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Mixed convection air cooling of protruding heat sources mounted in a horizontal channel
International Communications in Heat and Mass Transfer 36 (2009) 841–849
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International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Mixed convection air cooling of protruding heat sources mounted in a horizontal channel☆ Adel Hamouche, Rachid Bessaïh ⁎ Laboratoire d'Energétique Appliquée et de Pollution, Faculté des Sciences de l'Ingénieur, Département de Génie Mécanique, Université Mentouri-Constantine, Route de Ain El. Bey, Constantine 25000, Algeria
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Available online 27 May 2009 Keywords: Mixed convection Horizontal channel Electronics cooling
a b s t r a c t A numerical study of laminar mixed-convection heat transfer to air from two identical protruding heat sources, which simulate electronic components, located in a two-dimensional horizontal channel, is presented in this paper. The finite volume method and the SIMPLER algorithm are used to solve the conservation equations of mass, momentum, and energy for mixed convection. Results show that the heat transfer increases remarkably for Pr = 0.71 and 5 ≤ Re ≤ 30. It was also found that the increase of separation distance, the height and the width of the components has a considerable enhancement of the heat removal rate from the components, and therefore, on the improvement of the heat transfer inside the channel. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction With the advent of high-speed computers, heat dissipation at the microchip has become a source of concern. The high temperatures of the chip affect the reliability and the performance of the system. Therefore, effective heat removal is necessary to ensure better life of the system. A literature review shows that various studies have been done for thermal management of electronic devices. For example, Icoz and Jaluria [1] presented a methodology for the design and optimization of cooling systems for electronic equipment. In this approach, inputs from both experimentation and numerical modelling are to be used concurrently to obtain an acceptable or optimal design. It was shown that by using both approaches concurrently; the entire design domain is covered, leading to a rapid, convergent, and realistic design process. A numerical study of natural convection was conducted by Icoz and Jaluria [2], for protruding thermal sources located on a horizontal channel. They observed that the channel dimensions and the presence of openings have significant effects on the fluid flow. However, their effects on heat transfer are found to be relatively small. The increase in channel height is seen to lead to a less stable flow and, consequently, a decrease in the critical Grashof number. Da Silva et al. [3] presented an analytical and numerical study to discover the optimal distribution of discrete heat sources cooled by laminar natural convection. The results show that the optimal distribution is not uniform (the heat sources are not equidistant), and that as the Rayleigh number increases the heat sources placed near the tip of a boundary layer should have zero spacing. Wang and ☆ Communicated by J. Taine and A. Soufiani. ⁎ Corresponding author. E-mail address: [email protected] (R. Bessaïh). 0735-1933/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2009.04.009
Jaluria [4] studied numerically the three-dimensional conjugate heat transfer in a rectangular duct with two discrete flush-mounted heat sources in the context of cooling of electronic equipments. They have seen that the ratio of convection to the total heat transfer from the upstream heat source is larger than that from the downstream one, but the ratio of conduction to the total heat transfer is less than that of the downstream heat sources. A detailed investigation of forced convective cooling of a heated obstacle mounted upon a channel wall was presented by Young and Vafai [5]. It was shown that specific choices in obstacle size, shape and thermal conductivity can produce significant effects on the flow and heat transfer characteristics. Mixed convection heat transfer in a top and bottom heated rectangular channel with discrete heat sources has been investigated experimentally for air by Dogan et al. [6]. Results show that the face temperatures increase with increasing Grashof number. The row-averaged Nusselt numbers first decrease with the row number and then, due to the increase in the buoyancy affect secondary flow and the onset of instability. They also show an increase towards the exit as a result of heat transfer enhancement. Dogan et al. [7] investigated experimentally the mixed-convection heat transfer from arrays of discrete heat sources inside a horizontal channel. Their results show that changes in Grashof and Reynolds numbers affect significantly the buoyancy-driven secondary flow, the onset of instability, and the resulting heat transfer enhancement above the forced convection limit for the aspect ratio AR = 4. Furukawa and Yang [8] presented a numerical investigation of thermal-fluid flow behaviour in a bundle of parallel boards with heat generating blocks. They found that at a low Reynolds number flow, a developing flow may achieve a fully-developed flow state at certain block number from the entrance. Thermal conductivity of the board and thermal contact resistance between the chip and board have a considerable
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A. Hamouche, R. Bessaïh / International Communications in Heat and Mass Transfer 36 (2009) 841–849 Table 1 Results of the grid independence tests for Gr = 104 and Re = 20.
Nomenclature d Gr g H h k k⁎ L Nu Nu̅ ̅
separation distance between two components, m 2 Grashof number, = gβ(TS − T0)H3 / νair gravitational acceleration, m/s2 height of the channel, m height of the component, m thermal conductivity of the fluid, W/m K dimensionless thermal conductivity, k/kair length of the channel, m Aθ local Nusselt number, = − An R average Nusselt number, ¼ Nu dn exchange surface
n P Pr p0 p Re Ra Ri T t U0 U, V u, v w X, Y x, y
normal coordinate dimensionless pressure,= (p − p0) / ρU20 Prandtl number, =νair / αair external pressure, Pa pressure, Pa Reynolds number, =U0 H / νair Rayleigh number, =Gr.Pr Richardson number, =Gr/Re2 temperature, K time, s uniform inlet velocity, m/s dimensionless horizontal and vertical velocities, =(u,v)/ U0 horizontal and vertical velocities, m/s width of each heat source, m dimensionless horizontal and vertical Cartesian coordinates,=(x,y) / H horizontal and vertical Cartesian coordinates, m
Greek symbols α thermal diffusivity of the fluid, m2/s β thermal expansion coefficient of the fluid, 1/K θ dimensionless temperature ν kinematic viscosity, m2/s ν⁎ dimensionless kinematic viscosity,= ν / νair νair kinematic viscosity of the fluid, m2/s ρ density of the fluid, kg/m3 Ψ dimensionless stream function τ dimensionless time, =t / (H / U0)
Subscripts 0 evaluated at the reference temperature s source impact on thermal performance. Habchi and Acharya [9] studied the laminar mixed convection about a symmetrically or asymmetrically heated vertical plate with a block. They found that at low values of
Grid
110 × 42 nodes
220 × 52 nodes
320 × 62 nodes
420 × 72 nodes
Nu̅ ̅ ̅ Nu̅ ̅ 1̅ Nu̅ ̅ 2̅
2.400
2.694
2.662
2.735
1.528 0.872 2.256
1.711 0.983 2.254
1.705 0.957 2.257
1.771 0.964 2.258
Umax
Here, Nu̅ 1̅ ̅ and Nu̅ 2̅ ̅ are the average Nusselt numbers for the first and second component, respectively; and Nu̅ ̅ ̅ is the average Nusselt number for both components.
