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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

Isfahan University of TechnologyDepartment of Mechanical Engineering

Uniform Particle Motion

Mohammad Reza Tavakoli


Outline
Outline

  • Newton’s Resistance Law

  • Stokes’s Law

  • Settling Velocity & Mechanical Mobility

  • Slip Correction Factor

  • Nonspherical Particles

  • Aerodynamic Diameter

  • Settling at High Reynolds Number


Newton s resistance law
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.


Newton s resistance law1
Newton’s Resistance Law

  • Change of momentum= Force required to move the sphere through the gas

    CD= Constant for Re>1000: CD=0.44


Newton s resistance law2
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


Stokes s law
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


Stokes s law1
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


Newton s resistance law stokes s law
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.


Newton s resistance law stokes s law1
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)







Slip correction factor
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.


Slip correction factor1
Slip Correction Factor

For particle to below 0.01 micron: (2.1% error for all particle sizes)

Terminal Velocity:


Slip correction factor2
Slip Correction Factor

Slip Correction factor increases when the particle size decreases.


Slip correction factor3
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.


Slip correction factor4
Slip Correction Factor

Slip Correction factor increases when pressure decreases because the

mean free path increases.


Nonspherical particle dynamic shape factor
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)


Nonspherical particle dynamic shape factor1
Nonspherical Particle (Dynamic shape factor)

The Dynamic shape factor >1:

nonspherical particle settle more slowly than their equivalent volume spheres


Aerodynamic diameter
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


Aerodynamic diameter1
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



Settling at high reynolds number1
Settling at high Reynolds Number

- Table: 3.4 (d is known and V is unknown)

-(1945) Davies (up to Re=4):



Settling at high reynolds number2
Settling at high Reynolds Number

  • If settling velocity is known(V is known and d is unknown) (Table 3.5)

  • Empirical Equation:



Thank you questions
Thank YouQuestions?


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