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Theoretical prediction of plate height in serially coupled capillary columns
Journal oj’Chromatograph_v, 406 (1987) 3-10 Elsevier Science Publishers B.V.. Amsterdam CHROMSYMP.
Printed
1274
THEORETICAL PREDICTION OF COUPLED CAPILLARY COLUMNS
G. GUIOCHON*
in The Netherlands
PLATE
HEIGHT
IN
SERIALLY
and J. E. N. GUTIERREZ
Department of Chemistry, Georgetown University, Washington. DC 20057 (U.S.A.)
SUMMARY
An equation has been derived and tested for the prediction of the plate height on a serial combination of capillary columns in gas chromatography. The results show good agreement with experimental data. This equation, together with an equation for the apparent capacity factor, can be used to optimize the system parameters in resolving a mixture.
INTRODUCTION
Effective gas chromatographic (GC) separations depend on the column tivity (k’) and efficiency (N), as predicted by the resolution equation: R, = n+/4 [(a -
selec-
1)/a] [k’z/(l + k;)]
Proper selection of the stationary phase, column length and instrument operating parameters is therefore crucial in optimizing the separation. Serially coupled columns have experienced a recent revival owing to the lack of variety of commercially available stationary phases for capillary GClp5, i.e., serial coupling of two capillary columns with different stationary phases gives a new “single” column with an apparent intermediate phase. Previous work focused primarily on the prediction of apparent k’ values for several models1,2,4 displayed excellent solutes derived from coupled columns6-8; agreement with experimental data. Thus, solute selectivity can be varied and accurately predicted. Complete resolution, however, depends on both the capacity factor and’the plate number. To date, only Hildebrand and Reilley8 have attempted to predict apparent plate numbers. Their model, however, assumed the solutes’ plate number to be equal on both columns. The aim of this study was to extend the utility of serially coupled capillary GC by deriving and testing an equation which predicts the theoretical serial plate height from the individual column plate heights. THEORETICAL
We assume 0021-9673/87/$03.50
we have in series two open-tubular 0
1987 Elsevier Science Publishers
B.V.
columns
in the configuration
G. GUIOCHON,
4
P
P
P,
PO
0
Ll
L,+ L*
Y
u,
“0
%
”
Fig. 1. Pressure
and flow velocity
J. E. N. GUTIERREZ
along a column
series
shown in Fig. 1 with lengths Li and Lx. The columns are also assumed to have identical inner diameters, and therefore the same permeabilities. The peak variance stems from the integration of local variances associated with their finite distances: Jdo2
= 1 Hdx
(1)
where o2 is the peak variance, His the local plate height and I the abscisa along the column. The average theoretical plate height arises from this equation if we integrate it, assuming that H is independent of x and the variance of the injected band is zero:
(2)
cr2 =ji;L where L is the column length. length-average plate height:
For open-tubular three terms:
columns,
ln other words,
Golay
equated
the experimental
the column
plate height is the
plate height to the sum of
H = B/u + C,u + Clu
(4)
where H is the local plate height, u is the local velocity, and C, and Ci are the coefficient of resistance to mass transfer in the gas and liquid phase respectively. With thin-film coatings, the last term of the plate height due to resistance of mass transfer in the liquid phase can be neglected. The coefficients B and C are proportional to D, and l/D,, respectively, where D, is the solute diffusion coefficient in the gas phase. As D, is inversely proportional to the pressure and the product pu is constant along the column, the local plate height is constant along the column. It is important to emphasize here that the local plate height depends only on the mass flow velocity of carrier gas (mass flow rate divided by column cross-sectional area). Each elementary contribution do2 is produced at a certain pressure p and expands during elution. Due to carrier gas compressibility, Giddings’ correction factor1° must be included. The final plate height equation is
H = (Blue + C,uo)f
(54
where u. is the outlet flow velocity 9 (P f=
-
1) (P2 -
8 (P3 -
l)*
1)
and P is the inlet to outlet pressure
ratio.
