Keywords

1 Introduction

The miniaturization of modern electronic devices has resulted in a reduction of physical space available for the components, leading to a rise in power density. Since electronic components are temperature-sensitive and can be damaged or perform poorly if their temperature falls outside the nominal operating range. Therefore, thermal dissipation has become a critical aspect of their operation. However, implementing cooling systems in compact electronic systems is often challenging, mainly when the components generate different amounts of heat or are not uniformly distributed in electronic packaging. To ensure effective cooling in these systems, it is necessary to employ convection in its three forms (forced, mixed, and natural) in conjunction with thermal radiation. Researchers are actively pursuing new cooling system designs (usually simulated using cavities containing heating bodies) that offer a balance of compactness, high performance, and cost-effectiveness [1,2,3,4,5].

Mixed convection (MC) occurs as a result of a combination of both free and forced convection, where the flow is impacted by both a non-uniform density distribution of the fluid and an external forcing system. Numerous numerical studies have investigated MC in cavities that contain isothermal blocks. One such study conducted by Chamkha et al. [6], analyzed two-dimensional MC surrounding an isothermal square body placed at the center of a vented air-filled cavity. The cavity's right wall is fixed at a low temperature, while the remaining walls are kept adiabatic. The study assessed the impact of various factors such as the Richardson number, Reynolds number, outlet position, position, and aspect ratio of the block on the thermal fields and flow patterns. The findings indicated that the heat transfer was optimal when the outflow was positioned at the top of the right wall. In their research, Javadzadegan et al. [7] investigated the MC heat transfer and fluid flow inside a ventilated cavity containing a hot elliptical obstacle maintained at a constant temperature. The study's main objective was to examine the impact of using a porous medium on the velocity and thermal fields.

In comparison to an isothermal block, a heat-generating block (HGB) offers a more accurate portrayal of an electrical component. A numerical investigation was carried out by Ahammad et al. [8] to examine the impact of various factors, such as the Richardson number, the solid/fluid thermal conductivity ratio, and the diameter of the HGB, on a magnetohydrodynamic MC problem in a vented cavity. The study found that a smaller block size and a lower thermal conductivity ratio improved heat transfer in the system. Ahammad et al. [9] revisited the same configuration to investigate the impact of the Reynolds, Hartmann, and Prandtl numbers on the flow and thermal fields. It was observed that these parameters have a considerable impact on both the thermal and flow fields. MC in cavities containing simple heat-conducting bodies (obstacles) has been studied in the references [10,11,12].

MC in channels is usually used as a methodology for designing and optimizing cooling systems for electronic equipment. Pirouz et al. [13] numerically examined the MC in a horizontal channel with upper and lower bodies mounted on the horizontal walls and receiving uniform heat flux through their base. Ghaneifar et al. [14] investigated numerically the MC of a nanofluid in an insulated horizontal channel containing two central HGBs. Hamouche and Bessaïh [15] considered a 2D channel with two identical heaters on the bottom wall. The same problem was considered by Boutina and Bessaïh [16] with an inclined channel. Hssain et al. [17] studied numerically steady MC of a nanofluid in a horizontal channel with isothermal blocks to simulate extremely heat-conductive electronic components. All these studies are agreed with the effect of ventilation (Re effect) on the behavior of the considered system.

It is important to remember that thermal radiation (TR) is necessary for cooling electronic components [18]. Numerous studies have been conducted on the coupling of natural convection (NC) and TR in closed cavities containing heating blocks of various geometrical shapes, including rectangular [19], circular [20], and square [21, 22]. These investigations reveal that TR improves total heat exchange, enabling more effective cooling of the investigated heat sources. Mandal et al. [23] and Peiravi and Alinejad [24] examine the convective heat transfer and TR in a channel with blocks. Mandal et al. [23] investigated MC with TR in a channel containing discrete heat sources on the bottom wall. Their results show that the TR interaction between the surfaces decreases with an increase in the Reynolds number. Peiravi and Alinejad [24] evaluated the combined effects of TR and MC on heat transfer between a channel's working fluid and a square solid. The results reveal that when the radiation parameter rises, the heat transfer rate decreases.

The above literature review suggests that insufficient focus has been given to TR exchange, especially when heat-generating blocks are present in channels and cavities. This study aims to demonstrate the significance of MC and TR for cooling multiple HGBs located in a horizontal channel. This research is motivated by the search for cost-effective cooling systems for high-performance computers that are both compact and powerful. The findings are displayed in terms of isotherms, streamlines, maximum temperature profiles, and those of the Nusselt number.

