Dimensionless System and Local Equilibrium Dynamics

Key Insight

To analyze chromatography dynamics efficiently, dimensionless expressions for column mass balances are introduced. Dimensionless concentrations are defined as Xi = Ci / Cref and Yi = qˆi / qref (7.12), normalized by reference values like feed or initial concentrations. Dimensionless time τ = εvt / L and axial coordinate ζ = z / L (7.13) are utilized; τ represents the number of column volumes of mobile phase passed (CV = tQ / Vc). This transformation results in the dimensionless mass balance: k'ref ∂Yi/∂τ + ∂Xi/∂τ + (1/ε) ∂Xi/∂ζ = (1/(εPeL)) ∂^2Xi/∂ζ^2 (7.14). A reference retention factor is defined as k'ref = (1 - ε) qref / (ε Cref) (7.15), and the Peclet number, PeL = vL / DL (7.16), quantifies the ratio of convective to dispersive transport.

When the Peclet number approaches infinity (PeL → ∞), signifying negligible axial dispersion, a redefined dimensionless time τ1 = τ - ζ / k'ref (7.17) simplifies the material balances to ∂Yi / ∂τ1 + ∂Xi / ∂ζ = 0 (7.18). This variable τ1 serves as a throughput parameter. For example, if τ1 = 1 at the column outlet (ζ = 1), the corresponding physical time is t = (L / v) (1 + k'ref) (7.20). This specific time represents the elution time of a component with a retention factor k' equal to k'ref under isocratic elution with a linear isotherm. Alternatively, for frontal loading of a clean column with favorably adsorbed components, τ1 = 1 indicates the time of passage of the adsorption front or the point where the column has received enough feed to achieve complete adsorbent saturation at equilibrium with the feed concentration.

The simplest analysis, known as local equilibrium theory or the ideal model of chromatography, posits non-dispersed plug flow and instantaneous interphase equilibrium, thereby neglecting axial dispersion and assuming all rate factors are sufficiently fast. Solutes in the mobile phase move at the interstitial velocity, v, while those within adsorbent particles have an average axial velocity of zero, an assumption also valid for highly permeable particles. The chromatographic velocity (v_ci) of a solute is proportional to its fraction in the mobile phase. For isocratic elution with a linear isotherm, this velocity is given by v_ci = v / (1 + ((1 - ε)/ε) mi) (7.21), where mi is the slope of the linear isotherm. Consequently, the elution time (tRi) for a component is tRi = L / v_ci = (L / v) (1 + ((1 - ε)/ε) mi) = (L / v) (1 + k'i) (7.22), consistent with Equation 2.10. The general chromatographic velocity with an arbitrary isotherm is derived from ε ∂Ci/∂t + (1 - ε) ∂q*ˆi/∂t + εv ∂Ci/∂z = 0 (7.23), where ∂q*ˆi/∂t can be expressed as (dq*ˆi/dCi) ∂Ci/∂t (7.24).