# Turbulent Free Convection and Thermal Radiation in an Air-Filled Cabinet with Partition on the Bottom Wall

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{0.2}. Benyahia et al. [22] studied the complex heat transfer by turbulent natural convection, conduction and heat radiation in an enclosure having high aspect ratio. The enclosure had the shape of a parallelepiped and sides length was 0.076 × 2.18 × 0.52 m. The impact of surface radiation on heat transfer and airflow was investigated through various values of surface emissivity of internal walls. As a result of the research, it was found that the emissivity of the inner surfaces significantly affects the velocity patterns near the horizontal surfaces, without affecting the flow in the core of the enclosure. Fabregat and Pallares [23] investigated heat transfer via both laminar and turbulent natural convection in differently heated cube. Using the method of direct numerical simulation, the authors obtained correlations for the mean Nu depending on Ra ranged from 10

^{5}to 5.5 × 10

^{8}.

^{3}. Cold and hot enclosure walls were isothermal at temperatures 50 and 10 °C, respectively. Five baffles were installed on the hot isothermal wall made of a material with a higher thermal conductivity than that of the enclosure. The experiments carried out with a sufficiently high accuracy helped to establish experimental reference results and will undoubtedly be useful in testing numerical algorithms.

^{5}–10

^{8}) and aspect ratio (0.2–1.0). It was established that for a fixed Ra, the energy transport is higher for small values of the aspect ratio. However, if aspect ratio is fixed, then the intensity of energy transport can be raised with a growth of Ra. The influence of aspect ratio on energy transport in a cavity having mutually orthogonal warmed partitions was investigated by Kandaswamy et al. [27]. The enclosure had two isothermal thin partitions (horizontal and vertical). All surfaces of the cavity were kept at a constant temperature, namely, they were isothermal. It was found that energy transport in the enclosure can be significantly improved by raising the length of the vertical partition, regardless of its location. At the same time, a growth of the length of the horizontal partition increases the intensity of heat transfer only in the case when the partition is located below the center of the cavity.

## 2. Governing Equations and Numerical Method

_{1}, u

_{2}) are considered. The heat-conducting partition is placed vertically for various heights inside the cavity (h/L = 0.2, 0.5 and 0.7). A cavity with characteristic size of L is heated by setting a fixed temperature T

_{h}for the wall x = 0 and cooled by setting a fixed temperature T

_{c}for the wall x = 1 + 2l/L. Bottom and top walls at y = 0 and y = 1 + 2l/L respectively are considered adiabatic. Boussinesq approximation was employed, assuming an incompressible viscous Newtonian fluid. As assumptions used in modeling radiative heat transfer, the following can be noted, namely, the inner walls are gray, and the radiation is diffuse. Physical properties of air at reference temperature T

_{0}= (T

_{h}+ T

_{c})/2 presented in Table 2.

_{conv}is an average convective Nusselt number that illustrates a ratio of the heat transfer by convection to the heat transfer by conduction, while Nu

_{rad}is an average radiative Nusselt number that illustrates a ratio of the heat transfer by radiation to the heat transfer by conduction. Q

_{rad}is dimensionless net radiative heat flux. The dimensional form of Q

_{rad}is defined as the difference between the effective and incident radiation. N

_{rad}is a radiation parameter. The radiation of real bodies differs from the radiation of a completely black body. At equal temperatures, real bodies radiate less thermal energy than a completely black body. To describe the radiation of real bodies, the concept of the surface emissivity is introduced, which characterizes the ratio of the flux density of the own radiation of a real body to the flux density of its own radiation of an absolutely black body. The wall emissivity varies between 0 and 1. The radiation of real bodies in this work is modeled by the radiation of a gray body.

_{rad}, the following equations should be solved

^{9}, Pr = 0.7, λ

_{w,air}= 10,000, h/L = 0.2, $\tilde{\mathsf{\epsilon}}=0.9$. The resulting time series of the average Nu

_{rad}at the heated surface are shown in Figure 4. Moreover, the difference of outcomes between the grids of 120 × 120 elements and 180 × 180 elements is less than 2.5% for the average convective Nu and the 120 × 120 elements grid is chosen as the grid for numerical analysis in present research.

