Liquid solidification in low peclet number pipe flows

The combined effects of axial conduction and solidification on heat transfer and pressure drop in pipe flows are investigated by the use of a modified Galerkin finite element method. To allow for the upstream heat conduction, the domain of study is extended from X = ‐∞ to X = ‐∞ as has been done in previous analyses. As a preliminary study on the effect of axial conduction, the present investigation assumes a superheat ratio that is sufficiently large ( T o > T f or T w ≈ T f ) such that solidification begins at a location near X = 0. For numerical convenience, the infinite domain ‐∞ ≤ X ≤ ∞ is transformed onto a finite domain ‐1 ≤ z ≤ 1. The energy equation for the liquid‐phase is then solved by a modified Galerkin finite element method. For better numerical stability, a procedure is proposed for controlling the numerical error that might propagate from the singular point ( X, R ) = (O, R o ). The profile of the solid‐liquid interface, the heat transfer rate and the pressure drop are presented for various values of Peclet number, Pe = 1, 3, 5, 10 and 30, and for the modified superheat ratio, c = 0.1, 0.5, 5.0, and ∞.