Theoretical Considerations on Free Convection in Tubes

Methods are developed for predicting the flow and heat transfer due to free convection in heated vertical tubes, closed at the bottom and opening into a reservoir of cool fluid at the top. These include methods of predicting whether the flow is laminar or turbulent, and whether the boundary layer of heated fluid fills or does not fill the tube, or fills it with a stagnant region near the closed end; and six methods of analysis are needed for the various combinations of these cases. The flow found depends on a certain modified Grashof number Gα (seeList of symbols) and on the length-radius ratio l/a. For laminar flow, indeed, it depends only on the quotient of these quantities (Fig. 7), but laminar flow becomes hard to achieve for large Gα. For large entry disturbances turbulence is expected for Gα greater than about 104, and then the shape of the heat-transfer curve changes with l/a. The theory of the turbulent flow, which is based on an assumption relating the exchange coefficient to its value for ordinary pipe flow, indicated that turbulence decreases heat transfer when the boundary layer fills the tube and increases it otherwise. This leads to a peculiar ‘bucket’ in the heat-transfer curves (Fig.9). The Grashof number at which the heat transfer becomes most effective rises steeply with l/a (Fig. 10). The theory is extended to apply to a tube closed at both ends, PARTheated and PARTcooled. There is some discussion of a mechanical engineering application in which the external force is not gravity but centrifugal. The whole theory is tentative only, and a critical evaluation is postponed until a larger body of experimental information is available, but some reference to experiment is included in a ‘Note added in proof’ at the end of the paper.