Solution Of Conservation Equations Research Articles

Phase-field simulation of the columnar-to-equiaxed transition in alloy solidification

The columnar-to-equiaxed transition (CET) in directional solidification of alloys is simulated using the phase-field method. The method relies on the solution of a solute conservation equation and an equation for the propagation of the phase field on the scale of the developing microstructure. A parametric study is performed to investigate the effects of the applied temperature gradient and pulling speed, the seed spacing and nucleation undercooling for the equiaxed grains, and the crystalline anisotropy strength on the CET. The results qualitatively agree with a previously developed analytical model of the CET. At relatively high pulling speeds, a mixed columnar–equiaxed structure is found to be stable over a range of temperature gradients. Furthermore, the CET depends sensitively on the anisotropy strength. The simulations also reveal the presence of primary spacing adjustments during purely columnar growth due to nucleation of seeds, and deactivation of seeds by solutal interactions from nearby growing grains.

Modeling of macrosegregation caused by volumetric deformation in a coherent mushy zone

A two-phase volume-averaged continuum model is presented that quantifies macrosegregation formation during solidification of metallic alloys caused by deformation of the dendritic network and associated melt flow in the coherent part of the mushy zone. Also, the macrosegregation formation associated with the solidification shrinkage (inverse segregation) is taken into account. Based on experimental evidence established elsewhere, volumetric viscoplastic deformation (densification/dilatation) of the coherent dendritic network is included in the model. While the thermomechanical model previously outlined (M. M’Hamdi, A. Mo, and C.L. Martin: Metall. Mater. Trans. A, 2002, vol. 33A, pp. 2081–93) has been used to calculate the temperature and velocity fields associated with the thermally induced deformations and shrinkage driven melt flow, the solute conservation equation including both the liquid and a solid volume-averaged velocity is solved in the present study. In modeling examples, the macrosegregation formation caused by mechanically imposed as well as by thermally induced deformations has been calculated. The modeling results for an Al-4 wt pct Cu alloy indicate that even quite small volumetric strains (≈2 pct), which can be associated with thermally induced deformations, can lead to a macroscopic composition variation in the final casting comparable to that resulting from the solidification shrinkage induced melt flow. These results can be explained by the relatively large volumetric viscoplastic deformation in the coherent mush resulting from the applied constitutive model, as well as the relatively large difference in composition for the studied Al-Cu alloy in the solid and liquid phases at high solid fractions at which the deformation takes place.

A quantitative dendrite growth model and analysis of stability concepts

While a number of cellular automaton (CA) based models for dendrite growth have been proposed, none so far have been validated, casting doubt on their quantitative capabilities. All these models are mesh dependent and cannot correctly describe the influence of crystallographic orientation on growth morphology. In this work, we present an improved version of our previously developed CA based model for dendrite growth controlled by solutal effects in the low Peclet number regime. The model solves the solute and heat conservation equations subject to the boundary conditions at the interface, which is tracked with a new virtual front tracking method. It contains an expression equivalent to the stability constant required in analytical models, termed stability parameter, which is not a constant. The process determines its value, changing with time and angular position during dendrite formation. The article proposes solutions for the evaluation of local curvature, solid fraction, trapping rules, and anisotropy of the mesh, which eliminates the mesh dependency of calculations. Several tests were performed to demonstrate the mesh independence of the calculations using Fe-0.6 wt pct C and Al-4 wt pct Cu alloys. Computation results were validated in three ways. First, the simulated secondary dendrite arm spacing (SDAS) was compared with literature values for an Al-4.5 wt pct Cu alloy. Second, the predictions of the classic Lipton-Glicksman-Kurz (LGK) analytical model for steady-state tip variables, such as velocity, radius, and composition, were compared with simulated values as a function of melt undercooling for Al-4 wt pct Cu alloy. In this validation, it was found that the stability parameter approaches the experimentally and theoretically determined value of 0.02 of the stability constant. Finally, simulated results for succinonitrile-0.29 wt pct acetone (SCN-0.29 wt pct Ac) alloy are compared with experimental data. Model calculations were found to be in very good agreement with both the analytical model and the experimental data. The model is used to simulate equiaxed and columnar growth of Fe-0.6 wt pct C and Al-4 wt pct Cu alloys offering insight into microstructure formation under these conditions.

