A Novel Slip Correction Factor for Spherical Aerosol Particles
A 3D simulation study for an incompressible
slip flow around a spherical aerosol particle was performed.
The full Navier-Stokes equations were solved and the velocity
jump at the gas-particle interface was treated numerically by
imposition of the slip boundary condition. Analytical solution
to the Stokesian slip flow past a spherical particle was used as
a benchmark for code verification, and excellent agreement
was achieved. The Simulation results showed that in addition
to the Knudsen number, the Reynolds number affects the slip
correction factor. Thus, the Cunningham-based slip corrections
must be augmented by the inclusion of the effect of
Reynolds number for application to Lagrangian tracking of
fine particles. A new expression for the slip correction factor
as a function of both Knudsen number and Reynolds number
was developed.
[1] S.A. Schaaf, and P.L. Chambre, Flow of rarefied gases, Princeton
University Press, 1961.
[2] M. Gad-el-Hak, "The fluid mechanics of microdevicesÔÇöthe
freeman scholar lecture," J. of Fluids Engineering., vol. 121, pp. 5-33,
1999.
[3] C.T. Crowe, Multiphase Flow Handbook, 1rd Edition. Taylor and
Francis, Florida, 2006.
[4] J.H. Kim, G.W. Mulholland, S.R. Kukuck and D.Y.H. Pui, "Slip
correction measurements of certified PSL nanoparticles using a
nanometer differential mobility analyzer (Nano-DMA) for Knudsen
number from 0.5 to 8," J. Res. Natl. Inst. Stand. Technol., vol. 110, pp.
31-54, 2005.
[5] E. Cunningham, "On the velocity of steady fall of spherical particles
through fluid medium (Published Conference Proceedings style)," in
Proc. Royal Soc. (London) A, vol. 83, pp. 357-365, 1910.
[6] R.A. Millikan, "The isolation of an ion, a precision measurement of its
charge, and the correction of stokes's law." Science, vol. 32, pp. 436-
448, 1910.
[7] M. Knudsen and S. Weber, Ann. D. Phys., vol. 36, p. 981, 1911.
[8] W.C. Hinds, Aerosol Technology, Properties Behavior, and
Measurement of Airborne Particles, 2nd ed. John Wiley and Sons, New
York, 1998.
[9] R.A. Millikan, "The general law of fall of a small spherical body
through a gas, and its bearing upon the nature of molecular reflection
from surfaces," Physical Review, vol. 22, pp. 1-23, 1923.
[10] R.A. Millikan, The Electron: Its Isolation and Measurement and The
Determination of some of Its Properties, University of Chicago Press,
8th Ed, 1963.
[11] M.D. Allen and O.G. Raabe, "Slip correction measurements of spherical
solid aerosol particles in an improved Millikan apparatus." J. Aerosol
Sci. Tech., vol. 4, p. 269, 1985.
[12] D.J. Rader, "Momentum slip correction factor for small particles in nine
common gases," J. Aerosol Sci., vol. 21, pp. 161-168, 1990.
[13] D.K. Hutchins, M.H. Harper and R.L. Felder, J. Aerosol Sci. Tech., vol.
22, p. 202, 1995.
[14] M.D. Allen, and O.G. Raabe, "Re-evaluation of millikan's oil drop data
for the motion of small particles in air," J. Aerosol Sci., vol. 13, p. 537,
1982.
[15] C.N. Davies, "Definite equations for the fluid resistance of spheres
(Published Conference Proceedings style)," in Proc. Phys. Soc., vol. 57,
p. 18, 1945.
[16] R.W. Barber and D.R. Emerson, "Analytic solution of low Reynolds
number slip flow past a sphere (Report style)," Centre for Microfluidics,
Department of Computational Science and Engineering, CLRC
Daresbury Laboratory, Daresbury, Warrington, WA4 4AD, 2001a.
[17] R.W. Barber and D.R. Emerson, "Numerical simulation of low Reynolds
number slip flow past a confined microsphere (Published Conference
Proceedings style)," in 23rd International Symposium on Rarefied Gas
Dynamics, Whistler, Canada, pp. 20-25, 2002a.
