Is the Van Deemter equation only a model ? :-(

Chromatography Forum: GC Archives: Is the Van Deemter equation only a model ? :-(
By IL on Thursday, January 29, 2004 - 04:48 am:

Many years ago when I was introduced into the theory of capillary GC, the lecturer indicated various aspects (optimal lin velocity, min HETP) which can be obtained from a H versus u curve. Other more subtle things like radial diffusion and film and col. diametre was also discussed. This all sound fine at the time and appeared to work in practice. However, I recently read that the Van Deemter formula is only an approximation and it was realy developed only for packed or megabore open tubular columns where there is hardly any pressure drop between the inlet and outlet. The main problem with the Van Deemter formula, and thus the curve, is that it does not provide for the compressibilty of gasses - something which surely happens in cappilary GC columns. How trustworthy (or rather how useful) is the data generated in a Van Deemter experiment? Thanks

By Leon on Thursday, January 29, 2004 - 01:31 pm:

Dear IL,

Several different and incompatible formulae are frequently called as Van Deemter Equation. Therefore, the term “Van Deemter Equation” might mean different thing for different people. Capillary columns, are better described by Golay equation, but that too is known in several incompatible versions.

For capillary columns, you are most likely to find in the literature that Van Deemter–Golay equation is the formula

(*) H = B/u + C*u

where H is a column plate height (also known as height equivalent to one theoretical plate, H.E.T.P.), u is AVERAGE velocity of a carrier gas, and B and C are constants that depend on column dimensions, carrier gas type, and, to a smaller degree, on other factors.

The most amazing thing is that neither Van Deemter (with coauthors), nor Golay, nor anyone else ever derived theoretically or demonstrated experimentally that formula (*) is correct for GC with high pressure drop (higher than about 100 kPa or 15 psi), or for GC-MS with any inlet pressure.

Frequently (but not always) it is desirable to minimize H. Suppose you have a 20 m x 0.1 mm column with helium as a carrier gas. Get the values of B and C for your column from equation that Golay actually derived. Find from Eq. (*) the value of u – usually called optimal velocity of a carrier gas, and denoted as uopt – that leads to the minimum in H for those values of B and C. You will find that uopt should be about 100 cm/sec. With this value of uopt, you are for many surprises. First, the column head pressure should be about 1700 kPa (240 psi). Second, it is frequently recommended to use the gas velocity that is twice as higher than uopt (it is called optimum practical gas velocity, OPGV). For OPGV, the head pressure should be about 3400 kPa (500 psi). If you manage to get such high pressures (you need to customize your commercial GC instrument for that), be ready for another surprise. The separation that you get is so miserable that you most likely to scrap the whole project and to conclude that the Fast GC does not work. Wrong! It is the equation (*) that is wrong. It is not suitable for prediction of optimal conditions in majority of GC cases.

What are the practical implications?

If you need to experimentally find uopt for a particular column, you should not worry about Van Deemter equation or Golay equation or any other equation. Just experimentally find oupt that minimizes H in your column. Be careful, however, with OPGV. It might cause too great loss in the separation power of your column. What’s more, your experimentally found value of uopt might not be good for a column with the same diameter, but with different length. Again, the uopt value that you experimentally found might be only good for that particular column.

Theory and experimental data indicate that you will be better off controlling a carrier gas flow rate, F, rather than its average velocity, u. Commercial GC instruments allow to electronically control F.

Find experimentally the value of Fopt (optimal flow rate corresponding to minimal H) for your column. If you are looking for the best tradeoff between the separation power and the analysis time, use a speed-optimized flow rate (SOF) that is about 40% higher than Fopt. What is nice about Fopt or SOF is that either of these values is proportional to a column diameter and does not depend on the column length. For example, if you found SOF for your column, it is the same for a 10 times shorter and for a 10 times longer column of the same diameter. And, if you need to reduce internal diameter of your capillary column by, say, a factor of two, just reduce SOF by a factor of two as well. That’s all.

Moreover, you can save a lot of time if you calculating rather than experimentally find SOF for your capillary column. Use the following simple formula:

SOF(mL/min) = G*ID(mm)

where ID is internal diameter of a column (in millimeters) and G is a gas-dependent constant (G=10 for hydrogen, G=8 for helium, G=2.5 for nitrogen). For example, SOF in a 0.1 mm column (of any length) would be 1 mL/min for hydrogen and 0.8 mL/min for helium. If the column is 20 m long, these require the head pressure of about 430 kPa (62 psi) for hydrogen and 600 kPa (87 psi) for helium.

Leon

Literature:
L. M. Blumberg, J. High Resolut. Chromatogr. 1997, 20, 704.
L. M. Blumberg, J. High Resolut. Chromatogr. 1997, 20, 597-604.
L. M. Blumberg, J. High Resolut. Chromatogr. 1997, 20, 679-87.
L. M. Blumberg, J. High Resolut. Chromatogr. 1999, 22, 403-13.
M. J. E. Golay in D. H. Desty (Editor), Gas Chromatography 1958, Royal Tropical Institute, Amsterdam, May 19-23, 1958, Academic Press, New York, 1958, 36-55.
J. J. Van Deemter, F. J. Zuiderweg, A. Klinkenberg, Chem. Eng. Sci. 1956, 5, 271-89.

