顆粒間的碰撞是氣體–顆粒流中一項極為重要的物理機制，當顆粒的數目極大時，顯著的顆粒間碰撞將使得整個物理問題變得複雜難解。本研究針對碰撞主導流(collision-dominated flows)與接觸主導流(contact-dominated flows)兩種不可忽略顆粒間碰撞效應的流場，提出相應的數值模擬法。首先，以直接模擬蒙地卡羅法(direct simulation Monte Carlo method)結合雷諾平均那維爾史托克方程式(Reynolds-averaged Navier-Stokes equations)以來處理紊態碰撞主導流，並於具有實驗資料的完全發展之紊態具顆粒負載的渠道流中進行驗證。經測試顯示，蒙地卡羅直接模擬法與決定論法(deterministic method)相較，可獲得一致的結果。透過參數敏感度的分析，以較少的樣本顆粒進行蒙地卡羅直接模擬法的運算時，需要較長的採樣時間以達到有效的統計結果，但蒙地卡羅直接模擬法在計算上依然顯現出極佳的經濟效益，可紓解應用決定論法在模擬大量顆粒運動上的計算負荷。離散元素法(discrete element method)起源於模擬地質學上的岩塊運動，目前已廣泛的應用於計算接觸主導流場內的顆粒間接觸力，但其以決定論法搭配軟球模式的方式在計算上造成極大的負擔。本研究提出虛擬顆粒法，根據一個虛擬顆粒來代表一群實際顆粒的概念，推導虛擬系統與實際系統間符合物理特性的相對關係式。針對內含D型Geldart顆粒的噴流式(spouted)流體化床進行的驗證顯示，基本型離散元素法可獲得可靠的模擬結果。隨後應用基本型離散元素法驗證虛擬顆粒法於內含B型Geldart顆粒的氣泡式流體化床，模擬結果顯示虛擬顆粒法可獲得與實際顆粒模擬相同的大尺度流場運動特性，並顯著的節省計算上的負擔。

An important physical mechanism in the gas-particle flows is inter-particle collisions, which makes the entire problem complicated and difficult to be revolved while the number of particles becomes huge. Depending on the frequency of inter-particle collisions, gas-particle flows can be classified into three categories of collision-free flows, collision-dominated flows, and contact-dominated flows. The Lagrangian approach is used here to deal the effects of inter-particle collisions in collision-dominated flows and contact-dominated flows.
A Lagrangian modeling approach, which combines the direct simulation Monte Carlo (DSMC) method to account for inter-particle collisions and a Reynolds-averaged Navier-Stokes model for turbulence characteristics of the carrier fluid, is developed for the simulation of collision-dominated flows. Two wall-bounded turbulent particle-laden flows in which the experimental data are available are chosen as the test problems. Results obtained with the deterministic method accounting for inter-particle collisions are used as a basis for validating the proposed stochastic Lagrangian model. Good agreement between the predictions obtained separately with the deterministic and computational economic DSMC methods is achieved. Furthermore, the benefit of saving computational expenditure in use of the DSMC method becomes more remarkable than the deterministic method as the number of particles loaded in the flow is increased. Through a parametric sensitivity study, it is demonstrated that longer sample collecting time in the statistic calculations of the particles’ flow properties is required as less number of sampling particles are tracked in the DSMC process.
Discrete element method (DEM) is widely applied to account for the contact forces among particles in the simulation of contact-dominated flows. The DEM method is in nature of the deterministic method. In order to reduce the computation expenditure, a virtual particle method is proposed to reduce effectively the real particle number in the simulation. It is shown that the proposed virtual particle method is capable of yielding qualitatively accurate prediction on the bulk scale of flow motion, as what obtained in the computation made with tracking all real particles in a bubbling fluidized bed of Group-B particles. Moreover, the computational expenditure can be noticeably reduced using the virtual particle method.

Abe, K., Jang, Y. -J., & Leschziner, N. A., 2003. An investigation of wall-anisotropy expressions and length-scale equations for non-linear eddy-viscosity models. Int. J. Heat Fluid Flow, 24, 181-198.
