Showing papers on "Stefan number published in 2012"

On heat conduction with phase change: Accurate explicit numerical method

Abstract: The one phase Stefan problem is discussed using the Goodman HBI method and an explicit numerical method including modified boundary immobilization scheme. The main advantage of the HBI method lie in the remarkable association of simplicity, flexibility and acceptable accuracy which an error less than 2.5% in predicting the moving front location for Stefan number less than unity which covers most usual isothermal phase change material. An accurate explicit numerical model to track the moving front in Stefan-like problems is provided. The scheme is obtained using the variable space step method based on variable domain. The method is developed using central difference approximations to replace spatial and temporal derivatives. Furthermore, iterative procedure, in numerical calculation, is avoided by introducing simple assumptions. The numerical results show that the accuracy of the method has been considerably improved without additional computational cost.

Numerical experiment of cyclic layering in a solidified binary eutectic melt

Abstract: [1] In shallow magmatic intrusions, a characteristic layering structure (hereafter referred to as cyclic layering) can sometimes be observed. This cyclic layering is caused by double diffusion and crystallization kinetics, and different from what is observed as rhythmic layering caused by gravity. The cyclic layering is visualized as differential weathering in response to the differential stiffness caused by textural variations such as those in the volume fraction, number density, and size of vesicles or crystals. The spacing of layers seems to increase according to a geometric progression, like as in Liesegang bands of a diffusion-precipitation system. In order to understand the development condition for cyclic layering and the characteristics of textural variations, such as the spacing of layering in crystallized multi-component melts by conductive cooling, we carried out a numerical experiment on the 1D crystallization process of a binary eutectic melt. This simulation took into account the cooling from contact with country rock as well as the compositional and thermal diffusion and the kinetics of diffusion-limited crystallization. The governing equations include dimensionless control parameters describing the relative importance of thermal diffusion or compositional diffusion (Lewis number, Le) and the effective latent heat release (Stefan number, St). From the results of the numerical experiments, it was found that the layering develops through eutectic oscillation (compositional and thermal oscillation below the eutectic point), suggesting that the bi-activating condition, whereby both phases cooperatively activate their crystallization rates, is essential for the development of layering. No layering is observed at the margin, and the length of the region with no layering increases exponentially with decreasing St. The amplitude of textural oscillation decreases with decreasing St. Thus, practically no layering develops at small latent heat release. Three types of layering structure or oscillatory profiles of texture are observed (short, long and multiple types), depending mainly on Le. Realistic values of Le and St suggest that natural cyclic layering is the multiple or long type of layering. The common ratios of geometric progressions converge with increasing Le to constants in the range of approximately 1.02–1.05, which is similar to the range of the natural observations. Experiments with no latent heat release by the second-phase simulating vesicles show similar oscillatory behaviors, suggesting that the latent heat release of the first crystallizing phase is an essential factor for the development of vesicle layering.

Freezing and melting of a bath material onto a cylindrical solid additive in an agitated bath

Abstract: In melting and assimilation of a cylindrical shaped additive in an agitated hot melt bath during the process of preparation of cast iron and steel of different grades, an unavoidable step of transient conjugated conduction-controlled axisymmetric freezing and melting of the bath material onto the additive immediately after its dunking in bath occurs. Decreasing the time of completion of this step is of great significance for production cost reduction and increasing the productivity of such preparations. Its suitable mathematical model of lump-integral type is developed. Its nondimensional format indicates the dependence of this step upon independent nondimensional parameters- the bath temperature, θb the modified Biot number, Bim denoting the bath agitation, the property-ratio, B and the heat capacity-ratio, Cr of the melt bath-additive system, the Stefan number, St pertaining to the phase-change of the bath material. The model provides the closed-form expressions for both the growth of the frozen layer thickness, ξ onto the additive and the heat penetration depth, η in the additive. Both are functions of these parameters, but when they are transformed to the growth of the frozen layer thickness with respect to the heat capacity ratio per unit Stefan number; and the time per unit property-ratio, B, their expressions become only a function of single parameter, the conduction factor, Cof consisting of the parameters, B, Bim and θb. The closed-form expression for the growth of the maximum thickness of the frozen layer, its time of growth, the time of the freezing and melting; the heat penetration depth are also derived. When the heat penetration depth approaches the central axis of the cylindrical additive in case of the complete melting of the frozen layer developed Cof≤11/72. It is found that the decreasing Cof reduces both the time of this unavoidable step and the growth of the maximum frozen layer thickness and at Cof=0, the frozen layer does not form leading to zero time for this step. If the bath is kept at the freezing temperature of the bath material, only freezing occurs. To validate the model, it is cast to resemble the freezing and melting of the bath material onto the plate shaped additive. The results are exactly the same as those of the plate.

