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This paper concerns the mathematical modelling and numerical solution of thermoelectrical phenomena taking place in an axisymmetric induction heating furnace. We formulate the problem in a two-dimensional domain and propose a finite element method and an iterative algorithm for its numerical solution. We also provide a family of one-dimensional analytical solutions which are used to test the two-dimensional code and to predict the behaviour of the furnace under special conditions. Some numerical results for an industrial furnace used in silicon purification are shown.

Secondly, it is necessary to give a precise mathematical definition of the termappearing in Equation (19), because it does not make sense for any G ∈ Y. To attain this goal, for k = 1, . . . , m, let us denote by k the solution in H 1 ( k \ k ), unique up to a constant, of the following problem:). Since the current density J = E satisfies div J = 0 in and J · n = 0 on , we havewhere k denotes the boundary of k . Note that, while the right-hand side of the previous equality does not make sense for any G in Y, the left-hand side does.

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We present a coupled system of elliptic equations describing the steady state of the thermoelectrical behaviour of an aluminium electrolytic cell. The thermal model is a free boundary problem which consists of the heat equation with Joule heating as a source. We neglect the Joule heating in the ledge, and allow for temperature-dependent electrical conductivity. We also formulate a numerical approximation using a finite element method. An iterative algorithm and numerical results are presented. The existence of a weak solution is also proved.

Counterion competitive complexation is a background process currently ignored by using ionic hosts. Consequently, guest binding constants are strongly affected by the design of the titration experiments in such a way that the results are dependent on the guest concentration and on the presence of added salts, usually buffers. In the present manuscript we show that these experimental difficulties can be overcome by just considering the counterion competitive complexation. Moreover a single titration allows us to obtain not only the true binding constants but also the stoichiometry of the complex showing the formation of 1 : 1 : 1 (host : guest : counterion) complexes. The detection of high stoichiometry complexes is not restricted to a single titration experiment but also to a displacement assay where both competitive and competitive-cooperative complexation models are taken into consideration.