Gr the maximum velocity occurs near the adiabatic wall in the Re2 asymmetrically heated channel, but for the symmetrically heated channel, the velocity profiles are depressed at the center, and for both thermal conditions, the Nusselt numbers are smaller than the corresponding smooth duct Nusselt numbers. Bhowmik and Tou [10] performed experiments to study the single-phase transient forced convection heat transfer on an array of flush-mounted discrete heat sources in a vertical rectangular channel during the pump-on transient operation. The experimental results indicate that the heat transfer coefficient is affected strongly by the number of chips and the Reynolds number. Bhowmik et al. [11] performed steady-state experiments to study general convective heat transfer from four in-line simulated electronic chips in a vertical rectangular channel, using water as the working fluid. The results indicate that the heat transfer coefficient is strongly affected by the Reynolds number, and the fully-developed values of heat transfer coefficient are reached before the first chip. Young and Vafai [12] detailed a numerical simulation of forced convective incompressible flow with an array of heated obstacles attached to one wall. Extensive presentation and evaluation of the mean Nusselt numbers along the exposed faces of all obstacles in the array was fully documented. Kim et al. [13] investigated numerically the characteristics of a pulsating flow and the associated thermal transport from two heated blocks, representing energy dissipating electronic components with different heights in a channel. The results indicate that the recirculating flow behind the second block as well as that between block regions, are substantially affected by the pulsating frequency and heights of blocks. Ray and Srtnivasan [14] studied numerically the temperature variation in the insulation around an electronic component, mounted on a horizontal circuit board. The results indicate that the temperature variation within the insulation becomes important when the thermal conductivity of the insulation is less than ten times the thermal conductivity of the cooling medium. The present study deals with a numerical study of two-dimensional laminar mixed-convection heat transfer to air from two identical protruding heat sources, which simulate electronic components, located in a horizontal channel. Attention was focused on the effects of Re, height, width and separation distance of components on the fluid flow and heat transfer characteristics. Section 2 of this paper presents the mathematical formulation. Section 3 discusses the numerical method and techniques which have been used for the computation,
Fig. 1. Geometry of the problem. The horizontal channel contains two protruding heat sources, which simulate electronic components.
A. Hamouche, R. Bessaïh / International Communications in Heat and Mass Transfer 36 (2009) 841–849
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Fig. 2. Comparison between the present predictions and Habchi and Acharya's [9] predictions for Ra = 103, 105, and Gr/Re2 = 3, H / L = 0.2.
the grid independence study and code validation with numerical data. Section 4 presents the results and finally a conclusion is given. 2. Mathematical model The geometry considered is a two-dimensional channel with two identical protruding heat sources, which simulate electronic components, located at the bottom wall of the horizontal channel, as shown in Fig. 1. The one on the left is designated as the first component. Each heat source has a height h and a width w, and the distance between two heat sources is d. The distance between the channel inlet and the first heat source is L1 and between the second and the channel outlet is L2, H and L are the height and the length of the channel, respectively (Fig. 1). The top and bottom walls of the channel are insulated. At the channel inlet, a flow velocity U0 is imposed with a uniform temperature T0. The heating components are kept isothermal, at uniform temperature TS. Using the channel height H as the length scale, inlet velocity U0 as the velocity scale, H / U0 as the time scale, ρU20 as the pressure scale, and (TS − T0) for the temperature scale, the conservation equations of mass, momentum, and energy for a twodimensional laminar mixed convection and incompressible flow, with constant thermo physical properties, and with Boussinesq approximations, can be written in the following dimensionless form: AU AV + =0 AX AY
νS and νair are the kinematic viscosities of the heat source and of the fluid (air), respectively, and ν⁎ → ∞ in order to obtain U = V = 0 in a level of each component. kS and kair are the thermal conductivities of the solid and of the fluid (air), respectively, and k⁎ → ∞ in order to obtain a uniform temperature in a level of each heat source. The initial and boundary conditions, in dimensionless form, used for the components of velocity and for temperature are:
ð1Þ
AU AU AU AP 1 A AU A AU +U +V = − + m4 + m4 ð2Þ Aτ AX AY AX Re AX AX AY AY
AV AV AV AP 1 +U +V = − + Aτ AX AY AY Re A AV Gr m4 + + θ AY AY Re2
f AXA
AV m4 AX
ð3Þ
g
Aθ Aθ Aθ 1 Aθ Aθ Aθ Aθ +U +V = k4 + k4 Aτ AX AY Re Pr AX AX AY AY
ð4Þ
where, m4 =
∞ in each heat source f m1 in= mtheYfluid region S
air
and k4 =
kS = kair Y∞ in each heat source 1 in the fluid region:
Fig. 3. Dimensionless streamlines Ψ and isotherms in a horizontal channel at Gr = 104: (a) Re = 5, (b) Re = 10, and (c) Re = 30.
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Fig. 4. Average Nusselt number Nu̅ ̅ of each component (component 1 and component 2) at Gr = 104 and for different values of Re.
