Solution Of Conservation Equations Research Articles

Variability of travel time of solute particles in single fractures with spatially variable aperture

The arrival time of solutes from the source to a prescribed compliance boundary is a global transport quantity that is of practical interest in many applications, such as risk assessment for environmentally sensitive sites in regulatory analyses. Estimates of uncertainty are necessary and can play an important role in the decision-making process. The objective of this work is therefore to quantify the variability of solute travel time in individual fractures with spatially variable apertures. The variability of solute transport in fractures is largely determined by the heterogeneity of the apertures of naturally occurring rock fractures. To model solute transport, the fracture apertures and flow fields are considered as spatial random functions. Using the relationship between the two-dimensional depth-averaged solute conservation equation and the Fokker-Planck equation, it is possible to formulate the solute convection velocity in such a way that a general expression for the solute travel time variance can be developed by taking into account the effects of the variation in fracture aperture. A closed-form expression for the variance of travel time in the mean flow direction is derived for the case of advection-dominated solute transport. This derived expression allows the analysis of the effects of the variation of the fracture aperture on the variability of the solute travel time.

Modeling the Effect of Combined Electromagnetic Stirring Modes on Macrosegregation in Continuous Casting Blooms

Macrosegregation is one of the most frequently observed defects in continuous casting blooms, which causes nonconformity in ultrasonic flaw detection of rolled products. To investigate the influence of combined EMS modes (M-EMS + F-EMS) on macrosegregation, a 3D multiphase solidification model based on the volume-averaged Eulerian approach was established to simulate the electromagnetic field, fluid flow, microstructural evolution, and solute transport of heavy-rail steel blooms subjected to different EMS processes. In this model, a hybrid model of the mushy zone and a back-diffusion model were introduced into the momentum and solute conservation equations to realize the calculation of microstructural evolution and solute transport with electromagnetic stirring. The predicted magnetic induction intensity, macrostructure, and macrosegregation were verified with Tesla meter measurements, etched macrostructure analysis, and infrared carbon-sulfur analysis. The calculation results showed that M-EMS had little effect on the improvement of the positive centerline segregation, whereas F-EMS effectively reduced the positive centerline segregation. Moreover, a combination of these EMS modes could further reduce the positive centerline segregation in continuous casting blooms. The change in solute concentration caused by M-EMS could be inherited by the position of F-EMS, which could enhance the metallurgical effects of F-EMS. These results were also verified through an industrial application.

Growth of homogeneous mixed A1-xBx bulk crystals: Analytical prediction, phase-field modeling and experiment comparison

An analytical model is proposed to predict the growth behavior of homogeneous mixed A1-xBx bulk crystals by the traveling liquidus-zone (TLZ) method. The solidification and melting rates at the cold and hot solid/liquid (S/L) interfaces are determined by the solute conservation equations, coupled with the non-linear concentration distribution in the liquid zone. A novel time-dependent critical translation rate is derived, which can completely suppress the composition segregation in the grown crystal with an arbitrary liquid zone width. The proposed model is validated by the thin interface phase-field model for the TLZ-grown Si0.5Ge0.5 crystal, regarding the growth kinetics and concentration distributions in the crystal and liquid phase. A good agreement between the analytical predictions and phase-field simulations is achieved. The growth kinetics and composition profiles in the grown crystals predicted by the present analytical model agree well with the experimental measurements reported in the literature. It is found that after an initial transient, the non-linear liquid concentration distribution reaches the quasi-stationary condition. The average crystal growth rate increases with increasing temperature gradient. The proposed critical translation rate can precisely match the time-dependent solidification rate at the cold S/L interface, leading to a completely homogeneous TLZ-grown Si0.5Ge0.5 bulk crystal for an arbitrary initial liquid zone width. It is demonstrated that the present analytical model with the non-linear liquid concentration distribution is applicable to predict the growth behavior of homogeneous mixed crystals with a wide region of temperature gradient, translation rate and initial liquid zone width.

