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Study of isentropic sound speed of two-phase or multiphase flow has theoretical significance and wide application background. As is well known, the speed of sound in fluid containing particles in suspension differs from that in the pure fluid. In the particular case of bubbly liquids (gas liquid two-phase flow), the researches find that the differences can be drastic. Up to now, the isentropic speed of sound in the flow field with a small volume fraction of bubbles (less than 1%), has been investigated fully both experimentally and theoretically. In this paper, we consider another situation, as the case with solid particles in gas, which is the so-called gas particle two-phase flow. Although many results have been obtained in gas liquid two-phase flow, there is still a lot of basic work to do due to the large differences in the flow structure and flow pattern between gas particle two-phase flow and gas liquid two-phase flow. Treating the gas particle suspension as the relaxed equilibrium, thermodynamic arguments are used to obtain the isentropic speed of sound. Unlike the existing work, we are dedicated to developing the computational model under dense condition. The space volume occupied by particle phase and the interaction between particles are overall considered, then a new formula of isentropic sound speed is derived. The new formula includes formulae of the pure gas flow and the already existing dilute gas particle two-phase flow as a special case. On the one hand, the correctness of our formula is verified. On the other hand, the new formula is more general. The variations of sound speed with different mass fractions of particle phase are analyzed. The theoretical calculation results show that the overall physical law of sound speed change is that with the increase of the particle mass fraction, the sound speed first decreases and then increases. The velocity of sound propagation in gas particle two-phase flow is far smaller than in pure gas in a wide range, so it is easy to reach the supersonic condition. When the particle volume fraction is below 10%, the result is consistent with Prandtl theoretical analysis. In this range, the influences of the particle phase pressure modeling parameters can be neglected. When the particle volume fraction is more than 10%, the particle phase pressure modeling parameters produce influences. Furthermore the corresponding physical principles and the mechanisms are discussed and revealed. The new formula and physical understandings obtained in this paper will provide a theoretical support for the researches of dense gas particle two-phase compressible flow and related engineering applications.
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Keywords:
- dense gas particle two-phase flow /
- compressible fluid /
- isentropic speed of sound /
- computational modeling
[1] Liu D Y 1990 Chin. J. Theoret. Appl. Mech. 22 660 (in Chinese) [刘大有 1990 力学学报 22 660]
[2] Hu M B, Dang S C, Ma Q, Xia W D 2015 Chin. Phys. B 24 074502
[3] Nichita D V, Khalid P, Broseta D 2010 Fluid Phase Equilibr. 291 95
[4] Liu X Z, Wang Y X, Zhu S G, Li G H 2007 J. Engineer. Thermophys. 28 201 (in Chinese) [刘心志, 王益祥, 朱曙光, 李光辉 2007 工程热物理学报 28 201]
[5] Holloway W, Sundaresan S 2014 Chem. Eng. Sci. 108 67
[6] Valverde J M 2013 Soft Matter 9 8792
[7] Liu B, Fang D Y, Xia Z X, Wang L 2013 J. Propuls. Technol. 34 8 (in Chinese) [刘冰, 方丁酉, 夏智勋, 王林 2013 推进技术 34 8]
[8] Saito T 2002 J. Comput. Phys. 176 129
