Introduction
As in Chapter 1, from the large number of formulations of the conservation equations for multiphase flows, local volume averaging as founded by Anderson and Jackson, Slattery, and Whitaker was selected to derive rigorously the momentum equations for multiphase flows conditionally and divided into three velocity fields. The heterogeneous porous-media formulation introduced by Gentry et al., commented on by Hirt, and used by Sha, Chao, and Soo, is then implanted into the formalism as a geometrical skeleton. Beyond these concepts, I perform subsequent time averaging. This yields a working form that is applicable to a large variety of problems. All interfacial integrals are suitably transformed in order to enable practical application. Some minor simplifications are introduced in the finally obtained general equation and working equations for each of the three velocity fields are recommended for general use in multiphase fluid dynamic analysis.
This chapter is an improved and extended version of the work published in Kolev (1994b). The strategy followed is: We first apply the momentum equations for each of the velocity fields, excluding the interfaces by replacing their actions by forces. Then, we write a force balance at the interfaces, considering them as immaterial and therefore inertialess. This interfacial force balance links the momentum equations that are valid for the both sides of the interface.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Albring, W.: Angewandte Strömungslehre. Theodor Steinkopff, Dresden (1970)
Anderson, T.B., Jackson, R.: A fluid mechanical description of fluidized beds. Ind. Eng. Fundam. 6, 527 (1967)
Antal, S.P., Lahey Jr., R.T., Flaherty, J.E.: Analysis of phase distribution in a fully developed laminar bubbly two-phase flow. Int. J. Multiphase Flow 17(5), 635–652 (1991)
Auton, R.T.: The lift force on a spherical body in rotating flow. J. Fluid Mechanics 183, 199–218 (1987)
Barnea, D., Taitel, Y.: Interfacial and structural stability. Int. J. Multiphase Flow 20(suppl.), 387–414 (1994)
Bataille, J., Lance, M., Marie, J.L.: Bubble turbulent shear flows. In: ASME Winter Annular Meeting, Dallas (1990)
Bernemann, K., Steiff, A., Weinspach, P.M.: Zum Einfluss von längsangeströmten Rohrbündeln auf die großräumige Flüssigkeitsströmung in Blasensäulen. Chem. Ing. Tech. 63(1), 76–77 (1991)
Biberg, D.: An explicit approximation for the wetted angle in two-phase stratified pipe flow. The Canadian Journal of Chemical Engineering 77, 1221–1224 (1999)
Biesheuvel, A., van Wijngaarden, L.: Two-phase flow equations for a dilute dispersion of gas bubbles in liquid. J. Fluid Mechanics 168, 301–318 (1984)
Biesheuvel, A., Spoelstra, S.: The added mass coefficient of a dispersion of spherical gas bubbles in liquid. Int. J. Multiphase Flow 15, 911–924 (1989)
Bournaski, E.: Numerical simulation of unsteady multiphase pipeline flow with virtual mass effect. Int. J. Numer. Meth. Eng. 34, 727–740 (1992)
Boussinesq, J.: Essai sur la théorie des eaux courantes. Mem. Pŕs. Acad. Sci., Paris 23, 46 (1877)
Brackbill, J.U., Kothe, D.B., Zemach, C.: A continuum method for modeling surface tension. J. Comput. Phys. 100, 335–354 (1992)
Brauner, N., Maron, D.M.: Stability analysis of stratified liquid–liquid flow. Int. J. Multiphase Flow 18(1), 103–121 (1992)
Crowe, C.T., Pratt, D.T.: Analysis of the flow field in cyclone separators. Comp. Fluids 2, 249–260 (1974)
Cook, T.L., Harlow, F.H.: VORT: A computer code for bubble two-phase flow. Los Alamos National Laboratory documents LA-10021-MS (1983)
