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.