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266 B. Zalba et al. / Applied Thermal Engineering 23 (2003) 251–2833.1.2. Numerical solution considering only conductionAlthough the dominant heat transfer mechanisms are conduction and natural convection inliquid phase, the earliest literature analyses heat transfer basically by considering one-dimensionalconduction in pure substances. The first studies were by Lame and Clapeyron in 1831 and Stefanin 1891, concerning ice formation. In 1912 the results of F. Neuman were published, establishingprecise solutions to more general phase change problems. In 1943, London and Seban [98]analysed the process of ice formation for different geometries (cylinder, sphere, and flat plate).This study is very interesting although it was later disputed by Shamsundar [99] who asserted thatLondonÕs one-dimensional formulation led to errors increasing with the progress of the solidificationprocess and proposed a two-dimensional formulation for cylinders.In 1970, Lazaridis [100] put forward a study of the relative importance of conduction andconvection. Shamsundar and Sparrow [101] demonstrated the equivalence between the energyconversation equation applied in the three zones (solid, liquid, and solid/liquid) and the enthalpymodel. These are solved through a finite differential method and the solidification is analysed in asquare plate cooled by means of convection. This method is valid both for phase change in a singletemperature and for phase change in a range of temperatures (mixtures or alloys). Among thehypotheses in this work, it can be pointed out that the authors do not consider convection inthe melting phase and assume the solid and liquid densities to be equal and uniform. Later[102], they evaluated the effects of density change. In their conclusions, the relation between theheat transfer speed and the Biot number, linked with convection of the heat-carrier fluid, wasalready presented. Goodling [103] resolved a one-dimensional geometry outward solidification ina cylinder with a constant heat flow in the inner wall, the solution being given by the finite differencemethod applied to the temperature.Meyer [104] studied the problem of phase change conduction, establishing that the classicStefan problem implied the resolution of the temperature range and carrying out a review andcomparison of explicit and implicit methods. The same year, Marshall published a work [105]studying natural convection effects in an annular geometry using hydrated salts and waxes/paraffins.He showed that the existence of natural convection significantly reduced the time necessaryfor melting, and gave recommendations for making use of natural convection especially withsubstances of low thermal conductivity (paraffins).Bathelt [106] worked with heptadecane (C 17 H 36 ) and studied solidification around a horizontalcylinder. Using photographs, he showed the important role of natural convection, which increaseswith time and produces an increase in the average radius of solidification (non-concentric). Thistwo-dimensional behaviour for this geometry has also been confirmed by other authors [30,107].Shamsundar and Srinivasan [108] analysed a heat exchanger of shell and tubes, and proposed atwo-dimensional analysis [99], also taking into consideration the axial variation of the temperature,thus approaching a three-dimensional analysis. In 1981 Sparrow [109] analysed solidificationon the outside of a tube carrying coolant inside, and confirmed the influence of the axial temperaturevariation of the coolant. He proposed an analytical solution providing initial values forthe numerical method in which the moving boundary is immobilized by a transformation of coordinates.As a continuation of previous works, Shamsundar in 1981 [110] studied the influence ofchange in volume on phase change in the range of 10–20%. This involves the formation of cavitiesin the top part through which it is supposed there is no heat transfer (an additional adiabaticwall), so the usefulness of the methodology is therefore justified although there is density change
B. Zalba et al. / Applied Thermal Engineering 23 (2003) 251–283 267in the solidification. The same author in 1982 [111] developed analytical solutions and evaluatedthe resulting formulation using results obtained with numerical methods given in previous articles[109].In 1983 Achard et al. [112] carried out a thermal storage experimental study on a test bench,using an immersed tubular heat exchanger in the PCM, both with salt hydrates and with fattyacids. They also developed a theoretical study using the enthalpy method and solved by means offinite differences, disregarding the convection effect. In the conclusion the theoretical and experimentalresults were compared and significant discrepancies found in the melting. It was thereforededuced that it is necessary to consider natural convection in the liquid.Hunter in 1989 [113] and Amdjadi in 1990 [114] confirmed that the enthalpy method is the mostsuitable for real substances provided that there is no alteration to the numerical scheme in theboundary. Amdjadi added that if the material has hysteresis, it is necessary to rearrange or adjustthe method. In this later work the finite differences method is used with a variable time step,adjusting it at each moment according to the stage of the phase change process. In 1999, Banaszeket al. [115,116] studied experimentally and numerically solid–liquid phase change in a spiral TESunit.3.1.3. Numerical solution considering also convectionThe first publications that include the convection heat transfer mechanism are Sparrow et al.[117] and Bathelt et al. [118]. An interesting article about convection is by Gobin [119] whoseobjective is to determine the influence of natural convection on the melting process. To modelthese processes, some articles [120–122] include the influence of the convection considering aneffective thermal conductivity:k e¼ cRa nk l€Ozisik [123] includes a classification of the various solution methods:1. Exact solutions, limited to a few idealised situations.2. Integral method. One-dimensional: solution of an integral equation to localise the boundary[124].3. Heat moving source method.4. Perturbation method [124].5. Electrical analogy (this is being replaced by numerical methods owing to the availability ofpowerful computers).6. Finite differences method.7. Finite elements method.Completing this classification, Ismail et al. [125] compare the results obtained with four differentnumerical methods, that is, the continuous solid phase models, SchumannÕs model, thesingle phase models and the thermal diffusion models or models with thermal gradient insidethe particles. The authors evaluate the models in relation to the computational time consumedto solve a specific test problem and then compare them in relation to the influence to differentfactors. Also Ismail et al. in another publication [126] divide the numerical methods for the
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