Quantification of natural convective heat transfer within air-filled hemispherical cavities. Isothermal tilted disk with dome oriented upwards and wide Ra range☆
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 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 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 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 ratiowhere 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)
- et al.
Surface radiation effect on convection in a closed enclosure driven by a discrete heater
Int. Commun. Heat Mass Transfer
(2014) - et al.
Heat transfer in inclined rectangular receivers for concentrated solar radiation
Int. Commun. Heat Mass Transfer
(2008) - et al.
Numerical study of double diffusive natural convective heat and mass transfer in an inclined rectangular cavity filled with porous medium
Int. Commun. Heat Mass Transfer
(2012) - et al.
Heatlines based natural convection analysis in tilted isosceles triangular enclosures with linearly heated inclined walls: effect of various orientations
Int. Commun. Heat Mass Transfer
(2013) - et al.
Experimental and numerical analysis of buoyancy-induced flow in inclined triangular enclosures
Int. Commun. Heat Mass Transfer
(2012) Transient natural 2D convection in a cylindrical cavity with the upper face cooled by thermoelectric Peltier effect following an exponential law
Appl. Therm. Eng.
(2003)- et al.
Natural convection in hemispherical enclosure heated from below
Int. J. Heat Mass Transf.
(1994) - et al.
An experimental study of natural convection in a volumetrically heated spherical pool bounded on top with a rigid wall
Nucl. Eng. Des.
(1996) - et al.
Natural convection for in-vessel retention at prototypic Rayleigh numbers
Nucl. Eng. Des.
(2000) - et al.
Natural convection heat transfer within multilayer domes
Int. J. Heat Mass Transf.
(2001)
Numerical and experimental study of steady state free convection generated by constant heat flux in tilted hemispherical cavities
Int. J. Heat Mass Transf.
On thermal control of devices contained in inclined hemispherical cavities with dome oriented downwards and subjected to transient natural convection
Int. Commun. Heat Mass Transfer
Free convection in inclined hemispherical cavities with dome faced downwards. Nu-Ra relationships for disk submitted to constant heat flux
Int. J. Heat Mass Transfer
Calibrating pyrgeometers outdoors independent from the reference value of the atmospheric longwave irradiance, reference diffuse irradiance
J. Atmos. Terr. Phys.
Numerical and experimental study of natural convection in tilted parallelepipedic cavities for large Rayleigh numbers
Exp. Thermal Fluid Flow
Cited by (14)
Free convection heat transfer from a concave hemispherical surface: A numerical exercise
2021, International Communications in Heat and Mass TransferCitation Excerpt :They developed correlations between average Nu and Ra to quantify convective heat loss for various engineering applications. Bairi [19] extended his investigation in hemispherical cavities significantly. He studied natural convection in air-filled tilted hemispherical cavities with the isothermal disk connected with the dome for different disk orientations, and the cavities are filled with Cu-Water nanofluid.
Influence of dome shape on flow structure, natural convection and entropy generation in enclosures at different inclinations: A comparative study
2021, International Journal of Mechanical SciencesCitation Excerpt :Apart from these enclosure geometries, hemispherical, curved wall and dome-shaped cavities have also been considered by some researchers, nonetheless, these are not studied as frequently as the cavities with sharp corners despite having a large potential for heat transfer augmentation. In this regard, Bairi [52] performed a numerical work that focuses on the natural convection behavior of air in a hemispherical cavity by considering a wide range of Rayleigh numbers and different inclination angles. As a result of their study, various correlations are proposed which is claimed to be useful for different engineering applications in nuclear energy, solar systems, electronics and buildings.
Natural convection in inclined hemispherical cavities with isothermal disk and dome faced downwards. Experimental and numerical study
2014, Applied Thermal EngineeringCitation Excerpt :Relationships of the same type concerning vertical or inclined disk with the dome oriented downwards (90° ≤ α ≤ 180°) are proposed in Ref. [10] for 104 ≤ Raφ ≤ 5 × 108, based on survey results at steady state carried out in Ref. [11]. The Dirichlet condition concerning these hemispherical air-filled enclosures is examined in Ref. [12] at steady state and in Ref. [13] for transient regime. Convective heat transfer is quantified in these surveys by means of Nusselt–Rayleigh and Nusselt–Rayleigh–Fourier type correlations respectively, for the same RaT and α ranges.
A synthesis of correlations on quantification of free convective heat transfer in inclined air-filled hemispherical enclosures
2014, International Communications in Heat and Mass TransferCitation Excerpt :The aim of the present review is to synthesize the results obtained in the recent works [1–10] dealing with the quantification of the convective heat transfer occurring in hemispherical enclosures as sketched in Fig. 1(a).
Mixed Convection Heat Transfer From Swirling Open Spherical Cavity
2023, ASME Journal of Heat and Mass TransferThermofluidic analysis around a heated hollow spherical ring immersed in air
2023, Numerical Heat Transfer; Part A: Applications
- ☆
Communicated by Dr. W.J. Minkowycz.