Exact solution to entry-region laminar heat transfer with axial conduction and the boundary condition of the third kind

Abstract A mathematical scheme is devised for obtaining the exact thermal-entry-region temperature solutions for laminar flow (Poiseuille's parabolic velocity profiles) heat transfer subject to significant axial conduction and the boundary condition of the third kind. It is demonstrated that the heat transfer problem which ignores the effect of axial conduction, in reality, corresponds to a part of the generalized problem in which such effect is taken into consideration. Irrespective of the significance of axial conduction, the temperature solution can be represented by an unique mathematical expression in which only the magnitude of the eigenvalues and the associated constants vary with Peclet number. As Peclet number approaches infinity, the generalized temperature solution reduces to that for the case of negligible axial conduction. Determination of the first twelve eigenvalues, eigenfunctions and the series expansion coefficients was made, with the aid of a high-speed digital computer, for Peclet numbers of 1, 5, 10, 20, 30, 50, 100 and infinity, and for the convective parameter Δ (= κ r 0 / k ) of 2, 10 and 100. Examination of the thermal-entry-region temperature profiles and the local Nusselt numbers revealed that the effect of axial conduction is indeed significant if Peclet number is less than ≈ 100. The results of the present analysis showed good qualitative agreements with those of Schneider [1], who analysed a similar problem by assuming a slug-flow model.