By Leon on Thursday, January 29, 2004 - 01:54 pm:

Dear IL,

Several different and incompatible formulae are frequently called as Van Deemter Equation. Therefore, the term “Van Deemter Equation” might mean different thing for different people. Capillary columns, are better described by Golay equation, but that too is known in several incompatible versions.

For capillary columns, you are most likely to find in the literature that Van Deemter–Golay equation is the formula

(*) H = B/u + C*u

where H is a column plate height (also known as height equivalent to one theoretical plate, H.E.T.P.), u is AVERAGE velocity of a carrier gas, and B and C are constants that depend on column dimensions, carrier gas type, and, to a smaller degree, on other factors.

The most amazing thing is that neither Van Deemter (with coauthors), nor Golay, nor anyone else ever derived theoretically or demonstrated experimentally that formula (*) is correct for GC with high pressure drop (higher than about 100 kPa or 15 psi), or for GC-MS with any inlet pressure.

Frequently (but not always) it is desirable to minimize H. Suppose you have a 20 m x 0.1 mm column with helium as a carrier gas. Get the values of B and C for your column from equation that Golay actually derived. Find from Eq. (*) the value of u – usually called optimal velocity of a carrier gas, and denoted as uopt – that leads to the minimum in H for those values of B and C. You will find that uopt should be about 100 cm/sec. With this value of uopt, you are for many surprises. First, the column head pressure should be about 1700 kPa (240 psi). Second, it is frequently recommended to use the gas velocity that is twice as higher than uopt (it is called optimum practical gas velocity, OPGV). For OPGV, the head pressure should be about 3400 kPa (500 psi). If you manage to get such high pressures (you need to customize your commercial GC instrument for that), be ready for another surprise. The separation that you get is so miserable that you most likely to scrap the whole project and to conclude that the Fast GC does not work. Wrong! It is the equation (*) that is wrong. It is not suitable for prediction of optimal conditions in majority of GC cases.

What are the practical implications?

If you need to experimentally find uopt for a particular column, you should not worry about Van Deemter equation or Golay equation or any other equation. Just experimentally find uopt that minimizes H in your column. Be careful, however, with OPGV. It might cause too great loss in the separation power of your column. What’s more, your experimentally found value of uopt might not be good for a column with the same diameter, but with different length. Again, the uopt value that you experimentally found might be only good for that particular column.

Theory (that follows from a true Golay equantion) and experimental data indicate that you will be better off controlling a carrier gas flow rate, F, rather than its average velocity, u. Commercial GC instruments allow to electronically control F.

Find experimentally the value of Fopt (optimal flow rate corresponding to minimal H) for your column. If you are looking for the best tradeoff between the separation power and the analysis time, use a speed-optimized flow rate (SOF) that is about 40% higher than Fopt. What is nice about Fopt or SOF is that either of these values is proportional to a column diameter and does not depend on the column length. For example, if you found SOF for your column, it is the same for a 10 times shorter and for a 10 times longer column of the same diameter. And, if you need to reduce internal diameter of your capillary column by, say, a factor of two, just reduce SOF by a factor of two as well. That’s all.

Moreover, you can save a lot of time if you calculate rather than experimentally find SOF for your capillary column. Use the following simple formula:

SOF(mL/min) = G*ID(mm)

where ID is internal diameter of a column (in millimeters) and G is a gas-dependent constant (G=10 for hydrogen, G=8 for helium, G=2.5 for nitrogen). For example, SOF in a 0.1 mm column (of any length) would be 1 mL/min for hydrogen and 0.8 mL/min for helium. If the column is 20 m long, these flow rates require the head pressure of about 430 kPa (62 psi) for hydrogen and 600 kPa (87 psi) for helium.

Leon

Literature:
L. M. Blumberg, J. High Resolut. Chromatogr. 1997, 20, 704.
L. M. Blumberg, J. High Resolut. Chromatogr. 1997, 20, 597-604.
L. M. Blumberg, J. High Resolut. Chromatogr. 1997, 20, 679-87.
L. M. Blumberg, J. High Resolut. Chromatogr. 1999, 22, 403-13.
M. J. E. Golay in D. H. Desty (Editor), Gas Chromatography 1958, Royal Tropical Institute, Amsterdam, May 19-23, 1958, Academic Press, New York, 1958, 36-55.
J. J. Van Deemter, F. J. Zuiderweg, A. Klinkenberg, Chem. Eng. Sci. 1956, 5, 271-89.

By Leon on Thursday, January 29, 2004 - 01:59 pm:

Sorry, I unintentionally posted the first version of my message. Only the second version was to be posting. Leon