Experimental Challenges in Determining Heat Transfer Efficiency Scaling in Highly Turbulent Cryogenic Rayleigh-Benard Convection

Abstract

Cryogenic Rayleigh-Benard convection (RBC) at very high Rayleigh numbers (Ra) serves as a key system for understanding buoyancy-driven industrial and large scale natural flows and for testing theories of turbulent convective heat transport. Cryogenic helium experiments allow one to reach extremely high Ra under well-controlled laboratory conditions; however, interpretation of the resulting heat-transfer scalings remains sensitive to non-Oberbeck-Boussinesq (NOB) effects, experimental uncertainties, as well as a number of corrections that ought to be applied to raw data, including corrections for the adiabatic temperature gradient, parasitic heat leaks, or finite thermal conductivity of plates and sidewalls of RBC cells. We present an analysis of experimental uncertainties and data corrections procedures applicable to cryogenic RBC experiments, specifically to those performed in cylindrical RBC cells in Brno: measurement uncertainties, parasitic effects, choice of 4He working points in the p-T diagram and evaluation of relevant properties of the particular working fluid in connection with the available thermophysical property databases. In particular, our study highlights the necessity of rigorous uncertainty analysis for assessing experimental evidence suggesting either transition to the ultimate regime of RBC due to intrinsic ultimate-regime dynamics or as a manifestation of NOB effects and experimental imperfections.

Figures (9)

• Figure 1: a: Schematic drawing of the liquid-helium (LHe) subsystem of the Brno cryostat [17] with a cylindrical RBC cell of aspect ratio $\Gamma =1$. Positions 1–3 indicate locations of thermometers at the bottom of the LHe vessel, on the cell sidewall, and at the bottom of the cell shield, respectively. Positions 4, 6, and 10 indicate thermal anchoring points of the vent lines A and B to the LHe vessel. At positions 4–7, thermometers and heaters are installed for temperature monitoring and control. Position 8 marks the thermometer and heater on the 3.2 mm diameter filling tube supplying helium from the LHe vessel to the RBC cell when the cold valve inside the LHe vessel is open. The cell is mounted via the gaseous $^4$He filled heat exchange chamber (GHeCH) to the bottom of the liquid He vessel. b: The RBC cell with the holders of miniature temperature sensors; the central sensor Ge5 measuring the bulk temperature and its fluctuations, $T_{\rm{Ge5}}(t)$, is highlighted. c: Construction detail of the attachment of the thin stainless-steel sidewall to the copper bottom plate (the top plate is attached analogously). A 0.5 mm gap separates the plates from the sidewall at the cell corners. Copper parts are shown in red. d: Construction detail (dimensions in mm) of the vent line A connecting the RBC cell to the cryostat flange at room temperature.
• Figure 2: Photographs of the LHe subsystem of the current Brno RBC apparatus, containing the newest version of the $\Gamma=1$ RBC cell shown in Fig. \ref{['fig:conev']}b. The left image shows the cell when the helium shield (HES) is removed; vent line A and the filling tube connected to the cell are visible. The cell is connected to the LHe vessel via the GHeCH. b: The same assembly with the HES installed. Positions 4, 5, and 8 indicate locations of thermometers and heaters, as defined in Fig. \ref{['fig:conev']}. c: Construction detail showing the resistive heater and thermometer on vent line A at position 4. d: Detailed view of the siphon-shaped section of the filling tube with the attached resistive heater and thermometer at position 8.
• Figure 3: The $p-T$ diagram of $^4$He (a: shown in dimensionless relative units with respect to the critical pressure and temperature and b: in absolute units), with the working points (chosen far from the critical point) as used in recent $\Gamma=1$ Brno cryogenic RBC experiments. The crosses denote the corresponding temperatures of the top, $T_{\rm T}$, and bottom, $T_{\rm B}$, plates. The working point h-217 with corresponding $T_{\rm B}$ and $T_{\rm T}$ for which the time records are shown in Fig. \ref{['fig:fluctuations']} is highlighted - displayed as squares.
• Figure 4: Examples of time series of measured and derived quantities for ${\rm{Ra}}= 4.65 \times 10^{12}$ (recorded for the working point h-217 highlighted in Fig. \ref{['fig:PhaseDia']}) illustrate temporal stability and fluctuations typical for the steady state of RBC. a: Time series of the temperatures recorded in the middle of the top plate, $T_{\rm{T1}}$, bottom plate, $T_{\rm{B1}}$, central temperature $T_{\rm{Ge5}}$ (left axis), and the temperature difference $\Delta T1=T_{\rm{B1}}-T_{\rm{T1}}$ (right axis). b: closer views on fluctuations and long-term stability of $T_{\rm{T1}}$ and $T_{\rm{B1}}$. c: The mean plate temperature $T_{\rm{M1}}=(T_{\rm{B1}}+T_{\rm{T1}})/2$. d: the record of heating power into the bottom plate $Q_{\rm B}$. e: the record of the cell pressure $p$. f: the instantaneous Nusselt number, Nu. The displayed time records represent one of operating points used to construct the Nu(Ra) dependence. For this particular Ra, the time-averaged density and pressure of the working $^4$He gas were $\rho=19.9$ kg m$^{-3}$, $p = 141.189$ kPa.
• Figure 5: a: Measured temperature difference between the top and bottom plates during the calibration of the internal TTR-G miniature germanium sensors ($Q_{\rm B}$ = 0 W), recorded by four calibrated GR-200A-1500-1.4B reference sensors using the LS340 temperature controller. b: Temperature differences between the central and edge sensors on the top and bottom plates as a function of plate temperatures recorded during the calibration of the internal TTR-G miniature germanium sensors ($Q_{\rm B}$ = 0 W) and during the RBC experiment (RBC data). The sensors are embedded in the middle $\Delta T_1=T_{\rm{B1}}-T_{\rm{T1}}$ (blue) and at the edge of plates $\Delta T_2=T_{\rm{B2}}-T_{\rm{T2}}$ (red). c: Temperature differences between the central and edge sensors on the top and bottom plates as a function of all measured Ra. The right vertical axis shows the absolute temperatures of the top and bottom plates, $T_{\rm T}$ and $T_{\rm B}$, defined as averages of the two sensors on each plate. d: The same temperature differences, with the right vertical axis showing the pressure $p_{\rm{GHe}}$ in the helium heat-exchange chamber positioned between the RBC cell below and the liquid helium vessel above.