Page 25 - Subyek Encyclopedia - Encyclopedia of Separation Science

20 I / CENTRIFUGATION/ Derivatization

meters that govern settling velocity. They show that
the sedimentation rate (i.e. limiting velocity) of a par-
ticle in a centrifugal Reld:

increases as the square of the particle diameter and
rotor speed, i.e. doubling the speed or particle
diameter will lessen the run time by a factor of
four;
increases proportionally with distance from the
axis or rotation; and
is inversely related to the viscosity of the carrier
medium.
These are the fundamental premises that a practi-
tioner must know in order to develop a rational
approach to centrifugal separation.
Sedimentation Coef\cient
Since the terms r, P , M and as given in
Figure 1 Forces acting on a particle in a centrifugal field: F b ,
buoyancy; F f , frictional; F c , centrifugal; and F g , gravitational. eqns [13]}[15] are constant for a given particle in
a homogeneous medium, the sedimentation rate,
2
dx/dt, is proportional to x. This proportionality is
forces equals the centrifugal force: often expressed in terms of the sedimentation coef-
Rcient, S, which is simply a measure of the sedimenta-
[10]
F c "F b #F f
tion velocity per unit of centrifugal force. For a given
set of run conditions, the sedimentation coefRc-
Substituting eqns [2], [4] and [7] into eqn [10] gives:
ient, S r , may be calculated as:
2
2
m x"V P M x#6 r(dx/dt) [11]
S r "(dx/dt)/( x)"2r ( P ! M )/9 [16]
2
2
4
3
Assuming a spherical particle and substituting 3 r for
volume gives: The sedimentation coefRcient, S, has the dimen-
sions of seconds and is expressed in Svedberg units
13
3
4
4
2
3
2
( 3 r ) P x"( 3 r ) M x#6 r(dx/dt) [12] equal to 10 s. Its value is dependent on the particle
being separated, the centrifugal force and the proper-
Then solving for dx/dt: ties of the sedimentation medium. While adequate for
a given set of run conditions, it is sometimes useful to
2
dx/dt"[2r ( P ! M ) x]/9 [13] compare sedimentation coefRcients obtained un-
2
der differing conditions and/or sedimentation
Eqn [13] is more commonly expressed in terms of media by reference to the behaviour of the particle in
particle velocity, v, and particle diameter, d: water at 203C, S 20,w :
2
2
v"(d ( P ! M ) x)/18 ) [14]
S 20,w "S T,M T,M ( P ! 20,w )/ 20,w ( P ! T,M ) [17]
Eqn [14] may be integrated to determine the time where the subscripts T and M denote the experi-
required for a particle to traverse a radial distance mental temperature and medium, respectively.
from x 0 to x 1 :
Rotor Ef\ciency
2
2
t"[18 /(d ( P ! M ) )] ln (x 1 /x 0 ) [15]
The time required for a particle to traverse a rotor is
where x 0 is the initial position of the particle and x 1 is known as the pelleting efRciency or k-factor. The
the Rnal position of the particle. k- or clearing factor, which is calculated at the max-
While modiRcations can be made to eqns [13]}[15] imum rated rotor speed, is a function of rotor design
to account for speciRc rotor design, liquid}liquid, and is a constant for a given rotor. k-Factors provide
density-gradient separations, etc., these equations de- a convenient means of determining the minimum
scribe the relative impact of the more signiRcant para- residence time required to pellet a particle in a given