The two-domain model of solute transport in binary alloy

A mixed model for micro-macroscopic computer simulation of binary alloy solidification is proposed. It involves a two-domain approach to solute conservation equations in the liquid and solid phases, whereas transport of momentum and energy in the two-phase region is modelled using the phase mixture theory. To distinguish regions of columnar and equiaxed crystal structures evolving in a cast during solidification, the special front tracking technique on non-structural triangular grids is included in the model. In this two-domain approach, solute conservation equations are averaged across solid and liquid phases, and the solute transport at the phase interface is included. Additionally, the microstructure evolution is modelled to capture the development of various complex grain structures and more accurately describe the solute transport between the phases. The accuracy of the proposed model is first verified by a grid refinement analysis, and then the model is used to predict the solute concentration and macro-segregation in the example problem of Pb-48%wt Sn alloy solidification in a 2D mould. The results obtained are next compared with the relevant ones predicted by the fully single-domain model, earlier developed by authors. Thus, the role of finite diffusion in liquid and solid phases is identified and discussed.

Macrosegregation simulation model based on Lattice-Boltzmann method with high computational efficiency

A macrosegregation simulation model is developed by coupling solute and energy conservation equations with Lattice-Boltzmann Method (LBM), newly developing technique of computational fluid dynamics. Effect of the solidification shrinkage is taken into account in the present LBM as well as effects of the Darcy’s flow and thermos-solutal convection. The present LBM-coupled model is based on modified lattice Bhadnager-Gross-Krook method, the numerical stability of which is better than that of the standard LBM. Accordingly, the present LBM-coupled model can be applied to simulations of macrosegregation behaviors in metallic alloy systems that cannot be handled by the previous LBM-coupled model. The validity of the model was demonstrated by comparing the results for steady-state flows with those of analytical solutions and a conventional model based on the Navier-Stokes equation. In addition, the computational speed of the present model is compared with the one of conventional model in cases of lateral directional solidification of Sn-Bi alloy and continuous casting of a steel slab. It is shown that the present LBM-coupled model enables remarkably faster computation than the conventional model especially in the latter case.

Peritectic transformation with non-linear solute distribution in all three phases: Analytical solution, phase-field modeling and experiment comparison

An analytical model is derived to predict the kinetics of a peritectic transformation involving solute diffusion in the L-, δ- and γ-phases with non-zero constant diffusivities. A similarity method is adopted to solve the diffusion equations in the bulk L-, δ-, and γ-phases coupled with the solute conservation equations at the γ/L and γ/δ interfaces. The proposed model is applied to predict the kinetics of the isothermal peritectic transformation of Fe-C alloys for two conditions, zero and non-zero supersaturation, in the δ- and L-phases. An excellent match between the analytical solutions and phase-field simulations is achieved. In the case of zero supersaturation, the time evolution of γ-phase thickness and non-linear concentration distribution in the γ-phase predicted by the present analytical model agree well with the experimental data reported in the literature. When the holding temperature decreases, the parabolic rate constant at the γ/δ interface increases non-linearly, while it remains nearly unchanged at the γ/L interface. Additionally, the non-zero supersaturation in the parent phases increases the diffusion flux jump significantly at the γ/L interface but only slightly at the γ/δ interface. As a result, the growth velocity increases noticeably at the γ/L interface, but it is nearly unchanged at the γ/δ interface.

An asymptotic approach to solidification shrinkage-induced macrosegregation in the continuous casting of binary alloys

• Asymptotic analysis for continuous casting of a binary alloy using regular perturbation theory. • Micro-scale assumption for lever rule solidification feeds through to the leading order in the macroscale solution. • Boundary immobilization technique for determining the extent of the mushy zone and the level of macrosegregation. • Method implemented in Comsol Multiphysics. The modelling of macrosegregation in the continuous casting of alloys normally requires resource-intensive computational fluid dynamics (CFD). By contrast, here we develop an asymptotic framework for the case when macrosegregation is driven by solidification shrinkage; as a first step, a binary alloy is considered. Systematic asymptotic analysis of the steady-state two-dimensional mass, momentum, heat and solute conservation equations in terms of the shrinkage parameter indicates that the overall problem can be reduced to a hierarchy of decoupled problems: a leading-order problem that is non-linear, and a sequence of linear problems, with the actual macrosegregation of the solute then being determined by means of one-dimensional quadrature. A numerical method that solves this sequence is then developed and implemented, and yields realistic macrosegregation profiles at low computational cost.