(5b)
PREDICTION
OF PLATE
HEIGHT
5
As the processes that cause band broadening in the two columns are independent of each other, the total variance of the coupled columns is the sum of the variances due to each column:
a: = of + 0: Insertion
(6)
of eqn. 2 into eqn. 6 yields
H,L, = HlLl
+ HzLz
(7)
where HI, Hz and H, are the measured plate heights of the first column the second column and the apparent plate height for the coupled system, respectively. HI, Hz and H, are measured at the same column outlet velocity. Rearrangement of eqn. 7 and introduction of the pressure correction terms give the following equation: H
t
=
HlLl[fl(Pi, PJ/fi(Pi, PO)] + H2bdh(f’m Po)/f2(f’i,PO)] L1
+
L2
(8)
where fi and f2 are the pressure correction factors (eqn. 5b) for the first and the second column respectively, and Pa is the junction pressure (see Fig. I), given by’ P,” = (PFL2 + PGL,)/(L,
+ L,)
(9)
for two columns having the same internal diameter and operated at the same temperature. Then the gas has the same viscosity in both columns and the two columns have the same permeability. Pi and PO are the inlet and outlet pressures. Pi is chosen so that the outlet velocity at which H, is measured for the column couple is the same as the one at which HI and H2 have been measured for the first and second columns. The plate-height contribution of each column in the coupled system can be related to the individual plate heights of the uncoupled columns through the pressure correction factors. Owing to the inverse proportionality of pressure to D,, the terms B/u and C,u in the plate-height equation are constant along the column. Eqn. 6 permits the prediction of the apparent plate height of the column series from the measured plate height of each column at the same mass flow velocity. In order to use it we must relate the plate height of column 1 at the same mass flow rate with an outlet pressure Pa and with an outlet pressure atmospheric. Let HI,, be the plate height of the first column in the coupled system (inlet pressure PO, outlet pressure Pa) and HI,,, be the plate height of this column alone (same mass velocity, inlet pressure Pi, outlet pressure PO). Then
HI,, = (B/us + CgUa)fl(Pi,Pa)
(104
H l,al =
(lob)
(B/u0
+
Cguo).h(Pi,
PO)
Note that since D, is inversely proportional to pressure, since pu is constant along a column and u. is the same for the two experiments, the terms B/u, and B/u0 are equal, and so are the terms CIu, and C,u,. Dividing eqn. 10a by eqn. lob gives
H1.S= Ha,adfl(Pi,f’a)Lfl!f,Pi, PO)]
(104
G. GUIOCHON,
6
J. E. N. GUTTERREZ
This is valid because the mass-flow velocity in the column is the same in the two experiments. The final equation relates the apparent serial plate height to the individual column plate heights
H
=
{ffl,al[fl(Pi,
Pz~)if;(~i> pO)l}Ll + {H2,aifZ(Pa~ pO)l/lf2(pi~ pO)ll L2
(11)
t
-L
+
L2
Eqn. 11 is valid provided the measurements are made at the same outlet flow-rate. This requires the use of different inlet pressures for column 1 (Pi), column 2 (P,) and the column series (Pi). Otherwise, if efficiency measurements are made at different gas velocities, interpolation of the data on a Van Deemter curve must be carried out. EXPERIMENTAL
Data analysis Data analysis was carried out using an IBM (Danbury, CT, U.S.A.) CS 9000 computer. BASIC programs were written for the calculation of the apparent theoretical plate height. Retention times were obtained by fitting between 15 and 120 points over 90% of the peak height to a parabola and calculating the coordinates of the maximum of the parabola. Similarly, peak widths at half-height were obtained by fitting points between 40% and 60% of the peak height to a straight line. The constants obtained from fitting were used to calculate the desired parameter. Plots of In(signa1) vs. (t - t# were made to check whether the peaks were approximately Gaussian. The plots give validity to the peak fitting and the calculated values.
The chemicals used for the experiment were purchased from Sigma. The compounds were dissolved in HPLC-grade methylene chloride obtained from J. T. Baker (Phillipsburg, NJ, U.S.A.). Initial stock concentrations of 10 mg/ml were made. These were further diluted to 0.10 mg/ml. Instrumentation The instrument used was a double-oven IBM 9630 gas chromatograph equipped with dual flame ionization detectors. The instrument was operated in the split mode with a splitting ratio of 1:50. The detector gas flow-rate settings were set for optimal sensitivity according to the instrument manual. Inlet pressures were measured using a 4.5in. test gauge obtained from USG Industries. The column outlet pressure was assumed to be atmospheric and read from a mercury barometer. The columns were purchased from Quadrex (New Haven, CT, U.S.A.). Both had a l-pm film coating. The 10% phenylmethylsilicone column dimensions were 24.8 m x 0.34 mm I.D. and the Carbowax 20M had the column dimensions 19.1 m x 0.37 mm I.D. The two columns were connected via a zero dead volume connector purchased from Valco Instrument (Houston, TX, U.S.A.). Samples of 4 ~1 were injected using a IO-p1 Hamilton (Reno, NV, U.S.A.) syringe. In each injection, a small amount of natural gas was included to provide the unretained methane peak. All samples were fatty acid methyl esters.