Initially, at τ = 0, U = V = θ = 0. The dimensionless boundary conditions for our study are presented at τ N 0: at X = 0 and 0VYV1 ; U = 1; V = 0; θ = 0 ðinlet channelÞ AU Aθ = 0; V = 0; = 0 ðoutlet channelÞ AX AX Aθ = 0 ðadiabatic wallÞ at Y = 0 and 0V XVL ; U = 0; V = 0; Ay at X = L and 0VYV1 ;
at Y = 1 and 0V XVL ; U = 0; V = 0;
Aθ = 0 ðadiabatic wallÞ: Ay
the staggered locations, and the scalar quantities (P and θ) are stored in the center of these volumes. A fully implicit time marching scheme is employed. The numerical procedure called SIMPLER [15] is used to handle the pressure–velocity coupling. For treatment of the convection and diffusion terms in Eqs. (2)–(4), central difference scheme is adopted. Finally, the discretized algebraic equations are solved by the line-by-line tri-diagonal matrix algorithm (TDMA). Convergence at a given time step is declared when the maximum relative change between two consecutive iteration levels fall below 10− 4, for U, V and θ. At this stage, the steady-state solution is obtained. Four non-uniform grids are used in this work: 110 × 42, 220 × 52, 320 × 62, and 420 × 72 nodes. The discrepancy in the maximum values of horizontal Umax and average Nusselt numbers Nu̅ ̅ 1̅ ̅ (of the first heat source), Nu̅ ̅ 2̅ ̅ (of the second heat source), and Nu̅ ̅ ̅ (average Nusselt of the first and second heat source), by using the grids 320 × 62 and 420 × 72 nodes, is smaller than 4% [9] (see, Table 1). In order to optimize the CPU time and cost of computations, a fine grid downstream and above of the components in the x and y directions, corresponding to 320 × 62 nodes has been used for all simulations. The total CPU time on a PC Pentium 4 for the results presented here was about 2 h for each case. The numerical code was validated with the numerical results of Habchi and Acharya [9]. They obtained the velocity profiles due to the laminar mixed convection of air in a vertical channel containing a partial rectangular blockage on one channel wall. The wall containing the blockage is assumed to be heated, while the other wall is assumed to be adiabatic. The parameters L1/L and b/L have been assumed to be unity while the dimensionless blockage height H/L is assigned the value of 0.2. L2/L is chosen to be sufficiently far downstream so that the zero streamwise gradient condition is satisfied. As shown in Fig. 2, it is clear that the numerical results of the present work are in a good agreement with Habchi and Acharya's predictions [9].
3. Numerical procedure 4. Results and discussion The governing Eqs. (1)–(4), with the associated boundary conditions, are solved using a finite volume technique, as described by Patankar [15]. The components of the velocity (U and V) are stored at
A detailed numerical study has been carried out on mixedconvection heat transfer enhancement. The study deals with the
Fig. 5. Dimensionless streamlines Ψ in a horizontal channel at Re = 30, Gr = 104, and for different separation distances d: (a) d = w, (b) d = 2w, (c) d = 3w (d) d = 4w.
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effects of the Reynolds number Re, separation distance between two components d, the height h, and the width w components on the flow structure and heat removal rate from the components. In this study, dimensions are taken to be: L1 = 2H; L2 = 4H; d = w; h = H = 0: 25; and w = H = 0:25. In most practical applications of electronics systems cooling by air, the buoyancy strength is not very high [16] and the maximum of the Grashof number Gr is kept in the order of 104. The corresponding moderate temperature difference ΔT = TS − T0 and the channel height H are about 10 K and 20 mm, respectively. Here, all cases with low Reynolds numbers are studied in this paper.
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d = w, we notice a recirculating zone between two components. When the separation distance increases, the size of the recirculating zone increases remarkably downstream of the first component near the face C1D1. However, the streamlines upstream of the second component near the face A2B2 are straight. In this region, an important heat removal is provided because of the important mixing, which transports more thermal energy away from the components. When the separation distance increases, Fig. 6a–b show that for Re = 30 and Gr = 104, the heat removal is affected mostly from the faces C1D1 and
4.1. Effect of the Reynolds number We present the streamlines and isotherms in Fig. 3a–c at Gr = 104 (fixed value) and for different values of the Reynolds numbers (Re = 5, 10, 20, 30). For Re = 5 (Fig. 3a), the buoyancy effects are eminent, notably for the sizes of the first recirculating zone above the first component and near the top wall, and the second one downstream of the second component. For Re = 10 (Fig. 3b), the sizes decrease slightly, but when the Reynolds number increases more, the recirculating zone above the first component disappears completely and the size of the one downstream of the second component decreases considerably. This is due essentially to the increase of the effect of the Reynolds number, which reduces the buoyancy effects inside the channel. We observe from Fig. 3, which gives the isotherm contours as a function of the Reynolds number, when the Reynolds number is small the isotherms stretch out and occupy a large place in the channel. The regions where the effects of the components are felt tend to be localized to the neighborhood of the components. There is a large temperature distribution into the channel, upstream, downstream and above both components. The heated air is not well convected at the outflow because of the dominance of the buoyancy effects. However, when the Reynolds number is higher, the isotherms become approximately horizontal with a notable decrease in the temperature distribution into the channel, precisely to the neighborhood of the components. Consequently, the increase of the Reynolds number reduces considerably the buoyancy effects, and provides a good evacuation of the heated air outside the channel. In order to examine the effect of the Reynolds number on the heat removal rate from components, Fig. 4 displays the variations of the average Nusselt number Nu̅ ̅ of each component (component 1 and component 2) at Gr = 104 and for different values of Re. The results show that the increase of the Reynolds number increases the heat transfer on a level with the first component more than the second one. For the Reynolds number values varying from 5 to 30, the heat transfer results obtained above can be correlated by the following equations of the average Nusselt numbers Nu̅ ̅ 1̅ ̅ and Nu̅ ̅ 2̅ ̅ of the first and the second component respectively, versus the Reynolds number Re: 0:13
Nu1 ¼ 1:12 Re
0:12
Nu2 ¼ 0:64 Re
ð6aÞ :
ð6bÞ
The results indicate also that for Re = 5 and Re = 30, the difference between the corresponding Nusselt numbers for the second component is larger than the one of the first component. Consequently, when the Reynolds number is larger the heat transfer on a level with the first component is improved, but it is more important for the second one. 4.2. Effect of the separation distance The use of the spacing between components is in order to control their temperatures [1]. The variation of the fluid flow as the separation distance changes (d = w, 2w, 3w and 4w) is presented in Fig. 5a–d. For
Fig. 6. At Gr = 104, Re = 30, and for different values of the separation distance d: (a) local Nusselt number the along the faces of the first component (component 1), (b) local Nusselt number along the faces of the second component (component 2), (c) average Nusselt number Nu̅ ̅ of each component (component 1 and component 2).
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Fig. 7. Local Nusselt number along the faces for each component (component 1 and component 2) as a function of the height components h/H, for Re = 30 and Gr = 104.