2 Mathematical Model and Validation

2.1 Problem Description

Figure 1 presents a schematic representation of an open cavity, highlighting its various parameters. The air-filled rectangular cavity, with dimensions of \({\text{L}} = 3{\text{H}} = 15\,{\text{cm}}\), contains five HGBs, representing different electronic components. The positions of the HGBs are at \({\text{X}}_{{\text{i}}} = 0.5{\text{i}}\) and \({\text{Y}}_{{\text{i}}} = 0.5\) with \(1 \le {\text{i}} \le 5\). . The blocks having the same thermal conductivity (\({\text{k}}_{{\text{S}}}\)) and diameter (\({\text{d}}\)), and generate heat at varying rates, with \({\text{Q}}_{{\text{i}}} = {\uplambda }_{{\text{i}}} \times {\text{Q}} = 1720 \times {\uplambda }_{{\text{i}}}\), where \({\uplambda }_{{\text{i}}} = 2^{{3 - {\text{i}}}}\) is the internal heat generation parameter. All the cavity walls are kept adiabatic except the upper one, which is kept at the same temperature as the air inlet \({\text{T}}_{{\text{C}}} = {\text{T}}_{{{\text{in}}}} = 20\,^\circ {\text{C}}\). The inlet and outlet are placed, respectively, on the left wall and the top wall, with equal sizes \({\text{h}} = {\text{h}}^{*} \times {\text{H}}\). The fluid enters the cavity with a velocity of \({\text{u}}_{{{\text{in}}}}\). Simulations were also carried out for \({\text{h}}^{*} = 0\), which represents a closed cavity, to show that combining TR and MC instead of NC may significantly enhance the total heat transfer.

Fig. 1.
figure 1

Schematic configuration

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2.2 Mathematical Model

The flow under examination is assumed to be two-dimensional, laminar, Newtonian, and incompressible. Except for the density in the buoyancy component, which is addressed using the Boussinesq approximation, the thermophysical characteristics of the air are maintained constant. These assumptions permit the formulation of governing equations, which are expressed in dimensionless form, including the conservation of mass, momentum, energy, and the radiative transfer equation (RTE).

$$ \frac{{\partial {\text{U}}}}{{\partial {\text{X}}}} + \frac{{\partial {\text{V}}}}{{\partial {\text{Y}}}} = 0 $$
(1)
$$ \frac{{\partial {\text{U}}}}{{\partial {\uptau }}} + {\text{U}}\frac{{\partial {\text{U}}}}{{\partial {\text{X}}}} + {\text{V}}\frac{{\partial {\text{U}}}}{{\partial {\text{Y}}}} = - \frac{{\partial {\text{P}}}}{{\partial {\text{X}}}} + {\text{Pr}}\left( {\frac{{\partial^{2} {\text{U}}}}{{\partial {\text{X}}^{2} }} + \frac{{\partial^{2} {\text{U}}}}{{\partial {\text{Y}}^{2} }}} \right) $$
(2)
$$ \frac{{\partial {\text{V}}}}{{\partial {\uptau }}} + {\text{U}}\frac{{\partial {\text{V}}}}{{\partial {\text{X}}}} + {\text{V}}\frac{{\partial {\text{V}}}}{{\partial {\text{Y}}}} = - \frac{{\partial {\text{P}}}}{{\partial {\text{Y}}}} + {\text{Pr}}\left( {\frac{{\partial^{2} {\text{V}}}}{{\partial {\text{X}}^{2} }} + \frac{{\partial^{2} {\text{V}}}}{{\partial {\text{Y}}^{2} }}} \right) + {\text{RaPr}}\uptheta $$
(3)
$$ \frac{{\partial {\uptheta }}}{{\partial {\uptau }}} + {\text{U}}\frac{{\partial {\uptheta }}}{{\partial {\text{X}}}} + {\text{V}}\frac{{\partial {\uptheta }}}{{\partial {\text{Y}}}} = \left( {\frac{{\partial^{2} {\uptheta }}}{{\partial {\text{X}}^{2} }} + \frac{{\partial^{2} {\uptheta }}}{{\partial {\text{Y}}^{2} }}} \right) - \frac{{{\uptau }^{*} }}{{{\text{Pl}}}}\left[ {4\left( {\frac{{\uptheta }}{{{\text{T}}_{{\text{R}}} }} + 1} \right)^{4} - \mathop \smallint \nolimits_{0}^{{4{\uppi }}} {\text{I}}_{{\text{R}}} {\text{d}}\Omega } \right] $$
(4)
$$ \frac{{\partial {\uptheta }}}{{\partial {\uptau }}} = {\text{K}}\left( {\frac{{\partial^{2} {\uptheta }}}{{\partial {\text{X}}^{2} }} + \frac{{\partial^{2} {\uptheta }}}{{\partial {\text{Y}}^{2} }}} \right) + {\uplambda }_{{\text{i}}} $$
(5)
$$ {\upmu }\frac{{\partial {\text{I}}_{{\text{R}}} }}{{\partial {\text{X}}}} + {\upeta }\frac{{\partial {\text{I}}_{{\text{R}}} }}{{\partial {\text{Y}}}} + {\uptau }^{*} {\text{I}}_{{\text{R}}} = \frac{{{\uptau }^{*} }}{{4{\uppi }}}\left[ {{\upomega }\mathop \smallint \nolimits_{0}^{{4{\uppi }}} {\text{I}}_{{\text{R}}} {\upphi} {\text{d}}\rm{\Omega^{\prime}} + \left( {1 - {\upomega }} \right){\text{I}}_{{{\text{Rb}}}} } \right] $$
(6)