[18] H-C. F. Liu, A. Beskok, N. Gatsonis and G.E. Karniadakis, "Flow past a
microsphere in a pipe: effects of rarefaction," Micro-Electro-Mechanical
Systems (MEMS). ASME, DSC-Vol. 66, pp. 445-452, 1998.
[19] J.K. Fremerey, "Spinning rotor vacuum gauges," J. Vacuum, vol. 32, pp.
685-690, 1982.
[20] G. Reich, "Spinning rotor viscosity gauge: A transfer standard for the
laboratory or an accurate gauge for vacuum process control," J. Vacuum
Science and Technology, vol. 20, pp. 1148-1152, 1982.
[21] A. Li, G. Ahmadi, R.G. Bayer and M.A. Gaynes, "Aerosol particle
deposition in an obstructed turbulent duct flow," J. Aerosol Sci., vol. 25,
pp. 91-112, 1994.
[22] M. Soltani, G. Ahmadi, H. Ounis and B. McLaughlin, "Direct simulation
of charged particle deposition in a turbulent flow," Int. J. Multiphase
flow, vol. 24, pp. 77-92, 1998.
[23] S. Dhaniyala, P.O. Wennberg, R.C. Flagan, D.W. Fahey, M.J. Northway,
R.S. Gao and T.P. Bui, "Stratospheric aerosol sampling: effect of a
blunt-body housing on inlet sampling characteristics," J. Aerosol Sci.
and Technology, vol. 38, pp. 1080-1090, 2004.
[24] T.M. Peters and D. Leith, "Particle deposition in industrial duct bends,"
Ann. occup. Hyg., pp. 1-8, 2004.
[25] Z. Zhang, C. Kleinstreuer, J.F. Donohue and C.S. Kim, "Comparison of
micro- and nano-size particle depositions in a human upper airway
model," J. Aerosol Sci., vol. 36, pp. 211-233, 2005.
[26] X. Wang, A. Gidwani, S.L. Girshick, and P.H. McMurry, "Aerodynamic
focusing of nanoparticles: II. Numerical simulation of particle motion
through aerodynamic lenses," J. Aerosol Science and Technology, vol.
39, pp. 624-636, 2005.
[27] F.M. White, Viscous Fluid Flow. New York: Mc-Graw Hill, 2006, P. 47.
[28] B.P. Leonard, "Adjusted quadratic upstream algorithms for transient
incompressible convection (Published Conference Proceedings style),"
A Collection of Technical Papers. AIAA Computational Fluid Dynamics
Conference, AIAA Paper 79-1469, 1979.
[29] R.W. Davis, J. Noye and C. Fletcher, Finite Difference Methods for
Fluid Flow, Computational Techniques and Applications, Eds., Elsevier,
pp. 51-69, 1984.
[30] C.J. Freitas, R.L. Street, A.N. Findikakis, and J.R. Koseff, "Numerical
simulation of three-dimensional flow in a cavity," Int. J. Numerical
Methods Fluids, vol. 5, pp. 561-575, 1985.
[31] W. Ji and P.K. Wang, "Numerical simulation of three-dimensional
unsteady viscous flow past fixed hexagonal ice crystals in the airÔÇö
Preliminary results," Atmos. Res., vol. 25, pp. 539-557, 1990.
[32] B.P. Leonard and S. Mokhtari, "ULTRA-SHARP Nonoscillatory
convection schemes for high-speed steady multidimensional flow
(Report style)," NASA Lewis Research Center, NASA TM 1-2568
(ICOMP-90-12), 1990.
[33] W. Ji and P.K. Wang, "Numerical simulation of three-dimensional
unsteady viscous flow past finite cylinders in an unbounded fluid at low
intermediate Reynolds numbers," Theor. Comput. Fluid Dyn., vol. 3, pp.
43-59, 1991.
[34] J.M. Weiss, J.P. Maruszewski and W.A. Smith, "Implicit solution of the
Navier-Stokes equations on unstructured meshes (Published Conference
Proceedings style)," 13th AIAA CFD Conference, Snowmass, CO,
Technical Report AIAA-97-2103, 1997.