Approximate Analytical Solution for One-Dimensional Solidification Problem of a Finite Superheating Phase Change Material Including the Effects of Wall and Thermal Contact Resistances

Abstract: This work reports an analytical solution for the solidification of a superheating phase change material (PCM) contained in a rectangular enclosure with a finite height. The analytical solution has been obtained by solving nondimensional energy equations by using the perturbation method for a small perturbation parameter: the Stefan number, . This analytical solution, which takes into account the effects of the superheating of PCM, finite height of the enclosure, thickness of the wall, and wall-solid shell interfacial thermal resistances, was expressed in terms of nondimensional temperature distributions of the bottom wall of the enclosure and both PCM phases, and the dimensionless solid-liquid interface position and its dimensionless speed. The developed solution was firstly compared with that existing in the literature for the case of nonsuperheating PCM. The predicted results agreed well with those published in the literature. Next, a parametric study was carried out in order to study the impacts of the dimensionless control parameters on the dimensionless temperature distributions of the wall, the solid shell, and liquid phase of the PCM, as well as the solid-liquid interface position and its dimensionless speed.

Mathematical estimation of melt depth in conduction mode of laser spot remelting process

Abstract: A one-dimensional mathematical model based on the front tracking method was developed to predict the melt depth as a function of internal and external parameters of laser spot remelting process in conduction mode. Power density, pulse duration, and thermophysical properties of material including thermal diffusivity, melting point, latent heat, and absorption coefficient have been taken into account in the model of this article. By comparing the theoretical results and experimental welding data of commercial pure nickel and titanium plates, the validity of the developed model was examined. Comparison shows a reasonably good agreement between the theory and experiment. For the sake of simplicity, a graphical technique was presented to obtain the melt depth of various materials at any arbitrary amount of power density and pulse duration. In the graphical technique, two dimensionless constants including the Stefan number (Ste) and an introduced constant named laser power factor (LPF) are used. Indeed, all of ...

Thermal characteristics in latent heat energy storage system using paraffin wax

Abstract: An energy storage system has been designed to study the heat transfer characteristics of paraffin wax during melting and solidification processes in a vertical annulus energy storage system. In the experimental study, three important issues are focussed. The first one is temperature distribution in the phase change material (PCM) during the phase change processes. The second one is the thermal characteristics of the paraffin wax, which includes total melting and total solidification times, the nature of heat transfer phenomena in melted and solidified PCM and the effect of Reynolds number as inlet heat transfer fluid (HTF) conditions on the heat transfer parameters. The final one is to calculate heat transfer coefficient and effectiveness during solidification process. The experimental results proved that the PCM melts and solidifies congruently, and the melting front moved from top to the bottom of the PCM container whereas the solidification front moved from bottom to the top along the axial distances in the PCM. Experiment has been performed for different water flow rates at constant inlet temperature of heat transfer fluid for recovery and use of heat. Heat transfer characteristics are studied and the dimensionless phase transition time is related to Stefan number by a simple correlation.

Periodic Solidification in a Rectangular Duct Due to Velocity Modulation

Abstract: In the present study, we report on the solid phase dy- namic response due to time-varying duct flows when a portion of a duct wall is cooled to below the liquidus temperature, along which unidirectional solidification from the cooling duct wall, perpendicular to the flow direction, is assumed. A one-dimensional numerical model for the average solid phase thickness has been formulated employing the boundary tracking method. It is shown that a quasi-steady state temperature in the solid layer allows us to develop an analytical solution, making use of perturbation technique. The afore-mentioned perturbation analysis identifies important three nondimensional parameters, i.e. the Biot number based on the solid phase thickness at steady state, the Stefan number based on the temperature difference between the cooling wall and the liquidus temperatures, and the Stefan number based on the liquidus and the flowing liquid temperatures. Results ob- tained by both approaches agree well in general, and the time- variation trends of solid phase thickness and its phase delay have been obtained as a function of the non-dimensional angular fre- quency of the modulating duct flow velocity, with the above three non-dimensional parameters. Various applications in practical engineering and in engineering education have been identified and are being addressed by the developed Graphical Interface Frame- work for Educational and Engineering Support (GIFEES).