Table 1 summarizes the boundary conditions utilized to solve these equations (Eqs. 1–6).

Table 1. Boundary conditions.
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The non-dimensional parameters and variables that appear in Eqs. 1–6 are:

$$ {\uptau } = \frac{{{\upalpha }_{{\text{f}}} {\text{t}}}}{{{\text{H}}^{2} }},\left( {{\text{U}},{\text{V}}} \right) = \frac{{\left( {{\text{u}},{\text{v}}} \right){\text{H}}}}{{{\upalpha }_{{\text{f}}} }},{ }\Delta {\text{T}} = \frac{{{\text{QH}}^{2} }}{{{\text{k}}_{{\text{f}}} }},{ }\left( {{\text{X}},{\text{Y}}} \right) = \frac{{\left( {{\text{x}},{\text{y}}} \right)}}{{\text{H}}},{ }\uptheta = \frac{{\left( {{\text{T}} - {\text{T}}_{{\text{C}}} } \right)}}{{\Delta {\text{T}}}},{\text{ I}}_{{\text{R}}} = \frac{{{\text{i}}_{{\text{R}}} }}{{{\upsigma \text{T}}_{{\text{C}}}^{4} }} $$
(7)
$$ {\text{Pr}} = \frac{{\upupsilon }}{{{\upalpha }_{{\text{f}}} }},{\text{ Re}} = \frac{{{\text{u}}_{{{\text{in}}}} {\text{L}}}}{{\upupsilon }},{\text{ Ra}} = \frac{{{\text{g}}\upbeta {\text{L}}^{3} \Delta {\text{T}}}}{{\upupsilon \upalpha _{{\text{f}}} }},{\text{ Pr}} = \frac{{\upupsilon }}{{{\upalpha }_{{\text{f}}} }},{\text{ K}} = \frac{{{\text{k}}_{{\text{s}}} }}{{{\text{k}}_{{\text{f}}} }},{\text{ Pl}} = \frac{{{\text{k}}_{{\text{f}}} \Delta {\text{T}}}}{{{\upsigma \text{T}}_{{\text{C}}}^{4} {\text{H}}}},{\text{ T}}_{{\text{R}}} = \frac{{{\text{T}}_{{\text{C}}} }}{{\Delta {\text{T}}}} $$
(8)

\({\text{I}}_{{{\text{Rb}}}} = 4\left( {\frac{{\uptheta }}{{{\text{T}}_{{\text{R}}} }} + 1} \right)^{4}\) and \(\upphi\) correspond, respectively, to the non-dimensional emission and phase function of the blackbody. (μ, η) are the directional cosines and (\({\upomega }\), \({\uptau }^{*}\)) denote, respectively, the scattering albedo and optical thickness. These latter are set to zero in this study.