[35] S. Giors, F. Subba and R. Zanino, "Navier-Stokes modeling of a Gaede
pump stage in the viscous and transitional flow regimes using slip-flow
boundary conditions," J. Vac. Sci. Technol. A, vol. 23, No. 2, pp. 336-
346, 2005.
[36] V. Jain and C.X. Lin, "Numerical modeling of three-dimensional
compressible gas flow in microchannels," J. Micromech. Microeng., vol.
16, pp. 292-302, 2006.
[37] E.O.B. Ogedengbe, G.F. Naterer and M.A. Rosen, "Slip-Flow
irreversibility of dissipative kinetic and internal energy exchange in
microchannels," J. Micromech. Microeng., vol. 16, pp. 2167-2176,
2006.
[38] F.M. White, Viscous Fluid Flow. New York: Mc-Graw Hill, 2006, P.
175.
[39] H.P Kavehpour, M. Faghri and Y. Asako, "Effects of compressibility
and rarefaction on gaseous flows in microchannels," Numerical Heat
Transfer, Part A: Applications, vol. 32, pp. 677 - 696, 1997.
[40] G. Zuppardi, D. Paterna and A. Rega "Quantifying the effects of
rarefaction in high velocity, slip-flow regime (Published Conference
Proceedings style)," in Rarefied Gas Dynamics: 25-th International
Symposium, Russia, Novosibirsk, 2007.
[41] R.W. Barber and D.R. Emerson, "A numerical study of low Reynolds
number slip flow in the hydrodynamic development region of circular
and parallel plate ducts (Report style)," Centre for Microfluidics,
Department of Computational Science and Engineering, CLRC
Daresbury Laboratory, Daresbury, Warrington, WA4 4AD, 2002b.
[1] S.A. Schaaf, and P.L. Chambre, Flow of rarefied gases, Princeton
University Press, 1961.
[2] M. Gad-el-Hak, "The fluid mechanics of microdevicesÔÇöthe
freeman scholar lecture," J. of Fluids Engineering., vol. 121, pp. 5-33,
1999.
[3] C.T. Crowe, Multiphase Flow Handbook, 1rd Edition. Taylor and
Francis, Florida, 2006.
[4] J.H. Kim, G.W. Mulholland, S.R. Kukuck and D.Y.H. Pui, "Slip
correction measurements of certified PSL nanoparticles using a
nanometer differential mobility analyzer (Nano-DMA) for Knudsen
number from 0.5 to 8," J. Res. Natl. Inst. Stand. Technol., vol. 110, pp.
31-54, 2005.
[5] E. Cunningham, "On the velocity of steady fall of spherical particles
through fluid medium (Published Conference Proceedings style)," in
Proc. Royal Soc. (London) A, vol. 83, pp. 357-365, 1910.
[6] R.A. Millikan, "The isolation of an ion, a precision measurement of its
charge, and the correction of stokes's law." Science, vol. 32, pp. 436-
448, 1910.
[7] M. Knudsen and S. Weber, Ann. D. Phys., vol. 36, p. 981, 1911.
[8] W.C. Hinds, Aerosol Technology, Properties Behavior, and
Measurement of Airborne Particles, 2nd ed. John Wiley and Sons, New
York, 1998.
[9] R.A. Millikan, "The general law of fall of a small spherical body
through a gas, and its bearing upon the nature of molecular reflection
from surfaces," Physical Review, vol. 22, pp. 1-23, 1923.
[10] R.A. Millikan, The Electron: Its Isolation and Measurement and The
Determination of some of Its Properties, University of Chicago Press,
8th Ed, 1963.
[11] M.D. Allen and O.G. Raabe, "Slip correction measurements of spherical
solid aerosol particles in an improved Millikan apparatus." J. Aerosol
Sci. Tech., vol. 4, p. 269, 1985.
[12] D.J. Rader, "Momentum slip correction factor for small particles in nine
common gases," J. Aerosol Sci., vol. 21, pp. 161-168, 1990.
