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Thermal state of a concentric quarter spherical enclosure subjected to air free convection

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Abstract

The average temperature difference between the walls of a cavity consisting of two concentric quarter spheres and subjected to an air natural convective flow was determined. The work was performed numerically through volume control method based on the SIMPLE algorithm. The thermal state was determined for several combinations of the aspect ratio varying from 0.05 to 0.35 and Rayleigh number with elevated values reaching 6.76 × 1011. A new correlation has been proposed, allowing thermal sizing of this type of cavity which could be used in different fields of engineering such as electronics and building.

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Abbreviations

\(A\) :

Aspect ratio \(A = L/R_{{\text{i}}}\) (–)

\(a\) :

Thermal diffusivity (m2s−1)

\(C_{{\text{p}}}\) :

Specific heat at constant pressure (J kg−1 K−1)

\(\vec{e}_{{\text{g}}}\) :

Unit vector opposite to the gravity direction

\(g\) :

Gravity acceleration (m s−2)

\(k\) :

Coefficient defined in Eq. (11)

\(L\) :

Air layer thickness (m)

\(m\) :

Exponent defined in Eq. (11)

\(n\) :

Outgoing normal to surface \(S\)

\(\left( {\overline{{{\text{Nu}}}}_{{{L}}} } \right)_{{\left[ {28} \right]}}\) :

Average Nusselt number calculated from ref. [28] (–)

\({\text{Pr}}\) :

Prandtl number (–)

\(p\) :

Pressure (Pa)

\(p^{*}\) :

Dimensionless pressure (–)

\(r\) :

Radial direction (m)

\(R_{{\text{e}}}\) :

Outer radius (m)

\(R_{{\text{i}}}\) :

Internal radius (m)

\({\text{Ra}}_{{{L}}}^{{}}\) :

Rayleigh number (–)

\( S\) :

Surface (m2)

\(T\) :

Temperature (K)

\( T_{{\text{c}}}\) :

Temperature of the external wall (K)

\(T_{{\text{h}}}\) :

Local temperature of the internal wall (K)

\(\overline{T}_{h}\) :

Average temperature of the internal wall (K)

\(T^{*}\) :

Dimensionless temperature (–)

\(\vec{u}\) :

Vector velocity (m s−1)

\(\vec{u}^{*}\) :

Dimensionless vector velocity (–)

\(\beta\) :

Volumetric expansion coefficient (K−1)

\(\delta\) :

Deviation between \(\overline{\Delta T}\) and \(\left( {\overline{\Delta T} } \right)_{{\left[ {28} \right]}}\)

\(\overline{\Delta T}\) :

Average temperature difference (K)

\(\left( {\overline{\Delta T} } \right)_{{\left[ {28} \right]}}\) :

\(\overline{\Delta T}\) Calculated from Ref. \(\left[ {28} \right]\)

\(\nabla^{*2}\) :

Spherical Laplacian operator

\(\vec{\nabla }^{*}\) :

Nabla operator

\(\varphi\) :

Heat flux (Wm−2)

\(\lambda\) :

Thermal conductivity (Wm−1 K−1)

\(\mu\) :

Dynamic viscosity (Pa s)

\(\rho\) :

Density (kg m−3)

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Authors and Affiliations

  1. Laboratoire Thermique Interfaces Environnement (LTIE), Université de Paris, EA 4415, 50 Rue de Sèvres, 92410, Ville d’Avray, France

    A. Baïri & N. Alilat

  2. ENEDI Research Group, Department of Thermal Engineering, University of the Basque Country UPV/EHU, Plaza Europa 1, 20018, Donostia-San Sebastián, Spain

    A. Martín-Garín & J. A. Millán-García

  3. Université du Québec À Rimouski, 300, Allée des Ursulines, CP 3300, Rimouski, QC, Canada

    A. Ilinca

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  1. A. Baïri
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Baïri, A., Martín-Garín, A., Ilinca, A. et al. Thermal state of a concentric quarter spherical enclosure subjected to air free convection. J Therm Anal Calorim 147, 3703–3708 (2022). https://doi.org/10.1007/s10973-021-10739-w

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  • DOI: https://doi.org/10.1007/s10973-021-10739-w

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