Chromatography Handbook Of Hplc - Katz Eksteen Schocnmakers Miller wiley Sons -106

96\nTijssen\nalthough potentially affecting flow and dispersion phenomena [5,18,26; see under Sec. II.F.2.b], is of little practical interest under common-operating conditions.\n\nFirst, looking at the flow condition in open tubular channels, it is immediately clear that in the absence of obstacles all flow lines are straight and parallel to the column axis, strongly opposed to the situation in packed columns as described in the foregoing. The first obvious consequence of this is that molecular diffusion is unobstructed; hence, the obstruction factor γ equals unity, and the axial diffusion contribution to plate height is simply found from\n\nH_d,app = D_n / (u_avg) = R_app / (u_max)\n\nor\n\nh_d,app = H_d,app / t_d = 2 / v_app,\nwith v_app = (u_avg) / (D_m) = Re Sc\n\nEqually logical is the absence of an eddy diffusion term in the total plate height equation. The flow velocity profile, however, is very pronounced and yields a large C_m term contribution to plate height. The velocity profile properties, as a result of the radial symmetry of the column, can be calculated from first principles [15]. It is parabolic and known as the Poiseuille profile:\n\nu_avg(r) = u_max(1 - (r / R)^2)\nThe center line velocity u_max identifies the maximum velocity in the profile, which exactly equals 2 times the mean velocity. At the column wall (r = R), the velocity becomes zero, as slip of the fluid at the wall is not supposed to be present.\n\nWe are now interested in the zone-spreading of a thin solute zone introduced as a flat concentration profile evenly distributed throughout the cross section of the tube at the entrance (z = 0) of the column, through the action of the velocity differences in the profile. It is clear that molecules present at or near the center line, rapidly move away in the axial direction from the molecules near the column wall that have lower velocities, even down to zero at the exact position of the column wall. It is illustrative to take as an example an average velocity of 1 cm/s (low for GC but high for LC), which implies that the maximum velocity difference in the zone amounts to 2 cm/s. Hence, every second the zone widens by at least 2 cm, a growth rate that would make it impossible to obtain any useful efficiency of the column after a typical retention time of several minutes or longer: the sample zone fills as if it were the complete column over its entire length on elution of the first, fast-moving molecules on the center line. The strongly asymmetrical and tailing peak shapes (better called residence time distributions) under these conditions can be calculated [15], but have no practical meaning in chromatography.\n\nHence, a very strong zone-broadening would occur were it not that the molecular diffusion, which acts equally in all directions, also mutually exchanges molecules in the radial direction throughout the cross section, thereby more or less averaging out all velocity differences. The faster this happens, the less zone-broadening is originated. In GC, with gaseous mobile phases, relatively wide capillaries can be used in view of the fast molecular diffusion. The smaller the column ID, the more efficient radial diffusion maintains zone integrity, and the more separations can be provided. In LC, as a consequence of the very limited diffusional mixing, we can foresee that extremely small diameters should be used. This is all the more true because radial molecular diffusion is the only mass-transfer mechanism present in these open tubular systems. Convective mechanisms in the radial sense, such as eddy diffusion in packed systems, is absent. Thus, in the open systems we cannot, as with packed systems, make unlimited use of increased flow velocities to generate enhanced radial mixing and counteract zone-broadening owing to velocity profiles. The only effect of increased velocities would be a larger spreading as the velocity differences increase in proportion with the velocity.\n\nFor this purpose the earlier mentioned dimensionless time expressed as the Fourier number (Fo = D / r^2) is a good measure for the limiting velocities to be used: residence or velocity.