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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Isfahan University of TechnologyDepartment of Mechanical Engineering
Mohammad Reza Tavakoli
CD= Constant for Re>1000: CD=0.44
Stokes region: Re<1
Transition region: 1< Re< 1000 ( 3 < Re < 400 error < 2% )
( 400 < Re < 1000 error < 10%)
Newton region: Re>1000
In general Navier-Stokes equation
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
Total resisting force on a spherical particle due to its velocity V relative to the fluid:
(Re<1 & Err<10%)
Compare drag forces:
Stokes’s law contains viscosity but NOT inertia factors
Newton’s law contain rho but NOT viscosity.
Stokes’s law :
Newton’s law :
Flow in tubes: No normal force cd=16/Re
Validation of Stokes assumptions:
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.
For particle to below 0.01 micron: (2.1% error for all particle sizes)
Slip Correction factor increases when the particle size decreases.
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 factor increases when pressure decreases because the
mean free path increases.
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)
The Dynamic shape factor >1:
nonspherical particle settle more slowly than their equivalent volume spheres
- Table: 3.4 (d is known and V is unknown)
-(1945) Davies (up to Re=4):