Solidification in developing pipe flows

Abstract

A numerical solution is presented for the problem of transient freezing of laminar flows inside a circular pipe. Unlike other available solutions on similar subjects, the flow motion in the present study is determined as part of the solution where the fluid transport process is represented by the elliptic Navier-Stokes equations characterized by diffusion in the radial and axial directions. In the solid and liquid regions, thermal diffusion is accounted for in the axial direction as well as in the radial direction. The vorticity-stream function approach is used in the formulation of the flow problem. A Landau transformation is applied to map the variable-geometry physical domain into a fixed-geometry computational domain. A time-lag procedure is employed to treat the moving grids, and interpolation is used to determine the field variables at the new grid locations from those known at the old grid points. Three cases of flow motion are considered: flow with uniform velocity at the entrance, flow with fully developed profile at the entrance, and slug flow through the pipe. The position of the solid-liquid interface, the temperature distribution, heat flux at the walls, and the heat transfer coeffcient at the interface have been calculated for prescribed and calculated velocity profiles. Differences in solidification rates for the three kinds of flow configurations demonstrate that the computed flow profiles show significantly different results from the slug flow case, demonstrating the need for accurate treatment of the flow field. The influences of the Reynolds and Stefan numbers on the solidification rate are also investigated. The accuracy of the present method is verified by comparing the results with the closest analytical solution using the case of a slug velocity profile through a pipe.