Stefan problem

Conduction with Change of Phase A special type of transient problem (the Stefan problem) involves conduction of heat in a material when freezing or melting occurs. The liquid-solid interface moves with time, and in addition to conduction, latent heat is either generated or absorbed at the interface. Various problems of this type are discussed by Bankoff [in Drew et al. (eds.). Advances in Chemical Engineering, vol. 5, Academic, New York, 1964]. 

H. Kopetsch. A numerical method for the time-dependent Stefan-problem in Czochralski crystal growth. J Crystal Growth SS 71, 1988 H. Kopetsch. Phy-sico-Chem Hydrodyn 77 357, 1989 H. Kopetsch. J Cryst Growth 102 500, 1990. 

In Madejski s full model,l401 solidification of melt droplets is formulated using the solution of analogous Stefan problem. Assuming a disk shape for both liquid and solid layers, the flattening ratio is derived from the numerical results of the solidification model for large Reynolds and Weber numbers ... 

To use these models, the freezing constant, U, must be determined. One choice is the solution of the Stefan problem of solidification, as described by Madejski 401 ... 

To rationalize the isothermal assumption, Dykhuizen 39() discussed two related physical phenomena. First, heat may be drawn out of the substrate from an area that is much larger than that covered by asplat. Thus, the 1 -D assumption in the Stefan problem becomes invalid, and a solution of multidimensional heat conduction may make the interface between a splat and substrate closer to isothermal. Second, the contact resistance at the interface is deemed to be the largest thermal resistance retarding heat removal from the splat. If this resistance does not vary much with substrate material, splat solidification should be independent of substrate thermal properties. Either of the phenomena would result in a heat-transfer rate that is less dependent on the substrate properties, but not as high as that calculated by Madej ski based on the... 

If the boundary motion is controlled by an independent process, then the boundary motion velocity is independent of diffusion. This can happen if the magma is gradually cooling and crystal growth rate is controlled both by temperature change and mass diffusion. This problem does not have a name. In this case, u depends on time or may be constant. If the dependence of u on time is known, the problem can also be solved. The Stefan problem and the constant-w problem are covered below. 

If crystal growth or dissolution or melting is controlled by diffusion or heat conduction, then the rate would be inversely proportional to square root of time (Stefan problem). It is necessary to solve the appropriate diffusion or heat conduction equation to obtain both the concentration profile and the crystal growth or dissolution or melting rate. Below is a summary of how to treat the problems more details can be found in Section 4.2. 

Mathematically, diffusive crystal dissolution is a moving boundary problem, or specifically a Stefan problem. It was treated briefly in Section 3.5.5.1. During crystal dissolution, the melt grows. Hence, there are melt growth distance and also crystal dissolution distance. The two distances differ because the density of the melt differs from that of the crystal. For example, if crystal density is 1.2 times melt density, dissolution of 1 fim of the crystal would lead to growth of 1.2 fim of the melt. Hence, AXc = (pmeit/pcryst) where Ax is the dissolution distance of the crystal and Ax is the growth distance of the melt. 

The corresponding limiting problem is an instance of the one-phase Stefan problem of the form... 

L.I. Rubensstem. The Stefan Problem. Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1971. Translated from Russian by A.D. Solomon. 

This particular problem was first studied by Stefan (S10), and the general class of solutions of the diffusion equation subject to a free boundary condition, therefore, are sometimes called Stefan problems. An analogous problem in the freezing of moist soils was previously studied, however, by Lame and Clapeyron (LI), and in the Russian literature these problems are sometimes given the rather lengthy soubriquet of Larnd-Clapeyron-Stefan problems. 

Friazinov (F4) deals with a generalized Stefan problem involving finite depth of the two-phase layer, densities and thermal conductivities which are functions of position, and arbitrary initial and boundary conditions, by an approximate expansion in terms of appropriate Sturm-Liouville eigenfunctions. 

Models of Diffusion during LPE. An exact formulation of the LPE diffusion problem constitutes a nontrivial Stefan problem (80). An analytical first-order model capable of representing long-time experimental thickness data to within experimental error considers the following one-dimensional conservation equation ... 

Classical Stefan problems New mathematics required Cf. crystallization ancient glasses... 

For k —> oo the temperature at x = 0 has the value i) i) ]. This is the boundary condition in the Stefan problem discussed in 2.3.6.1. From (2.218) we obtain the first term of the exact solution (2.213), corresponding to Ph — oo, and therefore the time t according to (2.214). With finite heat transfer resistance (1 /k) the solidification-time is greater than i it no longer increases proportionally to s2. 

As a comparison with the exact solution of the Stefan problem shows, the quasisteady approximation discussed in the last section only holds for sufficiently large values of the phase transition number, around Ph > 7. There are no exact solutions for solidification problems with finite overall heat transfer resistances to the cooling liquid or for problems involving cylindrical or spherical geometry, and therefore we have to rely on the quasi-steady approximation. An improvement to this approach in which the heat stored in the solidified layer is at least approximately considered, is desired and was given in different investigations. 

Yih-o Tu, "Miltiphase Stefan Problem Describing the Swelling and Dissolution of Glassy Polymer", IBM Symposium on Mathematics and Computation, October 6-7, 1976. Yorktown Heights, N.Y. 

Equation (2.3-47), the pseudo steady-slate solution for the flux, could be used to predict the diflusion-conuojled growth or dissolution rate of the crystal in a manner analogous to the Stefan problem solution. The result would indicate that the sqnene of the particle sedius varies linearly with time. 

M. Davis, P. Kapadia, and J. Dowden, Solution of a Stefan Problem in the Theory of Laser Welding by Method of Lines, J. Comp. Phys, 60, pp. 534-548,1985. 

Laser heating and laser-induced phase transformations were simulated on the basis of numerical solution of the one-dimensional Stefan problem. 

In the frame of the vapor-hquid-sohd mechanism of a nanowhiskers growth the nucleation process in a catalytic droplet is considered. The expression for nucleation rate is obtained. Based on the Stefan problem mathematical model of nanowhiskers growth is derived. Results of simulation are presented. For nanowhiskers growth crucial role of substrate temperature is shown. 

It can be shown that we can neglect the heat transfer between a droplet and the gas phase in compare with the heat transfer through the nanowhisker. The equation for the nanowhisker growth derived from the Stefan problem is ... 

The model proposed by Tu and Ouano [43] for polymer dissolution assumes Fickian solvent penetration into the polymer. The polymer dissolution problem was modeled as a multi-phase Stefan problem [44], The key parameter in this model was the disassociation rate, R, which was defined as the rate at which the polymer transformed from a gel-like phase to a solution. It was proposed that the dissolution process was disassociation -controlled if the polymer diffusion rate in a liquid layer adjacent to the solvent-polymer interface was faster than the disassociation rate, or diffusion -controlled if the diffusion rate was slower than the disassociation rate. 

Equations (6) and (7) were solved with two sets of boundary conditions. The first set was source limited , i.e., disassociation rate-controlled and the second was flux limited , i.e., the concentration at the interface S was equal to an equilibrium value. The functions fi and f2 were assumed to be unity, Le., concentration-independent diffusion coefficients were used. The multi-phase Stefan problem was solved numerically [44] using a Crank-Nicholson scheme and the predictions were compared to experimental data for PS dissolution in MEK [45]. Critical angle illumination microscopy was used to measure the positions of the moving boundaries as a function of time and reasonably good agreement was obtained between the data and the model predictions (Fig. 4). 

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