui t hi ). All the lengths are normalized by the height of entry section ( x hi , y hi ). Velocities are normalized by the inlet velocity ( u ui , v ui ). The pressure is normalized as p / ρui 2 , where p is the modified pressure ( p+ ρ i gy ). The steady-state solutions can be obtained by setting the time dependence terms to zero in the above unsteady nondimensional equations. Fully-developed conditions are considered at the exit section of the enclosure [8–10], as follows: Hydrodynamic Boundary Conditions: No slip boundary conditions (U=V=0) on all the walls, V=0, U=1 at the entry section and V=0, ∂U / ∂X = 0 at the exit section of the enclosure. Thermal Boundary Conditions: θ = 0 at the entry section and adiabatic conditions ( ∂θ / ∂n = 0 ) for all the walls and at the exit section of the enclosure (n is the normal component to the surface). The Nusselt number, Nu, is presented as a measure of convective heat transfer coefficient at the surfaces. The Nusselt number can be determined from the temperature distribution obtained by solving the governing equations. Higher values of Nu, indicate higher heat transfer rates.
April 2008
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
195
Behzad Ghasemi and Saiied Mostafa Aminossadati
The local Nusselt number is defined as: ⎛h Nu = ⎜⎜ i ⎝k
Z D ⎞ qs = 3 ⎟⎟ θs ⎠ Ts − Ti
(14)
where, qs is the heat flux from the heat source ( qs = q Ve / As ). Ve is the volume of the heat source per its unit length. As is the heat transfer area of the heat generating part per its length ( As = a + 2t ). The dimensionless parameter D is defined as D = Ve /( As hi ) . θ s is the dimensionless temperature of the heat source surface. The average Nusselt number is obtained by integrating the Nusselt number over the heat source surface area as: Z D Nu m = 3 As
1
∫θ As
(15)
d As
s
where, A s = As / hi . The average Nusselt number, Num , can be used as an indication of the total heat transfer from the heat sources. However, in order to determine the effect of natural convection on the rate of heat transfer, Num / Num, F is studied, where, Nu m is the average Nusselt number and Num, F is the Nusselt number for only the forced convection; that is to say in other words, Num, F is the Nusselt number in the absence of natural convection flow. 4. NUMERICAL METHOD The system of governing equations (8–13) with the boundary conditions, stated previously, is solved through a control volume formulation of the finite difference method. The SIMPLE algorithm is used to handle the pressure– velocity coupling. The convective fluxes across the surfaces of the control volume are determined by the power law discretization scheme. The details of the algorithm are given by Patankar [22]. A program code in Fortran is developed to follow the algorithm. A non-equidistant grid with a concentration of grid lines near the walls of the enclosure is considered to cover the computational spatial domain. In order to select the appropriate grid, a systematic grid independence study is carried out and the effect of grid refinement on the flow parameters is examined. A sample of the results is presented in Figure 3. In this figure, for the enclosure with one heat source, B, the effect of grid refinement on the heat source average Nusselt number is presented. According to the results of grid independency study and for the sake of computation time, a non-equidistant grid with 7000 nodes is selected for the analysis. A maximum mass residual below 10-9 is considered as the convergence condition.
