Showing papers on "Stefan number published in 2010"

Freezing of a supercooled spherical droplet with mixed boundary conditions

Abstract: 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.

Numerical study of the transient behaviour of a thermal storage module containing phase-change material

Abstract: In this paper, the transient behaviour of a thermal storage module is studied numerically based on the enthalpy transforming method. The module is composed of a concentric tube, in which the annulus contains the phase-change material (PCM) and the inner tube carries the heat transfer fluid. The full set of governing equations accounting for heat and fluid flow in the inner tube, the change of phase of the PCM in the annulus, and heat conduction in the walls are solved simultaneously as a conjugate problem. A parametric study, based on three different PCMs, one with hypothetical constant properties and the other two with variable properties, is conducted to investigate the effects of outer wall thickness and variable thermo-physical properties of the PCM on the performance of the storage module during the phase-change process. It is observed that property variation effects become more pronounced as the Stefan number increases. Interestingly, more latent heat can be stored at a given time as the thermal diffusivity ratio is increased for both constant and variable property cases. This can prevent the over-sizing of the system based on constant property data. Another feature of considerable interests is the delay in the charging time: the time required for the PCM to store a certain amount of energy when the wall thermal resistance is included. A sample set of predicted data shows an error of at least 22 and/or 7.7 per cent when the property variation and/or wall thermal resistance is neglected, respectively. © 2010 Authors.

The multiple meanings of the Stefan-number (and relatives) in refrigeration.

Abstract: The Stefan-number is one out of the multitude of dimensionless numbers which predominantly are used in chemical engineering or heat and mass transfer problems. It relates sensible heat to latent heat and is a key number in solving the problem of heat transfer during solidification or melting. However, numbers similar to this show up also when several thermodynamic problems or relationships in refrigeration engineering are analyzed. Using the Stefan-number it can be distinguished if a refrigerant superheats or condenses during compression, it gives the size of the throttling loss, and it helps to decide if a suction line heat exchanger is beneficial or not. For absorption heat pumping, it allows to quantify the most important loss mechanism, the solution heat exchanger loss. Of course, all this information can be acquired in different ways – and maybe in more precise ways – also, but at least for teaching the basics of refrigeration the Stefan-number is simple to use and very easy to understand. In this paper, the different ways of how to use the Stefan-number in teaching and understanding thermodynamics of refrigerants are presented.

Perturbation solution to heat conduction in melting or solidification with heat generation

Abstract: The Stefan problem involving a source term is considered in this technical note. As an example, planar solidification with time-dependent heat generation in a semi-infinite plane is solved by use of a perturbation technique. The perturbation solution is validated by reducing the problem to the case without heat generation whose exact solution is available. An application to the case with constant heat generation is presented, for which a closed-form solution is obtained. The effects of heat generation and Stefan number on the evolution of solidification are examined using the perturbation solution.

Asymptotic Behavior of a Storage Unit Undergoing Cyclic Melting and Solidification Processes

Abstract: In this paper, the asymptotic behavior of a thermal latent-energy storage system undergoing periodic charge/ discharge cycles is numerically investigated. The system consists of a cylindrical tank, which is randomly packed with spheres having uniform sizes and encapsulating paraffin as phase change material. The working fluid flowing through the bed is pure air. In the main part of this study, the entrance air temperature is supposed to vary in a sinusoidal way with a one-day period. The related governing equations are solved by a control-volume-based finite difference method. First- and second-law-based efficiency indicators are used to characterize the performances of the system. The effects of the phase change material melting point temperature on the energy efficiency and the irreversibility of the system are investigated. It is shown that in all situations, an asymptotic regime of charge/ discharge cycles is reached. For a zero-cycle-average Stefan number, the bed proves to behave like a quasi-perfect reject band filter (i.e., it yields a quasi-constant outlet temperature signal). In such a case, the energy efficiency reaches its maximum value, which also corresponds to a maximum of irreversibility. Thus, this indicates that because of such opposite trends, the design of practical systems should be based on a sound compromise, on a case-by-case basis, and between energy and exergy efficiencies on one side and utilization requirements on the other side. Finally, the predictive capability of such a model is assessed in the situation in which a realistic inlet temperature of the working fluid is considered. The predicted time evolution of the outlet temperature of the working fluid proves to be in good agreement with that reported in the selected reference

Adaptive meshfree method for thermo-fluid problems with phase change

Abstract: In the present paper, the recently developed local meshfree method solution of thermo-fluid problems is modified from the collocation to the combined collocation and weighted least squares approach and upgraded with an h-adaptive strategy. A one domain enthalpy formulation is used for modelling the solid-liquid energy transport and the liquid phase is assumed to behave as an incompressible Newtonian fluid modelled by the Boussinesq hypothesis. The involved temperature, enthalpy, velocity and pressure fields are represented on overlapping local sub-domains through weighted least squares approximation (by a truncated Gaussian weight in the domain nodes) and collocation (at the boundary nodes) by using multiquadrics Radial Basis Functions (RBF). The transport equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively through a novel local pressure correction algorithm. The node adaptivity is established through a phase-indicator and a node refinement strategy that takes into account the dynamic number of neighbouring nodes. The proposed approach is used to solve the standard Gobin Le Quere melting benchmark with tin at Stefan number (Ste) 0.01, Prandtl number (Pr) 0.02, and Rayleigh number (Ra) 2.5e4. The node distribution changes through the simulation as the melting front advances. The solid is consequently computed at much lower node distribution density in comparison with the liquid, which speeds up the simulation and at the same time preserves accuracy. The latter issue has been demonstrated by comparison with the results of other combinations of numerical methods and formulations that attempted this benchmark in the past.

Linear spatio-temporal instability analysis of ice growth under a falling water film

Abstract: A linear spatio-temporal stability analysis is conducted for the ice growth under a falling water film along an inclined ice plane. The full system of linear stability equations is solved by using the Chebyshev collocation method. By plotting the boundary curve between the linear absolute and convective instabilities (AI/CI) of the ice mode in the parameter plane of the Reynolds number and incline angle, it is found that the linear absolute instability exists and occurs above a minimum Reynolds number and below a maximum inclined angle. Furthermore, by plotting the critical Reynolds number curves with respect to the inclined angle for the downstream and upstream branches, the convectively unstable region is determined and divided into three parts, one of which has both downstream and upstream convectively unstable wavepackets and the other two have only downstream or upstream convectively unstable wavepacket. Finally, the effect of the Stefan number and the thickness of the ice layer on the AI/CI boundary curve is investigated.

Analysis of Infiltration, Solidification, and Remelting of a Pure Metal in Subcooled Porous Preform

Abstract: The parts fabricated by selective laser sintering of metal powders are usually not fully densified and have porous structure. Fully densified parts can be obtained by infiltrating liquid metal into the porous structure and solidifying the liquid metal. When the liquid metal is infiltrated into the subcooled porous structure, the liquid metal can be partially solidified. Remelting of the partially solidified metal can also take place and a second moving interface can be present. Infiltration, solidification, and remelting of metal in a subcooled porous preform obtained by laser sintering of metal powders are analytically investigated in this article. The governing equations are nondimensionalized and the problem is described using six dimensionless parameters. The temperature distributions in the remelting and uninfiltrated regions were obtained by an exact solution and an integral approximate solution, respectively. The effects of porosity, Stefan number, subcooling parameter, and dimensionless infiltrati...