In solidification, one is primarily interested in the movement of the solidliquid interface and not in the details of the internal temperature distribution. This is parallelled in many other processes of practical interest. Because of the linearity of the governing equation, one may readily obtain an integral relationship between the system variables, as expressed by the moment equations. By further introducing physically reasonable assumptions, one obtains bounding or approximate solutions, some of which are simple but accurate analytic expressions. Approximate analytic solutions to problems which cannot be solved by rigid analysis are attractive for engineering purposes because they are easily incorporated in feasibility and optimization studies on complex processes. If two such solutions can be found which not only approximate but also bound the unknown true result, then a numerical or experimental verification becomes redundant as long as the relative difference between the two bounds is sufficiently small. The approach thus provides a built-in error criterion.When trying a systematic attack on certain classes of such problems, one immediately encounters a difficulty. Tight and hence useful bounds often require specific assumptions concerning the physical rather than the mathematical situation. In consequence, the results will not always be of the desired generality. For this reason, the present study deals only with linear diffusion types of problems containing a nonlinear boundary condition.More specifically, this paper considers the general problem of unidirectional solidification subject to a known heat flux from the melt and a boundary condition of the third kind (sometimes termed Newton's cooling) at the cold slab surface. Simplified versions of this problem have received continuing attention in the literature, but to our knowledge the above problem has only been dealt with by a few approximate methods, most of which require numerical evaluation and are difficult to assess in their accuracy. Our aim is to generate accurate analytic bounds for the general problem from which the simpler problems treated previously can be extracted as limiting cases.
CONCLUSIONS AND SIGNIFICANCEUsing various types of moments of the governing diffusion equation, one obtains integral relations which can be integrated analytically once suitable approximations to the integrands have been introduced. However, instead of mere approximations, we insert rigid inequalities so that after integration we have bounding solutions for the solidification time as a function of the solid thickness and the system parameters. These inequalities are suggested by physical reasoning, and it is obvious that the tighter the inequality the better will be the bound but the more involved may be the final integration. The limit is reached when we have to resort to numerical integration of the bounding equation.By comparison with previous numerical results, it is shown that analytic results can be generated which remain sufficiently accurate up to quit...
