Mario Primicerio

Numerical solution of phase-change problems

Abstract A three-time level implicit scheme, which is unconditionally stable and convergent, is employed for the numerical solution of phase-change problems, on the basis of an analytical approach consisting in the approximation of the latent heat effect by a large heat capacity over a small temperature range. Since the temperature dependent coefficients in the resulting parabolic equations are evaluated at the intermediate time level, the complication of solving a set of nonlinear algebraic equations at each time step is avoided. The numerical results thus obtained are satisfactorily compared with the available analytical solutions.

Free boundary problems for nonlinear parabolic equations with nonlinear free boundary conditions

In recent years a number of free boundary problems for parabolic equations have been brought to the attention of mathematicians. They have been suggested as mathematical schemes of several processes (change of phase, chemical reactions, fluid motion in porous media, problems in statistics, biomechanics, continuum mechanics, etc.). We refer to [l-5] and [6, Part r] for a bibliography. It should be noticed that in the quoted literature schemes of general type (i.e., gathering different classes of special problems) are not frequently considered. We recall that Stefan-like problems for semilinear parabolic equations having the heat operator as a principal part and with a rather general free boundary condition were studied in [3] under smoothness assumptions on the data. A detailed theory of a generalization of the classical Stefan problem is developed in [6j, where the free boundary condition is a linear relationship between the “temperature” gradient and the velocity of the free boundary with space and time dependent coefficients, and where the usual assumption on the sign of the data is omitted. Stefan problems with a nonlinear parabolic equation have been studied in the past (see [l-4] for references), but under special assumptions on the data and coefficients. In this paper we shall study a very general class of free boundary problems for parabolic equations in one space dimension, dealing with nonlinearities both in the differential equation and in the free boundary condition. To introduce the problem we shall investigate, let us consider a time interval (0, T) and for each t E (0, T] let us introduce the set Z(t) of the functions U(T) which are continuously differentiable in [0, t), continuous in [0, t], and such that D(T) E (b, , 4) for 7 E (0, t) and o(O) = b, for given b, > b > 6, > 0.

On the estimation of thermophysical properties in nonlinear heat-conduction problems

Abstract The influence of thermophysical property variations on the resulting temperature fields is investigated with reference to quasilinear heat-conduction problems. Both theoretical and experimental evidence is produced to show that errors in the calculation of temperature distributions, brought about by an inaccurate estimate of thermal properties, are small if approximate heat capacities, even exhibiting large local differences with respect to the actual ones in a small range of temperatures, retain enthalpy variations and if the integral across the whole working temperature interval of the absolute value of the difference between approximate and actual thermal conductivities is small. Then the practical relevance of these results is pointed out with reference to freezing and thawing processes of biological materials whose thermal coefficients vary sharply over the phase change zone, where only mean values of thermophysical properties can be easily measured.

On a Problem in the Polymer Industry: Theoretical and Numerical Investigation of Swelling

A recent model for the penetration of solvents into polymers leads to a parabolic free boundary problem with unusual boundary conditions. It is shown that the model equations are well posed, and some qualitative features of the free boundary are established. A numerical method for the free boundary problem is suggested and its convergence is proved. A numerical calculation is included to illustrate the theoretical results.

A two phase Stefan problem with temperature boundary conditions

RiassuntoSi studia un problema di Stefan a due fasi in uno strato piano indefinito, quando sia assegnata la temperatura sui piani che delimitano lo strato stesso.Viene dimostrata l’esistenza (in grande) e l’unicità della soluzione sotto ipotesi assai generali sui dati iniziali ed al contorno. Sî prova la dipendenza continua e monotona della soluzione dai dati iniziali ed al contorno.AbstractWe studied a two phase Stefan problem in a infinite plane slab, when the temperatures are prescribed on the two limiting planes.We proved global existence and uniqueness of the solution under minimal smoothness assumptions upon the initial and boundary data. Furthermore, we demonstrated the continuous and monotone dependence of the solution on the initial and boundary data.

BIOMASS GROWTH IN UNSATURATED POROUS MEDIA: HYDRAULIC PROPERTIES CHANGES

We present a model to describe the biomass growth process taking place in an unsaturated porous medium during a bioremediation process. We focus on the so-called column experiment. At the initial time biomass and polluted water is inoculated in the column. The subsequent changes of hydraulic properties are analyzed. We also show some preliminary simulations.

Remarks on the one-phase Stefan problem for the heat equation with the flux prescribed on the fixed boundary☆

where T is an arbitrarily fixed positive number. As is well known, the problem (l.l)-(1.5) is a mathematical description for the unidimensional heat conduction in a plane infinite slab of homogeneous thermally isotropic material with a phase occurring at one limiting plane and the thermal flux prescribed on the other. For sake of simplicity, in writing down (1. l)-( 1.5) we choose a system of variables such that the thermal coefficients (conductivity, heat capacity, density, latent heat) disappear.

EQUILIBRIUM OF TWO POPULATIONS SUBJECT TO CHEMOTAXIS

We consider a system of four partial differential equations modelling the dynamics of two populations interacting via chemical agents. Classes of nontrivial equilibrium solutions are studied and a rescaled total biomass is shown to play the role of a bifurcation parameter.

A two phase Stefan problem with flux boundary conditions

RiassuntoSi studia un problema di Stefan a due fasi in uno strato piano indefinito quando si suppongono assegnati i flussi termici sui piani che delimitano lo strato stesso.Viene dimostrata l’esistenza e l’unicità della soluzione con ipotesi assai generali sui dati iniziali ed al contorno del problema, nonchè la dipendenza continua e monotona della soluzione da tali dati.Si esaminano infine i casi in cui una delle due fasi può sparire ed il comportamento asintotico in caso di permanenza delle due fasi.AbstractWe studied a two phase Stefan problem in a infinite plane slab, when the thermal fluxes are assigned on the two limiting planes.We proved existence and uniqueness of the solution upon minimal smoothness assumptions upon the initial and boundary data, and we demonstrated the continuous and monotone dependence of the solution on the data.In sec. 5 we studied in which cases one of the two phases disappears and the asymptotic behavior in the cases in which the two phases exist for all time.

Determining calcium carbonate neutralization kinetics from experimental laboratory data

In the framework of a research aimed at estimating the performance and lifetime of porous filters filled with marble powder and used to neutralize acid waters, we propose a mathematical model for determining the calcium carbonate reaction kinetics from some experimental data. In particular we show how to determine the order of the reaction and the reaction rate when calcium carbonate is immersed in a HCl solution. These parameters are evaluated by means of a fitting procedure based on least square methods. The experiments are performed using CaCO3 in the form of a slab and powder and measuring (by means of BET analysis) the specific reaction surface.