Extended lumped formulation for laminar forced convection in tubes exposed to forced convection normal flows: Application to aerial and submerged pipelines

Abstract

The present study addresses forced convection heat transfer of an internal viscous fluid entering a tube with fully developed laminar velocity and uniform entrance temperature. The internal viscous fluid exchanges heat with an external viscous fluid flowing with constant velocity normal to the tube. Both fluids are assumed to have constant or nearly constant thermophysical properties. The description corresponds to a generalized Graetz problem with heat convection boundary condition. Contrary to the tradition in vogue, the differential formulation of the two-dimensional energy equation combined with the method of separation of variables and the ensuing Sturm–Liouville theory is not employed to solve analytically the dual convection problem. Rather, the objective of the study is to derive, develop and appraise a novel approximate, analytical method entitled extended lumped formulation, which involves a first order differential equation. The primary input in the formulation is the length averaged, mean Nusselt number distribution for a diminished problem consisting in a viscous fluid with fully developed velocity and uniform temperature with constant tube wall temperature, namely the classical Graetz problem. The approximate, analytical solution of the extended lumped equation is expressed by a single exponential function that contains the generalized Stanton number, which is easily evaluated by hand on the back-of-the-envelope with an inexpensive calculator. This step brings forth the developing mean bulk temperature of the internal viscous fluid for the entire spectrum of the modified Biot numbers (0 < Bi < ∞). Subsequently, the total heat transfer is easily determined with an algebraic equation that shares the same simplicity. The agreement between the approximate, analytical mean bulk temperature with the counterpart exact, analytical mean bulk temperature based on the generalized Graetz series is excellent for all Bi values ranging from 0.1 to 100.