Extension of Golay's plate height equation from laminar to turbulent flow I - Theory

•Golay's plate height equation was extended from laminar to turbulent flow.•The new plate height equation was validated from three independent theories.•Performance of open tubes in turbulent regime with carbon dioxide eluent is projected.•Resolution is improved as long as the film thickness is less than a few percent the tube radius.

The reduced plate height (RPH) equation of Golay derived in 1958 for open tubular columns (OTC) is extended from laminar to turbulent-like flow. The mass balance equation is solved under near-equilibrium conditions in the mobile phase for changing shapes of the velocity profile across the OTC diameter. The final expression of the general RPH equation is:(1)h=2νDa¯Dm+1+[n+4]k+n24+52n+5k24(n+2)(n+4)(1+k)2DmDr¯ν+23k(1+k)2DmDsdfD2νwhere ν is the reduced linear velocity, k is the retention factor, Dm is the bulk diffusion coefficient in the mobile phase, Da¯ is the average axial dispersion coefficient, Dr¯ is the average radial dispersion coefficient, Ds is the diffusion coefficient of the analyte in the stationary film of thickness df, D is the OTC inner diameter, and n ≥ 2 is a positive number controlling the shape of the flow profile (polynomial of degree n). The correctness of the derived RPH equation is verified for Poiseuille (n = 2), turburlent-like (n = 10), and uniformly flat (n → ∞) flow profiles.The derived RPH equation is applied to predict the gain in speed-resolution of a 180 μm i.d. × 20 m OTC (df = 2 μm) from laminar to turbulent flow in supercritical fluid chromatography. Using pure carbon dioxide as the mobile phase at 297 K, k = 1, and increasing the Reynolds number from 2000 (laminar) to 4000 (turbulent), the OTC efficiency is expected to increase from 125 to 670 (×5.4) while the hold-up time decreases from 19 to 9 s (×0.5). Despite the stronger resistance to mass transfer in the stationary phase, the projected improvement of the column performance in turbulent flow is explained by the quasi-elimination of the resistance to mass transfer in the mobile phase while axial dispersion remains negligible.