Abe, H., Kuwamura, H, & Matsuo, Y., 2001. Direct numerical simulation of a fully developed turbulent channel flow with respect to the Reynolds number dependence. J. Fluid Eng., 123, 382-393.
Abrahamson, J, 1975. Collision rates of small particles in a vigorously turbulent fluid. Chem. Eng. Sci., 30, 1371-1379.
Anderson, T. B. & Jackson, R., 1967. A fluid mechanical description of fluidised beds. I&EC Fundam., 6, 527-539.
Barrett, R., Berry, M., Chan, T. F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., & Van der Vorst, H., 1994. Templates for the solution of linear systems: Building blocks for iterative methods. SIAM, 27.
Bird, G. A., 1994, Molecular gas dynamics and the direct simulation of gas flows. Clarendon, Oxford.
Chang, K.C. & Jeng, S.S, 2002. Coefficients of time and length scales of turbulent eddies. Chinese J. Mech. Series A, 18, 193-197.
Chen, A., Grace, J.R., Epstein, N., & Lim, C. J., 2002. Steady state dispersion of mono-size, binary and multi-size particles in a liquid fluidized bed classifier. Chem. Eng. Sci., 57, 991-1002.
Comiti, J. & Renaud, M., 1989. A new model for determining mean structure parameters of fixed beds form pressure drop measurements: application to beds packed with parallerlepipedal particles. Chem. Emg. Sci., 44, 1539-1545.
Crowe, C. T., Sommerfeld, M., & Tsuji, Y., 1998. Multiphase Flows with Droplets and Particles. CRC Press.
Crowe, C.T., Troutt, T.R., & Chung J.N., 1996. Numerical models for two-phase turbulent flows. Annu. Rev. Fluid Mech., 28, 11-43.
Cundall, P. A. & Strack, O. D. L., 1979. A discrete numerical model for granular assemblies. Geotechnique, 29, 47-65.
Dennis, S.C.R., Singh, S.N., & Ingham, D.B., 1980. The steady flow due to a rotating sphere at low and moderate Reynolds number. J. Fluid Mech., 101, 257-279.
Di Felice, R., 1994. The voidage function for fluid-particle interaction systems. Int. J. Multiphase Flow, 20, 153-150.
Eisfeld, B. & Schnitzlein, K., 2001. The influence of confining walls on the pressure drop in packed beds. Chem. Eng. Sci., 56, 4321-4329.
Ergun, S., 1952. Fluid flow through packed columns. Chem. Eng. Prog.. 48, 89-94.
Geldart, D., 1973. Types of gas fluidisation. Powder Tech., 7, 285-292.
Gera, D., Gautam, M., Tsuji, Y., Kawaguchi, T., & Tanaka, T., 1998. Computer simulation of bubbles in large-particle fluidized beds. Powder Tech., 98, 38-47.
Gera, D., Syamlal, M., & O’Brien, T. J., 2004. Hydrodynamics of particle segregation in fluidized beds. Int. J. Multiphase Flow, 30, 419-428.
Hoffmann, A. C., Janssen, L. P. B. M., & Prins, J., 1993. Particle segregation in fluidised binary mixtures. Chem. Eng. Sci., 48, 1583-.
Hommans, B.P.B., Kuipers, J. A. M., & van Swaaij, W. P. M., 2000. Granular dynamics simulation of segregation phenomena in bubbling gas-fluidized beds. Powder Tech., 109, 41-48.
Horio, M. & Zhang, W., 2004. Four topics for further development of DEM to deal with industrial fluidization issues. 5th Int. Conference Multiphase Flow, ICMF’04, Yokohama, Japan, May 30 ~ June 4, 2004. Paper No. PL5.
Illner, R. & Neunzert, H., 1987. On simulation method for the Boltzmann equation. Transp. Theor. Stat., Phys., 19, 141-154.
Kafui, K.D., Thornton, C., & Adams, M. J., 2002. Discrete particle-continuum fluid modeling of gas-solid fluidised beds. Chem. Eng. Sci., 57, 2395-2410.
Kawaguchi, T., Kajiyama, S., Tanaka, T., & Tsuji, Y., 2002. DEM simulation of 2-D fluidized bed using similarity model. Proceedings 3rd Int. Conference Discrete Element Method, 161-171.