The following equation gives the local and mean heat transfer by convection and radiation on the upper wall.

$$ \begin{gathered} {\text{Nu}}_{{{\text{LC}}}} \left( {\text{X}} \right) = - \left. {\frac{{\partial {\uptheta }}}{{\partial {\text{Y}}}}} \right|_{{{\text{Y}} = 1}} ,\,{\text{Nu}}_{{{\text{LR}}}} \left( {\text{X}} \right) = \left. {\frac{{{\text{Q}}_{{\text{R}}} \left( {\text{X}} \right)}}{{{\text{Pl}}}}} \right|_{{{\text{Y}} = 1}} ,\, \hfill \\ {\text{Nu}}_{{\text{C or R}}} = \frac{1}{{3 - {\text{h}}^{*} }}\mathop \smallint \nolimits_{0}^{{3 - {\text{h}}^{*} }} {\text{Nu}}_{{\text{LC or LR}}} \left( {\text{X}} \right){\text{dX and Nu}}_{{\text{T}}} = {\text{Nu}}_{{\text{R}}} + {\text{Nu}}_{{\text{R}}} \hfill \\ \end{gathered} $$
(9)

2.3 Numerical Approach

The Finite Volume Method (FVM) is employed to discretize the previous equations and boundary conditions. The RTE is integrated using the Discrete Ordinate Method (DOM). A computer program written in FORTRAN is then used to solve the resulting system of equations. The solution is acceptable only if the following condition is satisfied on all variables: \({\text{max}}\left| {\left( {\frac{{\upphi_{{{\text{i}},{\text{j}}}}^{{{\text{new}}}} - \upphi_{{{\text{i}},{\text{j}}}}^{{{\text{old}}}} }}{{\upphi_{{{\text{i}},{\text{j}}}}^{{{\text{old}}}} }}} \right)} \right| \le 10^{ - 5}\), where ϕ represents a given dependence of the variables (\({\uppsi }\), θ), and i, j are the coordinates of space. A mesh test was done on the accuracy of the results and the computation time, it was found that the mesh 250× 80 gives good results.

To verify its accuracy, the numerical code underwent thorough testing using benchmark problems. The numerical results of Habchi and Acharya [25] in a horizontal channel that include a hot block fixed to the bottom wall were considered. The comparison of velocity obtained before and after the block (Fig. 2) shows a satisfactory matching. More validation of the numerical code with other previous studies can be found in the references [4, 5, 26, 27].

Fig. 2.
figure 2

Comparison of velocity profiles with Habchi and Acharya [25].

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3 Results and Discussion

The effect of TR on MC cooling of electronic components simulated by HGBs in a ventilated rectangular channel is demonstrated in this section. The flow structure and thermal field are investigated for various Reynolds number and emissivity. The parameters Ra, D, K, and Pr are set to \(2 \times 10^{6}\), \(0.4\), \(0.1\), and \(0.71\), respectively.

The effect of the Reynolds number on the obtained results is analyzed for \({\upvarepsilon } = 0\), \({\text{h}}^{*} = 0.2\). The streamlines (a) and isotherms (b) obtained for various values of Re are shown in Fig. 3. For the case \({\text{h}}^{*} = 0\), the flow pattern exhibits a bicellular structure. The first cell, (C1), is concentrated around block B1 which generates over 50% of the total heat. The second cell, (C2), surrounds the remaining blocks, but with less intensity. These two cells (C1, C2) rotate in opposite directions, indicating that block B1 establishes the flow direction within the cavity. This behavior can be explained by considering that block B1 generates a considerable amount of heat, creating a thermal plume that induces fluid flow in the cavity. When the mode is changed to MC (\({\text{h}}^{*} \ne 0\)), the cell (C2) is replaced by lines connecting the two openings via different paths. This change in the flow structure can be attributed to the presence of the cooling fluid, which alters the flow pattern and promotes more efficient heat transfer. Figure 3a also demonstrates that when the buoyancy force contribution reduces (Re rises), the number of lines flowing between the cylinders reduces.

According to Fig. 3b, the isotherms are tightly concentrated in the blocks that produce a significant amount of heat (\({\text{Q}}_{{\text{i}}} > Q\)). Conversely, the last two blocks (\({\text{Q}}_{{\text{i}}} < Q\)) appear to be nearly isothermal. The isotherms in the solids show a circular pattern, indicating that the temperature extremes (\({\uptheta }_{{{\text{i}},{\text{max}}}}\)) are located in the center of blocks Bi. This phenomenon can be explained by considering that the heat generated by the blocks is primarily conducted towards the center, where it accumulates, leading to the formation of a hot spot in the middle. At \({\text{h}}^{*} = 0\), a thermal plume appears over block B1, which generates a significant amount of heat. As the value of Re increases, the influence of buoyancy force decreases, resulting in a gradual reduction in the intensity of the thermal plume. This reduction can be explained by the fact that as Re rises, the inertia of the fluid flow becomes dominant over the buoyancy force, leading to a more streamlined flow pattern.