Cook, T.L., Harlow, F.H.: Virtual mass in multi-phase flow. Int. J. Multiphase Flow 10(6), 691–696 (1984)
de Crecy, F.: Modeling of stratified two-phase flow in pipes, pumps and other devices. Int. J. Multiphase Flow 12(3), 307–323 (1986)
Deich, M.E., Philipoff, G.A.: Gas dynamics of two phase flows, Energoisdat, Moskva (1981)
Delhaye, J.M.: Basic equations for two-phase flow. In: Bergles, A.E., et al. (eds.) Two-Phase Flow and Heat Transfer in Power and Process Industries. Hemisphere Publishing Corporation, McGraw-Hill Book Company, New York (1981)
Delhaye, J.M., Giot, M., Riethmuller, M.L.: Thermohydraulics of two-phase systems for industrial design and nuclear engineering. Hemisphere Publishing Corporation, McGraw Hill Book Company, New York (1981)
Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge Univ. Press, Cambridge (1981)
Drew, D.A., Lahey Jr., R.T.: The virtual mass and lift force on a sphere in rotating and straining flow. Int. J. Multiphase Flow 13(1), 113–121 (1987)
Erichhorn, R., Small, S.: Experiments on the lift and drag of spheres suspended in a Poiseuille flow. J. Fluid Mech. 20(3), 513 (1969)
Gray, W.G., Lee, P.C.Y.: On the theorems for local volume averaging of multi-phase system. Int. J. Multi-Phase Flow 3, 222–340 (1977)
Helmholtz, H.: Über diskuntinuirliche Flüssigkeitsbewegungen, Monatsberichte der Königlichen Akademie der Wissenschaften zu Berlin, pp. 215–228 (1868)
Hetstrony, G.: Handbook of multi-phase systems. Hemisphere Publ. Corp., McGraw-Hill Book Company, Washington, New York (1982)
Hirt, C.W., Nichols, B.D.: Volume of fluid (VOF) method for dynamics of free boundaries. J. of Comp. Physics 39, 201–225 (1981)
Ho, B.P., Leal, L.G.: Internal migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 78(2), 385 (1976)
Hwang, G.J., Schen, H.H.: Tensorial solid phase pressure from hydrodynamic interaction in fluid-solid flows. In: Proc. of the Fifth International Topical Meeting on Reactor Thermal Hydraulics, NURETH-5, Salt Lake City, UT, USA, September 21-24, vol. IV, pp. 966–971 (1992)
Ishii, M.: Thermo-fluid dynamic theory of two-phase flow. Eyrolles, Paris (1975)
Ishii, M., Michima, K.: Two-fluid model and hydrodynamic constitutive relations. NSE 82, 107–126 (1984)
Jeffrey, D.: Condition to a random suspension of spheres. Proc. R. Soc. London A335, 355–367 (1973)
Kendoush, A.A.: Modification of the classical theory of the virtual mass of an accelerated spherical particle. In: Proc. of the FEDSM 2006, 2006 ASME Joint US-European Fluids Engineering Summer Meeting, Miami, FL, July 17-20 (2006)
Klausner, J.F., Mei, R., Bernhard, D., Zeng, L.Z.: Vapor bubble departure in forced convection boiling. Int. J. Heat Mass Transfer 36, 651–662 (1993)
Kolev, N.I.: Two-phase two-component flow (air-water steam-water) among the safety compartments of the nuclear power plants with water cooled nuclear reactors during lose of coolant accidents, PhD Thesis, Technical University Dresden (1977)
Kolev, N.I.: Transiente Dreiphasen-Dreikomponenten Strömung, Teil 1: Formulierung des Differentialgleichungssystems, KfK 3910 (March 1985)
Kolev, N.I.: Transiente Dreiphasen-Dreikomponenten Strömung, Teil 3: 3D-Dreifluid-Diffusionsmodell, KfK 4080 (1986a)
Kolev, N.I.: Transient three-dimensional three-phase three-component non equilibrium flow in porous bodies described by three-velocity fields. Kernenergie 29(10), 383–392 (1986b)
Kolev, N.I.: A three field-diffusion model of three-phase, three-component Flow for the transient 3D-computer code IVA2/001. Nuclear Technology 78, 95–131 (1987)
