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Uniform Particle Motion

Isfahan University of Technology Department of Mechanical Engineering. Uniform Particle Motion. Mohammad Reza Tavakoli. Outline. Newton’s Resistance Law Stokes’s Law Settling Velocity & Mechanical Mobility Slip Correction Factor Nonspherical Particles Aerodynamic Diameter

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Uniform Particle Motion

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  1. Isfahan University of TechnologyDepartment of Mechanical Engineering Uniform Particle Motion Mohammad Reza Tavakoli

  2. Outline • Newton’s Resistance Law • Stokes’s Law • Settling Velocity & Mechanical Mobility • Slip Correction Factor • Nonspherical Particles • Aerodynamic Diameter • Settling at High Reynolds Number

  3. Newton’s Resistance Law • Resistance is a result of the acceleration of the air that has to be pushed aside to allow the sphere to pass through. • In 1 sec. a sphere of diameter d will push aside a volume of gas equal to the projected area of the sphere times its velocity V.

  4. Newton’s Resistance Law • Change of momentum= Force required to move the sphere through the gas CD= Constant for Re>1000: CD=0.44

  5. Newton’s Resistance Law Stokes region: Re<1 Transition region: 1< Re< 1000 ( 3 < Re < 400 error < 2% ) ( 400 < Re < 1000 error < 10%) Newton region: Re>1000 Cd=0.44

  6. Stokes’s Law In general Navier-Stokes equation Stokes assumptions: Inertial forces are negligibly small compare to viscous forces Fluid is incompressible There are no wall or other particle nearby Motion is constant Particle is rigid sphere No slip condition at particle’s surface Net force= normal force + tangential force Both of the forces acting in a direction opposite to particle motion

  7. Stokes’s Law Normal component: Tangential component: Total resisting force on a spherical particle due to its velocity V relative to the fluid: (Re<1 & Err<10%) See Appendix

  8. Newton’s Resistance Law & Stokes’s Law Compare drag forces: Stokes’s law contains viscosity but NOT inertia factors like rho Newton’s law contain rho but NOT viscosity.

  9. Newton’s Resistance Law & Stokes’s Law Stokes’s law : Newton’s law : Flow in tubes: No normal force cd=16/Re Validation of Stokes assumptions: • Fluid is incompressible • There are no wall or other particle nearby • Particle is rigid sphere ( 0.7% error for water drops) • No slip condition at particle’s surface (Slip Correction Factor) • Nonspherical particles (shape factor)

  10. Settling Velocity and Mechanical Mobility

  11. Settling Velocity and Mechanical Mobility

  12. Terminal velocity for other external forces

  13. Mechanical Mobility

  14. Example

  15. Slip Correction Factor The No Slip Condition is not valid for small particle whose size approaches the mean free path of the gas. 1910- Cunningham Correction factor (Cc): Cc >1 so, reduces the Stokes drag force by: For Particle of 0.1 micron.

  16. Slip Correction Factor For particle to below 0.01 micron: (2.1% error for all particle sizes) Terminal Velocity:

  17. Slip Correction Factor Slip Correction factor increases when the particle size decreases.

  18. Slip Correction Factor Slip Correction factor increases when pressure decreases because the mean free path increases. Pd: multiplying particle diameter by the pressure in atmospheres gives diameter of the particle that has the same slip correction factor at 1 atm pressure. Look at A12: compare particle of 1 micron at 2 atm pressure vs. particle at 2 micron and 1 atm pressure.

  19. Slip Correction Factor Slip Correction factor increases when pressure decreases because the mean free path increases.

  20. Nonspherical Particle (Dynamic shape factor) actual resistance force of nonspherical particle Dynamic shape factor = ---------------------------------------------------------------------------------- resistance force of the sphere with same volume and velocity de= equivalent volume diameter (diameter of the sphere having the same volume as the irregular particle)

  21. Nonspherical Particle (Dynamic shape factor) The Dynamic shape factor >1: nonspherical particle settle more slowly than their equivalent volume spheres

  22. Aerodynamic Diameter • Stokes diameter (ds): diameter of the sphere that has the same density and settling velocity as the particle • Aerodynamic diameter (da): diameter of the unit density sphere that has the same settling velocity as the particle

  23. Aerodynamic Diameter • If a particle has an aerodynamic diameter 1 micron it behaves in an aerodynamics sense like a 1 micron water droplet (density=1 g/cm^3) regardless of its shape, density or physical size • Rho_b: Bulk material • Rho_p: stokes particle • Rho_b==Rho_p

  24. Settling at high Reynolds Number

  25. Settling at high Reynolds Number - Table: 3.4 (d is known and V is unknown) -(1945) Davies (up to Re=4):

  26. Example

  27. Settling at high Reynolds Number • If settling velocity is known(V is known and d is unknown) (Table 3.5) • Empirical Equation:

  28. Example

  29. Thank YouQuestions?

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