THE GRAETZ PROBLEM FOR TWO-PHASE TWO-COMPONENT GAS-LIQUID FLOW IN VERTICAL TUBES
Recent measurements of heat transfer to two-phase, two-component fluids flowing vertically upwards in circular tubes show, under certain circumstances, next to the customary sharp decrease of Nusselt number near the entrance to the heated region, a gradual increase further away from the entrance point. An approximate theory is developed to account for this phenomenon, according to which the gradual increase in buoyant velocity of the bubbles causes a vertical velocity gradient of the liquid along the axis. Continuity requires a corresponding down-flow near the wall and a radial inflow, both of which cause increased convection cooling at the wall. It is shown that it is not necessary to solve the full Graetz problem in this case but that under certain simplifying assumptions it is sufficient to adjust the eigenvalues and eigenfunctions of the Graetz problem for single-phase fluids, by the addition of a parameter involving the product of Reynolds and Prandtl numbers and the non-dimensional vertical velocity gradient. It is shown that the theory qualitatively confirms the experimental observations.