Kolev, N.I.: IVA3: A transient 3D three-phase, three-component flow analyzer. In: Proc. of the Int. Top. Meeting on Safety of Thermal Reactors, Portland, Oregon, July 21-25, pp. 171–180 (1991a); Also presented at the 7th Meeting of the IAHR Working Group on Advanced Nuclear Reactor Thermal-Hydraulics, Kernforschungszentrum Karlsruhe, August 27-29 (1991)
Kolev, N.I.: A three-field model of transient 3D multi-phase, three-component flow for the computer code IVA3, Part 1: Theoretical basics: conservation and state equations, Numerics. KfK 4948, Kernforschungszentrum Karlsruhe (September 1991b)
Kolev, N.I., Tomiyama, A., Sakaguchi, T.: Modeling of the mechanical interaction between the velocity fields in three-phase flow. Experimental Thermal and Fluid Science 4(5), 525–545 (1991)
Kolev, N.I.: The code IVA3 for modeling of transient three-phase flows in complicated 3D geometry. Kerntechnik 58(3), 147–156 (1993)
Kolev, N.I.: IVA3 NW: Computer code for modeling of transient three-phase flow in complicated 3D geometry connected with industrial networks. In: Proc. of the Sixth Int. Top. Meeting on Nuclear Reactor Thermal Hydraulics, Grenoble, France, October 5-8 (1993)
Kolev, N.I.: Berechnung der Fluiddynamischen Vorgänge bei einem Sperrwasser-Kühlerrohrbruch, Projekt KKW Emsland, Siemens KWU Report R232/93/0002 (1993)
Kolev, N.I.: IVA3-NW A three phase flow network analyzer. Input description, Siemens KWU Report R232/93/E0041 (1993)
Kolev, N.I.: IVA3-NW components: relief valves, pumps, heat structures, Siemens KWU Report R232/93/E0050 (1993)
Kolev, N.I.: IVA4: Modeling of mass conservation in multi-phase multi-component flows in heterogeneous porous media. Kerntechnik 59(4-5), 226–237 (1994a)
Kolev, N.I.: The code IVA4: Modelling of momentum conservation in multi-phase multi-component flows in heterogeneous porous media. Kerntechnik 59(6), 249–258 (1994b)
Kolev, N.I.: The code IVA4: Second law of thermodynamics for multi phase flows in heterogeneous porous media. Kerntechnik 60(1), 1–39 (1995)
Kolev, N.I.: Three Fluid Modeling with Dynamic Fragmentation and Coalescence Fiction or Daily practice? In: 7th FARO Experts Group Meeting Ispra, October 15-16 (1996); Proceedings of OECD/CSNI Workshop on Transient Thermal-Hydraulic and Neutronic Codes Requirements, Annapolis, MD, November 5–8 (1996); 4th World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, ExHFT 4, Brussels, June 2-6 (1997); ASME Fluids Engineering Conference & Exhibition, The Hyatt Regency Vancouver, British Columbia, June 22-26 (1997); Invited Paper; Proceedings of 1997 International Seminar on Vapor Explosions and Explosive Eruptions (AMIGO-IMI), Aoba Kinen Kaikan of Tohoku University, Sendai-City, Japan, May 22-24 (1997)
Kolev, N.I.: Comments on the entropy concept. Kerntechnik 62(1), 67–70 (1997)
Kolev, N.I.: Verification of IVA5 computer code for melt-water interaction analysis, Part 1: Single phase flow, Part 2: Two-phase flow, three-phase flow with cold and hot solid spheres, Part 3: Three-phase flow with dynamic fragmentation and coalescence, Part 4: Three-phase flow with dynamic fragmentation and coalescence – alumna experiments. In: CD Proceedings of the Ninth International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-9), San Francisco, CA, October 3-8 (1999); Log. Nr. 315
Kreith, F., Sonju, O.K.: The decay of turbulent swirl in pipe. J. Fluid Mech. 2, part 2, 257–271 (1965)
Krepper, E., Lucas, D., Prasser, H.-M.: On the modeling of bubbly flow in vertical pipes. Nucl. Eng. Des. 235, 597–611 (2005)
