Sedimentation

Let us make some basic considerations about sedimentation to better understand

what is happening in a hydrocyclone. A characteristic parameter describing sedimentation

is the terminal settling velocity. It describes the state where a particle

moves at constant speed under the influence of frictional and gravitational forces.

Figure 6.20 shows the main forces which act on a settling particle; these are the

weight FW, the buoyancy FB, and the drag force FD.

vsettling

Weight force

Buoyancy force

Drag force

Fig. 6.20 Forces acting on a settling particle.

6.4 Centrifugal Cleaning Theory 581

Weight and buoyancy depend on the specific weight of the particle and the displaced

liquor, respectively. The drag force is a function of the particle movement

and the particle shape. It is always directed in the opposite direction of the velocity

vector. In the Earth’s gravitational field, the three forces are defined as follows:

FW _ _ S V g _12_

FB _ _ L V g _13_

FD _ cDAP _ L

v 2 s

2 _14_

where q S = density of the particle (kg m–3); q L = density of the liquid (kg m–3);

V = volume of the particle (m3); AP = area of the particle as seen in projection

along the direction of motion (m–2); g = acceleration due to gravity (9.81 m s–2);

cD = drag coefficient; and vS = settling velocity (m s–1).

In the steady state, the forces are in equilibrium, which means that

FW _ FB _ FD _ 0 _15_

Combining Eqs. (12–15) and solving for vS yields Newton’s law for the terminal

settling velocity in the Earth’s gravitational field:

vS _ _2_________________

cD

__ S _ _ L _

_ L

V

AP

_ g _16_

In the special case of a spherical particle with the diameter d (m), where

V = d 3p/6 and AP = d 2p/4, the above expression becomes:

vS _ __4________________

3 cD

__ S _ _ L _

_ L

_ d g _17_

Note that the settling velocity increases with the density difference between the

particle and the liquor and with the particle diameter. The drag coefficient

depends on the size and shape of the particle, on the viscosity and density of the

fluid, and on the settling velocity itself. When the sphere settles in a creeping,

laminar environment, Eq. (17) converts into Stokes’ law:

vS _

__ S _ _ L _ d 2

S g

18 l _18_

where l is the dynamic viscosity of the liquid (Pa·s).

582 6 Pulp Screening, Cleaning, and Fractionation

The solids contained in a pulp stream are very different in shape and size.

While sand particles may come close to spherical shape, pulp fibers obviously do

not. Likewise, there are wide ranges of particle densities from plastics to metals.

In addition, the reinforced gravitational field in the cyclone adds complexity to the

matter. Consequently, meaningful theoretical models for the settling of solids in a

pulp suspension during centrifugal cleaning are not available. We will therefore

use the general form of Newton’s law, as per Eq. (16), for the qualitative evaluation

of separation in a hydrocyclone.

vtangential

vradial

vsettling

r

Net gravitational force Drag force

Fig. 6.21 Forces acting on a particle in a hydrocyclone.

So, what is happening to a particle in a hydrocyclone? The tangential feed provokes

a tangential liquor velocity which makes the particle move along a circular

path around the axis of the cyclone (Fig. 6.21). A radial flow vector describes the

transport of the liquor from the feed inlet at the outer perimeter to the centrical

vortex finder. There is also an axial flow vector which is directed towards the apex

at the cyclone perimeter and towards the vortex finder around the axis. The forces

acting on the particle in a plain perpendicular to the axis are a drag force pointing

against the direction of the settling velocity, and gravitational forces as a function

of the different solid and liquid densities.

Clearly, the settling velocity must be larger than the radial velocity in the cyclone

for a particle to be separated to the underflow. Nevertheless, the tangential velocity

represents the most important flow vector in the hydrocyclone because it controls

the gravity forces acting on the particle.

The acceleration term is determined by the tangential velocity vT (m s–1) and the

distance between the particle and the center of rotation, r (m). When substituting

the acceleration due to the Earth’s gravity g by the centripetal acceleration vT

2/ r,

Eq. (16) can be rewritten to give:

vS _ vT _2_________________

cD

__ S _ _ L _

_ L

V

AP

r _ _19_

6.4 Centrifugal Cleaning Theory 583

Apparently, higher tangential flow velocities vT and smaller distances r increase

the settling velocity. This means that a cyclone of a smaller diameter is more efficient

for the removal of small particles than a large-diameter cyclone. Likewise,

higher tangential flow velocities improve the efficiency. Both the cyclone diameter

and the tangential velocity are physically limited by the necessity to maintain the

typical laminar flow pattern.

The density difference between the liquid and some particles (e.g., plastics or

light-weight wood components) may be very low. This means that high velocities

and small radii are needed for cleaning to be efficient. In a typical cleaner, the

centrifugal force is so much larger than the Earth’s gravity that it does not make

any difference whether the cleaner is installed vertically or horizontally.

For the cleaning of pulp, the relevant solids density q S is the apparent fiber density

– that is, the density of the swollen fiber consisting of the liquor-saturated

fiber wall and liquor-filled lumen. It has been suggested that for chemical pulp,

the influence of the fiber shape on the drag force and consequently on cD is not

significant [22].

The derivations described above are valid for particles which have a larger density

than the fluid. When a particle is lighter than the fluid, its weight becomes

smaller than the buoyancy, and the vector for the settling velocity shown in

Fig. 6.20 is directed upwards. This is when the particle begins to float to the surface

rather than settle to the bottom. Consequently, the drag force points downwards.

When then the terminal settling velocity is calculated in analogy to

Eq. (19), the solid and liquid densities in the numerator of the density term

change place:

vS _ vT _2_________________

cD

__ L _ _ S _

_ L

V

AP

r _ _20_

So, the separation of light-weight particles to the overflow is controlled by the

same factors as the separation of heavy-weight particles to the underflow, the difference

being that there is no need to overcome the radial velocity for separation

to occur. In theory, this circumstance facilitates the separation of light-weight particles

compared to heavy-weight particles. However, in practice the density difference

between light-weight material and liquor is often very small, and any support

for obtaining a reasonable separation efficiency is welcome.

6.4.4

Дата добавления: 2015-10-21