A2B2. Along the face B2C2, the heat removal rate decreases for d = w to d = 2w and increases for d = 3w and d = 4w. The variation of the average Nusselt number is displayed in Fig. 6c. It is depicted also that the heat removal from the first component is greater than from the second one. As it was seen in [1], streamlines presented in Fig. 5, show that secondary flow which gets stronger as the separation distance increases is created by the first component and between two components, and by the second component, behind it. These backflows influence the heat removal rate of the components, and its effect is more profound on the second component. For Re = 30 and Gr = 104, the enhancement in heat transfer rate from the second component is about 89.60%, however, from the first one it is about 45.66%. These
results confirm that the increase of the separation distance is more favourable for the heat removal rate from the second component than from the first one. In the separation distance range from 0.25 to 1.00, correlations of average Nusselt numbers Nu̅ ̅ 1̅ ̅ and Nu̅ ̅ 2̅ ̅ of the first and the second component, respectively, are proposed as: Nu1 = 1:71573 + 0:65618d − 0:2876d Nu2 = 0:8211 + 0:75596d for ð0:25VdV1:00Þ:
2
ð7aÞ ð7bÞ
A. Hamouche, R. Bessaïh / International Communications in Heat and Mass Transfer 36 (2009) 841–849
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numerical results, we propose correlations of average Nusselt numbers Nu̅ 1̅ and Nu̅ 2̅ of the first and the second component respectively, versus h = H: Nu1 = 1:17852 + 3:02631ðh = H Þ
ð8aÞ 2
Nu2 = 0:7296 + 0:68571ðh = HÞ + 2:64ðh=H Þ
ð8bÞ
for ð0:10Vh = HV0:30Þ:
4.4. Effect of the component width
Fig. 8. For Gr = 104 and Re = 30: (a) average Nusselt number Nu̅ ̅ ̅ of each component for different values of h/H, (b) average Nusselt number Nu̅ ̅ ̅ of each component for different values of w/H.
4.3. Effect of the height component The effect of varying the component height on heat transfer is demonstrated by comparing cases with a fixed width ðw = H = 0: 25Þ. As shown in Fig. 7, the greater mean value of the Nusselt number is noticed at the face A1B1. The face B1C1 which has a fixed length is characterized by a relatively lower value, but it increases when h = H = 0:30. For the faces C1D1 and A2B2, the mean Nusselt value decreases. It decreases also for the face B2C2, when h = H = 0:10 − 0:25, but it increases for h/H = 0.30. In the range of h = H from 0.10 to 0.20, the Nusselt number decreases for the face C2D2 and after increases for h/H = 0.25 and 0.30. At first sight, it appears that the heat transfer decreases when the component height increases, contrary to Fig. 8a, which indicates that the overall Nusselt numbers of the two components increase, as a function of the height components. However, the analysis of results indicates that for each component this is due essentially to the dominance and the preponderance of the sum height, represented by the faces A1B1, C1D1, A2B2, and C2D2 relative to the total length of the component faces. On the other hand, as the length of the component width w = H is fixed to 0. 25, it could be verified that the percentage of the increase of the length of the lateral faces in relation to the total length of the component face is equal to the percentage of the increase of the overall Nusselt number in a level with the component. For Re = 30 and Gr = 104, results indicate that increase of the height component involves an enhancement of the heat removal rate, especially, for the second component because it is about 53.65% and about 48.11% for the first one. Based on these
In order to consult the effect of the width component w = H on the heat removal rate, the width was varied in the range from 0.1 to 0.5, and the height component was fixed to 0.25. Fig. 9 shows that the values of the local Nusselt number for the first component are greater than the values of the second one with the exception of the faces C1D1 and C2D2. The profile size of the faces B1C1 and B2C2 increases as a function of the width component. On the other hand, the profile size for the other faces is the same because the height component is fixed in this case. However, the values of the mean Nusselt number decrease with the increase of w = H and it appears on the one hand that straightaway the heat removal rate decreases. Nevertheless, Fig. 8b indicates that the average Nusselt number for each component increases when w = H increases. This is due essentially to the dominance and the preponderance of the width represented by the faces B1C1 and B2C2, relative to the total length of the component faces. Moreover, the increased percentage of the width relative to the total length is equal to the increased percentage of the average Nusselt number on a level with the component. Hence, the increase of the component width has a considerable effect on the heat transfer, and results for Re = 30 and Gr = 104 indicate that the increase of the heat removal rate is about 39.70% for the first component and about 34.95% for the second one in the width range from 0.10 to 0.50. For all these numerical results, we propose correlations of average Nusselt numbers Nu̅ 1̅ ̅ and Nu̅ ̅ 2̅ ̅ of the first and the second component respectively, versus w = H: 2
Nu1 = 1:41627 + 1:8119ðw = HÞ − 0:29611ðw =H Þ
ð9aÞ 2
Nu2 = 0:86087 + 0:79579ðw = H Þ + 0:10603ðw=H Þ
ð9aÞ
for ð0:10Vw = HV 0:50Þ:
5. Conclusion Mixed convection air cooling of two identical protruding thermal sources, which simulate electronic components, located in a horizontal channel which is open on both sides, has been investigated numerically. The finite volume method has been used to solve the governing equations. The effects of the Reynolds number, the separation distance, the height and the width of the components on the flow structure and heat transfer inside the channel have been examined. Results show that the first component's average Nusselt number is always higher than the one of the second. However, the improvement of the heat removal rate is more important for the second component comparatively to the first one, notably for the effect of the Reynolds number. We have found that the overall heat transfer increases as the height
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Fig. 9. Local Nusselt number along the faces for each component (component 1 and component 2) as a function of the width components w/H, for Re = 30 and Gr = 104.
and width component increase too. It was also found that, the increase in the Reynolds number and separation distance can enhance considerably the cooling of electronic components inside the channel. Correlations were proposed to calculate the average Nusselt numbers of the first and second components. Acknowledgment The authors gratefully acknowledge the financial support of this work (Doctorate thesis) through the project N° J 0300920080073 provided by the Algerian Ministry of High Education and Scientific Research. The authors also take this opportunity to express sincere respect to the reviewers for their comments.
References [1] T. Icoz, Y. Jaluria, Design of cooling systems for electronic equipment using both experimental and numerical inputs, ASME Journal of Electronic Packaging 126 (2004) 465–471. [2] T. Icoz, Y. Jaluria, Numerical simulation of boundary conditions and the onset of instability in natural convection due to protruding thermal sources in an open rectangular channel, Numerical Heat Transfer. Part A, Applications 48 (2005) 831–847. [3] A.K. da Silva, S. Lorente, A. Bejan, Optimal distribution of discrete heat sources on a wall with natural convection, International Journal of Heat and Mass Transfer 47 (2004) 203–214. [4] Q. Wang, Y. Jaluria, Instability and heat transfer in mixed convection flow in a horizontal channel duct with discrete heat sources, Numerical Heat Transfer. Part A, Applications 42 (2002) 445–463. [5] T.J. Young, K. Vafai, Convective cooling of a heated obstacle in a channel, International Journal of Heat and Mass Transfer 41 (1998) 3131–3148.
