Quantification of natural convective heat transfer within air-filled hemispherical cavities. Isothermal tilted disk with dome oriented upwards and wide Ra range

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Abstract

This work quantifies the natural convective heat transfer occurring in hemispherical air-filled cavities whose disk is inclined at an angle varying between 0° and 90°. This active hot disk as well as the dome are maintained isothermal at different temperatures. The numerical approach by means of the control volume method allows the examination of the dynamical phenomena that occurs in many configurations obtained by varying the temperature difference between the two active walls and the radius of the hemisphere. Convective heat transfer at the hot wall is represented by the average Nusselt number associated to Rayleigh numbers varying between 104 and 2.55 × 1012. By taking into account all the studied configurations, correlations between these two dimensionless numbers are established for the set of considered inclination angles. Comparisons with results from other studies for the case of horizontal cavity show a good agreement. The relationships presented here cover the laminar, transitional and turbulent heat transfer regimes. They complement previous studies with the condition of heat flux imposed on the disk. The wide range of Rayleigh numbers considered in this survey and its association with the large inclination angle range allow the application of the correlations to various engineering fields such as nuclear technology, solar energy, building, embarked electronics, architecture, safety or domotics.

Introduction

This work quantifies the natural convective heat transfer that occurs in hemispherical air-filled cavities whose disk is horizontal with the dome oriented upwards or inclined with respect to the horizontal plane. The base and the dome constituting the active walls of the cavity are maintained isothermal and differentially heated. Radiation has a significant influence on flows in closed cavities [1]. This heat exchange phenomenon is particularly important in solar energy [2]. Works addressing closed hemispherical cavities are relatively rare, compared with those dealing with other geometries such as rectangular [3], triangular [4], [5] or cylindrical enclosures [6]. The numerical study [7] is one of the first works dealing with natural convection in inclined hemispherical cavities. In this study, a detailed description of the aerothermal phenomena is accompanied by illustrative explanations to understand the specific flows occurring in this enclosure. Unfortunately, the convective heat exchange is not quantified. Some studies are applied to nuclear engineering. They examine the natural convective heat transfer occurring in nuclear power plants for optimal system operation and to prevent accidents due to exceeding the thermal critical values. This is the case of [8], [9] who consider the particular case of a horizontal cavity whose base and dome are maintained isothermal at different temperatures. The numerical approach is complemented by measurements for a wide-range of Rayleigh numbers and for several Prandtl numbers Pr varying between 6 and 13,000. Studies show a low influence of Pr, a remark which is particularly interesting for applications. Flows, that are laminar for low Rayleigh numbers, become characterized by periodic thermal plumes and then develop into turbulent as the Rayleigh number increases. These studies lead to relationships of Nusselt–Rayleigh type for the ranges 106–109 and 109–5 × 1010. The study [10] also examines, from a numerical approach, the heat transfer that would occur in the case of a severe accident in nuclear power plants. In this case, the hemispherical cavity consists of a horizontal disk with the dome below and a volumetrically heated fluid. Local heat exchanges are examined on the cavity boundaries. Conclusions of the study that also provide Nusselt–Rayleigh correlations are consistent with those of [11], [12], [13], [14]. The influence of the Prandtl number is commented in [15] from a dimensionless approach. The 2D convective flows that take place in these cavities under different boundary conditions are presented in [16]. The numerical survey [17] provides correlations in which several parameters are included. This study is however limited to low Rayleigh numbers corresponding to steady state laminar natural convection. Other correlations from numerical and experimental studies are also proposed in the review [18]. The analytical solution proposed in [19] for the horizontal and isothermal disk is complemented with experimental data and confirms that the laminar heat transfer lead to Nusselt–Rayleigh correlations with an exponent of the Rayleigh number equal to 0.25. Correlations from experimental results performed for particular values of the Prandtl number lead to the same conclusions.

The main objective of the present work is to provide relationships to quantify the convective heat transfer according to the inclination angle of the disk and suitable for different Rayleigh numbers representing diverse sizes and flow regimes. Most of the works that quantify the heat transfer in hemispherical cavities consider the case of a horizontal disk maintained at a constant temperature. The results presented in [20], [21], [22] are the only ones examining the condition of constant heat flux imposed on the disk. They are performed numerically and confirmed experimentally in a wide range of Rayleigh number, covering areas of laminar, transitional and turbulent heat transfer, and lead to Nusselt–Rayleigh type correlations for the estimation of the heat transfer under different inclinations of the disk going from the horizontal to the vertical position. The correlations proposed in the present survey are new for the considered range of Rayleigh numbers, inclination angles and thermal boundary conditions. They are useful for the thermal sizing of facilities using closed hemispherical cavities in various application areas. This is the case for solar meteorological instrumentation (pyranometers, pyrgeometers) treated in several works such as [23]. Nuclear technology is also involved (confinement domes), as well as building (domes, integrated hemispherical thermal solar collectors), embarked electronic devices (radars, electrical and electronic boards), safety (detectors, camera, photographic equipments), or domotics (control systems).

Section snippets

The treated configurations. Governing equations. Numerical solution

The air-filled hemispherical cavity considered in this study is sketched in Fig. 1(a). The dome of radius R constituting the cold wall of the enclosure is maintained isothermal at temperature Tc. The hot active wall (disk) is insulated on its external face while its internal surface Sh is subjected to the convective flow and maintained at constant temperature Th. Two Cartesian systems represented in Fig. 1(b) are defined: (x,y,z) is tied to the inclined hemisphere while (x',y',z') is a fixed

Main results

Calculation results obtained for the average Nusselt number Nuc,αT¯ versus RaT are shown in Fig. 2 for the entire range 104  RaT  2.55 x 1012 and all the angles 0°  α  90°. The coefficients kαT and exponents nαT of the correlations NuαT¯=kαTRaTnαT were investigated for the considered angles and Rayleigh numbers. A careful examination of the results presented in Fig. 3 identifies three specific RaT ranges denoted by G1, G2 and G3, in which NuαT¯ shows distinguishable trends. Exact values of kαT and n

Comparison with other studies

The results obtained in [20], [21] for the thermal boundary condition of a heat flux imposed on the disk have been compared with those of the present study. The difference is represented by the ratioδTϕ=100NuαT¯Nuαϕ¯/NuαT¯where Nuαϕ¯ is the Nusselt number associated with the thermal boundary condition of heat flux imposed on the disk. This ratio is presented in Fig. 5. Under such thermal condition, the Rayleigh number is calculated with Raϕ = gβR4ρϕ/μλa whose numerical values must be

Conclusion

Correlations between Nusselt and Rayleigh numbers suitable for the calculation of convective heat transfer in inclined air-filled hemispherical cavities are proposed. Several inclinations of the isothermal disk going from the horizontal position with the dome upwards to the vertical position are considered. The relationships concern a wide range of Rayleigh numbers varying between 104 and 2.55 × 1012 covering the laminar, transitional and turbulent heat transfer zones. This work comes to complete

References (31)

Cited by (14)

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Communicated by Dr. W.J. Minkowycz.

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