7000
Num 4500
2000 0
5000
10000
15000
20000
Grid Points Figure 3. The effect of grid refinement on the average Nusselt number (AR=2, Ra = 10 5 , Re = 100 , Pr = 0.71 )
196
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
April 2008
Behzad Ghasemi and Saiied Mostafa Aminossadati
5. RESULTS In order to validate the present numerical code, the steady-state solution, obtained in a two-dimensional enclosure with two heat sources, is compared with the results of Papanicolaou and Jaluria [13]. The results (Table 1) show a maximum difference of 0.05 in terms of the maximum temperature in the entire enclosure for two different arrangements of heat sources and a range of Richardson numbers (Ric = Ra/Pr.Re2) between 0 and 10. Table 1. A Comparison Between the Present Study and Papanicolaou and Jaluria [13] Heat sources on the left and right walls Heat sources at the bottom and on the right wall Richardson Number θ max [13] θ max [present] Difference θ max [13] θ max [present] Difference 0 0.636 0.647 1.7% 0.589 0.597 1.4% 0.1 0.542 0.562 3.7% 0.540 0.551 2.0% 1 0.448 0.451 0.7% 0.487 10 0.337 0.347 3.0% 0.273 0.288 5.5% 5.1. Rayleigh Number For constant properties of air and silicon, Ra is a function of the inlet height (hi) and the heat generation per unit volume ( q ) only. Ra =
g β hi3 ∆T
αν
h 2 q & ∆T = i ke
⇒ Ra =
g β hi5 q α ν ke
(16)
For the air and silicon properties, Rayleigh number is Ra ≅ 6.2 × 105 hi5 q and will be only a function of the heat generation per unit volume ( q ) only if the inlet height is kept constant (hi=10 cm). Electronic devices such as Printed Circuit Boards (PBC) could generate heat up to 30 watts. For some electronic devices, this corresponds to a heat generation per unit volume of q = 2 × 105 W / m3 and a Rayleigh number of Ra = 1.6 × 106 [23]. That is why, in the present study, the Rayleigh number is considered to be in the range of 0 ≤ Ra ≤ 107 . The present analysis of the enclosure thermal behavior was first carried out by examining the number and then the arrangement of the heat sources. Table 2 presents the values of the flow and geometry parameters used in the analysis. Table 2. Values of the Flow and Geometry Parameters Pr 0.71 Re 100 0 ≤ Ra ≤ 107
L/ hi H/ hi AR=L/H
8 4 2
ho/hi t1/ hi t/ hi
1 0.05 0.25
a1/ hi a/hi
1.6 2
Z1 Z2 Z3
3.96 0.084 5630
5.2. Number of Heat Sources The flow and temperature patterns are examined to compare the behavior of the mixed convection heat transfer for different configurations of the enclosure with one (B), two (BB), and three (BBB) heat sources (Figure 1). The results of this comparison at Ra=106 are presented in Figure 4. 5.2.1. Flow Patterns Large recirculating zones are developed above the main air stream at Ra=106. This is in contrast to the simple diagonal airflow from the inlet to the exit at low Rayleigh numbers. The physical characteristics of the S-flow enclosure, the strength of inertia forces (Reynolds number), and the interaction between forced and natural convection flows determine the formation and size of the recirculating zones [20]. In addition, placing more heat sources in the enclosure increases the size of the recirculating zones and forces the main air stream to travel more horizontally along the heat sources before rising vertically towards the exit.
April 2008
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
197
Behzad Ghasemi and Saiied Mostafa Aminossadati
5.2.2. Temperature Patterns The interaction between the externally sourced airflow and the internal circulating vortex govern the temperature patterns. When there is only one heat source in the enclosure, the high temperature zones are concentrated near the heat source and the temperature distribution is more uniform in the rest of the enclosure. As the number of heat sources increases, the temperature contours move slightly inside the enclosure, showing a gradual growth of the high temperature zones. B
BB
BBB
Figure 4. The flow and temperature patterns for B, BB and BBB ( Ra = 10 6 , Re = 100 , Pr = 0.71 )
Figure 5 and Figure 6 show the maximum surface temperature of every heat source, θs, max, versus Reyleigh numbers for different enclosure configurations. A reduction in θs, max can be seen for all the heat sources as the Rayleigh number increases. Strengthening the natural convection improves the heat transfer process and provides better cooling.