Prandtl model for concentration polarization and osmotic counter-effects in a 2-D membrane channel

An accurate 2-D numerical model that accounts for concentration polarization and osmotic effects is developed for the cross-flow filtration in a membrane channel. Focused on the coupling between laminar hydrodynamics and mass transfer, the numerical approach solves the solute conservation equation together with the steady Navier-Stokes equations under the Prandtl approximation, which offers a simplified framework to enforce the non-linear coupling between filtration and concentration polarization at the membrane surface. The present approach is first validated thanks to the comparison with classical exact analytical solutions for hydrodynamics and/or mass transfer, as well as with approximated analytical solutions that attempted at coupling the various phenomena. The effects of the main parameters in cross-flow reverse osmosis (RO) or nanofiltration (NF) (feed concentration, axial flow rate, operating pressure and membrane permeability) on streamlines, velocity profile, longitudinal pressure drop, local permeate flux and solute concentration profile are predicted with the present numerical model, and discussed. With the use of data reported from NF and RO experiments, the Prandtl approximation model is shown to accurately correlate both average permeate flux and local solute concentration over a wide range of operating conditions.

A Phase Field Model for Binary Alloy Solidification with Boundary Interface Intersection

A phase field model for binary alloy solidification with boundary interface intersection is developed. In the phase field model, the heat and solute conservation equations are appropriately modified to account for the presence of heat and solute rejection inside the diffuse interface, and a relaxation boundary condition for the phase field variable is introduced to balance the interface energy and boundary surface energy in the multiphase contact region. The thin interface asymptotic analysis is applied on the phase field model to yield the free interface problem with dynamic contact point condition.

A meshfree interface-finite element method for modelling isothermal solutal melting and solidification in binary systems

In this paper, numerical modelling of isothermal solutal melting and solidification in binary systems is done using a new meshfree interface-finite element method (MI-FEM) where the implicitly represented liquid–solid interface is allowed to arbitrarily intersect the finite elements. A meshfree radial basis functions (RBFs) method is used for solving a distance-regularized level set (DRLS) equation such that re-initialization is completely eliminated and fast marching algorithms for interfacial velocity extension are not necessary resulting in a more efficient solution with excellent volume conservation. In the proposed method, intersection points between the mesh and the zero level set are used as meshfree nodes such that at the interface-embedded elements interpolants are constructed using meshfree RBFs ensuring both the partition of unity and Kronecker-delta properties are satisfied allowing for precise and easy imposition of Dirichlet boundary conditions (DBCs) on each side of the interface. A coupling of the MI-FEM with a new meshfree automata (MA) method is used to efficiently predict the microstructural evolution during solidification. Benchmark problems with strong discontinuities were solved where very good accuracy was obtained. The solute conservation and interfacial equilibrium equations describing solutal phase transformation in binary systems were solved using the newly developed method. Mathematical formulation and implementation followed by numerical results and analysis will be presented and discussed.

NUMERICAL AND EXPERIMENTAL INVESTIGATIONS OF EGR DISTRIBUTION IN A DI TURBOCHARGED DIESEL ENGINE

This paper presents the results of numerical and experimental investigations to evaluate the distribution of exhaust gas recirculation (EGR) between cylinders in a DI turbocharged diesel engine. The turbulent three-dimensional flow field was analyzed by the numerical solution of conservation equations with an appropriate turbulence model. EGR was applied to intake manifold with various rates at cooled and non-cooled states. The experiments were conducted on an MT4.244 turbocharged DI diesel engine under full load condition at 1900rpm. The model was validated by experimental data with a good agreement between experimental measurements and numerical predictions. Using this method, it is possible to control EGR distribution so as to reduce emissions formation as well as to improve performance.

Advanced Solute Conservation Equations for Dendritic Solidification Processes Part I: Experiments and Theory

