Modeling and dynamic analysis of liquid extraction columns are essential for the design,
control strategies and understanding of column behavior during start up
and shutdown. Because of the discrete character of the dispersed phase,
the population balance modeling framework is needed. Due to the
mathematical complexity of the full population balance model, it is
still not feasible for dynamic and online control purposes. In this
work, a reduced mathematical model is developed by applying the concept
of the primary and secondary particle method (Attarakih et al., 2009b,
Solution of the population balance equation using the one primary and
one secondary particle method (OPOSPM), Computer Aided Chemical
Engineering, vol. 26, pp. 1333–1338). The method is extended to solve
the nonhomogenous bivariate population balance equation, which describes
the coupled hydrodynamics and mass transfer in an RDC extraction
column. The model uses only one primary and one secondary particles,
which can be considered as Lagrangian fluid particles carrying
information about the distribution as it evolves in space and time. This
information includes averaged quantities such as total number, volume
and solute concentrations, which are tracked directly through a system
of coupled hyperbolic conservation laws with nonlinear source terms. The
model describes droplet breakage, coalescence and interphase solute
transfer. Rigorous hyperbolic analysis of OPOSPM uncovered the existence
of four waves traveling along the column height. Three of these are
contact waves, which carry volume and solute concentration information.
The dynamic analysis in this paper reveals that the dominant time
constant is due to solute concentration in the continuous phase. On the
other hand, the response of the dispersed phase mean properties is
relatively faster than the solute concentration in the continuous phase.
Special shock capturing method based on the upwind scheme with flux
vector splitting is used, with explicit wave speeds, as a time–space
solver. The model shows a good match between analytical and numerical
results for special steady state and dynamic cases as well as the
published steady state experimental data.
https://www.researchgate.net/publication/256679997_Modeling_and_dynamic_analysis_of_a_rotating_disc_contactor_RDC_extraction_column_using_one_primary_and_one_secondary_particle_method_OPOSPM