CHEE2945 – Lecture 13
Particles in fluid:
- As a particle moves down through a fluid, the fluid moves up. The fluid motion
can be laminar or turbulent.
- Surrounding particles can interrupt the flow of the fluid.
- When a fluid is forced through a bed of particles, there is a relationship
between fluid pressure, and velocity through the particles.
- If a fluid is forced up through a bed of particles, there may be a fluid flow that
will allow for particles to be entrained (lifted up), creating a fluidised bed.
Types of fluid flow:
- At low fluid flow rates, the fluid particles travel in roughly a straight line. This is
known as laminar flow.
- Laminar flow has a low flow rate with low energy. The particles travel parallel
to each other.
- At high fluid flow rates, in a higher energy system, the fluid particles do not
travel in a straight line and follow a far more chaotic path. This is known as
turbulent flow.
- Turbulent flow has a high flow rate, high energy, and the particles move in a
chaotic manner. It causes mixing of the particles and can be used to combine
two fluids.
- Particles can induce turbulence in a fluid flow.
Reynold’s number:
- Laminar flow is governed by viscous forces.
- Turbulent flow is dominated by inertia.
- The ratio of inertial to viscous forces determines the flow type, and is called
the Reynold’s number, ℜ.
inertial forces ρDW
ℜ= =
viscous forces μ
where:
ρ=¿ fluid density,
D=¿ characteristic length, e.g., diameter of pipe or particle,
W =¿ fluid characteristic velocity,
μ=¿fluid viscosity.
- Fluids with a low ℜ are laminar, and a high ℜ are turbulent.
, - The transition between laminar and turbulent flow is difficult to define.
- For flow in a pipe, turbulent flow occurs around ℜ=2000, whereas around a
particle, turbulent flow occurs around ℜ=500.
Resistance on spheres:
- For a particle, D is the particle diameter, and W is the relative particle/fluid
velocity.
- For a particle in an infinite fluid, settling under gravity, there is a buoyancy
force, and a force due to gravity. It will have a net force acting on it (unless
particle density equals fluid density), and so will accelerate. The drag force (
F d) acts in the direction opposite to velocity. At the point where there is no net
force acting on the particle, due to F b+ F d =F g, then the particle will reach its
terminal settling velocity.
- Assuming that the Reynold’s number is low, and the particle is spherical, we
can apply Stokes’ law:
F d=3 πμDW
- Combining the above equation with that for ℜ:
2
3πμ
F d= ℜ
ρf
- Experiments show that drag force is proportional to:
o W 2,
o ρf,
o A p (projected particle area).
- We can combine these, with a drag coefficient, to get:
Fd
C d=
1 2
A p ρf W
2
- Alternatively:
R'
C d=
1 2
ρW
2 f
' Fd
where R = .
Ap
- Combining the drag force and drag coefficient equations, assuming the
projected area is that of a sphere:
Particles in fluid:
- As a particle moves down through a fluid, the fluid moves up. The fluid motion
can be laminar or turbulent.
- Surrounding particles can interrupt the flow of the fluid.
- When a fluid is forced through a bed of particles, there is a relationship
between fluid pressure, and velocity through the particles.
- If a fluid is forced up through a bed of particles, there may be a fluid flow that
will allow for particles to be entrained (lifted up), creating a fluidised bed.
Types of fluid flow:
- At low fluid flow rates, the fluid particles travel in roughly a straight line. This is
known as laminar flow.
- Laminar flow has a low flow rate with low energy. The particles travel parallel
to each other.
- At high fluid flow rates, in a higher energy system, the fluid particles do not
travel in a straight line and follow a far more chaotic path. This is known as
turbulent flow.
- Turbulent flow has a high flow rate, high energy, and the particles move in a
chaotic manner. It causes mixing of the particles and can be used to combine
two fluids.
- Particles can induce turbulence in a fluid flow.
Reynold’s number:
- Laminar flow is governed by viscous forces.
- Turbulent flow is dominated by inertia.
- The ratio of inertial to viscous forces determines the flow type, and is called
the Reynold’s number, ℜ.
inertial forces ρDW
ℜ= =
viscous forces μ
where:
ρ=¿ fluid density,
D=¿ characteristic length, e.g., diameter of pipe or particle,
W =¿ fluid characteristic velocity,
μ=¿fluid viscosity.
- Fluids with a low ℜ are laminar, and a high ℜ are turbulent.
, - The transition between laminar and turbulent flow is difficult to define.
- For flow in a pipe, turbulent flow occurs around ℜ=2000, whereas around a
particle, turbulent flow occurs around ℜ=500.
Resistance on spheres:
- For a particle, D is the particle diameter, and W is the relative particle/fluid
velocity.
- For a particle in an infinite fluid, settling under gravity, there is a buoyancy
force, and a force due to gravity. It will have a net force acting on it (unless
particle density equals fluid density), and so will accelerate. The drag force (
F d) acts in the direction opposite to velocity. At the point where there is no net
force acting on the particle, due to F b+ F d =F g, then the particle will reach its
terminal settling velocity.
- Assuming that the Reynold’s number is low, and the particle is spherical, we
can apply Stokes’ law:
F d=3 πμDW
- Combining the above equation with that for ℜ:
2
3πμ
F d= ℜ
ρf
- Experiments show that drag force is proportional to:
o W 2,
o ρf,
o A p (projected particle area).
- We can combine these, with a drag coefficient, to get:
Fd
C d=
1 2
A p ρf W
2
- Alternatively:
R'
C d=
1 2
ρW
2 f
' Fd
where R = .
Ap
- Combining the drag force and drag coefficient equations, assuming the
projected area is that of a sphere:
