The classical two-phase Stefan problem. The case of a two-phase initial state with continuous agreement of boundary and initial conditions

2016 ◽  
pp. 141-189
Keyword(s):  
Stefan Problem ◽  
Initial Conditions ◽  
Initial State ◽  
Two Phase

The two-phase Stefan problem for parabolic equations

2019 ◽  
Vol 2019 (4) ◽  
pp. 54-64
Author(s):  
A.N. Elmurodov
Keyword(s):  
Stefan Problem ◽  
Parabolic Equations ◽  
Two Phase

scholarly journals Handling Initial Conditions in Vector Fitting for Real Time Modeling of Power System Dynamics

Energies ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2471
Author(s):  
Tommaso Bradde ◽  
Samuel Chevalier ◽  
Marco De Stefano ◽  
Stefano Grivet-Talocia ◽  
Luca Daniel
Keyword(s):  
Dynamical Systems ◽  
Power System ◽  
System Dynamics ◽  
Real Time ◽  
Initial Conditions ◽  
Test System ◽  
Initial State ◽  
Power System Dynamics ◽  
Vector Fitting ◽  
Time Modeling

This paper develops a predictive modeling algorithm, denoted as Real-Time Vector Fitting (RTVF), which is capable of approximating the real-time linearized dynamics of multi-input multi-output (MIMO) dynamical systems via rational transfer function matrices. Based on a generalization of the well-known Time-Domain Vector Fitting (TDVF) algorithm, RTVF is suitable for online modeling of dynamical systems which experience both initial-state decay contributions in the measured output signals and concurrently active input signals. These adaptations were specifically contrived to meet the needs currently present in the electrical power systems community, where real-time modeling of low frequency power system dynamics is becoming an increasingly coveted tool by power system operators. After introducing and validating the RTVF scheme on synthetic test cases, this paper presents a series of numerical tests on high-order closed-loop generator systems in the IEEE 39-bus test system.

scholarly journals CFD Simulation of Solid Suspension for a Liquid–Solid Industrial Stirred Reactor

2021 ◽  
Vol 11 (12) ◽  
pp. 5705
Author(s):  
Adrian Stuparu ◽  
Romeo Susan-Resiga ◽  
Alin Bosioc
Keyword(s):  
Chemical Reaction ◽  
Cfd Simulation ◽  
Simulation Analysis ◽  
Solid Phase ◽  
Initial Conditions ◽  
Chemical Reactor ◽  
Liquid Mixtures ◽  
Multiphase Model ◽  
Chemical Product ◽  
Two Phase

The present study examines the possibility of using an industrial stirred chemical reactor, originally employed for liquid–liquid mixtures, for operating with two-phase liquid–solid suspensions. It is critical when obtaining a high-quality chemical product that the solid phase remains suspended in the liquid phase long enough that the chemical reaction takes place. The impeller was designed for the preparation of a chemical product with a prescribed composition. The present study aims at finding, using a numerical simulation analysis, if the performance of the original impeller is suitable for obtaining a new chemical product with a different composition. The Eulerian multiphase model was employed along with the renormalization (RNG) k-ε turbulence model to simulate liquid–solid flow with a free surface in a stirred tank. A sliding-mesh approach was used to model the impeller rotation with the commercial CFD code, FLUENT. The results obtained underline that 25% to 40% of the solid phase is sedimented on the lower part of the reactor, depending on the initial conditions. It results that the impeller does not perform as needed; hence, the suspension time of the solid phase is not long enough for the chemical reaction to be properly completed.

scholarly journals On the Two-phase Fractional Stefan Problem

2020 ◽  
Vol 20 (2) ◽  
pp. 437-458 ◽  
Author(s):  
Félix del Teso ◽  
Jørgen Endal ◽  
Juan Luis Vázquez
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Free Boundary Problems ◽  
Classical Problem ◽  
Nonlocal Diffusion ◽  
Two Phase ◽  
Boundary Problems ◽  
Numerical Studies ◽  
The One ◽  
Evolution Type

AbstractThe classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest to us. Then we introduce the fractional Stefan problem and develop the basic theory. After that we center our attention on selfsimilar solutions, their properties and consequences. We first discuss the results of the one-phase fractional Stefan problem, which have recently been studied by the authors. Finally, we address the theory of the two-phase fractional Stefan problem, which contains the main original contributions of this paper. Rigorous numerical studies support our results and claims.