A. Hamouche, R. Bessaïh / International Communications in Heat and Mass Transfer 36 (2009) 841–849 [6] A. Dogan, M. Sivrioglu, S. Baskaya, Experimental investigation of mixed convection heat transfer in a rectangular channel with discrete heat sources at the top and at the bottom, International Communications in Heat and Mass Transfer 32 (2005) 1244–1252. [7] A. Dogan, M. Sivrioglu, 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) 2652–2662. [8] T. Furukawa, W.J. Yang, Thermal fluid flow in parallel boards with heat generating blocs, International Journal of Heat and Mass Transfer 46 (2003) 5005–5015. [9] S. Habchi, Y. Acharya, Laminar mixed convection in a partially blocked vertical channel, International Journal of Heat and Mass Transfer 29 (1986) 1711–1722. [10] H. Bhowmik, K.W. Tou, An experimental study of transient heat transfer from discrete heat sources in water cooled vertical rectangular channel, ASME Journal of Electronic Packaging 127 (2005) 193–199. [11] H. Bhowmik, C.P. Tso, K.W. Tou, F.L. Tan, Convection heat transfer from discrete heat sources in a liquid cooled rectangular channel, Applied Thermal Engineering 25 (2005) 2532–2542.
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International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Mixed convection air cooling of protruding heat sources mounted in a horizontal channel☆ Adel Hamouche, Rachid Bessaïh ⁎ Laboratoire d'Energétique Appliquée et de Pollution, Faculté des Sciences de l'Ingénieur, Département de Génie Mécanique, Université Mentouri-Constantine, Route de Ain El. Bey, Constantine 25000, Algeria
a r t i c l e
i n f o
Available online 27 May 2009 Keywords: Mixed convection Horizontal channel Electronics cooling
a b s t r a c t A numerical study of laminar mixed-convection heat transfer to air from two identical protruding heat sources, which simulate electronic components, located in a two-dimensional horizontal channel, is presented in this paper. The finite volume method and the SIMPLER algorithm are used to solve the conservation equations of mass, momentum, and energy for mixed convection. Results show that the heat transfer increases remarkably for Pr = 0.71 and 5 ≤ Re ≤ 30. It was also found that the increase of separation distance, the height and the width of the components has a considerable enhancement of the heat removal rate from the components, and therefore, on the improvement of the heat transfer inside the channel. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction With the advent of high-speed computers, heat dissipation at the microchip has become a source of concern. The high temperatures of the chip affect the reliability and the performance of the system. Therefore, effective heat removal is necessary to ensure better life of the system. A literature review shows that various studies have been done for thermal management of electronic devices. For example, Icoz and Jaluria [1] presented a methodology for the design and optimization of cooling systems for electronic equipment. In this approach, inputs from both experimentation and numerical modelling are to be used concurrently to obtain an acceptable or optimal design. It was shown that by using both approaches concurrently; the entire design domain is covered, leading to a rapid, convergent, and realistic design process. A numerical study of natural convection was conducted by Icoz and Jaluria [2], for protruding thermal sources located on a horizontal channel. They observed that the channel dimensions and the presence of openings have significant effects on the fluid flow. However, their effects on heat transfer are found to be relatively small. The increase in channel height is seen to lead to a less stable flow and, consequently, a decrease in the critical Grashof number. Da Silva et al. [3] presented an analytical and numerical study to discover the optimal distribution of discrete heat sources cooled by laminar natural convection. The results show that the optimal distribution is not uniform (the heat sources are not equidistant), and that as the Rayleigh number increases the heat sources placed near the tip of a boundary layer should have zero spacing. Wang and ☆ Communicated by J. Taine and A. Soufiani. ⁎ Corresponding author. E-mail address: [email protected] (R. Bessaïh). 0735-1933/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2009.04.009
Jaluria [4] studied numerically the three-dimensional conjugate heat transfer in a rectangular duct with two discrete flush-mounted heat sources in the context of cooling of electronic equipments. They have seen that the ratio of convection to the total heat transfer from the upstream heat source is larger than that from the downstream one, but the ratio of conduction to the total heat transfer is less than that of the downstream heat sources. A detailed investigation of forced convective cooling of a heated obstacle mounted upon a channel wall was presented by Young and Vafai [5]. It was shown that specific choices in obstacle size, shape and thermal conductivity can produce significant effects on the flow and heat transfer characteristics. Mixed convection heat transfer in a top and bottom heated rectangular channel with discrete heat sources has been investigated experimentally for air by Dogan et al. [6]. Results show that the face temperatures increase with increasing Grashof number. The row-averaged Nusselt numbers first decrease with the row number and then, due to the increase in the buoyancy affect secondary flow and the onset of instability. They also show an increase towards the exit as a result of heat transfer enhancement. Dogan et al. [7] investigated experimentally the mixed-convection heat transfer from arrays of discrete heat sources inside a horizontal channel. Their results show that changes in Grashof and Reynolds numbers affect significantly the buoyancy-driven secondary flow, the onset of instability, and the resulting heat transfer enhancement above the forced convection limit for the aspect ratio AR = 4. Furukawa and Yang [8] presented a numerical investigation of thermal-fluid flow behaviour in a bundle of parallel boards with heat generating blocks. They found that at a low Reynolds number flow, a developing flow may achieve a fully-developed flow state at certain block number from the entrance. Thermal conductivity of the board and thermal contact resistance between the chip and board have a considerable
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Nomenclature d Gr g H h k k⁎ L Nu Nu̅ ̅
separation distance between two components, m 2 Grashof number, = gβ(TS − T0)H3 / νair gravitational acceleration, m/s2 height of the channel, m height of the component, m thermal conductivity of the fluid, W/m K dimensionless thermal conductivity, k/kair length of the channel, m Aθ local Nusselt number, = − An R average Nusselt number, ¼ Nu dn exchange surface
n P Pr p0 p Re Ra Ri T t U0 U, V u, v w X, Y x, y
normal coordinate dimensionless pressure,= (p − p0) / ρU20 Prandtl number, =νair / αair external pressure, Pa pressure, Pa Reynolds number, =U0 H / νair Rayleigh number, =Gr.Pr Richardson number, =Gr/Re2 temperature, K time, s uniform inlet velocity, m/s dimensionless horizontal and vertical velocities, =(u,v)/ U0 horizontal and vertical velocities, m/s width of each heat source, m dimensionless horizontal and vertical Cartesian coordinates,=(x,y) / H horizontal and vertical Cartesian coordinates, m
Greek symbols α thermal diffusivity of the fluid, m2/s β thermal expansion coefficient of the fluid, 1/K θ dimensionless temperature ν kinematic viscosity, m2/s ν⁎ dimensionless kinematic viscosity,= ν / νair νair kinematic viscosity of the fluid, m2/s ρ density of the fluid, kg/m3 Ψ dimensionless stream function τ dimensionless time, =t / (H / U0)
Subscripts 0 evaluated at the reference temperature s source impact on thermal performance. Habchi and Acharya [9] studied the laminar mixed convection about a symmetrically or asymmetrically heated vertical plate with a block. They found that at low values of
Grid
110 × 42 nodes
220 × 52 nodes
320 × 62 nodes
420 × 72 nodes
Nu̅ ̅ ̅ Nu̅ ̅ 1̅ Nu̅ ̅ 2̅
2.400
2.694
2.662
2.735
1.528 0.872 2.256
1.711 0.983 2.254
1.705 0.957 2.257
1.771 0.964 2.258
Umax
Here, Nu̅ 1̅ ̅ and Nu̅ 2̅ ̅ are the average Nusselt numbers for the first and second component, respectively; and Nu̅ ̅ ̅ is the average Nusselt number for both components.