The effect of volume change during directional solidification of binary alloys

The effect of solidification contraction is examined through numerical simulations of hypoeutectic Pb–Sn alloys. In addition, particular attention is given to numerically approximating the solidification of the eutectic liquid. In order to predict the inverse segregation at the cooled surface accurately, the contribution of the boundary integral is retained in the weak formulation of the solute conservation equation. This procedure eliminates the false flux, which incorrectly causes a negative segregation at the cooled surface. In the case of vertical solidification under gravity, when channels form, the shrinkage turns the flow inside the channel downwards and towards the solidifying eutectic. Without including shrinkage, the flow in an entire channel is upwards.

Phase-field modeling of binary alloy solidification with coupled heat and solute diffusion.

A phase-field model is developed for simulating quantitatively microstructural pattern formation in solidification of dilute binary alloys with coupled heat and solute diffusion. The model reduces to the sharp-interface equations in a computationally tractable thin-interface limit where (i). the width of the diffuse interface is about one order of magnitude smaller than the radius of curvature of the interface but much larger than the real microscopic width of a solid-liquid interface, and (ii). kinetic effects are negligible. A recently derived antitrapping current [Phys. Rev. Lett. 87, 115701 (2001)]] is used in the solute conservation equation to recover precisely local equilibrium at the interface and to eliminate interface stretching and surface diffusion effects that arise when the solutal diffusivities are unequal in the solid and liquid. Model results are first compared to analytical solutions for one-dimensional steady-state solidification. Two-dimensional thermosolutal dendritic growth simulations with vanishing solutal diffusivity in the solid show that both the microstructural evolution and the solute profile in the solid are accurately modeled by the present approach. Results are then presented that illustrate the utility of the model for simulating dendritic solidification for the large ratios of the liquid thermal to solutal diffusivities (Lewis numbers) typical of alloys.

An explicit scheme for coupling temperature and concentration fields in solidification models

A numerical scheme for coupling temperature and concentration fields in a general solidification model is presented. A key feature of this scheme is an explicit time stepping used in solving the governing thermal and solute conservation equations. This explicit approach results in a local point-by-point coupling scheme for the temperature and concentration and avoids the multi-level iteration required by implicit time stepping schemes. The proposed scheme is validated by predicting the concentration field in a benchmark solidification problem. Results compare well with an available similarity solution. The simplicity of the proposed explicit scheme allows for the incorporation of complex microscale models into a general solidification model. This is demonstrated by investigating the role of dendrite coarsening on the concentration field in the solidification benchmark problem.

Modeling dendritic growth of a binary alloy

A two-dimensional model for simulation of the directional solidification of dendritic alloys is presented. It solves the transient energy and solute conservation equations using finite element discretizations. The energy equation is solved in a fixed mesh of bilinear elements in which the interface is tracked; the solute conservation equation is solved in an independent, variable mesh of quadratic triangular elements in the liquid phase only. The triangular mesh is regenerated at each time step to accommodate the changes in the interface position using a Delaunay triangulation. The model is tested in a variety of situations of differing degrees of difficulty, including the directional solidification of Pb–Sb alloys.