Krepper, E., Egorov, Y.: CFD-Modeling of subcooled boiling and application to simulate a hot channel of fuel assembly. In: 13th Int. Conference on Nuclear Engineering, Beijing, China, May 16-20 (2005)
Lahey Jr., R.T.: Void wave propagation phenomena in two-phase flow. AIChE Journal 31(1), 123–135 (1991)
Lahey Jr., R.T.: The analysis of phase separation and phase distribution phenomena using two-fluid models. NED 122, 17–40 (1990)
Lahey Jr., R.T., Lopez de Bertodano, M., Jones Jr., O.C.: Phase distribution in complex geometry conditions. Nucl. Eng. Des. 141, 177–201 (1993)
Lance, M., Bataille, J.: Turbulence in the liquid phase of a uniform bubbly air-water flow. J. Fluid Mech. 22, 95–118 (1991)
Lamb, M.A.: Hydrodynamics. Cambridge University Press, Cambridge (1945)
Lamb, H.: Hydrodynamics. Dover, New York (1945)
Laurien, E., Niemann, J.: Determination of the virtual mass coefficient for dense bubbly flows by direct numerical simulation. In: 5th Int. Conf. on Multiphase Flow, Yokohama, Japan, paper no 388 (2004)
Lopez de Bertodano, M.: Turbulent bubbly two-phase flow in triangular duct, PhD Thesis, Renssaelaer Polytechnic Institute, Troy, NY (1992)
Mamaev, W.A., Odicharia, G.S., Semeonov, N.I., Tociging, A.A.: Gidrodinamika gasogidkostnych smesey w trubach, Moskva (1969)
Mei, R.: An approximate expression for the shear lift force on spherical particle at finite Reynolds number. Int. J. Multiphase Flow 18(1), 145–147 (1992)
Mei, R., Klausner, J.F.: Shear lift force on spherical bubbles. Int. J. Heat Fluid Flow 15, 62–65 (1995)
Milne-Thomson, L.M.: Theoretical Hydrodynamics. MacMillan & Co. Ltd., London (1968)
Mokeyev, G.Y.: Effect of particle concentration on their drag induced mass. Fluid. Mech. Sov. Res. 6, 161 (1977)
Naciri, A.: Contribution à l’étude des forces exercées par un liquide sur une bulle de gaz: portance, masse ajoutée et interactions hydrodynamiques, Doctoral Dissertation, École Central de Lyon, France (1992)
Nigmatulin, R.I.: Spatial averaging in the mechanics of heterogeneous and dispersed systems. Int. J. Multiphase Flow 4, 353–385 (1979)
Nigmatulin, R.T.: Spatial averaging in the mechanics of heterogeneous and dispersed systems. Int. J. of Multiphase Flow 5, 353–389 (1979)
No, H.C., Kazimi, M.S.: Effects of virtual mass of the mathematical characteristics and numerical stability of the two-fluid model. In: NSE, vol. 89, pp. 197–206 (1985)
Pougatch, K., Salcudean, M., Chan, E., Knapper, B.: Modeling of compressible gas–liquid flow in convergent-divergent nozzle. Chemical Engineering Science 63, 4176–4188 (2008)
Prandtl, L.: Essentials of Fluid Dynamics, p. 342. Blackie & Son, Glasgow (1952)
Ransom, V.H., et al.: RELAP5/MOD2 Code manual, vol 1: Code structure, system models, and solution methods, NUREG/CR-4312, EGG-2396. rev. 1 (March 1987)
Ruggles, A.E., et al.: An investigation of the propagation of pressure perturbation in bubbly air/water flows. Trans. ASME J. Heat Transfer 110, 494–499 (1988)
Schlichting, H.: Boundary layer theory. McGraw-Hill, New York (1959)
Sha, T., Chao, B.T., Soo, S.L.: Porous-media formulation for multi-phase flow with heat transfer. Nuclear Engineering and Design 82, 93–106 (1984)
Shi, J.-M., Burns, A.D., Prasser, H.-M.: Turbulent dispersion in poly-dispersed gas–liquid flows in a vertical pipe. In: 13th Int. Conf. on Nuclear Engineering, ICONE, Beijing, China, May 16-20, vol. 13 (2005)
Slattery, J.C.: Flow of visco-elastic fluids through porous media. AIChE J. 13, 1066 (1967)
Slattery, J.C.: Interfacial transport phenomena. Springer, Heidelberg (1990)