AbstractThe macrosegregation formed in dendritic equiaxed structure during early stages of solidification of Al‐4.5%Cu alloy has been studied by experimental work and by metallurgical study of cast samples taken from the experimental work. An experimental work was conducted to study the coupled effect of natural convection streams, interdendritic strain and mushy permeability of Al‐4.5%Cu aluminum alloy solidified in horizontal rectangular parallelepiped cavity at different superheats. The metallurgical study includes macro‐microstructure evaluation, measurements of grain size of equiaxed crystals and macrosegregation analysis. This study shows that the level of surface segregation exhibiting as positive segregation varies with superheat whereas the rest of inner ingot areas show the light fluctuation in segregation values. In addition to experimental work, there is a mathematical study which contains a complete derivation of local solute redistribution equations based on Fleming's approach under different solute diffusion mechanisms in the dendritic solid. This derivation includes also the effects of interdendritic strain and mushy permeability on the local solute redistribution distribution. Owing to the length of the study, it is presented in two parts. The first part describes the experimental work and its results as well as a detail derivation of solute conservation equations. This part also involves comparison and discussion between existing and proposed solute conservation equations. The second part contains the mathematical analyses of a two dimensional mathematical model of fluid flow, heat flow, solidification, interdendritic strain and macrosegregation. Also, this part also contains the numerical simulations by using finite difference technique “FDT” to create convection patterns, heat transfer, interdendritic strain, and macrosegregation distributions. This part also includes comparisons between the available measurements and model predications as well as full discussion of different model simulations. The mechanism of interdendritic strain generation and macrosegregation formation during solidification of dendritic equiaxed structure under different diffusion mechanisms in dendritic solid has also been explained and discussed.

Advanced Solute Conservation Equations for Dendritic Solidification Processes Part II: Numerical Simulations and Comparisons

AbstractThe mathematical model of derived solute equations in part I for equiaxed dendritic solidification with melt convection streams and interdendritic thermo‐metallurgical strain is applied numerically to predict macrosegregation distributions with different diffusing mechanisms in dendritic solid. Numerical and experimental results are present for solidification of a Al–4.5% Cu alloy inside horizontal rectangular cavity at different superheats. The numerical simulations were performed by using simpler method developed by Patanker. The experiments were conducted to measure the cooling curves via thermocouples and the metallurgical examinations to measure the grain size and macrosegregation distributions in Part I. Preliminary validity of the model is demonstrated by the qualitative and quantitative agreements between the measurements and predications of cooling curves and predicted macrosegregation distributions including mushy permeability and interdendritic strain. In addition, several important features of macrosegregation in equiaxed dendritic solidification are identified through this combined experimental and numerical study. Also, quantitative agreements between the numerical simulations and experiments reveal several areas for future research work. The differences and errors between predicted macrosegregation results under different diffusing mechanisms have been discussed.

Effects of the Flow Speed on Dendritic Growth in Phase-Field Simulation of Binary Alloy with Convection

A phase-field approach which incorporates mass and momentum and solute conservation equations for simulation of Al-Si binary alloy solidification is studied. The effect of force flow on the dendrite growth and solute profile during the solidification of binary alloy were investigated. The results indicate that dendritic grows unsymmetrically under a forced flow, the growth velocity of the upstream tip is faster than the downstream tip. With the force flow, the upstream tip grows faster due the thinner solute boundary layer. The solute gradient in the solid/liquid interface regions of the upstream tip is higher than that of the downstream tip. The faster the flow velocity, the greater the solute gradients in the solid/liquid interface regions of the upstream tip, the thinner the diffusion layer before the upstream tip. The downstream tip is opposed to the upstream tip. The simulations agree qualitatively with the solidification theoretical results.

Modeling and simulation of dendrite growth in solidification of Al-Si-Mg ternary alloys

Based on a previously cellular automaton (CA) model for Al-Si binary alloys, a modified cellular automaton (MCA) model is developed to numerically simulate the dendrite growth and microsegregation of Al-Si-Mg ternary alloys. By direct coupling PanEngine, the MCA model can obtain thermodynamic data for dendrite growth of ternary alloys in real time, such as equilibrium liquidus temperature, partition coefficient and liquidus slope of two solute elements. In the model, the normal velocity of the interface cell can not be derived only by solving the solute conservation equation, and must be simultaneously coupled with dimensionless solute supersaturation equation for each alloying element and local equilibrium temperature equation at the S/L interface. All the equations of the MCA model are solved by finite difference method on a uniform mesh. By considering local curvature and constitutional undercooling, solute diffusion and solute redistribution at the S/L interface, the MCA model can simulate the columnar dendrite growth and the solidification path of the equiaxed dendrite growth.