Influences of nozzle flow and nozzle geometry on the shape and size of an air core in a hollow cone swirl nozzle

Theoretical and experimental studies have been carried out to determine the influences of nozzle flow and nozzle geometry on the shape and size of a fully developed air core in a hollow cone swirl nozzle. The theoretical study is based on the numerical solution of conservation equations for mass and momentum along with the volume fraction of the liquid phase. An interface capturing method has been adopted in the numerical simulation of free surface flow in the nozzle. Experiments have been carried out with a number of nozzles fabricated in Perspex material. The air core diameter has been measured from photographs taken by a camera outside the nozzle. It has been observed that the shape of the fully developed air core in a conical swirl nozzle is cylindrical with a considerable bulging at the entrance to the orifice, while in the case of a conical nozzle without a finite length of orifice, the air core is uniform throughout the nozzle. The values of the air core diameter in the swirl chamber ( da1) and in the orifice ( da2) of a nozzle increase sharply with an increase in inlet Reynolds number ( Re) below 1.1 × 104, but become almost independent of Re above this (up to 1.7 × 104). The air core diameter, both in the swirl chamber and in the orifice, increases with an increase in the value of the ratio of orifice to swirl chamber diameter and the cone angle of the swirl chamber and with a decrease in the value of the ratio of entry port to swirl chamber diameter and the ratio of orifice length to swirl chamber diameter.

Growth of solutal dendrites: A cellular automaton model and its quantitative capabilities

A model based on the cellular automaton (CA) technique for the simulation of dendritic growth controlled by solutal effects in the low Peclet number regime was developed. The model does not use an analytical solution to determine the velocity of the solid-liquid (SL) interface as is common in other models, but solves the solute conservation equation subjected to the boundary conditions at the interface. Using this approach, the model does not need to use the concept of marginal stability and stability parameter to uniquely define the steady-state velocity and radius of the dendrite tip. The model indeed contains an expression for the stability parameter, but the process determines its value. The model proposes a solution for the artificial anisotropy in growth kinetics valid at zero and 45° introduced in calculations by the square cells and trapping rules used in previous CA formulations. It also introduces a solution for the calculation of local curvature, which eliminates mesh dependency of calculations. The model is able to reproduce qualitatively most of the dendritic features observed experimentally, such as secondary and tertiary branching, parabolic tip, arms generation, selection and coarsening, etc. Computation results are validated in two ways. First, the simulated secondary dendrite arm spacing (SDAS) is compared with literature values. Then, the predictions of the classic Lipton-Glicksman-Kurz (LGK) theory for steady-state tip velocity are compared with simulated values as a function of melt undercooling. Both comparisons are found to be in very good agreement.

Influence of fuel volatility on combustion and emission characteristics in a gas turbine combustor at different inlet pressures and swirl conditions

The practical challenges in research in the field of gas turbine combustion mainly centre around a clean emission, a low liner wall temperature and a desirable exit temperature distribution for turboma-chinery applications, along with fuel economy of the combustion process. An attempt has been made in the present paper to develop a computational model based on stochastic separated flow analysis of typical diffusion-controlled spray combustion of liquid fuel in a gas turbine combustor to study the influence of fuel volatility at different combustor pressures and inlet swirls on combustion and emission characteristics. A κ-ɛ model with wall function treatment for the near-wall region has been adopted for the solution of conservation equations in gas phase. The initial spray parameters are specified by a suitable probability distribution function (PDF) size distribution and a given spray cone angle. A radiation model for the gas phase, based on the first-order moment method, has been adopted in consideration of the gas phase as a grey absorbing-emitting medium. The formation of thermal NO x as a post-combustion reaction process is determined from the Zeldovich mechanism. It has been recognized from the present work that an increase in fuel volatility increases combustion efficiency only at higher pressures. For a given fuel, an increase in combustor pressure, at a constant inlet temperature, always reduces the combustion efficiency, while the influence of inlet swirl is found to decrease the combustion efficiency only at higher pressure. The influence of inlet pressure on pattern factor is contrasting in nature for fuels with lower and higher volatilities. For a higher-volatility fuel, a reduction in inlet pressure decreases the value of the pattern factor, while the trend is exactly the opposite in the case of fuels with lower volatilities. The NOx emission level increases with decrease in fuel volatility at all combustor pressures and inlet swirls. For a given fuel, the NOx emission level decreases with a reduction in combustor pressure and an increase in inlet swirl number.

