Transient Transport Equation Research Articles

Dynamical simulation of quantum-well structures

For the design and development of optical semiconductor devices based on quantum-well structures, the investigation of saturation phenomena is necessary for high optical power operation. By applying stationary physical models, nonlinear effects cannot be described adequately; hence, transient models are important for an accurate analysis. By utilizing transient models, saturation phenomena, signal delays, and distortions can be investigated. For the analysis of integrated optoelectronic devices, such as lasers and modulators, transient transport or density matrix equations for carriers and photons and the Poisson equation have to be solved self-consistently. A transient model which is useful for the investigation of a wide range of optoelectronic applications is presented. Quantum optical phenomena are included by applying the interband density matrix formalism in real-space representation, where the Coulomb singularity is treated exactly in the limits of the discretization. As we focus on electroabsorption modulators, a drift-diffusion model adequately approximates the transport properties. Here, quantum effects are considered by a quantum correction, the Bohm potential. The model is applied to investigate transport effects in InP-based waveguide electroabsorption modulators including strained lattices.

Numerical Study of Convection in the Directional Solidification of a Binary Alloy Driven by the Combined Action of Buoyancy, Surface Tension, and Electromagnetic Forces

Directional solidification of a dilute electrically conducting binary alloy driven by the combined action of buoyancy, surface-tension, and electromagnetic forces is considered. A numerical methodology using a moving finite element technique is proposed for the simulation of the above phase change process. The melt is modeled as a Boussinesq fluid and the transient Navier–Stokes equations are solved simultaneously with the transient heat and solute transport equations. The location of the advancing solid–liquid interface is numerically determined using an energy preserving weak form of the Stefan condition. The standard SUPG/PSPG method for the simulation of incompressible fluid flow is here extended to flows driven by the combination of buoyancy, surface tension, and electromagnetic forces. A reference problem of directional solidification of a dilute germanium alloy in a horizontal open-boat configuration is considered. The relative influence of thermocapillary convection and buoyancy-driven convection on the solidification process is investigated by varying the Bond number. Thermocapillary convection is shown to have a significant influence on various solidification parameters, such as the shape of the solid–liquid interface and the solute segregation, especially under low gravity conditions. The influence of an external magnetic field on the reference solidification problem is investigated both in a normal and a reduced gravity environment. It is demonstrated that the application of an appropriate strong magnetic field significantly damps the melt flow and improves the solute segregation pattern. The relative influence of an external magnetic field on the solidification process is also studied by independently varying the orientation and magnitude of the applied magnetic field.

Theoretical derivation of the interaction effects with an eccentric cell model and void fraction propagation in two-phase flow

This paper was concerned with the dilute dispersion of gas bubbles in a continuous liquid phase. The starting point in the present study was the three-dimensional, transient averaged transport equations for inviscid incompressible and isothermal two-phase flow. For the evaluation of the averaged description, potential flow around bubbles was considered in order to get closure relationships, using an eccentric cell model approach. The resulting averaged equations are novel in the literature on two-phase flow. The closure averaged model was used to describe the features of the propagation of void fraction waves, and to investigate the linear stability of bubble flow regime. The hyperbolicity of the averaged wave equation was examined and demonstrated that the model is stable when the void fraction is smaller or equal to 2/7.

Two-fluid simulation on the mixed convection flow pattern in a nonisothermal water model of continuous casting tundish

A transient two-fluid model is applied to simulate fluid flow and heat transfer in a nonisothermal water model of continuous casting (CC) tundish. The original liquid in the bath is defined as the first fluid, and the inlet stream, with the temperature variation, is defined as the second fluid. The flow patterns and heat transfer are predicted by solving the three-dimensional (3-D) transient transport equations for each fluid. The results predicted by the two-fluid model make the effect of natural convection more clear compared with the generally used single fluid model k-ɛ turbulence model.