Gr the maximum velocity occurs near the adiabatic wall in the Re2 asymmetrically heated channel, but for the symmetrically heated channel, the velocity profiles are depressed at the center, and for both thermal conditions, the Nusselt numbers are smaller than the corresponding smooth duct Nusselt numbers. Bhowmik and Tou [10] performed experiments to study the single-phase transient forced convection heat transfer on an array of flush-mounted discrete heat sources in a vertical rectangular channel during the pump-on transient operation. The experimental results indicate that the heat transfer coefficient is affected strongly by the number of chips and the Reynolds number. Bhowmik et al. [11] performed steady-state experiments to study general convective heat transfer from four in-line simulated electronic chips in a vertical rectangular channel, using water as the working fluid. The results indicate that the heat transfer coefficient is strongly affected by the Reynolds number, and the fully-developed values of heat transfer coefficient are reached before the first chip. Young and Vafai [12] detailed a numerical simulation of forced convective incompressible flow with an array of heated obstacles attached to one wall. Extensive presentation and evaluation of the mean Nusselt numbers along the exposed faces of all obstacles in the array was fully documented. Kim et al. [13] investigated numerically the characteristics of a pulsating flow and the associated thermal transport from two heated blocks, representing energy dissipating electronic components with different heights in a channel. The results indicate that the recirculating flow behind the second block as well as that between block regions, are substantially affected by the pulsating frequency and heights of blocks. Ray and Srtnivasan [14] studied numerically the temperature variation in the insulation around an electronic component, mounted on a horizontal circuit board. The results indicate that the temperature variation within the insulation becomes important when the thermal conductivity of the insulation is less than ten times the thermal conductivity of the cooling medium. The present study deals with a numerical study of two-dimensional laminar mixed-convection heat transfer to air from two identical protruding heat sources, which simulate electronic components, located in a horizontal channel. Attention was focused on the effects of Re, height, width and separation distance of components on the fluid flow and heat transfer characteristics. Section 2 of this paper presents the mathematical formulation. Section 3 discusses the numerical method and techniques which have been used for the computation,
Fig. 1. Geometry of the problem. The horizontal channel contains two protruding heat sources, which simulate electronic components.
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Fig. 2. Comparison between the present predictions and Habchi and Acharya's [9] predictions for Ra = 103, 105, and Gr/Re2 = 3, H / L = 0.2.
the grid independence study and code validation with numerical data. Section 4 presents the results and finally a conclusion is given. 2. Mathematical model The geometry considered is a two-dimensional channel with two identical protruding heat sources, which simulate electronic components, located at the bottom wall of the horizontal channel, as shown in Fig. 1. The one on the left is designated as the first component. Each heat source has a height h and a width w, and the distance between two heat sources is d. The distance between the channel inlet and the first heat source is L1 and between the second and the channel outlet is L2, H and L are the height and the length of the channel, respectively (Fig. 1). The top and bottom walls of the channel are insulated. At the channel inlet, a flow velocity U0 is imposed with a uniform temperature T0. The heating components are kept isothermal, at uniform temperature TS. Using the channel height H as the length scale, inlet velocity U0 as the velocity scale, H / U0 as the time scale, ρU20 as the pressure scale, and (TS − T0) for the temperature scale, the conservation equations of mass, momentum, and energy for a twodimensional laminar mixed convection and incompressible flow, with constant thermo physical properties, and with Boussinesq approximations, can be written in the following dimensionless form: AU AV + =0 AX AY
νS and νair are the kinematic viscosities of the heat source and of the fluid (air), respectively, and ν⁎ → ∞ in order to obtain U = V = 0 in a level of each component. kS and kair are the thermal conductivities of the solid and of the fluid (air), respectively, and k⁎ → ∞ in order to obtain a uniform temperature in a level of each heat source. The initial and boundary conditions, in dimensionless form, used for the components of velocity and for temperature are:
ð1Þ
AU AU AU AP 1 A AU A AU +U +V = − + m4 + m4 ð2Þ Aτ AX AY AX Re AX AX AY AY
AV AV AV AP 1 +U +V = − + Aτ AX AY AY Re A AV Gr m4 + + θ AY AY Re2
f AXA
AV m4 AX
ð3Þ
g
Aθ Aθ Aθ 1 Aθ Aθ Aθ Aθ +U +V = k4 + k4 Aτ AX AY Re Pr AX AX AY AY
ð4Þ
where, m4 =
∞ in each heat source f m1 in= mtheYfluid region S
air
and k4 =
kS = kair Y∞ in each heat source 1 in the fluid region:
Fig. 3. Dimensionless streamlines Ψ and isotherms in a horizontal channel at Gr = 104: (a) Re = 5, (b) Re = 10, and (c) Re = 30.
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Fig. 4. Average Nusselt number Nu̅ ̅ of each component (component 1 and component 2) at Gr = 104 and for different values of Re.
