连铸中的自由边界问题的边界元解法

本文应用边界元法求解钢铁生产中连铸工艺出现的自由边界问题。首先,对较一般的连铸过程的数学模型进行简化并给出相应的边界积分方程,以及叙述了用边界元法求解该问题的步骤。然后,我们给出了一个计算实例,并对该方法的收敛快慢、对初值的敏感性和对区域形状的适应性等问题进行了探讨。最后,针对一种简化的模型,将数值解与解析解进行比较,两者吻合较好。     

On an inverse problem arising in continuous casting of steel billets

In continuous casting of steel, the control of the solidification front by means of the amount of water sprayed onto the strand is of great practical interest. We study the thermal history in a continuously cast cylindrical billet. The mathematical model is a two-dimensional nonlinear heat equation div[k(u)gradu] = ut subject to water-cooling and heat radiation boundary conditions. We establish existence, uniqueness and stability results for both the temperature field and the solidification front. We study the monotonicity behaviour of the temperature field and show that certain technically easy-to-realize cooling-strategies may generate double liquid fingers at the final stage of solidification. The inverse problem of determining the cooling strategy is an ill-posed problem. We therefore use Tikhonov regularization as a stable and convergent methodfor treating this problem.     

The use of artificial intelligence technique for the optimisation of process parameters used in the continuous casting of steel

The productivity and quality of a continuous caster depend mainly on process parameters, i.e. casting speed, casting temperature, steel composition and cleanliness of the melt, water flow rates in the different cooling zones, etc. This work presents the development of an algorithm, which incorporates heuristic search techniques for direct application in metallurgical industries, particularly those using continuous casting process for the production of steel billets and slabs. This is done to determine the casting objectives of maximum casting rate as a function of casting constraints. These constraints are evaluated with the aid of a heat transfer and solidification model based on the finite difference technique, which has been developed and integrated with a genetic algorithm. The essential parts of continuous casting equipment, which must be subjected to monitoring, as well as a methodology of mathematical model and physical settlements in each cooling region, are presented. The efficiency of the intelligent system is assured by the optimisation of the continuous casting operation by maximum casting rate and defect-free products. This approach is applied to the real dimension of a steel continuous caster, in real conditions of operation, demonstrating that good results can be attained by using heuristic search, such as: smaller temperature gradients between sprays zones, reduction in water consumption and an increase in casting speed.     

Parameters optimization of selected casting processes using teaching–learning-based optimization algorithm

In the present work, mathematical models of three important casting processes are considered namely squeeze casting, continuous casting and die casting for the parameters optimization of respective processes. A recently developed advanced optimization algorithm named as teaching–learning-based optimization (TLBO) is used for the parameters optimization of these casting processes. Each process is described with a suitable example which involves respective process parameters. The mathematical model related to the squeeze casting is a multi-objective problem whereas the model related to the continuous casting is multi-objective multi-constrained problem and the problem related to the die casting is a single objective problem. The mathematical models which are considered in the present work were previously attempted by genetic algorithm and simulated annealing algorithms. However, attempt is made in the present work to minimize the computational efforts using the TLBO algorithm. Considerable improvements in results are obtained in all the cases and it is believed that a global optimum solution is achieved in the case of die casting process.     

Casting solidification analysis and experimental verification

Finite element methods are being applied to a number of manufacturing processes such as rolling, forging, etc. This paper focuses on the effort to simulate turbine blade casting using commercially available software. Comparisons between experimental and analytical results are presented for simple geometries. A novel solution to the radiation boundary condition problem is discussed.     

Application of a heuristic search technique for the improvement of spray zones cooling conditions in continuously cast steel billets

The process of casting occupies an important place in the metallurgical industry, and the entire world of the metal user. In the past, the ingot casting–rolling (slabbing, blooming, or billeting) process was commonly used. The continuous casting process has largely replaced this earlier method because of the inherent advantages of energy savings, enhanced productivity, higher yield and reduced costs. However, continuous casting process is not without of its problems. Considerable effort has been made by many researches to establish adequate design, operation and maintenance of continuous casting machine to ensure metallurgical quality of the final product. One of the most severe defects in continuous casting products is concerned with the cracks provoked by improper design of the spray cooling system. The aim of this work is to develop a two dimensional heat transfer model based on the finite difference method in order to calculate the strand temperatures and the solid shell profile along the machine. An Artificial Intelligence heuristic search procedure interacts with the numerical model to determine the improved cooling conditions for the sprays zones of a real continuous caster for the production of quality billets.     

The variational inequality approach to the

Considering the one-phase Stefan problem, we present an account of some recent mathematical results within the framework of variational inequalities. We discuss several situations corresponding to different boundary conditions and different geometries, like the exterior problem, the continuous casting model, and the degenerate case of the quasi-steady model. We develop a few continuous-dependence results explaining their relevance to the stability properties of the solution and of the free boundary, including the asymptotic behaviour for large time, the stability for homogenization, and the perturbation of the Dirichlet boundary conditions.     

