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However, in this paper, continuous flow was assumed for each phase in the derivation of the equation of motion for obtaining the numerical solution. The model has been applied to the simulation of not only four phaseflows but also one to three phaseflows. Typical examplesfor application will be described for several processes.

The blast furnace process is a multi-phase chemical reactor whose main purpose is to reduce iron oxides producing hot metal. In the actual blast furnace operation several phases simultaneously interact with one another exchanging momentum, mass and energy. In this paper a three-dimensional multiphase mathematical model of the blast furnace is presented. This model treats the blast furnace process as a multiphase reactor in which all phases behave like fluids. Five phases are treated by this model, namely, gas, lump solids (iron ore, sinter, pellets and coke), pig iron, molten slag and pulverized coal. Conservation equations for mass, momentum, energy and chemical species for all phases are solved based on the finite volume method. In the discretized momentum equations, the covariant velocity projections are used, which is expected to give the best coupling between the velocity and pressure fields and improve the convergence of the calculations. This is a new feature of the present model regarding to the numerical procedures applied to the blast furnace modeling, which emphasizes its originality. In addition, gas and solid phases are treated as continuous phases possessing a pressure field and the SIMPLE algorithm is applied to extract the pressure field and ensure mass conservation. Hot metal, slag and pulverized coal are treated as discontinuous phases consisting of unconnected droplets. For such phases, momentum conservation is used to calculate the fields of velocity while the continuity equations are used to calculate the phase volume fractions.This model was applied to predict the three-dimensional blast furnace operation and predicted temperature distributions and operational parameters like productivity, coke rate and slag rate presented close agreement with the actual measured ones in the blast furnace process.

Currently low reducing agent operation of blast furnace has been actively pursued in ironmaking due to the global warming. The precise and reliable control of blast furnace aiming at the low reducing operation is required to attain smoothly the stable operation. It is considered that the mathematical model on the ironmaking process can play an important role to ensure the stable blast furnace operation. A mathematical model is a powerful tool to improve conventional processes and to design new processes in ironmaking. These tendencies and requirements are common to every country. Thus, the development of advanced mathematical models of blast furnace like DEM (Discrete Element Method) has attracted a special attention in ironmaking field. Moreover, the combined model of DEM with the continuum model is under development for the purpose of the accurate understanding of inner state of blast furnace. In this paper, the recent activity and progress on the development of advanced mathematical model of blast furnace and future perspective for desirable model are described.KEY WORDS: ironmaking; blast furnace; mathematical model; discrete element method; solid flow; particle method; contiuum model. Review Fig. 1. Transition of blast furnace operating condition in Japan.

An ironmaking blast furnace is a complex multiphase flow reactor involving gas, powder, liquid and solid phases. Understanding the flow behaviour of these phases is of paramount importance to the control and optimization of the process. Mathematical modelling, often coupled with physical modelling, plays an important role in this development. Yagi 1) gave a comprehensive review of the early studies in this area in 1993. Significant progress has since been made, partially driven by the needs in research but mainly as a result of the rapid development of computer and computational technologies. This paper reviews these developments, covering the formulation, validation and application of mathematical models for gas-solid, gas-liquid, gas-powder and multiphase flows. The need for further developments is also discussed.KEY WORDS: ironmaking; blast furnace; multiphase flow; two fluid model; discrete element method. Review development. Mathematical ModellingGenerally speaking, the existing approaches to modelling the multiphase flow in a BF can be classified into two categories: continuum approach at a macroscopic level and discrete approach at a microscopic level. In the continuum approach, phases are generally considered as fully interpenetrating continuum media and described by separate conservation equations with appropriate constitutive relations and interaction terms representing the coupling between phases. The general governing equations are based on the so-called two fluid model (TFM), originally developed for gas-particle flow, [29][30][31] given by:Conservation The so-called Model A and Model B result from the treatment of the fluid pressure. For example, for gas-solid flow, Model A assumes the pressure is shared between the two phases, while in Model B, the pressure is attributed to gas phase only. 31) Model A formulation has been widely used in multiphase modelling, especially in process metallurgy. Model B formulation was recently used in the so called combined continuum and discrete method for coupled flow of continuum fluid and discrete particles. 32) The continuum approach is suitable for process modelling and applied research because of its computational convenience and efficiency. Indeed, most of the BF modelling is based on this approach. However, its effective use heavily depends on constitutive or closure relations and the momentum exchange between phases. For Newtonian fluids (gas and liquid here), these relations can be readily determined. For non-Newtonian fluids such as solids and powder, general theories are not yet available. In the past, various theories have been devised for different flow regimes (three flow regimes have been identified: quasi-static regime, rapid flow regime and a transitional regime that lies inbetween). For example, models have been proposed to derive the constitutive equations for the rate-independent deformation of granular materials based on either the plasticity theory or the double shearing theory; the rapid flow of granular materials has been described by extend...

Temperature dependenceof thermal conductivities of dense pure iron oxides and iron.

Solid descending flow due to gravity is frequently applied to industrial processes. Several kinds of models have been proposed like the plug flow model, the potential flow model, the kinematic model, etc. In this study, a viscous flow model based on the Navier-Stokes equation has been developed and was compared with the potential flow and the kinematic models. Both two and three dimensional experimental apparatuses have been constructed to observe the velocity fields. The concept of solid viscosity was quoted to describe the friction between particles and the value was obtained from the experimental data. Slip boundary condition was used at the wall and the friction between particles and the wall or the dead zone was expressed by the Fanning equation. Navier-Stokes equation was applied to simulate the solid flow. Thegoodagreementbetweenthe observed and computed results was obtained in different scale apparatuses at different solid descending velocities. The computation results gave about 0.07 Pa's as the solid viscosity of sand particles with the diameter of 0.001-0.002m. The solid~~as countercurrent flow was also simulated by the Navier-Stokes equation with the same value of solid viscosity.