Initially, at τ = 0, U = V = θ = 0. The dimensionless boundary conditions for our study are presented at τ N 0: at X = 0 and 0VYV1 ; U = 1; V = 0; θ = 0 ðinlet channelÞ AU Aθ = 0; V = 0; = 0 ðoutlet channelÞ AX AX Aθ = 0 ðadiabatic wallÞ at Y = 0 and 0V XVL ; U = 0; V = 0; Ay at X = L and 0VYV1 ;
at Y = 1 and 0V XVL ; U = 0; V = 0;
Aθ = 0 ðadiabatic wallÞ: Ay
the staggered locations, and the scalar quantities (P and θ) are stored in the center of these volumes. A fully implicit time marching scheme is employed. The numerical procedure called SIMPLER [15] is used to handle the pressure–velocity coupling. For treatment of the convection and diffusion terms in Eqs. (2)–(4), central difference scheme is adopted. Finally, the discretized algebraic equations are solved by the line-by-line tri-diagonal matrix algorithm (TDMA). Convergence at a given time step is declared when the maximum relative change between two consecutive iteration levels fall below 10− 4, for U, V and θ. At this stage, the steady-state solution is obtained. Four non-uniform grids are used in this work: 110 × 42, 220 × 52, 320 × 62, and 420 × 72 nodes. The discrepancy in the maximum values of horizontal Umax and average Nusselt numbers Nu̅ ̅ 1̅ ̅ (of the first heat source), Nu̅ ̅ 2̅ ̅ (of the second heat source), and Nu̅ ̅ ̅ (average Nusselt of the first and second heat source), by using the grids 320 × 62 and 420 × 72 nodes, is smaller than 4% [9] (see, Table 1). In order to optimize the CPU time and cost of computations, a fine grid downstream and above of the components in the x and y directions, corresponding to 320 × 62 nodes has been used for all simulations. The total CPU time on a PC Pentium 4 for the results presented here was about 2 h for each case. The numerical code was validated with the numerical results of Habchi and Acharya [9]. They obtained the velocity profiles due to the laminar mixed convection of air in a vertical channel containing a partial rectangular blockage on one channel wall. The wall containing the blockage is assumed to be heated, while the other wall is assumed to be adiabatic. The parameters L1/L and b/L have been assumed to be unity while the dimensionless blockage height H/L is assigned the value of 0.2. L2/L is chosen to be sufficiently far downstream so that the zero streamwise gradient condition is satisfied. As shown in Fig. 2, it is clear that the numerical results of the present work are in a good agreement with Habchi and Acharya's predictions [9].
3. Numerical procedure 4. Results and discussion The governing Eqs. (1)–(4), with the associated boundary conditions, are solved using a finite volume technique, as described by Patankar [15]. The components of the velocity (U and V) are stored at
A detailed numerical study has been carried out on mixedconvection heat transfer enhancement. The study deals with the
Fig. 5. Dimensionless streamlines Ψ in a horizontal channel at Re = 30, Gr = 104, and for different separation distances d: (a) d = w, (b) d = 2w, (c) d = 3w (d) d = 4w.
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effects of the Reynolds number Re, separation distance between two components d, the height h, and the width w components on the flow structure and heat removal rate from the components. In this study, dimensions are taken to be: L1 = 2H; L2 = 4H; d = w; h = H = 0: 25; and w = H = 0:25. In most practical applications of electronics systems cooling by air, the buoyancy strength is not very high [16] and the maximum of the Grashof number Gr is kept in the order of 104. The corresponding moderate temperature difference ΔT = TS − T0 and the channel height H are about 10 K and 20 mm, respectively. Here, all cases with low Reynolds numbers are studied in this paper.
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d = w, we notice a recirculating zone between two components. When the separation distance increases, the size of the recirculating zone increases remarkably downstream of the first component near the face C1D1. However, the streamlines upstream of the second component near the face A2B2 are straight. In this region, an important heat removal is provided because of the important mixing, which transports more thermal energy away from the components. When the separation distance increases, Fig. 6a–b show that for Re = 30 and Gr = 104, the heat removal is affected mostly from the faces C1D1 and
4.1. Effect of the Reynolds number We present the streamlines and isotherms in Fig. 3a–c at Gr = 104 (fixed value) and for different values of the Reynolds numbers (Re = 5, 10, 20, 30). For Re = 5 (Fig. 3a), the buoyancy effects are eminent, notably for the sizes of the first recirculating zone above the first component and near the top wall, and the second one downstream of the second component. For Re = 10 (Fig. 3b), the sizes decrease slightly, but when the Reynolds number increases more, the recirculating zone above the first component disappears completely and the size of the one downstream of the second component decreases considerably. This is due essentially to the increase of the effect of the Reynolds number, which reduces the buoyancy effects inside the channel. We observe from Fig. 3, which gives the isotherm contours as a function of the Reynolds number, when the Reynolds number is small the isotherms stretch out and occupy a large place in the channel. The regions where the effects of the components are felt tend to be localized to the neighborhood of the components. There is a large temperature distribution into the channel, upstream, downstream and above both components. The heated air is not well convected at the outflow because of the dominance of the buoyancy effects. However, when the Reynolds number is higher, the isotherms become approximately horizontal with a notable decrease in the temperature distribution into the channel, precisely to the neighborhood of the components. Consequently, the increase of the Reynolds number reduces considerably the buoyancy effects, and provides a good evacuation of the heated air outside the channel. In order to examine the effect of the Reynolds number on the heat removal rate from components, Fig. 4 displays the variations of the average Nusselt number Nu̅ ̅ of each component (component 1 and component 2) at Gr = 104 and for different values of Re. The results show that the increase of the Reynolds number increases the heat transfer on a level with the first component more than the second one. For the Reynolds number values varying from 5 to 30, the heat transfer results obtained above can be correlated by the following equations of the average Nusselt numbers Nu̅ ̅ 1̅ ̅ and Nu̅ ̅ 2̅ ̅ of the first and the second component respectively, versus the Reynolds number Re: 0:13
Nu1 ¼ 1:12 Re
0:12
Nu2 ¼ 0:64 Re
ð6aÞ :
ð6bÞ
The results indicate also that for Re = 5 and Re = 30, the difference between the corresponding Nusselt numbers for the second component is larger than the one of the first component. Consequently, when the Reynolds number is larger the heat transfer on a level with the first component is improved, but it is more important for the second one. 4.2. Effect of the separation distance The use of the spacing between components is in order to control their temperatures [1]. The variation of the fluid flow as the separation distance changes (d = w, 2w, 3w and 4w) is presented in Fig. 5a–d. For
Fig. 6. At Gr = 104, Re = 30, and for different values of the separation distance d: (a) local Nusselt number the along the faces of the first component (component 1), (b) local Nusselt number along the faces of the second component (component 2), (c) average Nusselt number Nu̅ ̅ of each component (component 1 and component 2).
