Fluid Particle Interactions Basics
Outline
1. Single sphere: Stokes and
turbulent drag
2. Many spheres: Stokes flow
u
3. Many spheres: intermediate Re
4. Many spheres: terminal velocity
?
1. Single sphere: Stokes and turbulent drag
u
1. Single sphere: Stokes and turbulent drag
Stokes drag (Re Æ 0):
Turbulent drag (Re > 103 ):
General form of drag relation for a single sphere:
Re Æ 0
with drag coefficient
Re > 103
1. Single sphere: drag coefficient
1. Single sphere: drag coefficient
Re < 10-1:
Re > 103 :
General Re:
Most simple expression
Dallavalle (1948)
Turton &
Levenspiel
(1986)
1. Single sphere: drag coefficient
1. Single sphere: terminal velocity
First application: terminal velocity v of a sphere
Note:
For which v do the forces
exactly balance?
Solve:
with
for Re
1. Single sphere: terminal velocity
Solve:
1. Re < 10-1:
2. Re > 103:
3. General Re:
Solve numerically
1. Single sphere: Galileo experiment
Second application:
Galileo Galilei, dropping a large and small iron ball from the tower of Pisa.
Common conception:
Balls hit the ground at the same time
(= true without air effect)
Actual observation of Galileo:
“when the larger has reached the
ground, the other is short of it by two
fingerbreadths”
1. Single sphere: Galileo experiment
Equation of motion for the sphere:
Solving the differential equation for d = 0.22 m and d = 0.05 m gives that
when the larger has reached the ground, the other is short of it by 96 cm!
2. Many spheres: Stokes flow
u
Dimensioneless drag force
2. Many spheres: Stokes flow
What is known from theory?
Kim & Russel, 1985:
Accurate up to
= 0.1
Brinkman, 1947:
Diverges for
= 0.667
2. Many spheres: Stokes flow
2. Many spheres: Stokes flow
“Pragmatic” approach: make link with pipe flow
Laminair flow through circular pipe, with pressure gradient
L
(exact result)
R
P1
P2
volume of fluid
wet surface
Laminair flow through
arbitrary shape pipe
L
P1
R
P2
Can be well described by
the above expression
2. Many spheres: Stokes flow
Flow through a network of pipes:
u
X
k: kozeny constant
Experiments for the pressure drop over a wide range of porous media
shows that k = 5, independent of the type of medium:
Darcy’s law:
2. Many spheres: Stokes flow
Flow through an array of spheres:
u
Carman-Kozeny equation
Relation to F ?
volume of fluid
wet surface
2. Many spheres: Stokes flow
Relation pressure drop to drag force Fd that fluid exerts on a particle
Total force that fluid exerts
on a particle:
u
P2
P1
Force that each particle
exerts on the fluid:
S
L
Steady fluid flow: total
force on fluid is zero
2. Many spheres: Stokes flow
Carman-Kozeny:
2. Many spheres: Stokes flow
Accurate up to
Kim & Russel, 1985:
= 0.1
Brinkman, 1947:
Diverges for
Carman-Kozeny, 1937:
Does not approach 1 for
Van der Hoef, Beetstra & Kuipers, 2005:
= 0.667
2. Many spheres: Stokes flow
3. Many spheres: intermediate Re
Dimensioneless drag force
u
3. Many spheres: intermediate Re
What is known from theory?
Kaneda, 1986:
Accurate for Re <
< 0.01
Pragmatic approach: connection with pipe flow
Pressure drop for turbulent flow through circular pipe
For bed of spheres:
BurkePlummer eq.
3. Many spheres: intermediate Re
Limit of low Re: Carman-Kozeny
Re > 4000: Burke-Plummer:
For general Re: try equation of the form
Fit to 640 experimental data points: A = 150, B = 1.75 Æ Ergun equation
3. Many spheres: intermediate Re
3. Many spheres: intermediate Re
3. Many spheres: overview
Expressions for the dimensionless drag force
Stokes flow
in the limit
Drag for single particle at finite Re
given by
, and
Drag for low Re flow through dense
random arrays Æ Carman
Ergun equation from
pressure drop data
General form of “Ergun”
type equations
General form of “Wen & Yu”
type equations
4. Many spheres: terminal velocity
Richardson-Zaki
exponent
Experimentally
Steady state: force balance for one particle:
(mass displaced suspension)
(mass particle)
4. Many spheres: terminal velocity
Note that the terminal velocity is a function of
4. Many spheres: terminal velocity
Assuming a dimensioneless drag force of the form
gives:
Terminal velocity experiments are fitted to the form:
Hence the exponents are related as:
3. Many spheres: terminal velocity
Experimental results for the exponent n :
Wen & Yu (1966):
Di Felice (1994):
over entire Re range
4. Many spheres: overview
Expressions for the dimensionless drag force
Stokes flow
in the limit
Drag for single particle at finite Re
given by
, and
Drag for low Re flow through dense
random arrays Æ Carman
Ergun equation from
pressure drop data
Wen & Yu equation from
terminal velocity data
3. Many spheres: overview
Expressions for the dimensionless drag force
Stokes flow
in the limit
Drag for single particle at finite Re
given by
, and
Drag for low Re flow through dense
random arrays Æ Carman
Ergun equation from
pressure drop data
General form of “Ergun”
type equations
General form of “Wen & Yu”
type equations
4. Many spheres: terminal velocity
Richardson-Zaki
exponent
Experimentally
Steady state: force balance for one particle:
(mass displaced suspension)
(mass particle)
4. Many spheres: terminal velocity
Note that the terminal velocity is a function of
4. Many spheres: terminal velocity
Assuming a dimensioneless drag force of the form
gives:
Terminal velocity experiments are fitted to the form:
Hence the exponents are related as:
3. Many spheres: terminal velocity
Experimental results for the exponent n :
Wen & Yu (1966):
Di Felice (1994):
over entire Re range
4. Many spheres: overview
Expressions for the dimensionless drag force
Stokes flow
in the limit
Drag for single particle at finite Re
given by
, and
Drag for low Re flow through dense
random arrays Æ Carman
Ergun equation from
pressure drop data
Wen & Yu equation from
terminal velocity data