Carlos A. Dorao

A least squares method for the solution of population balance problems

A general framework is developed, using the least squares method (LSM), for the solution of a generalized population balance equation. The basic idea in the LSM is to minimize the integral of the square of the residual over the computational domain. The capability of the method for solving the PBE is evaluated by using case problems involving coalescence and breakage kernels having analytical solutions which allow the analysis of the method to be performed in a general way. By using the LSM to solve the PBE, the error in the properties of the distribution function depends on the order of the expansion, thus avoiding the introduction of heuristic rules to obtain sufficient accuracy in the values of a few of the physical moments. An interesting characteristic of the LSM applied to PBE is that a low number of equations are required to solve the problem.

Analysis of dynamic surfactant mass transfer and its relationship to the transient stabilization of coalescing liquid–liquid dispersions

In this work, experiments describing the behavior of the separation of a model liquid-liquid dispersion with various concentrations of a synthetic surfactant are presented which indicate that there is a dynamic stabilization of initially unstable emulsions that occurs when the initial surfactant concentration approaches the concentration that provides stable emulsions. A simple model is presented to suggest the mechanism for the dynamic stabilization of these emulsion systems that considers the redistribution of surfactant into the continuous phase after a coalescence event at the emulsion-bulk dispersed phase interface and the dynamic mass transport of surfactant in the continuous phase of the emulsion. The results indicate that coalescence at the interface between the emulsion layer and the bulk dispersed phase creates a local region in the vicinity of this interface where the concentration of the surfactant is much higher than the bulk surfactant concentration which could lead to a locally, dynamically stabilized emulsion at this interface. The extent to which the local excess surfactant concentration reduces the coalescence rate depends strongly on the rate of coalescence at the dense packed layer-bulk dispersed phase interface relative to the rate of surfactant diffusion through the dense packed layer and, of course, on the surfactant adsorption constant, the maximum adsorbed surfactant concentration, and the surface to volume ratio of the dispersed phase. Furthermore, the results indicate that coalescence can also act to significantly increase the local concentration of an initially very dilute surfactant in the vicinity of the interface between the emulsion layer and the bulk dispersed phase interface.

A least-squares method with direct minimization for the solution of the breakage-coalescence population balance equation

A least-squares method with a direct minimization algorithm is introduced to solve the non-linear population balance equation that consists of both breakage and coalescence terms. The least-squares solver, direct minimization solver together with a finite difference solver are implemented for comparisons. It is shown that the coalescence term introduces a strong non-linear behavior which can affect the robustness of the numerical solvers. In the comparison with the least-squares method, the direct minimization method is proved to be capable of producing equally accurate results, while its formulation is better conditioned. In the case of a non-linear population balance equation system, the direct minimization method converges faster than the standard least-squares method.

Least-Squares Spectral Method for the solution of a fractional advection-dispersion equation

Fractional derivatives provide a general approach for modeling transport phenomena occurring in diverse fields. This article describes a Least Squares Spectral Method for solving advection-dispersion equations using Caputo or Riemann-Liouville fractional derivatives. A Gauss-Lobatto-Jacobi quadrature is implemented to approximate the singularities in the integrands arising from the fractional derivative definition. Exponential convergence rate of the operator is verified when increasing the order of the approximation. Solutions are calculated for fractional-time and fractional-space differential equations. Comparisons with finite difference schemes are included. A significant reduction in storage space is achieved by lowering the resolution requirements in the time coordinate.

A parallel time–space least-squares spectral element solver for incompressible flow problems

Abstract The least-squares spectral element method (LSQ-SEM) is a very promising numerical method which combines the least-squares formulation with the spectral element method. In this article, a parallel space–time implementation of a least-squares spectral element method for incompressible flows is presented. In the space–time formulation, time is included as an extra dimension, providing an efficient framework for handling time dependent independent multidimensional problems. The algebraic solver is based on the Shur complement method. Numerical experiments are performed for showing the capability and computational efficiency of the solver. Besides, the scalability of the code is tested.

Population Balance Model for Batch Gravity Separation of Crude Oil and Water Emulsions. Part II: Comparison to Experimental Crude Oil Separation Data

The mathematical model presented in Part I of this article is compared to experimental data obtained from low-field NMR experiments on a heavy crude oil undergoing gravity separation with two different concentrations of a chemical demulsifier. Experimentally measured parameters are used in the model and include a) the water cut and drop size distribution of the emulsion (obtained directly from the NMR measurements), b) the densities and viscosities of the bulk liquids, and c) the interfacial tension; the results obtained from the model were used to analyze the experimental data in terms of these parameters. The model was formulated based on first-principle physical mechanisms, and thus, not only are the fitting parameters are kept to a minimum, good agreement between experiment and simulation results were obtained. The model reasonably predicts both sets of data for the different demulsifier concentrations with no change to the so-called fitting constants, but by the parameter that represents the magnitude of the interfacial force in the film drainage equations. Such a change in this physical parameter can be reasonably inferred by the action of the increased demulsifier concentration. Both the model and experimental NMR data indicate that the degree of poly-dispersity is a key factor in the rate of coalescence and subsequent rate of separation by sedimentation. This analysis illustrates the link between poly-dispersity and the coupling of coalescence and sedimentation rates; this is crucial for determining how different droplet size fractions will affect the overall efficiency of the separation and how physical properties of the fluids and phase interface can amplify or negate these phenomena. The separation model is a very helpful complement to the NMR technique as the model can output direct comparisons to the NMR data.

Solution of bubble number density with breakage and coalescence in a bubble column by Least-Squares Method

A steady-state model has been built for an air-water bubble column. The bubble number density constitutive equation has been formulated through integrating the bubble transport equation. Proper kernels for bubble breakage and coalescence rate have been chosen. The momentum balance of the gas phase is included in the model which leads to a set of non-linear differential equations. The model has been successfully solved by using the Least-Squares Method (LSM) with high accuracy and fast convergence. The successive iteration has been applied to the linearised equation set. The model shows excellent agreements with experimental data.

Simulation of thermal disturbances with finite wave speeds using a high order method

The classical heat diffusion theory based on the Fouriers model breaks down when considering transient heat flow, for short times, extreme thermal gradients or at low temperatures. The hyperbolic heat conduction equation based on the Cattaneo model for the heat flux incorporates a relaxation mechanism in order to gradually adjust to a change in the temperature gradient. A spectral element method is applied for solving the hyperbolic system treating the heat flux as an independent variable in addition to temperature. The numerical solution is based on the time-space least squares spectral method. Numerical examples are included for discussing the effects of the thermal waves.

Simulation of a natural circulation loop using a least squares hp-adaptive solver

In this work the implementation of a high-order method for the simulation of a natural circulation loop is discussed. An adaptive method is developed in order to improve both accuracy and computational time in the resolution of nonlinear problems. Finally, several numerical simulations are discussed in relation to the development of this kind of high-order adaptive methods for unstable thermo-hydraulic systems.

Numerical Study of Pressure Drop Oscillations in Parallel Channels

Pressure drop oscillations have been modeled and analyzed in the present study. The purpose of the present study is to understand the behavior of the individual channels in a parallel arrangement under different conditions such as balanced and unbalanced heat loads and operation mass flux during pressure drop oscillations. The differential equations have been solved using the least squares method. The cases with two parallel channels with balanced and unbalanced heat loads have been discussed. The modes of oscillations found in both cases are in phase and the unstable region can be divided in two characteristic regions. The so called “Region 1”, where the oscillations themselves take place, and a second region, “Region 2”, dominated by stable maldistributed solutions.Copyright