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Fig. 7. Local Nusselt number along the faces for each component (component 1 and component 2) as a function of the height components h/H, for Re = 30 and Gr = 104.
A2B2. Along the face B2C2, the heat removal rate decreases for d = w to d = 2w and increases for d = 3w and d = 4w. The variation of the average Nusselt number is displayed in Fig. 6c. It is depicted also that the heat removal from the first component is greater than from the second one. As it was seen in [1], streamlines presented in Fig. 5, show that secondary flow which gets stronger as the separation distance increases is created by the first component and between two components, and by the second component, behind it. These backflows influence the heat removal rate of the components, and its effect is more profound on the second component. For Re = 30 and Gr = 104, the enhancement in heat transfer rate from the second component is about 89.60%, however, from the first one it is about 45.66%. These
results confirm that the increase of the separation distance is more favourable for the heat removal rate from the second component than from the first one. In the separation distance range from 0.25 to 1.00, correlations of average Nusselt numbers Nu̅ ̅ 1̅ ̅ and Nu̅ ̅ 2̅ ̅ of the first and the second component, respectively, are proposed as: Nu1 = 1:71573 + 0:65618d − 0:2876d Nu2 = 0:8211 + 0:75596d for ð0:25VdV1:00Þ:
2
ð7aÞ ð7bÞ
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numerical results, we propose correlations of average Nusselt numbers Nu̅ 1̅ and Nu̅ 2̅ of the first and the second component respectively, versus h = H: Nu1 = 1:17852 + 3:02631ðh = H Þ
ð8aÞ 2
Nu2 = 0:7296 + 0:68571ðh = HÞ + 2:64ðh=H Þ
ð8bÞ
for ð0:10Vh = HV0:30Þ:
4.4. Effect of the component width
Fig. 8. For Gr = 104 and Re = 30: (a) average Nusselt number Nu̅ ̅ ̅ of each component for different values of h/H, (b) average Nusselt number Nu̅ ̅ ̅ of each component for different values of w/H.
4.3. Effect of the height component The effect of varying the component height on heat transfer is demonstrated by comparing cases with a fixed width ðw = H = 0: 25Þ. As shown in Fig. 7, the greater mean value of the Nusselt number is noticed at the face A1B1. The face B1C1 which has a fixed length is characterized by a relatively lower value, but it increases when h = H = 0:30. For the faces C1D1 and A2B2, the mean Nusselt value decreases. It decreases also for the face B2C2, when h = H = 0:10 − 0:25, but it increases for h/H = 0.30. In the range of h = H from 0.10 to 0.20, the Nusselt number decreases for the face C2D2 and after increases for h/H = 0.25 and 0.30. At first sight, it appears that the heat transfer decreases when the component height increases, contrary to Fig. 8a, which indicates that the overall Nusselt numbers of the two components increase, as a function of the height components. However, the analysis of results indicates that for each component this is due essentially to the dominance and the preponderance of the sum height, represented by the faces A1B1, C1D1, A2B2, and C2D2 relative to the total length of the component faces. On the other hand, as the length of the component width w = H is fixed to 0. 25, it could be verified that the percentage of the increase of the length of the lateral faces in relation to the total length of the component face is equal to the percentage of the increase of the overall Nusselt number in a level with the component. For Re = 30 and Gr = 104, results indicate that increase of the height component involves an enhancement of the heat removal rate, especially, for the second component because it is about 53.65% and about 48.11% for the first one. Based on these
In order to consult the effect of the width component w = H on the heat removal rate, the width was varied in the range from 0.1 to 0.5, and the height component was fixed to 0.25. Fig. 9 shows that the values of the local Nusselt number for the first component are greater than the values of the second one with the exception of the faces C1D1 and C2D2. The profile size of the faces B1C1 and B2C2 increases as a function of the width component. On the other hand, the profile size for the other faces is the same because the height component is fixed in this case. However, the values of the mean Nusselt number decrease with the increase of w = H and it appears on the one hand that straightaway the heat removal rate decreases. Nevertheless, Fig. 8b indicates that the average Nusselt number for each component increases when w = H increases. This is due essentially to the dominance and the preponderance of the width represented by the faces B1C1 and B2C2, relative to the total length of the component faces. Moreover, the increased percentage of the width relative to the total length is equal to the increased percentage of the average Nusselt number on a level with the component. Hence, the increase of the component width has a considerable effect on the heat transfer, and results for Re = 30 and Gr = 104 indicate that the increase of the heat removal rate is about 39.70% for the first component and about 34.95% for the second one in the width range from 0.10 to 0.50. For all these numerical results, we propose correlations of average Nusselt numbers Nu̅ 1̅ ̅ and Nu̅ ̅ 2̅ ̅ of the first and the second component respectively, versus w = H: 2
Nu1 = 1:41627 + 1:8119ðw = HÞ − 0:29611ðw =H Þ
ð9aÞ 2
Nu2 = 0:86087 + 0:79579ðw = H Þ + 0:10603ðw=H Þ
ð9aÞ
for ð0:10Vw = HV 0:50Þ:
5. Conclusion Mixed convection air cooling of two identical protruding thermal sources, which simulate electronic components, located in a horizontal channel which is open on both sides, has been investigated numerically. The finite volume method has been used to solve the governing equations. The effects of the Reynolds number, the separation distance, the height and the width of the components on the flow structure and heat transfer inside the channel have been examined. Results show that the first component's average Nusselt number is always higher than the one of the second. However, the improvement of the heat removal rate is more important for the second component comparatively to the first one, notably for the effect of the Reynolds number. We have found that the overall heat transfer increases as the height
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Fig. 9. Local Nusselt number along the faces for each component (component 1 and component 2) as a function of the width components w/H, for Re = 30 and Gr = 104.
and width component increase too. It was also found that, the increase in the Reynolds number and separation distance can enhance considerably the cooling of electronic components inside the channel. Correlations were proposed to calculate the average Nusselt numbers of the first and second components. Acknowledgment The authors gratefully acknowledge the financial support of this work (Doctorate thesis) through the project N° J 0300920080073 provided by the Algerian Ministry of High Education and Scientific Research. The authors also take this opportunity to express sincere respect to the reviewers for their comments.
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