A Fourier-Laplace transform finite element method (FLTFEM) for the analysis of contaminant transport in porous media

In many practical situations, a simplified two-dimensional model is often sufficiently accurate for practical purposes, however on some occasions it is still necessary to consider the full three-dimensional situation. Conventional numerical techniques such as the finite difference and finite element methods are able to deal with contaminant transport under 3D conditions, however the computational effort involved in using them is often very large. Under some circumstances it is possible to perform 3D analyses more efficiently. A particular case is when the geometry is 2D but the problem is actually 3D. In such cases the analysis can be simplified by application of a Fourier transform. In this paper, the dimensions of the contaminant transport problem will be reduced by one and the transient mass transport equation will be transformed into a time-independent mass transport equation through the use of a combination of integral transforms (Laplace and Fourier transforms) and a finite element technique. The mathematical model developed here is called Fourier–Laplace transform finite element method (FLTFEM). By using the FLTFEM a three-dimensional transient problem can be analysed as a two-dimensional time-independent problem and the computational cost will thus greatly be reduced. Copyright © 1999 John Wiley & Sons, Ltd.

Nonstandard methods for the convective-dispersive transport equation with nonlinear reactions

A new nonstandard Eulerian{Lagrangian method is constructed for the one-dimensional, transient convective-dispersive transport equation with nonlinear reaction terms. An exact difference scheme is applied to the convection-reaction part of the equation to produce a semi-discrete approximation with zero local truncation errors with respect to time. The spatial derivatives involved in the remaining dispersion term are then approximated using standard numerical methods. This approach leads to significant, qualitative improvements in the behavior of the numerical solution. It suppresses the numerical instabilities that arise from the incorrect modeling of derivatives and nonlinear reaction terms. Numerical experiments demonstrate the scheme’s ability to model convection-dominated, reactive transport problems. c 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 617{624, 1999

Nonstandard methods for the convective‐dispersive transport equation with nonlinear reactions

A new nonstandard Eulerian-Lagrangian method is constructed for the one-dimensional, transient convective-dispersive transport equation with nonlinear reaction terms. An “exact” difference scheme is applied to the convection-reaction part of the equation to produce a semi-discrete approximation with zero local truncation errors with respect to time. The spatial derivatives involved in the remaining dispersion term are then approximated using standard numerical methods. This approach leads to significant, qualitative improvements in the behavior of the numerical solution. It suppresses the numerical instabilities that arise from the incorrect modeling of derivatives and nonlinear reaction terms. Numerical experiments demonstrate the scheme's ability to model convection-dominated, reactive transport problems. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 617–624, 1999

An object oriented implementation of a front tracking finite element method for directional solidification processes

This paper focuses on the numerical simulation of phase-change processes using a moving nite element technique. In particular, directional solidi cation and melting processes for pure materials and binary alloys are studied. The melt is modelled as a Boussinesq uid and the transient Navier–Stokes equations are solved simultaneously with the transient heat and mass transport equations as well as the Stefan condition. The various streamline-upwind=Petrov–Galerkin-based FEM simulators developed for the heat, ow and mass transport subproblems are reviewed. The use of classes, virtual functions and smart pointers to represent and link the particular simulators in order to model a phase change process is discussed. The freezing front is modelled using a spline interpolation, while the mesh motion is de ned from the freezing front motion using a trans nite mapping technique. Various twoand three-dimensional numerical tests are analysed and discussed to demonstrate the e ciency of the proposed techniques. Copyright ? 1999 John Wiley & Sons, Ltd.

Nonstandard methods for the convective transport equation with nonlinear reactions

A new nonstandard Lagrangian method is constructed for the one-dimensional, transient convective transport equation with nonlinear reaction terms. An “exact” time-stepping scheme is developed with zero local truncation error with respect to time. The scheme is based on nonlocal treatment of nonlinear reactions, and when applied at each spatial grid point gives the new fully discrete numerical method. This approach leads to solutions free from the numerical instabilities that arise because of incorrect modeling of derivatives and nonlinear reaction terms. Algorithms are developed that preserve the properties of the numerical solution in the case of variable velocity fields by using nonuniform spatial grids. Effects of different interpolation techniques are examined and numerical results are presented to demonstrate the performance of the proposed new method. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 467–485, 1998

Time-resolved analysis of self-heating in power VDMOSFETs using backside laserprobing

A laserprobe technique for characterization of transient self-heating in power devices is proposed and its applicability is demonstrated on VDMOSFETs. The change in the silicon lattice temperature in the device active region under pulsed operation is sensed by an infrared laser beam focused on a single VDMOS cell from the substrate side. The temperature-induced change in the silicon refractive index causes an optical phase change which is detected interferometrically. The transient phase measurements are carried out under various VDMOSFET bias conditions and for heating pulse widths in the μs range. The laser beam modulation is numerically modeled using a solution of the transient heat transport equation and a transmission line approach to model the beam propagation. Calculated optical phase signals show a very good agreement with the experiments. The phase change calculated by using a simple geometric optic approximation was compared with the phase variation obtained by the rigorous transmission line approach. The results show that the geometric optic approach can accurately be used in the analysis of transient self-heating of VDMOSFETs.

