Page 24 - Subyek Encyclopedia - Encyclopedia of Separation Science

Sepsci*11*TSK*Venkatachala=BG
I / CENTRIFUGATION 19

where is the viscosity of the medium in poise, attractive forces between the particles being separ-
1
1
P(g cm s ); r is the radius of the particle (cm); and ated and/or the medium in which they are suspended.
1
(dx/dt) is the velocity of the moving particle (cm s ). Often, gravitational force alone is insufRcient to
Eqn [4] shows that the frictional force is propor- provide the minimum force necessary to disrupt such
tional to the particle velocity and its diameter. At low attractions. The use of centrifugal settling addresses
velocities and pressures, the frictional force is again the shortcomings of gravity settling by shortening the
negligible in a gas. However, at higher velocities, even time required for sample recovery at a given purity,
in gases, this force becomes substantial, combining providing a greater force for disrupting particle/par-
with the buoyancy force eventually to exactly oppose ticle or particle/media interactions and, within limits,
the gravitational force, resulting in no further acceler- lessening the detrimental effects of diffusion.
ation of the particle. This condition is known as the
limiting or terminal velocity. Mathematically, the Sedimentation in a Centrifugal Field
conditions for attaining terminal velocity are met A particle moving in a circular path continuously
when: experiences a centrifugal force, F c . This force acts in
the plane described by the circular path and is di-
[5]
F g "F b #F f rected away from the axis of rotation. The centrifugal
force may be expressed as:
The above discussion would imply that with suf-
Rcient time completely pure phases can be obtained F c "ma"m x [7]
2
by gravity sedimentation alone. While this may be
true for the sedimentation of large particles in a me- where m is the particle mass (g); a is the acceleration
dium with a signiRcantly higher or lower density than (cm s ); is the angular velocity (radians
2
the particle, this is not the case for smaller particles, s "2 rpm/60); and x is the radial distance from
1
which are impacted by diffusional forces that the axis of rotation to the particle (cm).
ultimately limit the separation efRciency as well Thus, centrifugal force is proportional to the
as to other nonideality effects (see below).
square of the angular velocity and to the radial dis-
tance from the axis of rotation. The force generated
Diffusion Random Brownian motion results in
the net movement of solute or suspended particles during centrifugation can be compared to the gravi-
from regions of higher concentration to regions of tational force by the relative centrifugal force, RCF,
lower concentration, a process called diffusion. often referred to as the g force:
Thus, diffusion works in opposition to centrifu- 2 2
gal sedimentation, which tends to concentrate par- RCF"F c /F g "(m x)/(mg)"( x)/g [8]
ticles. The rate of diffusion of a particle is given
by Fick’s law: Converting to rpm and substituting values for the
acceleration due to gravity, eqn [8] can be rewritten
dP/dt"!DA(dP/dx) [6] in a more convenient form as:

5
2
where D is the diffusion coefRcient which RCF"1.119 10 (rpm) x [9]
varies for each solute and particle; A is the cross-
sectional area through which the particle dif- While RCF is a ratio, and therefore unitless, it is
fuses; and dP/dx is the particle concentration frequently expressed in units of g to indicate the
gradient. number of times that the force of the applied centrifu-
The precise impact of diffusion can be dif- gal Reld is greater than the force of gravity.
Rcult and cumbersome to calculate for complex sys- The forces acting on a particle suspended in
tems. It is often sufRcient to keep in mind that the a liquid medium in a centrifugal Reld are illustrated in
rate of diffusion is generally more pronounced Figure 1. Within the centrifugal plane, the centrifugal
for smaller particles than for larger ones, it increases force acts to move particles away from the axis of
with temperature, and its effects are lessened by rotation, while the buoyancy and frictional forces
higher centrifugal forces. oppose this movement. The effect of the Earth’s
Aside from theoretical considerations, in a more gravity can generally be regarded as negligible. Ana-
practical sense, the time required for the settling of logous to the conditions for attaining terminal velo-
small to medium size particles in a gravitational Reld city in a gravitational Reld (eqn [5]), the particle will
is often prohibitive. Additional obstacles to obtaining reach a limiting or terminal velocity in a centrifugal
pure phases during gravity settling can also arise from Reld when the sum of the frictional and buoyancy