Numerical simulation of deformation-induced segregation in continuous casting of steel

The deformation-induced macrosegregation in continuous casting of steel has been simulated using a finite-volume scheme. For that purpose, a two-dimensional heat-flow computation was first performed in a Eulerian reference frame attached to the mold, assuming a unique solidification path, i.e., a unique relationship between temperature and enthalpy. This gave the stationary enthalpy field in the longitudinal section of the slab. On the other hand, bulging of the slab between two rolls was calculated in the same section, assuming plane-strain deformation and using the ABAQUS code. The Lagrangian reference frame was attached to the slab, and the rolls were moved at the surface until a stationary, bulging deformation profile was reached. The bulging of the surface was then used as an input condition for the calculation of the velocity and pressure fields in the interdendritic liquid. Using a fairly simple hypothesis for the deformation of the solid skeleton, the mass conservation and Darcy equations were solved in a Eulerian reference frame. This calculation was performed in an iterative loop, within which the solute conservation equation was also solved. At convergence and using the enthalpy field, this calculation allowed to obtain the temperature, the volume fraction of solid, and the average concentration fields, in addition to the fluid velocity and pressure. It is shown that the positive centerline segregation of carbon in the slab is well reproduced with this model. The effects of shrinkage and soft reduction were also investigated.

Arc Modeling

ABSTRACT: This paper is an account of how arc modeling has increased our predictive capability of properties of arc plasmas. Basic material functions of, for example, thermal and electrical conductivities, can be calculated for any plasma, given the component elements of the plasma. Then solution of conservation equations gives properties of the plasma. Examples are given of successes of arc modeling. (1) A one dimensional calculation for an arc confined in a cylindrical tube gives the principal electrical properties of an arc lamp. (2) Two dimensional calculations including the arc and the electrodes gives an explanation of the transition from globular to spray modes of metal transfer in arc welding. (3) Also predictions of plasma and electrode temperatures in tungsten arc welding are in good agreement with experiment. (4) Time dependent calculations of the development of electron avalanches in lightning, explain the development of the stepped leader.

Assessment of mathematical models for the flow in directional solidification

In a binary solution unidirectionally solidified from below, the bulk melt and the eutectic solid is separated by a dendritic mushy zone. The mathematical formulation governing the fluid motion shall thus consist of the equations in the bulk melt and the mushy zone and the associated boundary conditions. In the bulk melt, assuming that the melt is a Newtonian fluid, the governing equations are the continuity equation, the Navier-Stokes equations, the heat conservation equation, and the solute conservation equation. In the mushy layer, however, the formulation of the momentum equation and the associated boundary conditions are diversified in previous investigations. In this paper, we discuss three mathematical models, which had been previously applied to study the flow induced by the solidification of binary solutions cooling from below. The assessment is given on the bases of the stability characteristics of the convective flow and the comparison between the numerical and experimental results.

The energy and solute conservation equations for dendritic solidification

The energy equation for solidifying dendritic alloys that includes the effects of heat of mixing in both the dendritic solid and the interdendritic liquid is derived. Calculations for Pb-Sn alloys show that this form of the energy equation should be used when the solidification rate is relatively high and/or the thermal gradients in the solidifying alloy are relatively low. Accurate predictions of transport phenomena in solidifying dendritic alloys also depend on the form of the solute conservation equation. Therefore, this conservation equation is derived with particular consideration to an accounting of the diffusion of solute in the dendritic solid. Calculations for Pb-Sn alloy show that the distribution of the volume fraction of interdendritic liquid (gL) in the mushy zone is sensitive to the extent of the diffusion in the solid. Good predictions ofgL are necessary, especially when convection in the mushy zone is calculated.

