Analysis of centrifugal convection in rotating pipes

New exact solutions, obtained for centrifugal convection of a compressible fluid in pipes and annular pipes, explain axially elongated counterflow and energy separation-poorly understood phenomena occurring in vortex devices, e.g., hydrocyclones and Ranque tubes. Centrifugal acceleration (which can be up to 10⁶ times gravity in practical vortex tubes), combined with an axial gradient of temperature (even small), induces an intense flow from the cold end to the hot end along the pipe wall and a backflow near the axis. To account for large density variations in vortex devices, we use the axial temperature gradient as a small parameter instead of the Boussinesq approximation. For weak pipe rotation, the swirl is of solid-body type and solutions are compact: v_z/v_za=1 -4y² + 3y⁴ and (T-T_w)/(T_a-T_w) = (1-y²)³; where y = r/r_w, the subscripts w and a denote values of axial velocity v_z, temperature T, and radial distance r, at the wall and on the axis. The axial gradient of pressure, being proportional to 3y²-1, has opposite directions near the wall, y = 1, and near the axis, y = 0; this explains the counterflow. With increasing pipe rotation, the flow starts to converge to the axis. This causes important new effects: (i) the density and swirl velocity maxima occur away from the wall (vortex core formation), (ii) the temperature near the axis becomes lower than near the wall (the Ranque effect), (iii) the axial gradient of temperature drops from the wall to the axis, and (iv) the total axial heat flux (Nu) reaches its maximum Nu_max ≈4000 and then decreases as swirl increases. These features can be exploited for the development of a micro-heat-exchanger, e.g., for cooling computer chips.