HPLC Column Cross-Sectional Area and Scaling

Here is a simple formula to use when scaling up or down Internal Column Diameter to maintain retention values (under constant linear velocity). Flow rate must be adjusted to account for any changes made to the column's cross-sectional area. We usually refer to these types of changes as the "Scaling Factor". To determine the scaling factor, we need to know the internal column diameters of the two columns we are scaling from (actually, we need to know the radius, but once we have the diameter, we simply divide the diameter by 2 to obtain the radius). *In this discussion, changes in cross-sectional area are the only parameters we are concerned with as column length does not affect scaling.

• Scaling Factor = (S);
• Column #1 Radius =  (R1);
• Column #2 Radius =  (R2).

     S = R2² / R1²

Example #1: 250 x 4.60 mm column scaled down to a 250 x 2.10 mm column. 

          Answer = 0.208. 

• If the original flow rate was 1.000 mL/min, the the scaled down flow rate would be 0.208 of the original or 0.208 mL/min for the 2.10 mm ID column. *For practical use and application, we often use either 200 ul/min or 210 ul/min to simplify the value.

Example #2: 250 x 4.60 mm column scaled up to a 250 x 10.00 mm ID semi-prep column.

          Answer  = 4.726. 

• If the original flow rate used was 1.000 mL/min with the 4.60 mm ID column, then we would increase the flow rate to 4.726 mL/min on the 10.00 mm ID column to maintain the same relative velocity (and relative retention). *For practical use and application, we often use 5 mL/min to simplify (round off) the value. 

Notes:

1. Flow rate optimization should always be carried out by running a standard at different flow rates and plotting the plate height (N) vs the flow rate. Test flow rates that are slightly below the predicted linear velocity and up to 2 times higher than that rate to find and optimize the flow rate for your sample (it must be determined through experimentation for your specific method). 
    
2. HPLC Columns packed with sub 2 micron supports may have optimum flow rates 2 to 5 times more than the predicted std linear flow rate so actual testing is critical to determining the most efficient flow rate. I recommend optimizing the flow rate used with analysis methods which use any particles which are 2.5 microns or smaller in diameter.
   

Number of theoretical plates (N), Calculation Formulas

Number of theoretical plates (N):

Often used to quantify the efficiency (performance) of a column (HPLC or GC). "Plates" are expressed per meter of column length and should be calculated based on a retained peak with ideal peak shape or symmetry. 

   N = Plates; tr = Retention Time of Peak; w = Peak width;  w_0.5 = Peak width measured at half height.

Two popular formulas are:

Tangent: USP (United States Pharmacopeia / ASTM)

Best for Gaussian peaks. Peak width is often determined at 13.4% of the peak height (w). Inaccurate for peaks which are non-Gaussian, poorly resolved or tail.

   N = 16 (tr / w)²

Half Peak Height: (European Pharmacopeia)

For peaks which are less Gaussian in appearance, using a slightly different formula with the peak width measurement made at the half-height (W_0.5). Less accurate for peaks which are poorly resolved or tail.

   N = 5.54 (tr / w_0.5)²

 

• Other formulas, not included, for calculating Plate numbers include: Half Width, Variance Method, Area / Height & Exponential Modified Gaussian (EMG).
• Caution. HPLC column "Plate" values should not be used for a final determination of efficiency unless you are comparing all results on the same exact HPLC system, which is setup and run under identical conditions each time. Since the result is based on many possible variables, including how your HPLC system is plumbed (dwell volume & tubing ID), the peak's Kprime and symmetry, the detector used, sampling rate, integration quality, flow cell volume, flow rate, actual column used (to name a few), it can easily be manipulated to be very large or small.

USP Guideline Note: HPLC Column Diameter Changes to Maintain Flow Rate Linear Velocity

USP Allowed Variations in HPLC Column Diameter (*USP 32, Second Supplement, Dec 1, 2009). In the previous USP version, a change of up to 50% of the flow rate was allowed. This has been changed in the more recent version. A wide range of column diameter changes are now allowed, provided that the linear velocity is kept constant. *We addressed the effect of changing column diameter on flow rate in a previous blog post, but this time I have also expanded on the calculation by including the extra variable for column length (L1 and L2) as well.

*Adjusting the Column Flow Rate for Changes in the HPLC Column Diameter.

Linear Velocity Formula:

   New Linear Flow Rate₂ = Flow Rate₁ x (L2 x D2²) / (L1 x D1²)

Flow Rates are in ml/min.

L1 = Column Length (original) in mm.

L2 = Column Length (proposed) in mm.

D1 = Column Diameter (original) in mm.

D2 = Column Diameter (proposed) in mm.

Example #1:

Original column is a 150mm x 4.6mm (L x ID) used at 1.000 ml min. We would like to determine what the equivalent flow rate (F₂) would be for a column which is 150mm x 2.1mm (L x ID) to maintain the same linear velocity. This is a proposed change in column diameter of > 50% so it would not have been allowed under the old guidelines. The newer guidelines take into account that with the same particle size, changing the column diameter will not change the chromatography if the linear velocity is maintained as before. Let’s calculate the new flow rate using the formula above.

1.000 x (150 x 2.1²) / (150 x 4.6²) = F₂

                    1 x (661.50 / 3,174) = F₂

                              0.208 ml/min = F₂