Numerical simulation of the retention mechanism of stationary phase in counter-current chromatography

The high separation efficiency of counter-current chromatography (CCC) strongly depends on the robust retention of the stationary phase, which is maintained by a hydrodynamic equilibrium between the two liquid phases. However, developing reliable computational fluid dynamics (CFD) models to simulate this process is challenging, mainly because the hydrodynamic mechanisms governing the equilibrium has not been fully clarified. This study proposes a modeling strategy based on a stationary spiral-tube domain subjected to a time-dependent rotational force field. This formulation provides a reduced two-dimensional framework for exploring the two-phase hydrodynamic equilibrium process in CCC. The model represents the force field through three categories of body-force terms: the rotational gravitational force, the rotational inertial force, and a unidirectional spiral force associated with the Archimedean screw effect. Under the conditions examined, the combined force representation produced a sustained retained-phase distribution in the two-dimensional domain. Moreover, the predicted increase in stationary-phase retention with rotational speed, decrease in retention with increasing flow rate, and the near-constant pressure drop over the examined flow-rate range were qualitatively consistent with published observations. Analysis of the simulated phase distributions and velocity fields provides a reasonable mechanistic interpretation of stationary phase retention. With this interpretation, retention may arise from three coupled behaviors: radial stratification, head-tail driving, and cyclic compensation. The proposed framework offers a tractable starting point for subsequent experimentally validated, higher-fidelity three-dimensional CFD studies of stationary phase retention in CCC coils.