## 3. Results

^{9}, 1.5∙10

^{9}, and 10

^{10}), four values of thermal conductivity ratio (i.e., λ

_{w,air}= 10, 100, 1000, and 10,000). For all these runs, Pr = 0.7 and L

_{1}= 0.7L are used. The thermal conductivity of materials λ

_{w,air}chosen for this study corresponds to a wide range of materials with various industrial applications. Insulating materials such as foam concrete, aerated concrete or wood modeling by selecting λ

_{w,air}≈ 10. Sandstone and granite are among the materials that have λ

_{w,air}≈ 100. Aluminum nitride, aluminum, silicon have λ

_{w,air}≈ 1000.

^{9}, Pr = 0.7, λ

_{w,air}= 10,000, h/L = 0.2 is shown in Figure 5. The obtained patterns are considered in the steady state, which corresponds to the dimensionless time $\tau =10,000$. In the central part of the cavity, the isotherms take a horizontal position. Near solid vertical walls, one can see an appearance of a heat boundary layer (thickening of temperature isolines close to walls).

_{rad}can be seen in Figure 6. Numerical results show that, as expected, a growth of the values of $\tilde{\epsilon}$ results to a significant intensification of the radiative mechanism of energy transfer. At $\tau =10,000$ the mean radiative Nu raises up to 3.46 times at changing of surface emissivity values from 0.3 to 0.9. It should be noted that when solving the boundary-value problem, the internal medium (air) was considered diathermic, that is, transparent to radiation.

^{9}, Pr = 0.7, λ

_{w,air}= 10,000, h/L = 0.2. According to the presented tabular data, it can be found that with a growth of the surface emissivity values, a slight reduction of the strength of convective energy transport is shown. The mean convective Nu diminishes up to 0.16% as the surface emissivity changes from 0 to 0.9. Maximum absolute value of Ψ also decreases with increasing surface emissivity values. This is due to a reduction of the temperature gradient, and a growth of the radiative thermal flux. A similar effect of the parameter $\tilde{\mathsf{\epsilon}}$ on convective energy transport in closed cavities for the case of laminar flow was studied by Martyushev and Sheremet [43] and Miroshnichenko et al. [44].

_{w,air}= 10,000, h/L = 0.2 can been seen in Figure 7. A single convective cell is appeared within the cavity, regardless of the Rayleigh number. The occurrence of this recirculation air flow is due to the geometric features of the problem, as well as the formulated boundary conditions. A growth of Ra manifests itself in a characteristic increase in the thickness of the heat boundary layer near the inner vertical borders. Visually, the isotherms are similar, but the difference in the distribution of temperature isolines can be traced by the isotherm of $\Theta =-0.2$.

^{10}, and decreases up to 12.2% at 10

^{9}. The mean radiative Nu also reduces up to 13.6% and 11.2% respectively. An increment of Ra from 10

^{9}to 10

^{10}results in an increment of the mean convective Nu up to 31.4% at h/L = 0.7 and up to 34.12% at h/L = 0.2.

^{9}, Pr = 0.7, h/L = 0.7. A growth of the heat conductivity ratio by a factor of 10 due to a raise of the heat conductivity of the solid wall material results to both more intensive cooling and heating of the analyzed area. At $\tau =10,000$ and λ

_{w,air}= 100 the flow in the cavity does not yet correspond to a stationary position. This can be seen from the temperature patterns within the heat-conducting walls.

^{9}, Pr = 0.7, h/L = 0.2. According to the values of the mean convective Nu depending on Ra, two facts can be noted, namely, with a growth of heat conductivity ratio, the time required to achieve a steady flow in the cavity decreases significantly, and a growth of λ

_{w,air}results to a raising the values of the average convective Nusselt number. A similar dependence is also found for the average radiative Nu at the characteristic boundary.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

F_{k–i} | view factor between k-th and i-th elements of a cavity |

L | size (m) |

g | gravity acceleration (m/s^{2}) |

k | turbulence kinetic energy (m^{2}/s^{2}) |

h | partition heights (m) |

K | non-dimensional turbulent kinetic energy |

${G}_{k}=-\frac{{\nu}_{t}}{{\mathit{Pr}}_{t}}\frac{\partial T}{\partial y}$ | generation/destruction of buoyancy turbulent kinetic energy |

E | non-dimensional dissipation rate of turbulent kinetic energy |

$Ra=g\beta \left({T}_{h}-{T}_{c}\right){L}^{3}/\nu {\alpha}_{air}$ | Rayleigh number |

$N{u}_{con}$ | average convective Nusselt number |

${N}_{rad}=\sigma {T}_{h}^{4}L/\left[{\lambda}_{air}\left({T}_{h}-{T}_{c}\right)\right]$ | radiation parameter |

