Solidification model

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 ... 

Madejski s solidification model did not account for partial solidification of a droplet prior to impact. San Marchi et al.[157]... 

Modiiying the Scheil Solidification Model 11.3.3.3 More Explicit Methods of Accounting for 447... 

IL3.3.I Using the Scheil solidification model Although it is realised that some back-diffusion will occur, results on the calculation of solidification using the Scheil simulation have proved to be successful in a number of cases. For Al-alloys it appears to be particularly successful, allowing not only very accurate predictions for fiaction solid transformed (/,) as a function of temperature but also for predicting the phases which appear during solidification, which in Al-alloys, is a complex phenomenon. 

II.3.3.2 Modifying the Scheil solidification model The Scheil equation can be modified to allow some back diffusion into the solid during a small isothermal increment of solidification. The composition of the liquid is then modified to a new... 

For a semi-crystalline polymer the solidification model of Fischer [11] predicts that the chain dimensions are frozen in at Tm if crystallization occurs sufficiently fast so as to prevent the disentanglement of chains by unravelling and/or their segregation according to their mass. For a series of polyethylene samples Michler [12] has accomplished an instructive comparison between the dimensions of molecular coils and the thickness of crystalline lamellae (Fig. 4). 

FIGURE 11.6 Solidification model illustrating crystallization from the melt. (Adapted from Dettenmaier, M. et al.. Colloid Polym. Sci., 228, 343,1980. With permission from Dr Dietrich Steinkopff Verlag, Darmstadt.)... 

Fig. 5.7 The solidification model of the crystallisation process, showing how a chain can be incorporated into a lamellar structure without significant change of overall shape. (Reproduced by permission of lUPAC.)...

The findings above led to two different models. In 1980 Dettenmaier et al. (128) proposed their solidification model, whereby it was assumed that crystallization occurred by a straightening out of short coil sequences without a long-range diffusion process. Thus these sequences of chains crystallized where they stood, following a modified type of switchboard model (53). This was the first model to illustrate how Rg values could remain virtually unchanged during crystallization. 

The blowup ratio, BUR, is the ratio of the final bubble radius to the initial radius and is an analog of the draw ratio in spinning. Primes denote differentiation with respect to the dimensionless axial length, which is normalized with respect to Ro. The differential equations are more complex for viscoelastic liquids, and the stress equations must be solved in parallel with the momentum equations, r = w = 1 at the exit from the die, and r = 0 at the freezeline. The last boundary condition assumes that the viscosity becomes infinite at the freezehne and that there is no further deformation. (This condition is approximate at best and need not be used when a solidification model in which the sohd phase evolves and locks in structure is employed.) Heat transfer is very important, although it has usually been handled with rather simplistic assumptions about the heat transfer coefficient. 

Figure 1.13 Chain conformation (a) in the melt and (b) in the crystal according to the solidification model. (Adapted from Stamm, M., Fischer, E.W., Dettenmaier, M. and Convert, P. (1979) Chain conformation in the crystalline state by means of neutron scattering methods. Discuss. Chem. Soc. (London), 68, 263. Copyright (1979) RSC.)...

Although a number of models have been proposed based on the variable domain method, the single-region method remains the most popular. This is because of its ease of implementation and, unlike the variable domain method it can be readily extended to multidimensional problems. In addition, alloy solidification models can be conveniently developed using the single-region method. 

Studies on solidification modeling have been largely directed towards macroscopic phenomena. A variety of numerical techniques have been used for such modeling studies. Among these are the finite difference method (FDM) with or without the alternate direction implicit (ADI) time-stepping scheme, the FEM, the boundary element method (BEM), the direct finite difference method (DFDM), and the control volume element (VFM) method. 

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