Transient 1D transport equation simulated by a mixed Green element formulation

New discrete element equations or coefficients are derived for the transient 1D diffusion–advection or transport equation based on the Green element replication of the differential equation using linear elements. The Green element method (GEM), which solves the singular boundary integral theory (a Fredholm integral equation of the second kind) on a typical element, gives rise to a banded global coefficient matrix which is amenable to efficient matrix solvers. It is herein derived for the transient 1D transport equation with uniform and non-uniform ambient flow conditions and in which first-order decay of the containment is allowed to take place. Because the GEM implements the singular boundary integral theory within each element at a time, the integrations are carried out in exact fashion, thereby making the application of the boundary integral theory more utilitarian. This system of discrete equations, presented herein for the first time, using linear interpolating functions in the spatial dimensions shows promising stable characteristics for advection-dominant transport. Three numerical examples are used to demonstrate the capabilities of the method. The second-order-correct Crank–Nicolson scheme and the modified fully implicit scheme with a difference weighting value of two give superior solutions in all simulated examples. © 1997 John Wiley & Sons, Ltd.

Transient 1D transport equation simulated by a mixed Green element formulation

Transient 1D transport equation simulated by a mixed Green element formulation

Coupled estimation of flow and solute transport parameters

A methodology for solving the coupled linear flow and transport inverse problem is presented. It allows the estimation of aquifer parameters (transmissivity, storativity, recharge, boundary heads and flows, leakage, dispersivity, molecular diffusion, porosity, retardation, linear decay, boundary concentrations), using their prior estimates as well as head and concentration measurements as basic information. Transient flow and transport equations are solved using the finite element method in space and a weighted finite difference scheme in time. Flow and transport domains can be one‐, two‐, or quasi three‐dimensional, applying linear one‐dimensional elements, linear two‐dimensional triangles, or bilinear two‐dimensional rectangles mixed at will. Time regimes can be transient or steady state in any equation (flow or transport). Parameters are estimated on the basis of maximum likelihood theory, which allows us to obtain information about parameters' uncertainty. Minimization of the negative log likelihood is performed using Marquardt's method. The applicability of the method is shown by means of two real examples. They illustrate that the proposed approach can successfully address model selection, parameter uncertainty, and nontrivial identifiability problems. Estimation of time‐dependent sources and the role of prior estimates are also addressed in the examples. While the formulation of the problem leaves room for both stochastic and deterministic treatment of hydraulic parameters, the latter has been adopted in both examples. Limitations of the approach are also apparent.

Natural-convection heat transfer in a nonuniform finite annulus with concentric heat source

In this paper, the mechanisms of natural-convection heat transfer inside a nonuniform finite annulus have been numerically investigated. The system is actually a streamlined, water-filled latex balloon with a coaxial cylindrical heating element. The balloon can be applied as a local hyperthermia treatment device for the removal of undesirable tissue if sufficiently high temperatures and preferably uniform surface heat fluxes can be maintained. A validated control-volume-based method has been employed to solve the coupled transient three-dimensional transport equations for laminar free convection. The effects of heat-source temperature distributions and device orientations on the heat transfer have been studied. Possible design improvements of this device are discussed.

Analysis of passive cooling in a vertical finite channel using a falling liquid film and buoyancy-induced gas-vapor flow

A fundamental study of the coupled effects of a falling, partially vaporizing liquid film and buoyancy-induced gas-vapor flow in a vertical one-sided heated channel is presented for different film Reynolds numbers, liquid film inlet temperatures, wall heat fluxes and channel spacings. The transient (elliptic) gas-phase transport equations are solved numerically in conjunction with the laminar (parabolic) liquid film equations. The validated computer simulation model is used to depict transient developments of streamlines, gas-phase parameters and interfacial quantities as well as steady-state velocity profiles, temperature profiles and averaged Nusselt number distributions. The present analysis may contribute to the solution of passive cooling problems associated with electronic equipment operations, manufacturing processes, modern nuclear power plant designs and numerous environmental phenomena.

