Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body

A note on the solution of singular boundary value problems arising in engineering and applied sciences: Use of OHAM

The freezing of a supercooled droplet occurs in two steps: recalescence, that is, a rapid return to thermodynamic equilibrium at the freezing temperature leading to a liquid–solid mixture and a longer stage of complete freezing. The second freezing step can be modelled by the one-phase Stefan problem for an inward solidification of a sphere, assuming the droplet to be spherical. A convective heat transfer with the ambient immiscible fluid is modelled by a mixed boundary condition on the outer surface of the droplet. This condition depends on the Biot number (ratio of the heat transfer resistances inside the droplet and at its surface). A novel asymptotic solution is developed for a small Stefan number and an arbitrary Biot number. Applying the method of matched asymptotic expansions, uniformly valid solutions are obtained for the temperature profile and freezing front evolution in the whole stage of complete freezing. For an infinite Biot number, that is, for a fixed temperature at the droplet outer boundary, known solutions are recovered. In parallel, numerical results are obtained for an arbitrary Stefan number using a finite-difference scheme based on the enthalpy method. The asymptotic and numerical solutions are in good agreement.

Freezing of a supercooled spherical droplet with mixed boundary conditions

Simplified analysis of heat and mass transfer model in droplet evaporation process

Transient heat transfer problems with phase-changes, also known as the “Stefan problems” or “moving-boundary problems,” are practically significant in many engineering and technological fields. Injection molding, one of the most widely used plastics processing techniques, mainly consists of filling, packing, and cooling, and the cooling stage is crucial since it considerably affects the productivity and quality of the molded parts. Thus, solutions for transient phase-change heat conduction problems during injection molding will be instructive. In this article, the enthalpy transforming scheme proposed by Cao and Faghri, which could handle the Stefan problems for generalized multidimensional phase-change structures, is applied coupled with the control-volume/finite-difference techniques. Considering the polydispersity and hierarchical structures, the polymer extended phase change temperature range or mushy zone was included in the two-dimensional enthalpy formulation to forecast the transient phase-change heat conduction during the cooling stage for injection-molded high density polyethylene (HDPE) parts. Experiments were performed and good agreement has been achieved, which reveals that the enthalpy transforming model gives good prediction, especially for the cooling analysis for the injection molding of thick-walled parts of crystalline polymers. The understanding of the phase-change heat conduction characteristics may facilitate the optimal designs of polymer injection molding process for industrial applications. POLYM. ENG. SCI., 2008. © 2008 Society of Plastics Engineers

Numerical prediction of phase-change heat conduction of injection-molded high density polyethylene thick-walled parts via the enthalpy transforming model with mushy zone

B.D. Haynes

Papers

Perturbation solution for spherical and cylindrical solidification by combined convective and radiative cooling