0 .3 6 B1
θs, m ax
B2
B2
B
0 .3 0 B
B1 0 .2 4 1 03
1 04
1 05
1 06
1 07
Ra Figure 5. Maximum dimensionless temperature versus Rayleigh number for B and BB ( Re = 100, Pr = 0.71 )
198
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
April 2008
Behzad Ghasemi and Saiied Mostafa Aminossadati
B1 B2 B3
0.40
θs,max
B
B3
0.32 B2 B B1
0.24 103
104
105
106
107
Ra Figure 6. Maximum dimensionless temperature versus Rayleigh number for B and BBB ( Re = 100, Pr = 0.71 )
5.2.3. Position of the Heat Sources on the Bottom Wall At low Rayleigh numbers, forced convection generated by the externally sourced airflow is the primary reason for taking heat away from the heat sources. The temperature of the airflow increases as it flows over the heat source located nearer to the inlet. Therefore, the heat source located further comes into contact with the airflow with relatively higher temperature. As a result, θs, max increases with the distance of the heat source from the inlet. At high Rayleigh numbers, the heat transfer is considerably due to the dominant natural convection. Therefore, the position of the heat sources with respect to the entrance of externally sourced airflow has little influence on θs, max. 5.2.4. Heat Sources Located at Hot Spots Figure 7 shows that at low Rayleigh numbers, the heat source located at the hot spot experiences higher θs, max as the number of heat sources located on the bottom wall increases. This can be explained by examining the position of the hotspot heat source relative to the inlet, which is the origin of the dominated forced convection. At high Rayleigh numbers, the influence of the position is less significant as the natural convection dominates the forced convection in the heat transfer process.
0.40
B
θs,max
BB
BB
BBB BBB
0.32 B
0.24 103
104
105
106
107
Ra Figure 7. Heat sources temperature located at the hot spots on the bottom wall ( Re = 100 , Pr = 0.71 )
April 2008
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
199
Behzad Ghasemi and Saiied Mostafa Aminossadati
5.2.5. Thermal Performance The average Nusselt number presented in Figure 8 is an indication of the total heat transfer per surface area of the heat sources placed in each enclosure. Increasing the Rayleigh number improves the heat transfer process by strengthening the natural convection and, therefore, increases the average Nusselt number. At any specific Rayleigh number, placing more heat sources in the enclosure increases the total surface area, which in turn decreases the average Nusselt number. The enclosure with one heat source on the bottom wall has the best thermal performance amongst all the three configurations.
B
BB
BBB
7 500
N um B
5 000
BB BBB
2 500 1 03
1 04
1 05
1 06
1 07
Ra Figure 8. Average Nusselt number versus Rayleigh number for B, BB and BBB ( Re = 100 , Pr = 0.71 )
5.3. Arrangement of Heat Sources The aim of this part of study is to examine the effects of heat sources arrangement on the enclosure thermal performance. Figure 2 shows four different configurations of the three heat sources in the enclosure. The flow and temperature patterns at various Rayleigh numbers of 0, 105 and 107 are plotted in Figures 9–12 for the configurations BBB, BBL, BBR and BLR respectively. 5.3.1. Flow Patterns In the absence of natural convection (Ra=0), all configurations of the enclosure experience similar flow behavior. The externally sourced airflow diagonally travels through the enclosure (S-flow) from the inlet to the exit with a recirculating zone above the airflow. As the Rayleigh number increases, the flow field in various configurations behave differently and the flow patterns move more closely and parallel to the heat sources. Increasing the Rayleigh number increases the contribution of natural convection in the heat transfer process. Therefore, the arrangement of the sources of natural convection influences the flow behavior. At very high Rayleigh numbers, the single circulation in the enclosure breaks up to two or more recirculation zones as a result of the strong interaction between the inertia and buoyancy forces and the interference of the heat sources with the forced and natural convection flow. 5.3.2. Temperature Patterns The position of the heat sources influences the temperature patterns in the enclosure. At zero Rayleigh number, the heat transfer process is basically due to the forced convection flow generated by the externally sourced airflow. That is why relatively high temperature regions are developed in the vicinity of the heat sources while the rest of the enclosure experiences lower temperatures. Increasing the Rayleigh number improves the heat transfer and, therefore, the temperature lines move much closer to the heat sources providing a larger region in the enclosure with relatively lower temperature. At Ra=107, the concentration of the temperature patterns considerably increases near the heat sources, indicating that the enclosure is significantly affected by the interaction of forced and natural convection flows.