A correlation for the optimum collector flow rate in one tank forced circulation solar water heaters

A mathematical model has been developed, which simulates the operation of one tank forced circulation solar water heaters, based on a finite-difference solution of the one-dimensional transient heat transport equation by convection and conduction along the prevailing direction of the storage tank. By applying the developed procedure over a wide range of conditions, a large number of predictions has been obtained, which showed that the optimum ratio of collector to mean load mass flow rates ( m ̇ c / ▪ 1) opt , i.e. that which maximizes the performance of the system, varies from 0.4 to 1.4 and can be expressed as a function of the ratio of the mean load mass flow rate to collector area, ▪ 1/A c , only. By correlating the numerous predictions obtained, the function ( m ̇ c / ▪ 1) opt = f( ▪ 1/A c ) is developed and presented. The proposed correlation provides a simple and quick way of estimating, with an accuracy which is satisfactory for practical applications, the optimum collector flow rate in one tank forced circulation solar water heaters, using as the only data the mean load mass flow rate and the collector area.

A two-dimensional integrated device/circuit numerical model for GaAs DCFL gate simulation

A two-dimensional integrated numerical device/circuit model for GaAs/MESFET DCFL gate simulation is presented. It is different from previous publications in that (a) transient carrier transport equations of the FETs are calculated with consideration for the interaction between interconnected FETs in a circuit, (b) the numerical calculations are carried out for the coupled equations to provide accuracy and (c) with the new algorithm used, the numerical calculation can be carried out with high efficiency. A “dynamic method” and a “single gate method” have been developed to simulate the steady state transfer and ring oscillator transient characteristics, respectively, which have proved to be much more efficient than the previously used methods. Based on the model, a gate level two-dimensional numerical simulator, LADES2G, has been developed and successfully applied to GaAs DCFL circuit design.

Determination of Optimal Upstream Weighting Parameter for the Finite Element Solution of Transient Transport Equation

AbstractIn this study the results of a numerical experiment involving the finite element solution of the convection‐dominated transport equation are presented. It is shown that, for large Peclet numbers, the standard Galerkin approach produces unacceptable oscillations. Although the upstream weighting residual method eliminates this problem for steady state problems, its application to transient flow problems has received considerable criticism, because false dispersion and/or the smearing of steep gradients are observed. The numerical experiments performed for transient problems revealed that the upstream weighting residual method can be improved considerably if the upstream weighting parameter is selected as a function of the Courant number. Based on the results of these numerical experiments, a relationship between the optimal weighting parameter and the Courant number is proposed so that oscillations are eliminated and false dispersion is reduced to an acceptable level.

An upstream finite element method for solution of transient transport equation in fractured porous media

A finite element method for the solution of two‐dimensional transient dispersive‐convective transport of nonconservative solute species in fractured porous media is presented. A two‐nodal point one‐dimensional transport element for fractures is developed which provides a number of advantages relative to conventional fracture representation by two‐dimensional continuum elements. To eliminate the oscillatory behavior of convective‐dominated transport which is a more likely occurrence in fracture, a very efficient one‐dimensional upstreaming method along with a two‐dimensional method is implemented. Validity of the numerical scheme is established by comparison with existing one‐ and two‐dimensional analytic solutions.

Solution of transient transport equation using an upstream finite element scheme

An upstream finite element scheme is presented for overcoming the problem of numerical oscillations associated with the transient transport equation for convective dominated or purely convective flow. This scheme differs from the standard Galerkin scheme in that spatial discretization is performed via a general weighted residual technique which employs asymmetric weighting functions and yields upstream weighting of the convective term in the transport equation. The time derivative term of this equation is weighted using the standard basis functions which lead to a consistent mass matrix. The proposed finite element scheme is compared with the conventional upstream finite difference scheme both in one and two dimensions. Numerical results indicate that the new scheme is superior to its finite difference counterpart and is capable of producing reliable solutions for cases which cannot be handled satisfactorily by the standard Galerkin technique.