$\mathit{Pr}=\nu /{\alpha}_{air}$ | Prandtl number |

$N{u}_{rad}$ | average radiative Nusselt number |

${P}_{k}={\nu}_{t}\left[2{\left(\frac{\partial u}{\partial x}\right)}^{2}+2{\left(\frac{\partial v}{\partial y}\right)}^{2}+{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^{2}\right]$ | shearing production |

${\mathit{Pr}}_{t}={\nu}_{t}/{\alpha}_{t}$ | turbulent Prandtl number |

R_{k} | non-dimensional radiosity of the k-th element of an enclosure |

Q_{rad} | non-dimensional net radiative heat flux |

T_{h} | temperature at the left border (K) |

t | time (s) |

T_{c} | temperature at the right border (K) |

T | temperature (K) |

Θ_{f} | non-dimensional temperature of fluid |

Θ | non-dimensional temperature |

u_{1}, u_{2} | velocity components for x and y axis (m/s) |

U_{1}, U_{2} | non-dimensional velocity components for X and Y axis |

Θ_{w} | non-dimensional temperature of walls |

X, Y | non-dimensional Cartesian coordinates |

ε | dissipation rate of turbulent kinetic energy (m^{2}/s^{3}) |

$\zeta ={T}_{c}/{T}_{h}$ | temperature parameter |

β | factor of volumetric heat expansion (1/K) |

${\alpha}_{w}$ | thermal diffusivity of the wall material (m^{2}/s) |

${\alpha}_{air}$ | air thermal diffusivity (m^{2}/s) |

${\alpha}_{i,j}={\alpha}_{i}/{\alpha}_{j}$ | thermal diffusivity ratio |

$\tilde{\mathsf{\epsilon}}$ | surface emissivity of inner surfaces |

${\lambda}_{air}$ | air heat conductivity (W/mK) |

${\lambda}_{w}$ | heat conductivity of the wall material (W/mK) |

${\lambda}_{i,j}={\lambda}_{i}/{\lambda}_{j}$ | heat conductivity ratio |

ν | kinematic viscosity (m^{2}/s) |

ψ | stream function (m^{2}/s) |

ω | vorticity (s^{−1}) |

${\nu}_{t}={c}_{\mu}{k}^{2}/\epsilon $ | turbulent viscosity (m^{2}/s) |

Ψ | non-dimensional stream function |

$\begin{array}{l}\mathsf{\xi}=a+\frac{b-a}{2}\left\{1+\mathrm{tg}\left[\frac{\pi \mathsf{\varpi}}{b-a}\left(X-\frac{a+b}{2}\right)\right]/\mathrm{tg}\left[\frac{\pi}{2}\mathsf{\varpi}\right]\right\},\\ \eta =a+\frac{b-a}{2}\left\{1+\mathrm{tg}\left[\frac{\pi \mathsf{\varpi}}{b-a}\left(Y-\frac{a+b}{2}\right)\right]/\mathrm{tg}\left[\frac{\pi}{2}\mathsf{\varpi}\right]\right\}\end{array}$ | new dimensionless independent variables |

Ω | non-dimensional vorticity |

τ | non-dimensional time |

σ | Stefan–Boltzmann constant (W/m^{2}K^{4}) |

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**Figure 4.**Dependences of the average radiative Nusselt number at left wall vs. various grids and dimensionless time.

**Figure 5.**Isolines of stream function Ψ (

**a**) and temperature Θ (

**b**) at Ra = 10

^{9}, Pr = 0.7, λ

_{w,air}= 10,000, h/L = 0.2: $\tilde{\mathsf{\epsilon}}=0$—dashed lines, $\tilde{\mathsf{\epsilon}}=0.9$ —solid lines.

**Figure 6.**Profiles of the average radiative Nu vs. time and surface emissivity at Ra = 10

^{9}, Pr = 0.7, λ

_{w,air}= 10,000, h/L = 0.2.

**Figure 7.**Isolines of Ψ and Θ at $\tilde{\mathsf{\epsilon}}=0.9$, Pr = 0.7, λ

_{w,air}= 10,000, h/L = 0.2: Ra = 10

^{9}(

**a**), Ra = 1.5 × 10

^{9}(

**b**), Ra = 10

^{10}(

**c**).

**Figure 8.**Dependences of the mean convective (

**a**) and radiative (

**b**) Nusselt numbers at the left border vs. Rayleigh number, heat-conducting partition heights and dimensionless time at $\tilde{\mathsf{\epsilon}}=0.9$, Pr = 0.7, λ

_{w,air}= 10,000.