Thermosolutal convection during dendritic solidification of alloys: Part i. Linear stability analysis

This paper describes the simulation of thermosolutal convection in directionally solidified (DS) alloys. A linear stability analysis is used to predict marginal stability curves for a system that comprises a mushy zone underlying an all-liquid zone. In the unperturbed and nonconvecting state .e.}, the basic state), isotherms and isoconcentrates are planar and horizontal. The mushy zone is realistically treated as a medium with a variable volume fraction of liquid that is con-sistent with the energy and solute conservation equations. The perturbed variables include tem-perature, concentration of solute, and both components of velocity in a two-dimensional system. As a model system, an alloy of Pb-20 wt pct Sn, solidifying at a velocity of 2 X 10-3 cm s-1 was selected. Dimensional numerical calculations were done to define the marginal stability curves in terms of the thermal gradient at the dendrite tips,G L ,vs the horizontal wave number of the perturbed quantities. For a gravitational constant of 1g,0.5 g, 0.1g, and 0.01g, the marginal stability curves show no minima; thus, the system is never unconditionally stable. Nevertheless, such calculations quantify the effect of reducing the gravitational constant on reducing convection and suggest lateral dimensions of the mold for the purpose of suppressing convection. Finally, for a gravitational constant of 10-4 g, calculations show that the system is stable for the thermal gradients investigated (2.5 ≤G L ≤ 100 K-cm-1).

Calculation methods for reacting turbulent flows: A review

The purpose of this review is to describe and appraise components of calculation methods, based on the solution of conservation equations in differential form, for the velocity, temperature and concentration fields in turbulent combusting flows. Particular attention is devoted to the combustion models used within these methods and to gaseous-combustion applications. The differential equations are considered first and the implications of conventional (i.e., unweighted) and density-weighted averaging discussed in the contexts of solution methods and physical interpretation. In general, it is concluded that equations should be solved with dependent variables in density-weighted form and the interpretation of measurements requires special care to distinguish between conventionally averaged and density-weighted properties. Finite difference approximations contained within numerical procedures for solving the equations relevant to two- and three-dimensional recirculating flows, such as are to be found in combustion chambers, are considered briefly. It is concluded that computer storage requirements very often preclude the possibility of reducing the numerical error to entirely negligible proportions everywhere. Considerable care must therefore be taken in both specifying a sufficient number and the distribution of mesh points to be used and also in the interpretation of computational results. Turbulence models are also discussed briefly and deficiencies noted. Since many flows are partly controlled by mechanisms other than diffusion and turbulence transport, these deficiencies are of major importance in a limited range of circumstances which are discussed. The various recent methods proposed to represent reaction in turbulent flames are reviewed in relation to diffusion and premixed flames and to flames in which an element of both is present. The application of laminar flame sheet models, chemical equilibrium assumptions, probability density functions of different forms, and truncated series expansion of reaction rate expressions are considered together with the use of probability density function transport equations and their (Monte Carlo) solution. The appraisal is made in relation to presently available results and future requirements and possibilities.

Velocity Characteristics of a Confined Coaxial Jet

The velocity characteristics of a turbulent, confined, coaxial-jet flow have been determined by measurement and by the solution of conservation equations in differential form. The ratios of maximum annulus to pipe velocity were 3 and 1 and, in both cases, the profiles were fully developed in the exit plane. The geometric arrangement corresponded to a model furnace and the investigation was undertaken to provide information, for isothermal flow, relevant to furnace flows. The measurements were obtained with a hot-wire anemometer and include distributions of the axial mean velocity and the components of the Reynolds-stress tensor. They show, for example, that the larger velocity ratio results in a larger region of recirculation, larger velocity gradients and larger turbulence intensities in the mixing region and downstream of the region of reverse flow. Numerical solutions of the time-averaged forms of the equations of conservation of mass and momentum, together with equations for turbulence energy and dissipation rate, provided results which are in close agreement with the measurements except in regions of more than one major component of the velocity-gradient tensor where they are substantially in error. The reasons for the discrepancies and the consequential value of the calculation method are discussed.

