Terminal Settling Velocity of a Solid Particle in Fluid
Introduction
This application calculates the terminal velocity of a solid particle settling in a fluid.
Deriving the Settling Velocity
restart:
Drag force
Fd≔12⋅π⋅Dia24⋅CD⋅ρ__f⋅v2:
Buoyancy force
Fb≔ρ__p−ρ__f⋅g⋅π⋅Dia36:
The terminal settling velocity is reached when the drag force equals the buoyancy force. Hence, the settling velocity is given by the following equation.
res≔v__terminal=solveFb=Fd,v1
res≔v__terminal=2−3CDρ__fgDiaρ__f−ρ__p3CDρ__f
Parameters
Gravity:
g≔9.81:
Density of the particle:
ρ__p≔1800:
Fluid density:
ρ__f≔994.6:
Fluid viscosity:
μ≔0.0008931:
Particle diameter:
Dia≔0.000208:
Governing Equations
Drag coefficient:
CD≔Rey→{24ReyRey<0.124Rey1+0.14⋅Rey0.70.1≤Rey≤10000.441000≤Rey≤3500000.19−8⋅104ReyRey>350000:
ReynoldsNumber≔Rey=Dia ρ__f v__terminalμ:
TerminalVelocity≔v__terminal=4gρ__p−ρ__fDia3CDRey⋅ρ__f:
Solution
fsolveReynoldsNumber,TerminalVelocity,v__terminal,Rey;
Rey=3.656696458,v__terminal=0.01578618582
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