[13] D.K. Hutchins, M.H. Harper and R.L. Felder, J. Aerosol Sci. Tech., vol.
22, p. 202, 1995.
[14] M.D. Allen, and O.G. Raabe, "Re-evaluation of millikan's oil drop data
for the motion of small particles in air," J. Aerosol Sci., vol. 13, p. 537,
1982.
[15] C.N. Davies, "Definite equations for the fluid resistance of spheres
(Published Conference Proceedings style)," in Proc. Phys. Soc., vol. 57,
p. 18, 1945.
[16] R.W. Barber and D.R. Emerson, "Analytic solution of low Reynolds
number slip flow past a sphere (Report style)," Centre for Microfluidics,
Department of Computational Science and Engineering, CLRC
Daresbury Laboratory, Daresbury, Warrington, WA4 4AD, 2001a.
[17] R.W. Barber and D.R. Emerson, "Numerical simulation of low Reynolds
number slip flow past a confined microsphere (Published Conference
Proceedings style)," in 23rd International Symposium on Rarefied Gas
Dynamics, Whistler, Canada, pp. 20-25, 2002a.
[18] H-C. F. Liu, A. Beskok, N. Gatsonis and G.E. Karniadakis, "Flow past a
microsphere in a pipe: effects of rarefaction," Micro-Electro-Mechanical
Systems (MEMS). ASME, DSC-Vol. 66, pp. 445-452, 1998.
[19] J.K. Fremerey, "Spinning rotor vacuum gauges," J. Vacuum, vol. 32, pp.
685-690, 1982.
[20] G. Reich, "Spinning rotor viscosity gauge: A transfer standard for the
laboratory or an accurate gauge for vacuum process control," J. Vacuum
Science and Technology, vol. 20, pp. 1148-1152, 1982.
[21] A. Li, G. Ahmadi, R.G. Bayer and M.A. Gaynes, "Aerosol particle
deposition in an obstructed turbulent duct flow," J. Aerosol Sci., vol. 25,
pp. 91-112, 1994.
[22] M. Soltani, G. Ahmadi, H. Ounis and B. McLaughlin, "Direct simulation
of charged particle deposition in a turbulent flow," Int. J. Multiphase
flow, vol. 24, pp. 77-92, 1998.
[23] S. Dhaniyala, P.O. Wennberg, R.C. Flagan, D.W. Fahey, M.J. Northway,
R.S. Gao and T.P. Bui, "Stratospheric aerosol sampling: effect of a
blunt-body housing on inlet sampling characteristics," J. Aerosol Sci.
and Technology, vol. 38, pp. 1080-1090, 2004.
[24] T.M. Peters and D. Leith, "Particle deposition in industrial duct bends,"
Ann. occup. Hyg., pp. 1-8, 2004.
[25] Z. Zhang, C. Kleinstreuer, J.F. Donohue and C.S. Kim, "Comparison of
micro- and nano-size particle depositions in a human upper airway
model," J. Aerosol Sci., vol. 36, pp. 211-233, 2005.
[26] X. Wang, A. Gidwani, S.L. Girshick, and P.H. McMurry, "Aerodynamic
focusing of nanoparticles: II. Numerical simulation of particle motion
through aerodynamic lenses," J. Aerosol Science and Technology, vol.
39, pp. 624-636, 2005.
[27] F.M. White, Viscous Fluid Flow. New York: Mc-Graw Hill, 2006, P. 47.
[28] B.P. Leonard, "Adjusted quadratic upstream algorithms for transient
incompressible convection (Published Conference Proceedings style),"
A Collection of Technical Papers. AIAA Computational Fluid Dynamics
Conference, AIAA Paper 79-1469, 1979.
[29] R.W. Davis, J. Noye and C. Fletcher, Finite Difference Methods for
Fluid Flow, Computational Techniques and Applications, Eds., Elsevier,
pp. 51-69, 1984.
