Steady natural convection in a vertical cylindrical envelope with adiabatic lateral wall

The natural convection heat transfer and fluid flow in a vertical cylindrical envelope with constant but different temperatures of the two end surfaces and an adiabatic lateral wall was numerically investigated. Apart from some other applications, it serves as a simplified model of the pulse tube of a pulse tube refrigerator. The simulation was conducted for two end wall temperature differences: ΔT_w=10 and 220 K. For the cases of ΔT_w= 10 K, it is found that the variation patterns of Nu_L vs. Ra_L within the range of L/D=3-10 are in good consistency with the experimental and theoretical results provided by Catton and Edward for L/D=0-2.5. The C-E chart is thus extended from L/D=2.5 to 10. Within the range of Ra_L=1.1×10⁵ to 4 × 10⁷ the fluid isothermals in both longitudinal and cross-sections exhibit some laminated character, and the convective heat transfer rate is in the same order or one-order larger than that of pure heat conduction. For the case of large temperature difference (ΔT_w=220 K) the natural convection in the enclosure is quite strong in that the convective heat transfer rate is about two-orders larger than that of pure heat conduction which occurs when the cold end is placed down. To reduce the loss of cooling capacity of the pulse tube refrigerator, the pulse tube should be positioned with cold end down. Numerical simulation also revealed that the ratio of the axial length, L, to the diameter, D, has effect on the average heat transfer rate of the envelope under the same other conditions. Within the range of L/D=1-9, the increase in L/D leads to the decrease in heat transfer rate.