200
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
April 2008
Behzad Ghasemi and Saiied Mostafa Aminossadati
R a= 0
5 R a= 1 0
7 R a= 1 0
Figure 9. The flow and temperature patterns for BBB ( Re = 100,Pr = 0.71 )
R a= 0
5 R a= 1 0
7 R a= 1 0
Figure 10. The flow and temperature patterns for BBL ( Re = 100, Pr = 0.71 )
April 2008
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
201
Behzad Ghasemi and Saiied Mostafa Aminossadati
R a= 0
5 R a= 1 0
7 R a= 1 0
Figure 11. The flow and temperature patterns for BBR ( Re = 100,Pr = 0.71 )
R a= 0
5 R a= 1 0
7 R a= 1 0
Figure 12. The flow and temperature patterns for BLR ( Re = 100 , Pr = 0.71 )
Figures 13–15 show the maximum surface temperature, θs, configurations BBL, BBR, and BLR respectively.
max,
of the heat sources in the enclosure with
5.3.3. Position of Heat Sources on the Bottom Wall of BBL and BBR Figures 13 and 14 show that in BBL and BBR configurations, B1 has always a lower θs, max than B2. An explanation for this could be the influence of the position of the heat sources with respect to the inlet and the temperature of the main air stream being in contact with the heat source.
202
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
April 2008
Behzad Ghasemi and Saiied Mostafa Aminossadati
5.3.4. Heat Source on the Left Wall of BBL Figure 13 also shows that at low Rayleigh numbers, the heat source located on the left wall, L, has the highest θs, max (hot spot). The heat source L is being exposed to the recirculating flow with a relatively higher temperature than the main airflow which is flowing over the heat sources located on the bottom wall. As the Rayleigh number increases, the flow behavior in the enclosure changes and the intensity of the airflow with relatively low temperature in the vicinity of the heat source L increases. As a result, θs, max decreases rapidly with Rayleigh number and becomes even lower than θs, max for B2 at a specific range of Rayleigh numbers. 5.3.5. Heat Source on the Right Wall of BBR Figure 14 shows that θs, max for the heat source located on the right wall, R, (hot spot) is always higher than θs, max for the two heat sources located on the bottom wall (B1 and B2). This is probably because of the heat source R being in contact with the airflow, which has a relatively higher temperature after flowing over the heat sources B1 and B2. 5.3.6. θs, max in BLR Figure 15 shows that at low Rayleigh numbers, the hot spot for the BLR arrangement is the location of heat source L. This could be because of the relatively high temperature recirculating flow over the heat source L. Increasing the Rayleigh number changes the flow behavior in the enclosure and therefore, beyond Ra=105, the heat sources L and R are almost subjected to the same temperature. 5.3.7. Thermal Performance Figure 16 shows that at low Rayleigh numbers, amongst all different heat sources arrangements, BBB has the highest and BLR has the lowest average Nusselt number. In addition, BBR has a higher average Nusselt number than BBL. At high Rayleigh numbers, the average Nusselt number is not indicative to be a function of the heat sources arrangement and all different configurations of the enclosure have almost the same thermal performance. 0.4 0