[30] C.J. Freitas, R.L. Street, A.N. Findikakis, and J.R. Koseff, "Numerical
simulation of three-dimensional flow in a cavity," Int. J. Numerical
Methods Fluids, vol. 5, pp. 561-575, 1985.
[31] W. Ji and P.K. Wang, "Numerical simulation of three-dimensional
unsteady viscous flow past fixed hexagonal ice crystals in the airÔÇö
Preliminary results," Atmos. Res., vol. 25, pp. 539-557, 1990.
[32] B.P. Leonard and S. Mokhtari, "ULTRA-SHARP Nonoscillatory
convection schemes for high-speed steady multidimensional flow
(Report style)," NASA Lewis Research Center, NASA TM 1-2568
(ICOMP-90-12), 1990.
[33] W. Ji and P.K. Wang, "Numerical simulation of three-dimensional
unsteady viscous flow past finite cylinders in an unbounded fluid at low
intermediate Reynolds numbers," Theor. Comput. Fluid Dyn., vol. 3, pp.
43-59, 1991.
[34] J.M. Weiss, J.P. Maruszewski and W.A. Smith, "Implicit solution of the
Navier-Stokes equations on unstructured meshes (Published Conference
Proceedings style)," 13th AIAA CFD Conference, Snowmass, CO,
Technical Report AIAA-97-2103, 1997.
[35] S. Giors, F. Subba and R. Zanino, "Navier-Stokes modeling of a Gaede
pump stage in the viscous and transitional flow regimes using slip-flow
boundary conditions," J. Vac. Sci. Technol. A, vol. 23, No. 2, pp. 336-
346, 2005.
[36] V. Jain and C.X. Lin, "Numerical modeling of three-dimensional
compressible gas flow in microchannels," J. Micromech. Microeng., vol.
16, pp. 292-302, 2006.
[37] E.O.B. Ogedengbe, G.F. Naterer and M.A. Rosen, "Slip-Flow
irreversibility of dissipative kinetic and internal energy exchange in
microchannels," J. Micromech. Microeng., vol. 16, pp. 2167-2176,
2006.
[38] F.M. White, Viscous Fluid Flow. New York: Mc-Graw Hill, 2006, P.
175.
[39] H.P Kavehpour, M. Faghri and Y. Asako, "Effects of compressibility
and rarefaction on gaseous flows in microchannels," Numerical Heat
Transfer, Part A: Applications, vol. 32, pp. 677 - 696, 1997.
[40] G. Zuppardi, D. Paterna and A. Rega "Quantifying the effects of
rarefaction in high velocity, slip-flow regime (Published Conference
Proceedings style)," in Rarefied Gas Dynamics: 25-th International
Symposium, Russia, Novosibirsk, 2007.
[41] R.W. Barber and D.R. Emerson, "A numerical study of low Reynolds
number slip flow in the hydrodynamic development region of circular
and parallel plate ducts (Report style)," Centre for Microfluidics,
Department of Computational Science and Engineering, CLRC
Daresbury Laboratory, Daresbury, Warrington, WA4 4AD, 2002b.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:54342", author = "Abouzar Moshfegh and Mehrzad Shams and Goodarz Ahmadi and Reza Ebrahimi", title = "A Novel Slip Correction Factor for Spherical Aerosol Particles", abstract = "A 3D simulation study for an incompressible
slip flow around a spherical aerosol particle was performed.
The full Navier-Stokes equations were solved and the velocity
jump at the gas-particle interface was treated numerically by
imposition of the slip boundary condition. Analytical solution
to the Stokesian slip flow past a spherical particle was used as
a benchmark for code verification, and excellent agreement
was achieved. The Simulation results showed that in addition
to the Knudsen number, the Reynolds number affects the slip
correction factor. Thus, the Cunningham-based slip corrections
must be augmented by the inclusion of the effect of
Reynolds number for application to Lagrangian tracking of
fine particles. A new expression for the slip correction factor
as a function of both Knudsen number and Reynolds number
was developed.", keywords = "CFD, Cunningham correction, Slip correction factor,Spherical aerosol.", volume = "2", number = "10", pages = "1121-7", }
