Flow rate, linear velocity and residence time

A pump is set in mL/min. Linear velocity in cm/h is that flow divided by the column cross-section, so the same value means the same flow per unit area on any diameter. Dynamic binding capacity can depend on residence time. These are the same flow seen three ways, linked by the column geometry.

The formulas

For a column of inner diameter d and bed height h, both in cm:

• Cross-sectional area: A = π (d/2)², in cm².
• Column volume: CV = A · h = π (d/2)² · h, in mL (1 cm³ = 1 mL).
• Linear velocity: u = Q / A × 60, in cm/h, with Q in mL/min. The factor 60 converts minutes to hours.
• Residence time: τ = h / u × 60 = CV / Q, in min.
• Column volumes per hour: CV/h = 60 / τ.

Linear velocity here is superficial velocity: flow divided by the empty column cross-section. Liquid moves faster between the beads, because only part of the bed is open to flow. Residence time is likewise the empty-bed value, packed bed volume divided by flow. Some papers use the void volume instead, which gives a shorter time. State which one you use.

Worked example

A column with 1.6 cm inner diameter and a 10 cm bed: A = π × 0.8² = 2.01 cm², and CV = 20.1 mL. At Q = 2 mL/min, u = 2 / 2.01 × 60 = 59.7 cm/h. τ = 20.1 / 2 = 10.05 min, and CV/h = 60 / 10.05 = 5.97. These are the calculator defaults.

Which number to hold constant

• Linear velocity compares runs on columns of different diameter but the same bed height, because it is divided by the cross-section. For the same reason, a flow limit given in cm/h applies to any diameter, while one given in mL/min applies only to the column it was written for. Check whether your resin and column instructions state the limit in cm/h, in mL/min or as a pressure.
• Residence time is the contact time between liquid and bed. Dynamic binding capacity can depend on it, because uptake into porous beads takes time; how much is specific to the resin and feed. Measure it as in measuring dynamic binding capacity.
• CV/h is residence time in another form. It is convenient when step lengths are written in CV.

To move a method to a larger column, see scale up at constant residence time and bed height.

Limits this tool does not check

Pressure drop over a packed bed grows with linear velocity, bed height and liquid viscosity. It also depends on the resin particle size, how the bed was packed and the column hardware. Cold buffers and additives such as glycerol raise viscosity and pressure. Use the resin and column instructions for velocity and pressure limits under your conditions. If the limit is in cm/h, enter it above to get a warning when the computed velocity exceeds it; convert a limit in mL/min with the converter first.

The example geometries are generic inner diameter × bed height pairs for common lab formats. Prepacked columns list their exact bed volume and dimensions on the data sheet; use those values.

Sources and scope

• Cytiva™, Ion Exchange Chromatography: Principles and Methods (handbook), Appendix 5 — the conversion between flow velocity (cm/h) and volumetric flow rate (mL/min) through the column cross-sectional area, with the worked example of a 1.6 cm column at 150 cm/h giving 5.03 mL/min. Our tests reproduce that example.
• Cytiva™, Develop your column packing methods based on pressure-flow measurements (procedure) — pressure drop depends on flow velocity, viscosity, bed height, particle size, bed compression, column hardware and the flow path, and must be measured for compressible resins.
• Chen et al. (2021), Linear flow-velocity gradient chromatography — distinguishes empty-bed from void-volume residence time and discusses how capacity and productivity trade off with residence time. Open access, CC BY-NC-ND; cited for facts only.

The formulas are geometry; the page text is original. Results are not saved and are not sent anywhere.