**Figure 9.**Isolines of Ψ and Θ at $\tilde{\mathsf{\epsilon}}=0.9$, Ra = 10

^{9}, Pr = 0.7, h/L = 0.7: λ

_{w,air}= 100 (

**a**), λ

_{w,air}= 1000 (

**b**).

**Figure 10.**Time profiles of the average convective Nu vs. thermal conductivity ratio at $\tilde{\mathsf{\epsilon}}=0.9$, Ra = 10

^{9}, Pr = 0.7, h/L = 0.2.

Authors | Number of Partitions | Flow Regime | Aspect Ratio | Medium |
---|---|---|---|---|

Ampofo [24] | 5 | Turbulent | 2 | Air |

Priam et at. [25] | 1 | Laminar | 1 | Air and water |

Saha and Gu [26] | 1 | Laminar | 0.2, 0.5, 1 | Air |

Kandaswamy et al. [27] | 2 | Laminar | 1 | Air |

Khatamifar et al. [31] | 1 | Laminar | 1 | Air |

Wu and Ching [32] | 2 | Laminar | 0.6–0.1 | Air |

Costa [33] | 2 | Laminar | 1 | Air |

Said et al. [34] | 1 | Turbulent | 10 | Air |

Xu [35] | 1 | Laminar | 1 | Water |

Famouri and Hooman [36] | 1 | Laminar | 1 | Air |

Khatamifar et al. [37] | 1 | Laminar | 1 | Air |

Al-Farhany et al. [38] | 1 | Laminar | 1 | Nanofluid |

Physical Properties | Symbol | Value |
---|---|---|

Thermal expansion coefficient | $\beta $ | 3.67 × 10^{–3} K^{−1} |

Kinematic viscosity | $\nu $ | 14.61 × 10^{–6} m^{2}∙s^{−1} |

Thermal diffusivity | $\alpha $ | 20.72 × 10^{–6} m^{2}∙s^{−1} |

Density | $\rho $ | 1.226 kg∙m^{−3} |

$\partial \Theta /\partial \overline{n}=0$ | $Y=0,Y=1+2l/L.$ |

$\Theta =0.5$ | $X=0.$ |

$\Theta =-0.5$ | $X=1+2l/L.$ |

$\Psi =0,\text{}\frac{\partial \Psi}{\partial Y}=0,\text{}{\Theta}_{1}={\Theta}_{2},\frac{\partial {\Theta}_{w}}{\partial \overline{n}}=\frac{{\lambda}_{f}}{{\lambda}_{w}}\frac{\partial {\Theta}_{f}}{\partial \overline{n}}-{N}_{rad}{Q}_{rad}$ | at internal solid-fluid interfaces |

Surface Emissivity Value | ${\left|\mathit{\psi}\right|}_{\mathbf{max}}$ | $\mathit{N}{\mathit{u}}_{\mathit{c}\mathit{o}\mathit{n}\mathit{v}}$ |
---|---|---|

$\tilde{\mathsf{\epsilon}}=0.0$ | 0.011325 | 56.02 |

$\tilde{\mathsf{\epsilon}}=0.3$ | 0.011316 | 55.99 |

$\tilde{\mathsf{\epsilon}}=0.6$ | 0.011301 | 55.96 |

$\tilde{\mathsf{\epsilon}}=0.9$ | 0.011292 | 55.93 |

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**MDPI and ACS Style**

Miroshnichenko, I.V.; Sheremet, M.A.
Turbulent Free Convection and Thermal Radiation in an Air-Filled Cabinet with Partition on the Bottom Wall. *Axioms* **2023**, *12*, 213.
https://doi.org/10.3390/axioms12020213

**AMA Style**

Miroshnichenko IV, Sheremet MA.
Turbulent Free Convection and Thermal Radiation in an Air-Filled Cabinet with Partition on the Bottom Wall. *Axioms*. 2023; 12(2):213.
https://doi.org/10.3390/axioms12020213

**Chicago/Turabian Style**

Miroshnichenko, Igor V., and Mikhail A. Sheremet.
2023. "Turbulent Free Convection and Thermal Radiation in an Air-Filled Cabinet with Partition on the Bottom Wall" *Axioms* 12, no. 2: 213.
https://doi.org/10.3390/axioms12020213