Numerical study of two-dimensional freezing in an annulus

Numerical and experimental study on the solidification of PCM around a vertical axially finned isothermal cylinder

Abstract: This paper presents the results of a numerical and experimental investigation realized on finned tubes with the objective of using them in thermal storage systems. The model is based upon the pure conduction mechanism of heat transfer, the enthalpy formulation approach and the control volume method. The finite difference approximation and the alternating direction scheme are used to discretize the basic equations and the associated boundary and initial conditions. The model was validated by comparison with available results and additional experimental measurements realized by the authors. The number of fins, fin length, fin thickness, the degree of super heat and the aspect ratio of the annular spacing are found to influence the time for complete solidification, solidified mass fraction and the total stored energy. The results confirm the importance of the fins in delaying the undesirable effects of natural convection during the phase change processes. Also, this study indicates the strong influence of the annular space size, the radial length of the fin and the number of fins on the solidified mass fraction and the time for complete phase change. Based upon experimental observations and the tendencies of the numerical results, a metallic tube fitted with four–five fins of constant thickness equal to the tube wall thickness and of radial length around twice the tube diameter should be a compromise solution between efficiency, increase in the heat flow rate and the loss of available storage capacity.

An experimental study on ice formation around horizontal long tubes

Abstract: The results of an experimental study are presented where the growth rate of ice on the outside of cooled copper tubes was studied. The tubes, which were immersed in water in an insulated vessel, were internally cooled by circulating glycol through them. It was found that axial growth rate of ice is distinct at low values of the coolant Reynolds number and short freezing times. The slope of the ice thickness with axial distance showed moderate dependency on time but varied with coolant flow rate, and with Stanton and Biot numbers. A key result from the experiments is the abrupt ice thickness enlargements on the surface of tube bends. This anomaly may be attributed to internal flow disturbances of the coolant, and creation of local eddies inside the bends that enhance growth of ice. The effect was evident for a low Reynolds number (Re = 251.9 and Bi < 1), and fades out for large Reynolds number flows.

Experimentally validated two dimensional numerical model for the solidification of PCM along a horizontal long tube

Abstract: This paper presents the results of a numerical study validated by experimental measurements on the solidification of PCM along a horizontal tube by using the boundary immobilization technique. The two dimensional model (r,z) of the phase change problem is formulated based on the energy equation and the Landau transform which transforms the moving irregular interfaces to fixed parallel interfaces. The finite volume method is used to discretize the system of equations and the associated boundary and initial conditions. A computer program was elaborated and the time and space grids were optimized to make the numerical solution insensitive to the grid size. The model was validated against experimental and numerical results available in the literature and good agreement was found. Additional results were obtained and the numerical predictions were found to agree well with the new measurements showing that the immobilization technique is adequate to handle phase change problems.

Parametric study of solidification of PCM around a cylinder for ice-bank applications

Abstract: This paper presents the results of a numerical parametric study of the solidification of a phase change material (PCM) around a cylinder carrying a heat-transfer fluid (HTF) inside. A pure conduction model is used for the PCM and tube wall, the finite volume method is used together with the interface immobilization technique for treating the phase-change process. The convection problem inside the tube is solved by an energy balance with a Nusselt number value, obtained from the steady-state values for constant wall heat-flux conditions. The effects of the HTF entry temperature, the initial PCM temperature and the thermal conductivity of the tube material on the evolution of the solidification front are studied. Results for the temperature distribution during the process, phase-change interface velocity and thermal energy stored in the system are also presented.

Ice formation around isothermal radial finned tubes

Abstract: The present study presents a thermal numerical model for the solidification of Phase Change Material around a radially finned tube with a constant wall temperature. The model is based upon a pure conduction formulation and the enthalpy method. The finite difference approach and the alternating direction implicit scheme are used to discretize the system of equations and the associated boundary, initial and final conditions. Numerical experiments were realized to optimise the numerical code. Numerical simulations were performed to investigate the effects of the number of fins, fin thickness, fin material, aspect ratio of the tube arrangement and the tube wall temperature. Graphical results were presented, discussed and equations relating the effect of each of the variables on the time for complete solidification are also presented.

Free and moving boundary problems

Abstract: 1. Moving boundary problems: formulation 2. Free boundary problems: formulation 3. Analytical solutions 4. Front-tracking methods 5. Front-fixing methods 6. Fixed-domain methods 7. Analytical solution of seepage problems 8. Numerical solution of free boundary problems References Author index Subject index

Numerical Method for Multi-Dimensional Freezing Problems in Arbitrary Domains

Abstract: This paper presents a simple numerical method for solving two and three-dimensional freezing problems with arbitrary geometries. The change of variable method introduced by Landau for the one-dimensional problem is extended to the multi-dimensional using an independent variable which takes constant values at the boundary and the freezing front. Example calculations were performed for the Stefan type freezing problem in regular squares, triangles, and ellipses. Then some of the results were compared with the experimental ones that were obtained for the constant cooling rate.

Analysis of Two-Dimensional Diffusion-Controlled Moving Boundary Problems

Abstract: This paper presents a technique for the analysis of unsteady, two-dimensional diffusive heat- or mass-transfer problems characterized by moving irregular boundaries. The technique includes an immobilization transformation and a numerical scheme for the solution of the transformed equations. Specifically, the immobilization consists of transforming the governing partial differential equations into a coordinate system where the phase boundaries correspond to fixed coordinate surfaces. An example problem involving the solidification or melting of a finite cylinder is analyzed, and results for a range of conditions are presented.

Numerical solution of moving boundary problems by boundary immobilization and a control-volume-based finite-difference scheme

Abstract: A methodology is set forth for the numerical solution of transient two-dimensional diffusion-type problems (e.g. Heat conduction) in which one of the boundaries of the solution domain moves with time. The moving boundary is immobilized by a coordinate transformation, but the transformed coordinates are, in general, not orthogonal. Furthermore, with respect to a given control volume in the new coordinate system, mass appears to pass through the control surface which bounds the volume, and this mass movement brings about a convection-like transport of energy. The energy equation for a moving, nonorthogonal control volume is derived in general and then specialized to the transformed coordinate system associated with the immobilization of the moving boundary. A fully implicit scheme is used to discretize the control volume energy equation. The spatial derivatives are discretized by either of two schemes depending on the size of the pseudo-convection relative to the diffusion. The energy balance at the moving boundary of the solution domain is also transformed and discretized. A numerical procedure is then developed for solving the discretized energy equations. The use of the control volume formulation and the solution methodology will be illustrated for a specific physical situation in a companion paper that follows this paper in the journal.

Analysis of two-dimensional freezing on the outside of a coolant-carrying tube

Abstract: The solution methodology developed in the preceding paper has been amplified, adapted, and then employed to solve the conjugate phase change—convection problem which results when a coolant passes through a tube situated in a liquid phase-change medium. The axial temperature increase experienced by the coolant gives rise to two-dimensional freezing about the tube. In the first part of the paper, the procedures used to incorporate the coolant energy equation and the various boundary conditions into the solution methodology are described. A closed-form analytical solution is then derived to start the main numerical solutions. The numerical work was focused on gaseous coolants because they give rise to much larger axial variations than do liquid coolants. Results were obtained for the thickness of the frozen layer, the coolant bulk temperature, the tube wall temperature, and the energy extracted from the phase-change medium, with the coolant Stanton number, the Biot number, and the solid-phase Stefan numbers as parameters. Among the parameters, the results were not very sensitive to the Stanton and Stefan numbers but were quite responsive to the Biot number.