L B1
L
BBL
B2
θs, m a x 0.3 2 B2 B1
0.2 4 1 03
1 04
Ra
1 05
1 06
1 07
Figure 13. Maximum dimensionless temperature versus Rayleigh number for BBL ( Re = 100 , Pr = 0.71 )
0.40
B1
θs,max
B2
R
BBR
R B2
0.32 B1 0.24 103
104
Ra
105
106
107
Figure 14. Maximum dimensionless temperature versus Rayleigh number for BBR ( Re = 100 , Pr = 0.71 )
April 2008
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
203
Behzad Ghasemi and Saiied Mostafa Aminossadati
L
0.40
θs,max
BLR
R
B
L R
0.32 B
0.24 103
104
Ra
105
106
107
Figure 15. Maximum dimensionless temperature versus Rayleigh number for BLR ( Re = 100 , Pr = 0.71 ) 6000
N um
BBL
4000 BBB BBR BLR BBL
2000 103
104
Ra
105
106
107
Figure 16. Average Nusselt number versus Rayleigh number for BBB, BBR, BBL, BLR( Re = 100, Pr = 0.71 )
5.3.8. Natural Convection Contribution Figure 17 shows the effects of Rayleigh number on Num/Num, F which indicates the natural convection contribution in the enclosure thermal performance. At Ra=103, Num/Num, F is nearly one for all different arrangements. This means that the heat transfer is only due to the forced convection. A rise in the Rayleigh number strengthens the buoyancy forces and increases the contribution of natural convection for all different arrangements. For BBL and BLR, however, Num/Num, F seems to be less than unity in the range of Ra=103 and 104, which is possibly an indication of negative effects of buoyancy forces on the heat transfer process. At high Rayleigh numbers, natural convection shows a different contribution to the heat-transfer process. Natural convection has the lowest influence in BBB and the highest influence in BLR on the thermal performance of the enclosure. However, the contribution of natural convection in the heat transfer process is almost the same for BBR and BBL. 2.3
Num\Num,F
BL R
BB R BB B B BL
1.0
103
104
105
106
107
Ra Figure 17. Contribution of natural convection in the thermal Performance ( Re = 100 , Pr = 0.71 )
204
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
April 2008
Behzad Ghasemi and Saiied Mostafa Aminossadati
5.3.9. Oscillating Results For BBL at Ra=107, the governing unsteady equations are solved to determine the average Nusselt number, Num. Figure 18 shows that Num decreases with time until it reaches an asymptotic value. The scale of the Num axis is expanded in Figure 19. Once the asymptote region is reached, it can be seen that the value of Num fluctuates in a sinusoidal manner. The steady-state values of Num for BBL and at Ra=107, presented in Figures 16 and 17, are obtained from the average of maximum and minimum values of Num in the asymptotic zone. The fluctuating results were also observed by Papanicolaou and Jaluria [24] and D’Orazio et al. [25]. They argued that the instability in the physical system can explain the fluctuating results at particular geometries and Rayleigh numbers. Other researchers [26–28] also found the fluctuating results in the closed and opened cavities and discussed that the Hopf bifurcations lead to oscillating flows.