Simulation of dendritic growth for ternary alloys based on modified cellular automaton model

Based on the binary cellular automaton method, a modified cellular automaton model for ternary alloys is developed to simulate dendrite growth controlled by solutal effects and microsegregation in the low Peclet number regime by coupling PanEngine, which is a multicomponent thermodynamic and equilibrium calculation engine. The model can be used to calculate the interfacial equilibrium composition by considering the influence of Gibbs-Thomson effect induced curvature undercooling, and multicomponents contributed constitutional undercooling. Meanwhile, the growth velocity of interface is determined by solving the solute conservation equation simultaneously with dimensionless solute supersaturation equation for each alloying element. Moreover, equilibrium liquidus temperature and equilibrium solid concentration at the interface are derived by PanEngine. Free dendrite growth of Al-7%Si-xMg ternary alloys is simulated by the present model, which shows that the increase of solute Mg can suppress the growths of both primary and secondary dendrite arms. Meanwhile, constrained columnar dendrite growth of Al-7%Si-0.5%Mg with the increases of pulling velocity and constant thermal gradient during directional solidification is calculated. The results reveal the competitive growth of columnar dendrites, and demonstrate that the primary dendrite arm spacing would decrease as the pulling velocity increases, which accords well with the Hunt model.

A fixed-grid method for transient simulations of dopant segregation in VGF-RMF growth

In this work a fixed-grid, virtual-front tracking model originally developed for modeling dendritic growth has been adopted for transient simulations of dopant segregation in vertical gradient freeze (VGF) melt growth of Ga-doped germanium under the influence of a rotating magnetic field (RMF). The interfacial Stefan conditions for temperature and solute are formulated in volumetric terms in energy and solute conservation equations, which allow the interface to be tracked implicitly with no need to calculate the growth velocity. The model and the code are validated against an analytical solution for the transient solidification of a binary alloy at constant velocity. The numerical results show the strong relationship between the melt flow pattern and the dopant concentration in the crystal grown. The better melt mixing during growth under the influence of RMF is found to have a significant impact on the axial and radial macrosegregation of dopants. Simulation results are in good qualitative agreement with previous experimental observations of the dopant segregation in VGF-RMF growth, which now are seen ass a direct consequence of the mixing state of the melt.

Phase-Field Simulation of Double Dendritic Growth in Solidification of Binary Alloy with Forced Convection

A phase-field approach which incorporates mass and momentum and solute conservation equations for simulation of Al-Cu binary alloy solidification is studied. The effect of force convection on the double dendrite growth and solute profile during the solidification of binary alloy were investigated. The results indicate that dendritic grows unsymmetrically under a forced flow, the growth velocity of the upstream tip is faster than the downstream tip. The downstream tip of the first dendrite and the upstream tip of the second dendrite are influenced each other, the upstream tip of the second dendrite will Coarsen, and the concentration at the boundary between them is the highest. Moreover, the interaction between the two dendrites is more and more obvious with the increasing of the flow speed.

3D macrosegregation simulation with anisotropic remeshing

The article presents a three-dimensional coupled numerical solution of momentum, mass, energy and solute conservation equations, for binary alloy solidification. The interdendritic flow in the mushy zone is assumed to obey the Darcy's law. Microsegregation is governed by the lever rule, assuming local equilibrium at phase interfaces. The resulting energy and solute advection–diffusion equations are solved using the Streamline-Upwind/Petrov–Galerkin (SUPG) finite element method. A SUPG-PSPG velocity-pressure formulation is applied for the momentum equation. The full algorithm was implemented in the 3D code THERCAST, together with an anisotropic remeshing method. Two applications have been considered: a small ingot of Pb-48wt%Sn alloy and a large steel ingot. The numerical results of these two cases are presented with the evolution of temperature, liquid velocity, and solute concentration fields during solidification. To cite this article: S. Gouttebroze et al., C. R. Mecanique 335 (2007).

A Modified Cellular Automaton Method for the Modeling of the Dendritic Morphology of Binary Alloys

A cellular automaton (CA)-based model for the precise two-dimensional simulation of the dendritic morphology of cast aluminum alloys was developed. Compared with previous CA models, the new model considers the solidification process in more detail, solving the solute and heat conservation equations in the modeling domain, including calculation of the solid fraction, the tip velocity, and the solute diffusion process, all of which have significant influence on the dendrite evolution. The rotating grids technique was used in the simulation to avoid anisotropy introduced by the square grid. Dendritic grain profiles for different crystallographic orientations show the existence of a great number of regular and parallel secondary and tertiary arms. The simulation results for the secondary arm spacing and grain size were compared with experimental data and with results reported in the literature. A good agreement was found between the simulated results and the experimental data. It can be concluded that the model can be used to predict the dendritic microstructure of aluminum alloy in a quantitative manner.