Axially Dispersed Plug Flow Model and Conservation Equations

Key Insight

The fundamental dynamics of chromatographic columns, applicable to virtually all transient chromatography, are addressed by the axially dispersed plug flow model. This model assumes uniform flow across the column diameter, incorporating any deviations from ideal flow through a Fickian, axial dispersion term. It is suitable for describing real columns provided flow deviations are not extreme; for large variations, detailed multidimensional models are needed, though these conditions are generally avoided in industrial chromatography which aims for uniform flow even in large-diameter columns. The solution of conservation equations is considered in two steps: initially neglecting axial dispersion and assuming local equilibrium ('ideal chromatography'), which is approximated in well-packed columns at low flow rates with small particles, before later considering dispersion effects.

A simplified model for chromatography assumes plug-flow through a uniformly packed sorption bed. The general mass balance for M components is given by ε ∂Ci/∂t + (1 - ε) ∂qˆi/∂t + εv ∂Ci/∂z = εDL ∂^2Ci/∂z^2 (7.1). Here, qˆi represents the particle-average adsorbate concentration, comprising both adsorbed molecules (qi) and those within the pore fluid (εp ci), quantified by (3/rp^3) ∫ (qi + εp ci)r^2 dr from 0 to rp (7.4); for dilute solutions with favorable adsorption, qˆi approximates qi. DL is the axial dispersion coefficient. The system incorporates Danckwerts boundary conditions: z = 0: Ci = CFi + (DL/v) ∂Ci/∂z and z = L: ∂Ci/∂z = 0 (7.3).

When axial dispersion (DL = 0) is neglected, the mass balance simplifies to ε ∂Ci/∂t + (1 - ε) ∂qˆi/∂t + εv ∂Ci/∂z = 0 (7.5) with the boundary condition z = 0: Ci = CFi (7.6). Rate equations (7.2) describe intra-particle transport and kinetic processes, utilizing multi-component isotherms (q*j = q*j (C1, C2, ..., CM)) to relate concentrations at the particle-fluid interface. Column entrance conditions vary for classical modes: Elution chromatography includes isocratic-analytical (pulse injections, CFi = (Mi/Q) δ(t)), isocratic-preparative (finite injection, CFi = CFeed,i for 0 < t < tFeed, else 0), and gradient-analytical (linear, CFi = (Mi/Q) δ(t) for M-1 components, CFM = C0M + βt). Frontal analysis uses CFi = CFeed,i × H(t), and displacement development involves CFi = CFeed,i for initial components, then CF1 = CD and others zero. δ(t) is the delta function, and H(t) is the unit step function; delta function inputs simplify analysis and are practical when the feed volume is a small fraction of the eluted peak volume.