Three-dimensional graphing representing six variables for speed and separation performance in liquid chromatography

•Landscape of number of theoretical plates generated by velocity and column length.•Number of theoretical plates per unit length to show performance of packing support.•Pressure and time are transformed by logarithmically rotating coordinate system.•Number of theoretical plates increases monotonically with pressure or time.•Number of theoretical plates has an asymptotic upper limit at pressure or time. The two variables, flow rate and column length, enable naive determination of the number of theoretical plates (N) in isocratic elution; this, in turn, enables the formation of a three-dimensional graph with N as the z-axis. An alternate three-dimensional graph with N as the z-axis can be drawn, then, with the alternate basal plane illustrating the pressure drop and hold-up time. In this article, the pressure drop and hold-up time are formulated so as to be represented unitarily in the former graph, because the flow rate and column length interact simultaneously as operational variables. This formulation manipulates both the pressure drop and the hold-up time as logarithmic axes, to evaluate the landscape. Also of use is the representation, in the same graph, of the height equivalent to a theoretical plate, as the fundamental property of the packing supports. For this purpose, the number of theoretical plates per unit length are here introduced as the sixth variable, instead of the height equivalent to a theoretical plate. Representing the six variables in three-dimensional graphs enables a clear understanding both of the separation condition optimization methods and the relation among variables for the speed and separation performance. The linear velocity, column length, N, velocity-length product, hold-up time, and number of theoretical plates per unit length, are here selected as the six elementary variables for the three-dimensional graphs; and, based on the packing supports of 2, 3, and 5-μm particle and monolithic columns. Finally, the usage of logarithmic three-dimensional graph is illustrated for understanding the speed and separation performance.