Slattery, J.C.: Advanced transport phenomena. Cambridge University Press, Cambridge (1999)
Staffman, P.G.: The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22, Part 2, 385–400 (1965)
Staffman, P.G.: Corrigendum to “The lift on a small sphere in a slow shear flow”. J. Fluid Mech. 31, 624 (1968)
Steenberger, W.: Turbulent flow in a pipe with swirl, PhD thesis Eindhoven University of Technology (1995)
Stokes, G.G.: On the theories of the internal friction of fluids in motion and of the equilibrium and motion of elastic solids. Trans. Cambridge Phil. Soc. 8, 287–305 (1845)
Stuhmiller, J.H.: The influence of the interfacial pressure forces on the character of the two-phase flow model. In: Proc. of the 1977 ASME Symp. on Computational Techniques for Non-Equilibrium Two-Phase Phenomena, pp. 118–124 (1977)
Thomas Jr., G.B., Finney, R.L., Weir, M.D.: Calculus and analytic geometry, 9th edn. Addison-Wesley Publishing Company, Reading (1998)
Tomiyama, A.: Struggle with computational bubble dynamics. In: Proc. of the 3rd Int. Conf. on Multiphase Flow ICMF-1998, France (1998)
Tomiyama, A., et al.: Transverse migration of single bubbles in simple shear flows. Chem. Eng. Sci. 57, 1849–1858 (2002)
Trent, D.S., Eyler, L.L.: Application of the TEMPTEST computer code for simulating hydrogen distribution in model containment structures, PNL-SA-10781, DE 83 002725 (1983)
Truesdell, C.: Essays in the history of mechanics. Springer, New York (1968)
van Wijngaarden, L.: Hydrodynamic interaction between the gas bubbles in liquid. J. Fluid Mechanics 77, 27–44 (1976)
van Wijngaarden, L.: On pseudo turbulence. Theor. Comp. Fluid Dyn. 10, 449–458 (1998)
Vasseur, P., Cox, R.G.: The lateral migration of spherical particles in two-dimensional shear flows. J. Fluid Mech. 78, Part 2, 385–413 (1976)
Wang, S.K., Lee, S.J., Jones, O.C., Lahey Jr., R.T.: 3-D turbulence structure and phase distribution measurements in bubbly two-phase flows. Int. J. Multiphase Flow 13(3), 327–343 (1987)
Wellek, R.M., Agrawal, A.K., Skelland, A.H.P.: Shapes of liquid drops moving in liquid media. AIChE J. 12, 854 (1966)
Whitaker, S.: Diffusion and dispersion in porous media. AIChE Journal 13, 420 (1967)
Whitaker, S.: Advances in theory of fluid motion in porous media. Ind. Engrg. Chem. 61(12), 14–28 (1969)
Whitaker, S.: A Simple geometrical derivation of the spatial averaging theorem. Chem. Eng. Edu., 18–21, 50-52 (1985)
Winatabe, T., Hirano, M., Tanabe, F., Kamo, H.: The effect of the virtual mass force term on the numerical stability and efficiency of the system calculations. Nucl. Eng. Des. 120, 181–192 (1990)
Wallis, G.B.: One-dimensional two-phase flow. McGraw-Hill, New York (1969)
Yamamotto, Y., Potthoff, M., Tanaka, T., Kajishima, Tsui, Y.: Large-eddy simulation of turbulent gas-particcle flow in a vertical channel: effect of considering inter-particle collisions. J. Fluid Mech. 442, 303–334 (2001)
Zuber, N.: On the dispersed two-phase flow in the laminar flow regime. Chem. Eng. Science 49, 897–917 (1964)
Zun, I.: The transferees migration of bubbles influenced by walls in vertical bubbly flow. Int. J. Multiphase Flow 6, 583–588 (1980)
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Kolev, N.I. (2011). Conservation of Momentum. In: Multiphase Flow Dynamics 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20605-4_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-20605-4_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-20604-7
Online ISBN: 978-3-642-20605-4
eBook Packages: EngineeringEngineering (R0)