Page 26 - Subyek Encyclopedia - Encyclopedia of Separation Science

Sepsci*11*TSK*Venkatachala=BG
I / CENTRIFUGATION 21

rotor and are useful for comparing sedimentation term, . In Stokes’ equation, the term 6 r describes
times for different rotors. The k-factor is derived the frictional coefTcient, f 0 , for a spherical par-
from the equation: ticle. The correction term, , is calculated as the ratio
of the frictional resistance, f, encountered by a par-
2
13
k"ln (r max !r min ) 10 /(3600 ) ticle of nonspherical geometry to that encountered by
a sphere of the same volume, or:
11
"2.53 10 ln (r max !r min )/rpm 2 [18]
"f /f 0 "f /6 r [22]
where r max and r min are the maximum and minimum
distances from the centrifugal axis, respectively. The equation describing the terminal velocity for
Eqn [18] shows that the lower the k-factor, the nonspherical particles in a centrifugal Reld may be
shorter the time required for pelleting. If the sedi- rewritten as:
mentation coefRcient of a particle is known, then dx/dt"[d e ( P ! M ) x]/18 [23]
2
2
the rotor k-factor can also be calculated from the
relation: where d e is the diameter of a sphere whose volume
equals that of the sedimenting particle (d e /2 is the
k"TS [19] Stokes radius).
The net result of this modiRcation is that non-
where T is the time in hours required for pelleting spherical particles are predicted to sediment more
and S is the sedimentation coefRcient in Svedberg slowly, which is a more accurate depiction of their
units. real-world behaviour.
When k is known (normally provided by the manu- In addition to deviations from spherical-particle
facturer), then eqn [19] may be rearranged to calcu- geometry, there are other effects that can lead to
late the minimum run time required for particle departure from predicted behaviour (nonideality)
pelleting. during sedimentation. For example, many biological
For runs conducted at less than the maximum rated particles interact with the medium via hydration, the
rotor speed, the k-factor may be adjusted according extreme case being for those particles with osmotic
to: properties, which can result in drastic changes in
particle density and, in turn, sedimentation coef-
k adj "k(rpm max /rpm act ) 2 [20] Rcients. Interparticle attractions, e.g. charge or hy-
drophobic effects, may increase the effective
where rpm max and rpm act are the maximum rated viscosity of the medium. In more severe cases such
rotor speed and actual run speed, respectively. attractions can lead to poor separations where the
k-Factors are also useful when switching from a ro- centrifugal energy is insufRcient to disrupt the
tor with a known pelleting time, t 1 , to a second rotor attractions between particles that are targeted for
of differing geometry by solving for t 2 in the separation. This latter effect is aggravated by the
relation: fact that the larger or denser particle will lead as the
particle pair migrates toward the rotor wall while
[21] the smaller or lighter attached particle follows in its
t 1 /t 2 "k 1 /k 2
wake, and therefore experiences less frictional drag.
where t 1 , t 2 , k 1 and k 2 are the pelleting times and Particles may also concentrate locally to increase the
k-factors for rotors 1 and 2, respectively. effective medium density, or form aggregates
that yield complicated sedimentation patterns. Be-
Deviation from Ideal Behaviour cause of such deviations from ideal behaviour, equiv-
alent sedimentation coefRcients, SH,deRned as
Eqns [13] and [14] showed the relative impact on
the sedimentation coefRcient of an ideal spherical
settling velocity of the more important and control-
lable experimental parameters. However, there are particle, are often reported for a given set of experi-
other effects that are more difRcult to char- mental conditions.
acterize and which can result in signiRcant deviations
from the settling velocities predicted by these equa- Filtration
tions. The most common of these effects occurs A mathematical description of liquid drainage from
when the particles are nonspherical, as these equa- a packed bed by centrifugal forces is essentially the
tions are derived from Stokes’ equation assuming same as that used to describe more conventional
spherical particles. For nonspherical particles, gravity or differential-pressure Rltration, the pri-
eqns [13] and [14] may be modiRed with a correction mary differences being that the centrifugal force