Fig. 3.
figure 3

Streamlines (a) and isotherms (b) obtained for different values of Re with \({\upvarepsilon } = 0\).

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The profiles of the average convective Nusselt number (NuC) and that of the maximum temperature (Tmax) are plotted in Fig. 4. This figure shows that NuC decreases significantly with increasing Re number. For Re = 700, more than 40% of the overall heat generated (Qtot = 7.75Q) is removed through the outlet, implying that the cooling method is effective in dissipating a considerable amount of heat from the system. In addition, a decrease of 4.4% is observed in the maximum temperature of block B1. This value reaches its lowest point at \({\text{Re}} \approx 300\) before experiencing a slight increase at higher values of Re. These findings suggest that the cooling method (MC) is effective in dissipating heat from multiple components (Bi) that generate varying levels of heat, as long as the Ri number is chosen appropriately (in this case, \({\text{Ri}} = {\text{Ra}}/{\text{Re}}^{2} {\text{Pr}} \approx 31\)). In other words, the cooling method can effectively dissipate heat in the presence of heat sources if the flow regime is appropriately selected.

Fig. 4.
figure 4

Maximum temperature and mean convective Nusselt number for various values of Re with \({\upvarepsilon } = 0\) and \({\text{h}}^{*} = { }0.2\).

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The impact of emissivity on the maximum temperature (Tmax) and the total heat transfer (NuT) within the cavity was studied for \({\text{h}}^{*} = { }0.2{ }\left( 0 \right)\) and Re = 300 (Fig. 5). The results shown in Fig. 5a reveal that Tmax decreases with \({\upvarepsilon }\) for both MC \(({\text{h}}^{*} = { }0.2)\) and NC \(({\text{h}}^{*} = { }0)\) modes. The recorded temperature drop was 3.4% and 5.5% for \({\text{h}}^{*} = { }0.2\) and \({\text{h}}^{*} = { }0\), respectively, indicating the effectiveness of radiative cooling in both modes. This effect can be explained by considering that an increase in emissivity leads to a higher rate of radiation heat transfer, which can dissipate more heat from the system, reducing the maximum temperature. On the other hand, Fig. 5b demonstrates that NuT does not exhibit a significant change with an increase in emissivity. For NC mode, the total heat transfer is not dependent on emissivity as all the heat produced is removed by both radiation and convection via the active wall. However, in the MC mode, only a portion of the produced heat is removed through the cold wall, with \({\text{Nu}}_{{\text{T}}} \left( {{\text{h}}^{*} = 0.2} \right)/{\text{Nu}}_{{\text{T}}} \left( {{\text{h}}^{*} = 0} \right) = 71{\text{\% }}\left( {78{\text{\% }}} \right)\) for ε = 0(1).

Overall, these results provide insights into the impact of emissivity on the system's thermal behavior and can be useful for designing cooling systems that incorporate radiative cooling. The results suggest that the effectiveness of radiative cooling can vary depending on the cooling mode and the emissivity of the surfaces.

Fig. 5.
figure 5

Maximum temperature (a) and total Nusselt number (b) for \({\text{Re}} = 300\) and different values of ε.

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4 Conclusion

In this paper, coupled thermal radiation and mixed convection in an elongated horizontal ventilated cavity with five heat-generating blocks that generate varying levels of heat have been investigated. The outcomes of the mixed convection mode are contrasted to those found in the case of a closed unventilated cavity (natural convection). The key findings demonstrating the impact of surface emissivity and the Reynolds number are as follows:

  • The cavity’s maximum temperature decreases by 4.4% as the Re number increases and stabilizes around \({\text{Re}} \approx 300\).

  • An increase in Re reduces lines flowing between the blocks, indicating a decrease in buoyancy contribution.

  • The maximum temperature in the cavity reduces by 3.4% and 5.5% for \({\text{h}}^{*} = 0.2\) and \({\text{h}}^{*} = 0\), respectively, with an increase in emissivity from 0 to 1, demonstrating the effectiveness of radiative cooling in both mixed and natural convection.