3.4
Theory and Mathematical Models
Counter-current distribution has gradually evolved into a preferred multi-stage
contacting method for aqueous two-phase partitioning of biological materials
[71]. However, scale-up of this technique beyond the laboratory has been
hampered by the lack of a comprehensive model that accounts for the influence
of non-idealities on the partitioning process. Counter-current distribution of a
mixture of solutes has been modeled as an equilibrium staged process which
generates a solute concentration profile that can be represented by a Gaussian
error distribution. However, aqueous two-phase CCD is an inherently nonequilibrium process and can result in non-ideal behavior. This is mainly due to
factors such as non-alignment of the phase interface and phase cut location,
incomplete phase demixing, incomplete solute mass transfer, and perturbation
of the phase system composition as a result of solute loading and system dilution. In general, a perturbation of the phase composition in the feed stage due
to these and other factors will result in a stage-to-stage variation in the solute
partition coefficient (K), the phase volume ratio (R), and the total system
volume (V T ). This can translate into a significant deviation from ideal CCD
performance. Guinn [95] has developed a comprehensive material balance
model for non-ideal CCD and tested it using the ORSEP multi-staged extractor
described earlier.
In stage-independent CCD operation K, R, and V T do not vary from stage to
stage. However, stage-independent CCD operation is approached only for those
limited cases for which solute loading has a negligible impact on phase system
composition, total system volume, and phase volume ratio and for which mass
transfer and phase demixing have reached equilibrium. Our objective is to
begin with the assumption of stage-independent partitioning in order to
develop the general equations which describe solute partitioning in CCD but
take into account the effect of liquid interface and phase cut nonalignment on
CCD performance. The model will then be systematically generalized for nonideal operation by applying an empirical demixing model to account for nonequilibrium phase demixing and a model to account for the effect of solute
partitioning and feed stage dilution on the perturbations of the phase volume
ratio. Perturbation of the feed stage due to any of these non-equilibrium events
will be conveyed as a deviation from ideal CCD operation to each successive
stage.
In a CCD apparatus, phase transfer is accomplished by slicing the two-phase
system along the plane joining the upper and lower chamber halves. In the
unlikely event that the phase interface is aligned with this phase cut plane,
Eq. (37) can be applied to predict the solute composition in each phase. A more
general treatment takes into account the non-alignment of the liquid interface
and phase cut.
Three distinct volume regions are defined as shown in Fig. 17 for two
adjacent extraction stages of a typical CCD apparatus containing a total of n
extraction stages. Light phase is transferred from the left adjacent cavity (n – 1)
to the right cavity (n) by slicing the phase system at the junction of the two
Multistage Magnetic and Electrophoretic Extraction of Cells, Particles and Macromolecules
179
Theory and Mathematical Models
Counter-current distribution has gradually evolved into a preferred multi-stage
contacting method for aqueous two-phase partitioning of biological materials
[71]. However, scale-up of this technique beyond the laboratory has been
hampered by the lack of a comprehensive model that accounts for the influence
of non-idealities on the partitioning process. Counter-current distribution of a
mixture of solutes has been modeled as an equilibrium staged process which
generates a solute concentration profile that can be represented by a Gaussian
error distribution. However, aqueous two-phase CCD is an inherently nonequilibrium process and can result in non-ideal behavior. This is mainly due to
factors such as non-alignment of the phase interface and phase cut location,
incomplete phase demixing, incomplete solute mass transfer, and perturbation
of the phase system composition as a result of solute loading and system dilution. In general, a perturbation of the phase composition in the feed stage due
to these and other factors will result in a stage-to-stage variation in the solute
partition coefficient (K), the phase volume ratio (R), and the total system
volume (V T ). This can translate into a significant deviation from ideal CCD
performance. Guinn [95] has developed a comprehensive material balance
model for non-ideal CCD and tested it using the ORSEP multi-staged extractor
described earlier.
In stage-independent CCD operation K, R, and V T do not vary from stage to
stage. However, stage-independent CCD operation is approached only for those
limited cases for which solute loading has a negligible impact on phase system
composition, total system volume, and phase volume ratio and for which mass
transfer and phase demixing have reached equilibrium. Our objective is to
begin with the assumption of stage-independent partitioning in order to
develop the general equations which describe solute partitioning in CCD but
take into account the effect of liquid interface and phase cut nonalignment on
CCD performance. The model will then be systematically generalized for nonideal operation by applying an empirical demixing model to account for nonequilibrium phase demixing and a model to account for the effect of solute
partitioning and feed stage dilution on the perturbations of the phase volume
ratio. Perturbation of the feed stage due to any of these non-equilibrium events
will be conveyed as a deviation from ideal CCD operation to each successive
stage.
In a CCD apparatus, phase transfer is accomplished by slicing the two-phase
system along the plane joining the upper and lower chamber halves. In the
unlikely event that the phase interface is aligned with this phase cut plane,
Eq. (37) can be applied to predict the solute composition in each phase. A more
general treatment takes into account the non-alignment of the liquid interface
and phase cut.
Three distinct volume regions are defined as shown in Fig. 17 for two
adjacent extraction stages of a typical CCD apparatus containing a total of n
extraction stages. Light phase is transferred from the left adjacent cavity (n – 1)
to the right cavity (n) by slicing the phase system at the junction of the two
Multistage Magnetic and Electrophoretic Extraction of Cells, Particles and Macromolecules
179