Kawaguchi, T., Sakamoto, M., Tanaka, T., & Tsuji, Y., 2000. Quasi-three-dimensional numerical simulation of spouted beds in cylinder. Powder Tech., 109, 3-12.
Kawaguchi, T., Tanaka, T., & Tsuji, Y., 1998. Numerical simulation of two-dimensional fluidized beds using the discrete element method (comparison between the two- and three-dimensional models). Powder Tech., 96, 129-138.
Kazari, M., Roko, K., Kawaguchi, T., Tanaka, T., & Tsuji, Y., 1995. A study on conditions for similarity of particle motion in numerical simulation of dense gas-solid two phase flow. Proceedings 2nd Int. Conference Multiphase Flow, FB2-9 ~ FB 2-15.
Khan, A. R. & Richardson, J. F., 1990. Pressure gradient and friction factor for sedimentation and fluidisation of uniform spheres in liquids. Chem. Eng. Sci., 45, 255-265.
Koch, D. L. & Hill, R. J., 2001. Inertial effects in suspension and porous-media flows. Annu. Rev. Fluid Mech., 33, 619-647.
Kulick, J. D., Fessler, J. R., & Eaton, J. K., 1994. Particles response and turbulence modification in fully developed channel flow. J. Fluid Mech., 277, 108-134.
Kuwagi, K, Takeda, H, & Horio, M., 2004. The similar particle assembly (SPA) model, an approach to large-scale discrete element (DEM) simulation. Fluidization XI, Italy, AMS9.
Lain, S., Sommerfeld, M., & Kussin, J., 2002. Experimental studies and modeling of four-way coupling in particle-laden horizontal channel flow. Int. J. Heat Fluid Flow, 23, 647-656.
Liu, Z., Kawaguchi, T., Tanaka, T., & Tsuji, Y., 2002. The effect of temperature on the minimum fluidization velocity calculated by distinct element method. JSME Int. J. B, 45, 66-71.
Lu, P. J., Pan, D. Z., & Yeh, D. Y., 1995. Transonic flutter suppression using active acoustic excitation. AIAA J., 33, 694-702.
Mei, R., 1992. An approximate expression for the shear lift force on a spherical particle at finite Reynolds number. Int. J. Multiphase Flow, 18, 145-147.
Michaelides, E. E., 1997. Review - The transient equation of motion for particles, bubbles, and droplets. Trans. ASME I: J. Fluid Eng., 119. 233-247.
Miller, T.F. & Schmidt, F.W., 1988. Use of a pressure-weighted interpolation method for the solution of the incompressible Navier-Stokes equations on a nonstaggered grid system. Num. Heat Trans., 14, 213-233.
Mito, Y. & Hanratty, T. J., 2002. Use of a modified Langevin equation to describe turbulent dispersion of fluid particles in a channel flow. Flow, Turbulence and Combustion, 68, 1-26.
Mostafa, A. A. & Mongia, H. C., 1987. On the modeling of turbulent evaporating sprays: Eulerian versus Lagrangian approach. Int. J. Heat Mass Trans., 30, 2583-2593.
Nienow, A. W., Naimer, N. S., & Chiba, T., 1987. Studies of segregation / mixing in fluidised beds of different size particles. Chem. Eng. Commun., 62, 52-.
Oesterle, B. & Bui Dinh, T., 1998. Experiments on the lift of a spinning sphere in a range of intermediate Reynolds number. Exp. Fluids, 25, 16-22.
Omori, Y, 1987. Blast furnace phenomena and modeling. Elsevier Applied Science, London.
Pan, Y., Tanaka, T., & Tsuji, Y., 2002. Turbulence modulation by dispersed solid particles in rotating channel flows. Int. J. Multiphase Flow, 28, 527-552.
Pasquill, F. & Smith F. B., 1983. Atmospheric diffusion, 3rd edition. John Wiley & Sons.
Patankar, S. V., 1980. Numerical heat transfer and fluid flow. Hemisphere, New York.
