Fabio Rosso

A New Model for the Dynamics of Dispersions in a Batch Reactor

A new model for coalescence and breakage of liquid-liquid dispersion is presented. The main features are: (i) the introduction of an efficiency factor which controls the time rate of the various processes affecting the size distribution function of droplets, (ii) a new effect — that we call volume scattering — which is consistent with the experimentally observed circumstance of the existence of a top size limit for droplets depending on the general dynamical conditions. The model is proved to be mathematically and physically correct by proving existence and uniqueness of a regular solution to the Cauchy problem.

Instability of Poiseuille flow of two immiscible liquids with different viscosities in a channel

Abstract We study the stability of plane Poiseuille flow of two immiscible liquids of different viscosities and equal densities. The problem is like one considered by C. S. Yih who found that flow in two layers of equal thickness was always unstable. We find regions of stability when there are three layers with one of the fluids centrally located. We view our contribution as a study of selection of stable steady flow from a nonunique continuum of Poiseuille flows all of which satisfy the steady Navier-Stokes and which differ from another in the number and thickness of layers of different viscosity. Experiments have shown that there is a tendency for the less viscous fluid to encapsulate the more viscous one. This arrangement of components, with the more viscous fluid in the center of the channel maximizes the mass flux for a fixed pressure gradient. A linear stability analysis of centrally located configuration to long waves is carried out by the analytic methods introduced by Yih [1]. The stability results depend on the viscosity and volume ratio in a fairly complicated way. The flow with the high viscosity fluid centrally located is always stable. Centrally located layers of less viscous fluid, called fingering flows, are always unstable.

Retrieving the Bingham model from a bi-viscous model: Some explanatory remarks

Abstract In this note we prove some analytical results on the Bingham model. In particular we show how to derive some constitutive and kinematical properties through a limit procedure in which the visco-plastic model is retrieved from a linear bi-viscous model. We also prove that, assuming a no-slip condition at the interface separating the two viscous fluids, no source of entropy can be present on such interface.

On quasi-steady axisymmetric flows of Bingham type with stress-induced degradation

We consider a quasi steady laminar axisymmetric flow of Bingham type under the assumption that the yield stress increases at a rate proportional to the internal power dissipation. Such a behaviour has been observed in coal-water slurries. Many different cases can occur (including the appearance of a rigid shell at the boundary of the pipeline), which are analyzed both theoretically and numerically.ZusammenfassungWir betrachten die quasistationäre, laminare, achsensymmetrische Strömung vom Bingham-Typus unter der Annahme, daß die Fließspannung mit einer zur inneren Dissipationsleistung proportionalen Rate zunimmt. Ein derartiges Verhalten ist in Kohle-Wasser-Gemischen beobachtet worden. Viele unterschiedliche Fälle können auftreten (einschließlich des Auftretens einer starren Schale am Rand der Rohrleitung); sie werden sowohl theoretisch als auch numerisch analysiert.

Stability of a viscous liquid between sliding pipes

A numerical investigation of the eigenvalue problems related to both the energy and the linear theory of stability of a Newtonian liquid between sliding pipes is presented. An enlargement of the region of nonlinear stability is shown as the radii ratio is increased. On the other hand, the flow always turns out to be linearly stable.

A New Model for the Dynamics of Dispersions in a Batch Reactor: Numerical Approach

In this paper, we develop some considerations concerning the structure of coalescence, breakage and volume ‘scattering’ kernels appearing in the evolution equation related to a new model for the dynamics of liquid–liquid dispersions and show some numerical simulations. The mathematical model has been presented in [3, 4], where a proof of the existence and uniqueness for a classical solution to the integro–differential equation describing the physical phenomenon is provided as well as a complete analysis of the general characteristics of the integral kernels. Numerical simulations agree with experimental data and with the expected asymptotical behavior of the solution.

The Constant Flow Rate Problem for Fluids with Increasing Yield Stress in a Pipe

Experiments show that the degradation effect observed during both stirring and pipelining tests of some coal-water slurries is mainly to be ascribed to the increase of the yield stress. Regardless of the particular mathematical model adopted to investigate the dynamics of these fluids, engineering applications force us to consider the problem of how long a constant flow rate can be maintained during the pipelining process. We choose a Bingham model where the yield stress is assumed to increase with the dissipated energy as in [5]. It is first shown that the constant flow rate problem is equivalent to solving a nonlinear functional equation in the unknown pressure gradient that generalizes the classical algebraic Buckingham equation for the same problem with constant rheological parameters. By means of a fixed-point argument we also prove that the functional equation has one and only one solution which is local in time. We finally find an estimate from below of the interval of the interval of existence. Numerical results are rather good and agree with those expected from the engineering point of view.

Sedimentation in Coal-Water Slurry Pipelining

In this chapter we present an overview of recent investigations on the problem of sedimentation related to the pipelining of a coal-water slurry. The two main aspects of the problem are the determination of the sedimentation velocity and the understanding and modeling of the dynamics of the sedimentation bed that accumulates on the bottom of the pipe. The analysis is carried out using a combination of suggestions dictated by experimental evidence and suitable mathematical techniques. The result is a model that appears to be both easily manageable and flexible. Predictions of the model are compared with experiments finding a remarkable agreement with the available data.

Estimation of rate coefficients from the overall amount of material reacted at various scan rates

A method to evaluate the temperature dependence of the reaction rate from the variation of the amount reacted at the end of a number of experiments at various scan rates is deduced from the fundamental equations of DTA. The method was applied to the synthesis of Paakkonenite and gave parameters in good agreement with those determined for the same reaction by using other experimental data and the differential method of Sharp and Wentworth. Furthermore, an empirical procedure previously used to get the same results is compared with the present elaboration. The parameters evaluated are reported and discussed.

Modeling Degradating Dispersions in a Three-Dimensional Finite Container under General Boundary Conditions

This paper deals with a mathematical model widely used to analyze the evolution of a dispersion of bubbles in a liquid. The model takes into account both diffusion and buoyancy in a finite three-dimensional container. Rather general Dirichlet as well as Neumann boundary conditions are allowed. We prove the well-posedness of the initial-boundary value problem. The unique solution is classical and global in time.