Heat transfer in cryogenic helium gas by turbulent Rayleigh–Bénard convection in a cylindrical cell of aspect ratio 1

Abstract

We present experimental results on the heat transfer efficiency of cryogenic turbulent Rayleigh–Bénard convection (RBC) in a cylindrical cell 0.3 m in both diameter and height which has improvements with respect to various corrections connected with finite thermal conductivity of sidewalls and plates. The heat transfer efficiency described by the Nusselt number ${\rm{Nu}}={\rm{Nu}}({\rm{Ra}},{\rm Pr} )$ is investigated for the range of Rayleigh number ${{10}^{6}}<{\rm{Ra}}<{{10}^{15}}$ , with the Prandtl number varying such that $0.7\leqslant {\rm Pr} <15$ , using cryogenic $^{4}$ He gas with well-known and in situ tunable properties as a working fluid. For $7.2\times {{10}^{6}}<{\rm{Ra}}<{{10}^{11}}$ our data (both corrected and uncorrected) agree with suitably corrected data from similar cryogenic experiments and are consistent with ${\rm{Nu}}\propto {\rm{R}}{{{\rm{a}}}^{2/7}}$ . Up to ${\rm{Ra}}\simeq {{10}^{12}}$ , our data could be treated as Oberbeck-Boussinesq data. For ${\rm{Ra}}\gt{{10}^{12}}$ , the heat transfer efficiency becomes affected by non-Oberbeck–Boussinesq (NOB) effects, causing asymmetry of the top and bottom boundary layers. For ${{10}^{12}}{\leqslant \rm{Ra}\leqslant }{{10}^{15}}$ , the Nusselt number closely follows ${\rm{Nu}}\propto {\rm{R}}{{{\rm{a}}}^{1/3}}$ if ${\rm{Nu}}$ and ${\rm{Ra}}$ are evaluated on the basis of the working fluid properties at the directly measured bulk temperature, ${{T}_{{\rm{c}}}}$ , and suitable corrections are taken into account. In contrast, if the mean temperature is determined as an arithmetic mean of the bottom and top plate temperatures, ${\rm{Nu}}({\rm{Ra}})\propto {\rm{R}}{{{\rm{a}}}^{\gamma }}$ displays spurious crossover to higher γ that might be misinterpreted as a transition to the ultimate Kraichnan regime. The second step of our analysis, reported here for the first time, is to ignore the NOB effects affecting the top half of the RBC cell. We replace it by the inverted nearly OB bottom half in order to eliminate the boundary layer asymmetry. This leads to the effective temperature difference $\Delta {{T}_{{\rm{eff}}}}=2({{T}_{{\rm{b}}}}-{{T}_{{\rm{c}}}})$ , where ${{T}_{{\rm{b}}}}$ denotes the bottom plate temperature, and to effective ${\rm{N}}{{{\rm{u}}}_{{\rm{eff}}}}$ and ${\rm{R}}{{{\rm{a}}}_{{\rm{eff}}}}$ values. The effective heat transfer efficiency obtained, showing no tendency of crossover to the ultimate regime up to $2\times {{10}^{15}}$ in ${\rm{R}}{{{\rm{a}}}_{{\rm{eff}}}}$ , is reported and discussed.

Keywords

• Rayleigh-Bénard convection
• heat transfer efficiency
• cryogenic helium
• 44.25.+f
• 47.55.pb