9000
Num 7000
5000
0
50
100
τ
150
200
250
Figure 18. Average Nusselt number versus time for BBL ( Ra = 107 , Re = 100 , Pr = 0.71 ) 5735
Num 5730
5725 240
242
244
τ
246
248
250
Figure 19. Sinusoidal fluctuations of the average Nusselt number for BBL ( Ra = 107 , Re = 100 , Pr = 0.71 )
6. CONCLUSION A numerical analysis has been developed to simulate the mixed convection heat transfer of a two-dimensional enclosure with an emphasis on the number and arrangement of the heat sources placed in the enclosure. The results of the numerical analysis lead to the following conclusions. • Increasing the Rayleigh number decreases the maximum surface temperature of the heat sources, increases their average Nusselt number and improves the enclosure heat transfer process. • At any Rayleigh number, placing less heat sources on the bottom wall results in a higher Nusselt number and a better cooling performance. The heat sources, which are placed on the enclosure bottom wall and are closer to the inlet, encounter lower maximum surface temperature than the other heat sources. At high Rayleigh numbers, the influence of the position of the heat sources on the maximum surface temperature is less significant. • The BBB arrangement for the enclosure with three heat sources has the highest Nusselt number and the best cooling performance at low Rayleigh numbers. At high Rayleigh numbers, the arrangement has negligible April 2008
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
205
Behzad Ghasemi and Saiied Mostafa Aminossadati
influence on the Nusselt number and cooling performance. In addition, at high Rayleigh numbers, the process of natural convection makes the lowest contribution to the heat transfer process for the BBB arrangement. • The flow and temperature distributions show positions within the enclosure that have been identified as having better cooling performances. Thus, electronic components with highly thermal sensitivities can be placed in these positions. • The flow conditions and the arrangements of the electronic components which result in fluctuating system behavior can also be identified as part of this analysis. • The results of this study show promise as an efficient thermal design of electronic devices where the arrangement of electronic components with different thermal sensitivities is of interest. ACKNOWLEDGMENT The authors would like to acknowledge the support of Shahrekord University during the numerical analysis of this research. REFERENCES
206
[1]
A. Bar-Cohen, A.D. Kraus, and S.F. Davidson, “Thermal Frontiers in the Design and Packaging of Microelectronic Equipment”, Mechanical Engineering, 105 (1983), pp. 53–59.
[2]
G. P. Peterson and A. Ortega, “Thermal Control of Electronic Equipment and Devices”, Advanced Heat Transfer, 20 (1990), pp. 181–314.
[3]
S. Sathe and B. Sammakia, “A Review of Recent Developments in Some Practical Aspects of Air-Cooled Electronic Packages”, (ASME) Journal of Heat Transfer, 120 (1998), pp. 830–839.
[4]
E. Papanicolaou and S. Gopalakrishna, “Natural Convection in Shallow Horizontal Air Layers Encountered in Electronic Cooling”, Journal of Electronic Packaging, 117 (1995), pp. 307–316.
[5]
Y. Liu and N. P. Thien, “An Optimum Spacing Problem for Three Chips Mounted on a Vertical Substrate in an Enclosure”, Numerical Heat Transfer, Part A, 37 (2000), pp. 613–630.
[6]
Q. H. Deng, G. F. Tang, Y. Li, and M. Y. Ha, “Interaction Between Discrete Heat Sources in Horizontal Natural Convection Enclosures”, International Journal of Heat and Mass Transfer, 45 (2002), pp. 5117–5132.
[7]
S. H. Chuang, J. S. Chiang, and Y. M. Kuo, “Numerical Simulation of Heat Transfer in a Three-Dimensional Enclosure with Three Chips in Various Position Arrangements”, Heat Transfer Engineering, 24 (2003), pp. 42–59.
[8]
J. H. Bae and J. M. Hyun, “Time Dependent Buoyant Convection in an Enclosure with Discrete Heat Sources”, International Journal of Thermal Sciences, 43 (2004), pp. 3–11.
[9]
A. K. da Silva, G. Lorenzini, and A. Bejan, “Distribution of Heat Sources in Vertical Open Channels with Natural Convection”, International Journal of Heat and Mass Transfer, 48 (2005), pp. 1462–1469.
[10]
A. Bazylak, N. Djilali, and D. Sinton, “Natural Convection in an Enclosure with Distributed Heat Sources”, Numerical Heat Transfer, Part A, 49 (2006), pp. 655–667.
[11]
K. J. Kennedy and A. Zebib, “Combined Free and Forced Convection Between Horizontal Parallel Planes: Some Case Studies”, International Journal of Heat and Mass Transfer, 26 (1983), pp. 471–474.
[12]
B. S. Yilbas, S. Z. Shuja, and M. O. Iqbal, “Energy and Entropy Analysis in a Square Cavity with Protruding Body: Effects of Protruding Body Aspect Ratio”, International Journal of Energy Research, 26 (2002), pp. 851–866.