Constraining inverse stefan design problems

A new formulation possessing stable numerical characteristics is presented for inverse Stefan design processes. In such processes, the goal of the analysis is to design transient boundary conditions which produce the desired interfacial surface motion. This subclass of mildly ill-posed mathematical problems is amenable to the proposed solution methodology. This investigation presents a fixed-front differential formulation from which a weighted residual statement is developed. Orthogonal collocation is used to obtain numerical results illustrating the merit of imposing physical constraints in the mathematical model. These mathematical constraints can be viewed as design specifications and are available to the designer or experimentalist. The proposed methodology is flexible and can be generalized to problems involving continuous casting or crystal growth. Finally, symbolic manipulation is used for augmenting the computational methodology.     

Mathematical and physical modeling of systems for metal delivery in the continuous casting of steel and DC casting of aluminum

In the continuous casting of steel, various nozzles have been used (e.g., bifurcated nozzles with ports inclined at various angles to the horizontal) to deliver metal from the tundish into the caster. An even greater variety of devices is used in the case of semi-continuous (direct chill (DC) or electromagnetic (EM)) casting of aluminum, for example, nozzles delivering metal into bags of various designs. The paper describes a physical (water) model whereby particle image velocimetry has been used to measure velocities. These measured velocities are compared to ones predicted using computational fluid dynamics. Conclusions are reached concerning the validity of the computations and recommendations made about improvement in casting operations by modification of melt flow.     

On the explicit resolution of the mushy zone in the modelling of the continuous casting of alloys

In the continuous casting of alloys, it is well-known that the mushy zone is decisive for the final properties of the casting. Most numerical models for the process use enthalpy-based methods on fixed grids which determine the extent the mushy zone implicitly. Here, on the other hand, we develop a methodology for explicitly resolving the geometrical extent of the mushy zone; this involves a sharp-interface formulation to solve a dual moving boundary problem to locate the solidus and liquidus isotherms. The results compare favourably with those from enthalpy-based methods, and the advantages of our approach with respect to future multiphysics calculations are discussed.     

一个具有内部物质对流的两项连铸Stefan问题

考虑了一个具有内部物质对流和非线性边界热交换的多维连铸Stefan问题,并得到了这个问题整体弱解的存在性、唯一性和对初边界条件的连续依赖性。本项工作改进和推广了J.F.Fodri-gues＆F.Yi的结果,放宽了他们对内部流和边界条件的一些不太符合实际的限制。     

A three-dimensional steady state thermal fluid model of jumbo ingot casting during electron beam re-melting of Ti–6Al–4V

A 3-D coupled thermal-fluid model describing mass, momentum and energy transport within a Ti–6Al–4V rolling ingot cast in an (Electron Beam Cold Hearth Remelting) EBCHR process has been developed to describe steady state casting conditions. The model incorporates a number of the physical phenomena inherent to the industrial process, including a metal inlet in the center of one of the narrow faces, complex boundary conditions based on industrial practice, buoyancy driven flow within the liquid and flow attenuation using a Darcy momentum source term within the mushy zone. The model ignores turbulence in the liquid pool and Marangoni (surface tension) driven surface flows. The model has been validated against liquid pool depth and profile measurements made on an experimental casting seeded with insoluble dense markers and doped with dense alloy additions. Comparisons have also been made to video images taken of the top surface during casting. The results indicate that the model is able to quantitatively predict the steady state sump depth and profile and is able to qualitatively predict aspects of the top surface temperature distribution. The model has also been used to conduct a process heat balance and sensitivity analyses. The process heat balance conducted on the model domain indicates that at steady state the liquid metal inlet contributes 88% of the total power input, while the electron beam provides net 12% after accounting for radiation losses from the top surface; 62% of the heat is lost through the ingots sides and the balance is lost via bulk transport of sensible heat through the bottom of the domain. The results of the sensitivity analysis on pool depth indicate that casting rate has the largest effect followed by metal inlet superheat. The thermal, flow and pressure fields predicted by the steady state model serves as the initial conditions for a transient hot-top model, which is the subject of a forth-coming paper.     

MHD analysis in hydromagnetic casting process of clad steel slabs

A novel continuous casting process for clad steel slabs has been developed by suppressing the mixing of molten steels in the mold pool of continuous casting strand with a level DC magnetic field (LMF) installed in the mold. In this process, two molten steels of different chemical composition are discharged by two nozzles into the upper and the lower pools respectively to solidify in the outer and the inner layers as a clad steel slabs. The mechanism of separation into two layers has been elucidated by using a three dimensional MHD analysis. The numerical prediction employing Maxwell's equation, Ohm's law, and the turbulent flow model shows that the mixing of the different type of steels is suppressed by the electromagnetic dividing of the upper and the lower recirculating flows. The principle of the new process has also been verified by steel casting trials of the stainless-steel clad steel slabs with an 8-ton scale pilot continuous casting machine.     