Regularization of a two-dimensional two-phase inverse Stefan problem

1997 ◽  
Vol 13 (3) ◽  
pp. 607-619 ◽  
Author(s):  
D D Ang ◽  
A Pham Ngoc Dinh ◽  
D N Thanh
Keyword(s):  
Stefan Problem ◽  
Two Dimensional ◽  
Two Phase

Entrainment of Air at the Transoms of Full-Scale Surface Ships

2015 ◽  
Vol 59 (01) ◽  
pp. 49-65
Author(s):  
Eric J. Terrill ◽  
Genevieve R.L. Taylor
Keyword(s):  
Initial Conditions ◽  
Air Entrainment ◽  
Full Scale ◽  
Far Field ◽  
Two Phase ◽  
Cfd Models ◽  
Surface Ship ◽  
Full Speed ◽  
Bubbly Wake ◽  
Scale Surface

We report on the results from a series of full-scale trials designed to quantify the air entrainment at the stern of an underway vessel. While an extremely complex region to model air entrainment due to the confluence of the breaking transom wave, bubbles from the bow, turbulence from the hull boundary layer, and bubbles and turbulence from propellers, the region is a desirable area to characterize and understand because it serves as the initial conditions of a ship's far-field bubbly wake. Experiments were conducted in 2003 from R/V Revelle and 2004 from R/VAthena II using a custombuilt conductivity probe vertical array that could be deployed at the blunt transom of a full-scale surface ship to measure the void fraction field. The system was designed to be rugged enough to withstand the full speed range of the vessels. From the raw timeseries data, the entrainment of air at speeds ranging from 2.1 to 7.2 m/s is computed at various depths and beam locations. The data represent the first such in-situ measurements from a full-scale vessel and can be used to validate two-phase ship hydrodynamic CFD codes and initialize far-field, bubbly wake CFD models.

Numerical Solution of the Two-Phase Stefan Problem in the Enthalpy Formulation with Smoothing the Coefficients

2021 ◽  
pp. 4-23
Author(s):  
V.I. Vasilyev ◽  
M.V. Vasilyeva ◽  
S.P. Stepanov ◽  
N.I. Sidnyaev ◽  
O.I. Matveeva ◽  
...  
Keyword(s):  
Numerical Solution ◽  
Stefan Problem ◽  
Large Diameter ◽  
Discontinuous Coefficients ◽  
Numerical Calculations ◽  
Smooth Functions ◽  
Two Phase ◽  
Time Layer ◽  
Dirac Delta ◽  
Selection Of

To simulate heat transfer processes with phase transitions, the classical enthalpy model of Stefan is used, accompanied by phase transformations of the medium with absorption and release of latent heat of a change in the state of aggregation. The paper introduces a solution to the two-phase Stefan problem for a one-dimensional quasilinear second-order parabolic equation with discontinuous coefficients. A method for smearing the Dirac delta function using the smoothing of discontinuous coefficients by smooth functions is proposed. The method is based on the use of the integral of errors and the Gaussian normal distribution with an automated selection of the value of the interval of their smoothing by the desired function (temperature). The discontinuous coefficients are replaced by bounded smooth temperature functions. For the numerical solution, the finite difference method and the finite element method with an automated selection of the smearing and smoothing parameters for the coefficients at each time layer are used. The results of numerical calculations are compared with the solution of Stefan’s two-phase self-similar problem --- with a mathematical model of the formation of the ice cover of the reservoir. Numerical simulation of the thawing effect of installing additional piles on the existing pile field is carried out. The temperature on the day surface of the base of the structure is set with account for the amplitude of air temperature fluctuations, taken from the data of the Yakutsk meteorological station. The study presents the results of numerical calculations for concrete piles installed in the summer in large-diameter drilled wells using cement-sand mortars with a temperature of 25 °С. The distributions of soil temperature are obtained for different points in time

A Combined Eulerian and Lagrangian Method for Prediction of Evaporating Sprays

2002 ◽  
Vol 124 (3) ◽  
pp. 481-488 ◽  
Author(s):  
M. Burger ◽  
G. Klose ◽  
G. Rottenkolber ◽  
R. Schmehl ◽  
D. Giebert ◽  
...  
Keyword(s):  
Two Phase Flow ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Lagrangian Method ◽  
Phase Flow ◽  
Two Phase Flows ◽  
Two Phase ◽  
Lagrangian Methods ◽  
Eulerian Method ◽  
Lagrangian Modeling

Polydisperse sprays in complex three-dimensional flow systems are important in many technical applications. Numerical descriptions of sprays are used to achieve a fast and accurate prediction of complex two-phase flows. The Eulerian and Lagrangian methods are two essentially different approaches for the modeling of disperse two-phase flows. Both methods have been implemented into the same computational fluid dynamics package which is based on a three-dimensional body-fitted finite volume method. Considering sprays represented by a small number of droplet starting conditions, the Eulerian method is clearly superior in terms of computational efficiency. However, with respect to complex polydisperse sprays, the Lagrangian technique gives a higher accuracy. In addition, Lagrangian modeling of secondary effects such as spray-wall interaction enhances the physical description of the two-phase flow. Therefore, in the present approach the Eulerian and the Lagrangian methods have been combined in a hybrid method. The Eulerian method is used to determine a preliminary solution of the two-phase flow field. Subsequently, the Lagrangian method is employed to improve the accuracy of the first solution using detailed sets of initial conditions. Consequently, this combined approach improves the overall convergence behavior of the simulation. In the final section, the advantages of each method are discussed when predicting an evaporating spray in an intake manifold of an internal combustion engine.

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