[13]
E. Papanicolaou and Y. Jaluria, “Mixed Convection from Simulated Electronic Components at Varying Relative Positions in a Cavity”, ASME, Journal of Heat Transfer, 116 (1994), pp. 960–970.
[14]
S. Chen, Y. Liu, S. F. Chan, C. W. Leung, and T. L. Chan, “Experimental Study of Optimum Spacing Problem in the Cooling of Simulated Electronic Package”, Heat and Mass Transfer, 37 (2001), pp. 251–257.
[15]
E. Kehoe, M. Davies, and D. Newport, “Mixed Convection Cooling of Horizontally Mounted Printed Circuit Board”, IEEE Transactions on Components and Packaging Technologies, 26(1) (2003), pp. 126–133.
[16]
A. K. da Silva, G. Lorente, and A. Bejan, “Optimal Distribution of Discrete Heat Sources on a Plate with Laminar Forced Convection”, International Journal of Heat and Mass Transfer, 47 (2004), pp. 2139–2148.
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
April 2008
Behzad Ghasemi and Saiied Mostafa Aminossadati
[17]
T. J. Young and K. Vafai, “Convective Flow and Heat Transfer in a Channel Containing Multiple Heated Obstacles”, International Journal of Heat and Mass Transfer, 41 (1998), pp. 3279–3298.
[18]
Y. P. Cheng, T. S. Lee, and H. T. Low, “Numerical Analysis of Mixed Convection in Three-Dimensional Rectangular Channel with Flush-Mounted Heat Sources Based on Field Synergy Principle”, International Journal for Numerical Methods in Fluids, In Press (2006).
[19]
A. Dogan, M. Sivrioglu, and S. Baskaya, “Investigation of Mixed Convection Heat Transfer in a Horizontal Channel with Discrete Heat Sources at the Top and at the Bottom”, International Journal of Heat and Mass Transfer, 49 (2006), pp. 2652–2662.
[20]
S. Singh and M. A. R. Sharif, “Mixed Convective Cooling of a Rectangular Cavity with Inlet and Exit Openings on Differentially Heated Side Walls”, Numerical Heat Transfer, Part A, 44 (2003), pp. 233–253.
[21]
J. Rhee and G. Wong, “Characterization of Airflow Impedance in Two Types of Telecommunications Chassis”, 20th IEEE Semiconductor Thermal Measurement and Management Symposium–Proceedings, 20, 2004, pp. 161–167.
[22]
S. V. Patankar, Numerical Heat Transfer and Fluid Flow. Washington, D.C.: Hemisphere, 1980.
[23]
Y. A. Cengel, Heat Transfer: A Practical Approach. McGraw-Hill Inc. 1998, p. 868.
[24]
E. Papanicolaou and Y. Jaluria “Transition to a Periodic Regime in Mixed Convection in a Square Cavity”, Journal of Fluid Mechanic, 239 (1992), pp. 480–509.
[25]
M. C. D’Orazio, C. Cianfrini, and M. Corcione, “Rayleigh–Benard Convection in Tall Rectangular Enclosures”, International Journal of Thermal Sciences, 43 (2004), pp.135–144.
[26]
C. C. Jahnke, V. Subramanyan, and D. T. Valentine, “On the Convection in an Enclosed Container with Unstable side Wall Temperature Distributions”, International Journal of Heat and Mass Transfer, 41 (1998), pp. 2307–2320.
[27]
A. Javam and S. W. Armfield, “Stability and Transition of Stratified Natural Convection Flow in Open Cavities”, Journal of Fluid Mechanics, 445 (2001), pp. 285–303.
[28]
Y. Li, P. Xu, H. Qian, Q. H. Deng, and J. Wu, “Flow Bifurcation Due to Opposing Buoyancy in Two Vertically Connected Open Cavities”, International Journal of Heat and Mass Transfer, 49 (2006), pp. 3298–3312.
April 2008
The Arabian Journal for Science and Engineering, Volume 33, Number 1B
207