Reichelt, W., 1972. Zur Berechnung des Druckverlustes einphasig durchstromter Kugel- und Zylinderschuttungen. Chemie-In-genieur-Technik, 44, 1068-1071.
Rhodes, M, 2001. Fluidization of particles by fluids. Educ. Reso. for Part. Techn. 012Q-Rhodes, http://www.erpt.org/012Q/rhod-00.htm.
Sakano, M., Yaso, T., & Nakanishi, H., 2000. Numerical simulation of two-dimensional fluidized bed using discrete element method with imaginary sphere model. Japanese. J. Multiphase Flow, 14, 66-73.
Schiller, P. R. & Naumann, A., 1933. Uber die grundlegenden Berechungen bei der Schwerkraftaufbereitung. Ver. Deut. Ing., 77, 318-320.
Sommerfeld, M., 2001. Validation of a stochastic Lagrangian modeling approach for inter-particle collisions in homogeneous isotropic turbulence. Int. J. Multiphase Flow, 27, 1829-1858.
Sommerfeld, M., 2003. Analysis of collision effects for turbulent gas-particle flow in a horizontal channel: Part I: Particle transport. Int. J. Multiphase Flow, 29, 675-699.
Takagi, H., 1977. Viscous flow induced by slow rotation of a sphere. J. Phys. Soc. Japan, 42, 319-325.
Takeda, H. & Horio, M., 2000. Granular flow simulation by a hybrid model of DEM and continuous model. Proceedings 6th SCEJ Symp. Fluidization, 162-167.
Tanaka, T. & Tsuji, Y., 1991. Numerical simulation of gas-solid two-phase flow in a vertical pipe: On the effect of inter-particle collision. In Gas-Solid Flows, FED-121, 123-128, ASME.
Tanaka, T., Yonemura, S., Kiribayashi, K., & Tsuji, Y., 1993. Cluster formation and particle-induced instability of gas-solid flow (Numerical simulation of flow in vertical channel using DSMC method). Trans. Jpn. Soc. Mech. Eng. B, 59, 2982-2989.
Tanaka, T., Yonemura, S., Kiribayashi, K., & Tsuji, Y., 1996. Cluster formation and particle-induced instability in gas-solid flows predicted by the DSMC method. JSME Int. J., Ser. B, 39, 239-245.
Tsuji, Y., 2000. Activities in discrete particle simulation in Japan, Powder Tech., 113, 278-286.
Tsuji, Y., Kawaguchi, T., & Tanaka, T., 1993. Discrete particle simulation of two-dimensional fluidized bed. Powder Tech., 77, 79-87.
Tsuji, Y., Tanaka, T., & Ishida, T., 1992. Lagrangian numerical simulation of plug flow of collisionless particles in a horizontal pipe. Powder Tech., 71, 239-250.
Van Doormaal, J. P. & Raithby, G. D., 1984. Enhancement of the SIMPLE method for predicting incompressible fluid flows. Num. Heat Trans., 7, 147-163.
Yamamoto, Y., Potthoff, M., Tanaka, T., Kajishima, T., & Tsuji, Y., 2001. Large-eddy simulation of turbulent gas-particle flow in a vertical channel: Effect of considering inter-particle collisions. J. Fluid Mech., 442, 303-334.
Yonemura, S., Tanaka, T., & Tsuji, Y., 1995. Numerical simulations of cluster formation in gas-solid flow (Effects of particle size and particle concentration on the structure of clusters). Trans. Jpn. Soc. Mech. Eng. B, 61, 3671-3678.
Yu, A. B. & Xu, B. H., 2003. Particle-scale modeling of gas-solid flow in fluidisation. J. Chem. Tech. Biotech., 78, 111-121.
Yuu, S., Nishikawa, H., & Umekage, T., 2001. Numerical simulation of air and particle motions in group-B particle turbulent fluidized bed. Powder Tech., 118, 32-44.
Yuu, S., Umekage, T., & Johno, Y., 2000. Numerical simulation of air and particle motions in bubbling fluidized bed of small particles. Powder Tech., 110, 158-168.
Wen, C. Y. & Yu, Y. H., 1966. Mechanics of fluidization. Chem. Eng. Prog. Symp. Ser., 62, 100-111.