Verified reduction of a model for a continuous casting process

The casting of metals is known to involve the complex interaction of turbulent momentum and heat transfer in the presence of solidification, and it is believed that computational fluid dynamical (CFD) techniques are required to model it correctly. Here, using asymptotic methods, we demonstrate that the key quantities obtained in an earlier CFD model for a particular continuous casting process – ostensibly for a pure metal, but equally for an alloy of eutectic composition – can be recovered using a much simpler model that takes into account just the heat transfer, requiring the numerical solution of a two-phase Stefan problem. Combining this with a more recent asymptotic thermomechanical model for the same continuous casting process, we postulate that it should be possible, with the additional help of algebraic manipulation, to reduce a model that takes into account turbulent momentum and heat transfer in the melt and the thermomechanics in the solid shell to one formulated in terms of only heat transfer, without adversely affecting model predictions.     

Large Scale Oscillations in the Continuous Casting Process

Long-term oscillations of the flow in a continuous casting tundish are investigated numerically and experimentally. The numerical model is based on the Reynolds averaged Navier-Stokes equations (URANS). A comparison between numerical results and experimental data from a water model is presented. Frequencies due to long-term oscillations are resolved in the simulation. These frequencies are in satisfactory agreement with the experimental results of LDA measurements. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Perturbation analysis of unsteady magnetohydrodynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction

An analytical study for the problem of unsteady mixed convection with thermal radiation and first-order chemical reaction on magnetohydrodynamics boundary layer flow of viscous, electrically conducting fluid past a vertical permeable plate has been presented. Slip boundary condition is applied at the porous interface. The classical model is used for studying the effect of radiation for optically thin media. The non-linear coupled partial differential equations are solved by perturbation technique. The results obtained show that the velocity, temperature and concentration fields are appreciably influenced by the presence of chemical reaction, thermal stratification and magnetic field. It is observed that the effect of thermal radiation and magnetic field decreases the velocity, temperature and concentration profiles in the boundary layer. Also, the effects of the various parameters on the skin-friction coefficient and the rate of heat transfer at the surface are discussed.     

Two-dimensional radiative equilibrium

An exact formulation for the radiative flux and the emissive power is presented for a two-dimensional, finite, planar, absorbing-emitting, gray medium in radiative equilibrium. Exact expressions are obtained for a medium subjected to the following types of boundary conditions: (A) cosine varying collimated radiation, (B) a strip of collimated radiation, (C) cosine varying diffuse radiation, and (D) a uniform temperature strip. The solution for the cosine varying collimated radiation model is used to construct the solutions for the other boundary conditions. The two-dimensional equations are reduced to one-dimensional equations by the method of separation of variables.     

Two-dimensional isotropic scattering

An exact formulation for the source function, the radiative flux, and the intensity is presented for a two-dimensional, finite, planar, isotropically scattering medium. Exact expressions are obtained for both collimated and diffuse radiation. Cosine varying collimated and diffuse radiation is considered in detail. The solution for the cosine varying collimated radiation model is used to construct the solutions for the other boundary conditions. The two-dimensional integral equations are reduced to one-dimensional equations by the method of separation of variables.     

Simulation of direct chill casting under the influence of a low-frequency electromagnetic field

A comprehensive, multiphysics, meshless, numerical model is developed for the simulation of direct chill casting under the influence of a low-frequency electromagnetic field. The model uses mixture-continuum-mass, momentum and energy-conservation equations to simulate the solidification of axisymmetric aluminium-alloy billets. The electromagnetic-induction equation is coupled with the fluid flow and used to calculate the Lorentz force. The involved partial-differential equations are solved with the meshless-diffuse-approximate method by employing second-order polynomial shape functions and a 13-noded local support. An explicit time-stepping scheme is used. The boundary conditions for the heat transfer involve the effects of hot-top, mould chill and direct chill. The use of a meshless method and the automatic node-arrangement generation made it possible to investigate the complicated flow structures in geometrically complex inflow conditions, including sharp and curved edges, in a straightforward way. A time-dependent adaptive computational node arrangement is used to decrease the calculation time. The model is demonstrated by casting an Al-5.25wt%Cu aluminium alloy billet with a radius of 120 mm. Results on simplified and realistic inflow geometry are considered and compared. The effect of the low-frequency electromagnetic force on the temperature, liquid fraction and fluid flow are investigated under different current densities and frequencies.     

THE BIDIMENSIONAL EVOLUTIONARYBOUSSINESQ-STEFAN PROBLEM WITH CONTINUOUS EXTRACTION

1IntroductionTheaimofthispaperistoextelldtotheevolutionarycasesomeresultobtainedbyJ.F.RDdriguesin[l]aboutthesteady-stateBoussinesq-Stefanproblem.AtthesametimeitistoextendtothecontinuouscastingproblemsomeresultobtainedbyJ.R.Cannonetal.in[2]abouttheStefanproblemwithconvection.TheevolutionaryBossinesq-Stefanproblemisamodeldescribingthesolidfficationwithconvectionofamaterial(ingotOfsteelisanex~le)beingcastcontinuouslywithaprexcribedvelocityinacylindricaldomainQ=rx(0,l)eR",n=2,3,where0相似文献