[9] Liu L, Wang Y S, Zhou F D 1999 Chin. J. Appl. Mechan. 16 22 (in Chinese) [刘磊, 王跃社, 周芳德 1999 应用力学学报 16 22]
[10] Wang P, Sun H Q, Shao J L, Qin C S, Li X Z 2012 Acta Phys. Sin. 61 234703 (in Chinese) [王裴, 孙海权, 邵建立, 秦承森, 李欣竹 2012 物理学 报 61 234703]
[11] Nguyen D L, Winter E R F, Greirer M 1981 Int. J. Multiphas. Flow 7 311
[12] Zhao J F, Li W 1999 J. Basic Sci. Engineer. 7 321 (in Chinese) [赵建福, 李炜 1999 应用基础与工程科学学报 7 321]
[13] Zeng D L, Zhao L J, Xiao Y 2001 Proceedings of The International Conference on Energy Conversion and Application Wuhan, China 200 65
[14] Huang F, Bai B F, Guo L J 2004 Prog. Nat. Sci. 14 344
[15] Chaudhuri A, Osterhoudt C F, Sinha D N 2012 ASME J. Fluid. Eng. 134 101301
[16] Zhao L J, Li B, Gao H, Li D S, Yuan Y X, Zeng D L 2007 J. Engineer. Thermophys. 28 388 (in Chinese) [赵良举, 李斌, 高虹, 李德胜, 袁悦祥, 曾丹苓 2007 工程热物理学报 28 388]
[17] Temkin S 1992 Phys. Fluid A 4 2399
[18] Fang D Y 1988 Two-phase Fluid Dynamics (Changsha: National University of Defense Technology Press) p14-22 (in Chinese) [方丁酉 1988 两相流 动力学(长沙: 国防科技大学出版社) 第 14-22 页]
[19] Li W X 2003 One-Dimensional Nonsteady Flow and Shock Waves (Beijing: National Defence of Industry Press) pp40-49, 206 (in Chinese) [李维新 2003 一维不定常流与冲击波(北京: 国防工业出版社) 第 40-49, 206页]
[20] Snider D M 2001 J. Comput. Phys. 170 523
[21] Harris S E, Crighton D G 1994 J. Fluid Mech. 266 243
[22] Auzerais F M, Jackson R, Russel W B 1988 J. Fluid Mech. 195 437
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[1] Liu D Y 1990 Chin. J. Theoret. Appl. Mech. 22 660 (in Chinese) [刘大有 1990 力学学报 22 660]
[2] Hu M B, Dang S C, Ma Q, Xia W D 2015 Chin. Phys. B 24 074502
[3] Nichita D V, Khalid P, Broseta D 2010 Fluid Phase Equilibr. 291 95
[4] Liu X Z, Wang Y X, Zhu S G, Li G H 2007 J. Engineer. Thermophys. 28 201 (in Chinese) [刘心志, 王益祥, 朱曙光, 李光辉 2007 工程热物理学报 28 201]
[5] Holloway W, Sundaresan S 2014 Chem. Eng. Sci. 108 67
[6] Valverde J M 2013 Soft Matter 9 8792
[7] Liu B, Fang D Y, Xia Z X, Wang L 2013 J. Propuls. Technol. 34 8 (in Chinese) [刘冰, 方丁酉, 夏智勋, 王林 2013 推进技术 34 8]
[8] Saito T 2002 J. Comput. Phys. 176 129
[9] Liu L, Wang Y S, Zhou F D 1999 Chin. J. Appl. Mechan. 16 22 (in Chinese) [刘磊, 王跃社, 周芳德 1999 应用力学学报 16 22]
[10] Wang P, Sun H Q, Shao J L, Qin C S, Li X Z 2012 Acta Phys. Sin. 61 234703 (in Chinese) [王裴, 孙海权, 邵建立, 秦承森, 李欣竹 2012 物理学 报 61 234703]
[11] Nguyen D L, Winter E R F, Greirer M 1981 Int. J. Multiphas. Flow 7 311
[12] Zhao J F, Li W 1999 J. Basic Sci. Engineer. 7 321 (in Chinese) [赵建福, 李炜 1999 应用基础与工程科学学报 7 321]
[13] Zeng D L, Zhao L J, Xiao Y 2001 Proceedings of The International Conference on Energy Conversion and Application Wuhan, China 200 65
[14] Huang F, Bai B F, Guo L J 2004 Prog. Nat. Sci. 14 344
[15] Chaudhuri A, Osterhoudt C F, Sinha D N 2012 ASME J. Fluid. Eng. 134 101301
[16] Zhao L J, Li B, Gao H, Li D S, Yuan Y X, Zeng D L 2007 J. Engineer. Thermophys. 28 388 (in Chinese) [赵良举, 李斌, 高虹, 李德胜, 袁悦祥, 曾丹苓 2007 工程热物理学报 28 388]
[17] Temkin S 1992 Phys. Fluid A 4 2399
[18] Fang D Y 1988 Two-phase Fluid Dynamics (Changsha: National University of Defense Technology Press) p14-22 (in Chinese) [方丁酉 1988 两相流 动力学(长沙: 国防科技大学出版社) 第 14-22 页]
[19] Li W X 2003 One-Dimensional Nonsteady Flow and Shock Waves (Beijing: National Defence of Industry Press) pp40-49, 206 (in Chinese) [李维新 2003 一维不定常流与冲击波(北京: 国防工业出版社) 第 40-49, 206页]
[20] Snider D M 2001 J. Comput. Phys. 170 523
[21] Harris S E, Crighton D G 1994 J. Fluid Mech. 266 243
[22] Auzerais F M, Jackson R, Russel W B 1988 J. Fluid Mech. 195 437
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