On the mixing of a low Reynolds number biological jet with a quiescent outer bathing solution

Existing analyses of the very low Reynolds number Stokes jet, e.g. Birkhoff & Zarantonello (1957) and Förste (1963), have been confined to a single-component fluid satisfying the usual zero-slip boundary conditions at a solid surface. In contrast, the low Reynolds number biological jets which emerge from the pores and channels at the secreting surfaces of numerous human, animal and insect organs entail the movement of water and solute subject to novel boundary conditions that arise from the local osmotic driving forces at the secreting surfaces. These boundary conditions introduce a nonlinear coupling between the fluid momentum and solute conservation equations. This paper first discusses the fundamental dimensionless groups and length scales that characterize these biological jet flows and then examines in detail, using the technique of matched inner and outer expansions, one flow situation important in epithelial membrane transport, namely, the two-dimensional mixing of an inhomogeneous jet with a quiescent outer bathing solution which is bounded by a semi-permeable mem brane in the plane of the exit. The paper concludes with a discussion of other physiologically relevant problems which arise from different orderings of the length scales, and different overall geometrical configurations and flow conditions.

Flame structure and flame reaction kinetics - V. Investigation of reaction mechanism in a rich hydrogen+nitrogen+oxygen flame by solution of conservation equations

A powerful combination of two computational methods has been used to investigate the reaction mechanism in a fuel-rich hydrogen+nitrogen+oxygen flame. The first of these involves the solution of the time-dependent heat conduction and diffusion equations by finite difference methods. It allows a preliminary assessment of reaction mechanisms and rate constants which must be used to reproduce the observed flame velocity. However, the transport fluxes are only represented approximately in this time-dependent model, so that a precise calculation of flame profiles cannot be made. The second computational method uses a Runge–Kutta procedure to calculate the steady-state flame profiles, and is an extension of the methods discussed by Dixon-Lewis (1968). It incorporates detailed transport property calculations, and thus allows computation of detailed flame profiles for comparison with experiment. Application of the methods to the rich hydrogen+nitrogen+oxygen flame and subsequent comparison with experiment has established the participation of hydroperoxyl in the flame mechanism, and has shown the principal reactions in the flame to be: OH + H2= H2O + H, (i) H + O2=OH + O, (ii) O + H2=OH + H, (iii) H + O2+ M = HO2+ M, (iv) H + HO2= OH + OH, (vii) H + HO2= H2+ O2, (xii) H+ H + M = H2+ M. (xv) It was found that the interplay between these reactions is such that it is impossible to use the atmospheric pressure flame for an independent, precise determination of the hydrogenoxygen chain branching-rate constantk2. Another property of the mechanism is that the hydrogen atom concentration profile in the flame is not very dependent on the precise rate constants employed, so that the profile itself can be computed probably to better than ±10%. The reaction zone of the very rich flame commences at about 550 K, the maximum overall reaction rate is at about 900 K, and the maximum hydrogen atom concentration is at 1030 to 1040 K. The rate constant ratiok7/k12is found to lie in the range 5±1, assumed independent of temperature over the reaction zone. Assuming equal efficiencies of all the molecules in the flame as third bodies in the hydrogen atom recombination, the rate constantk15is estimated to lie in the range 4.5±1.5 x 1015cm6mol-2s-1.

Flame structure and flame reaction kinetics I. Solution of conservation equations and application to rich hydrogen-oxygen flames

The time dependent heat conduction and diffusion equations have been set up for some simplified reaction mechanisms representing a fuel-rich hydrogen-nitrogen-oxygen flame, and have been solved by finite difference methods. The effects on the flame properties of changes in some of the reaction parameters and the transport properties of the gas mixture have been examined and some comparisons made with experimental results. The validity of the frequently made assumption regarding equilibration of H, OH and O between themselves in the burnt gas or radical recombination region of such flames has been verified for H and OH in a flame having a final temperature of 1072 °K. The effects on the flame model of some assumed contributions of reactions of hydroperoxyl to the flame mechanism are also described.