This paper reviews recent research on the numerical simulations of solid-liquid and three-phase flows. The numerical simulations are classified into three types, namely, non-model, half-model and model type. Typical research of each type and constitutive equations for the motion of solid in liquid are briefly introduced.

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In solid-fluid two phase flows, the interaction between particles and particle-wall play important roles. They are deeply related not only to individual motions of the particle but to the formation of meso-scale structure such as particle clouds. Therefore the foundamental study on the interaction of particles has been required for modeling the solid-fluid flow. In this paper, theoretical models of short-range interaction between solids in fluid are briefly reviewed. Then we describe the direct simulation of the sedimentation of a solid particle toward a wall, and discuss the fluid force acting on the particle at finite Reynolds number. Furthermore, it is shown that the fluid effects due to the unsteady motion of the particle are significant role on short-range interaction in liquid.

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The mathematical simulation of blast furnace operation, which uses theories of reaction kinetics and transport phenomena, has been being developed in past four decades. Although blast furnace was once treated as a black-box-reactor, a lot of experimental and numerical efforts have revealed that multiple regions exist within a furnace. The requirements for more detailed information of their functions and characteristics has been increased, as they revealed. Such demands improves the mathematical simulation model of blast furnace from one-dimensional to multi-dimensional, from steady to transient, and from single phase to multiple phases. This paper explains the outline of the latest mathematical model of blast furnace operation, which is based on multi-fluid theories, reaction kinetics and transport phenomena, and its applications. This model treats four materials, that have different flow characteristics and thermo-physical properties, as fluid. These materials are gas (blast and reaction gases), lump solids (coke and ore), liquids (molten metal and slag) and fine particles (unburnt pulverized coal and fine coke). The equations of motion of these four phases have the same form and are solved by same technique. This way of modeling allows efficient process simulation including heat and mass transfer, and reaction analyses. The model has been applied to a variety of operating conditions, and revealed the in-furnace status of blast furnace in detail.

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Computational methods to simulate gas-solid-liquid flows in chemical reactors such as agitators, bubble columns and fluidized beds are reviewed. In addition, their problems are discussed, keeping in mind the present technical level and future problems of related commercial software.

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Slit nozzle gas-particle two-phase jet is numerically simulated by a vortex method, which has been proposed for gas-solid two-phase free turbulent flow by the current authors in a prior paper. The method can take account of the interaction between the two phases by calculating the behavior of the particle and the vortex element of gas-phase through the Lagrangian approach. The mean velocities and turbulent intensities of the two phases are in good agreement with the measured values. The effects of the particle on the velocity decay and momentum diffusion of the gas-phase are also favorably compared with the measurement. These indicate the applicability of the present vortex method to the gas-particle two-phase jet.

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Fundamental forms of correlations are proposed for large bubble lengths, liquid slug lengths and slug unit lengths, and slug periods of gas-liquid two-phase slug flow in vertical pipes. The empirical correlations for each length are derived using 337 experimental data obtained from water-air, water solution containing NaOH, K₃Fe(CN)₆ and K₄Fe(CN)₆- air and water-carbon dioxide two-phase slug flows in pipes of their diameter 8-50.8mm, and volumetric fluxes J_G=0.0256−3.42m/s, J_L=0.0390−1.50m/s, J_T=0.0847−4.40 m/s. They are shown below. For large bubble length L_LB: L_LB=0.121D(J_G/J_T)^1.70(J_L/J_T)^-0.408(J_T/(gD)^0.5)^0.582(ρ_LDJ_T/μ_L)^0.149(ρ_LDJ_T²/σ)^-0.207(μ_G/μ_L)^0.291(ρ_G/ρ_L)^-0.831+0.00186. For liquid slug length L_LS: L_LS=534000D(J_G/J_T)^0.415(J_L/J_T)^0.143(J_T/(gD)^0.5)^0.143(ρ_LDJ_T/μ_L)^0.0983(ρ_LDJ_T²/σ)^-0.119(μ_G/μ_L)^1.34(ρ_G/ρ_L)^0.753−0.124. For slug unit length L_SU: L_SU=19700D(J_G/J_T)^0.305(J_L/J_T)^-0.340(J_T/(gD)^0.5)^0.393(ρ_LDJ_T/μ_L)^0.0275(ρ_LDJ_T²/σ)^-0.154(μ_G/μ_L)^0.944(ρ_G/ρ_L)^0.346−0.291. The similar method is applied to derive the correlation for the slug period. The following correlation is obtained using 205 experimental data. T_B=16100(D/J_T)(J_G/J_T)^1.38(J_L/J_T)^-0.166(J_T(gD)^0.5)^-0.317(ρ_LDJ_T/μ_L)^1.61(ρ_LDJ_T²/σ)^-0.564(μ_G/μ_L)^0.333(ρ_G/ρ_L)^3.04+0.0870. These empirical correlations are applicable for gas-liquid two-phase slug flow under nearly room temperature and at atmospheric pressure.

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