PREDICTION
OF PLATE
HEIGHT
I
Measurements were made succesively on each column and on the two possible serial combinations. As far as possible, we tried to achieve the same gas flow-rate in all the experiments. RESULTS
AND DISCUSSION
Experimental and theoretical plate heights and the capacity factors are reported in Tables I-III, together with the relative differences. There seems to be a systematic difference between the experimental and predicted plate heights. This is attributable to a certain lack of consistency of the mass flow regulator in the system. Hence the experimental values will be directionally lower or higher than the predicted values. The capacity factors indicate good agreement with a previously proposed model’. The relative difference, ranging from 3 to 9%, are well within experimental error. Variability of the apparent capacity factors due to column order is barely seen. This stems from the similar lengths of the two columns, which affect equally the intermediate pressure Pa in eqn. 9. The reversal of the elution order of oleic and stearic acids on the two column series, (1 + 2j and (2 + l), is worth noting, however. TABLE
I
CHARACTERISTICS Column
OF THE COLUMNS
1 = 10% phenylmethylsilicone;
USED
column
2 = Carbowax
20M.
Colllmn
Pi(kPa)
PO(kPa)
rofmm)
L(m)
d,(v)
1 2 1+2 2+1
132.2 114.8 194.0 194.0
101.2 101.0 100.8 100.6
0.17 0.19
24.8 19.1
1 1
TABLE
II
COLUMN
CAPACITY
Cf., ref. 1: k&, = Compound
Caproate Caprylate Caprate Laurate Myristate Palmitate Stearate Oleate
FACTORS
k; (P: - P,“) + k; (P,3 -
k;
0.229 0.519 1.138 2.443 5.182 10.88 22.66 20.76
P;)
(P: - Pa)
k;
0.197 0.374 0.714 1.370 2.625 5.012 9.532 10.39
Column 2 -+ column I
Column I + column 2
0.232 0.487 1.02 2.12 4.39 9.04 18.5 17.7
0.217 0.467 0.985 2.06 4.26 8.77 17.9 17.0
Difference (%)
k&
6.47 4.11 3.43 2.83 2.96 2.99 3.24 3.95
0.224 0.469 0.975 2.02 4.18 8.58 16.9 17.6
kbred.
Difference I%)
0.1213 0.445 0.923 1.90 3.88 7.90 15.5 16.0
5.03 5.1 5.33 5.98 7.18 1.93 8.28 9.09
8
G. GUIOCHON,
TABLE
III
PLATE
HEIGHTS
Compound
H (I) H (2) (x lo--“) (x IfIF)
J. E. N. GUTIERREZ
Column 2 + column I
Column 1 + column 2
Difference HexP.
Caproate Caprylate Caprate Ldurate Myristate Palmitate Stearate Oleate
8.0 7.1 5.9 5.0 4.2 3.1 3.1 3.1
11 IO 11 11 11 9.3 8.6 8.2
12.3 10.6 9.9 9.1 8.2 7.8 7.5 7.6
9.5 8.6 8.1 7.7 6.9 6.2 5.8 5.7
22.8 18.9 18.2 15.4 15.9 20.5 22.0 24.5
6.2 6.3 6.7 7.0 7.0 6.9 6.8 6.7
9.5 8.6 8.1 7.7 6.9 6.2 5.8 5.7
53.2 36.5 20.8 10.0 1.40 10.1 14.7 14.9
The plate heights show relative differences ranging from about 10 to 25%. The error between the predicted and the experimental values may be deemed high. HOWever, typical plate height experiments on a column usually yield a lo-15% variability. Hence the predicted values from the derived equation show good agreement with the experimental data. The predicted plate heights for either column combination are the same. This appears anomalous. However, this should be expected as the contributions of column A or B to the apparent system plate height are affected more by the length ratios [i.e., L,/(La + Lb) than by the pressure correction factor ratios. The pressure corection factor varies from 1.O to 1.125. Therefore, the pressure correction ratios are effective only when a high inlet to outlet pressure ratio is used. Otherwise, at pressure ratios of less than 2, the effect of the pressure correction factor on the plate height seems negligible. In an effort to decrease the error in the predicted plate height, a study of the propagated errors was carried out. Much of the error obtained in the predicted values was attributed to the accuracy of the inlet and outlet pressure readings. For the apparent capacity factors, variances in compound retention times affected the individual column capacity factors. Retention time variances ranging from 0.15 s for the first eluted to 16 s for last eluted compounds show their cumulative effects on the a.ccuracy of the apparent capacity factors given in the tables. Further, the mobile phase holdup time, t,, contributes markedly when broken down into its theoretical components. For later eluting compounds the intermediate pressure P, also begins to decrease the accuracy of k&,,, as it weights the P, error by a factor proportional to ki - k;. The intermediate-pressure term is also subject to errors arising from both the inlet and outlet pressure readings. Other variables such as column length, viscosity and column diameter produce smaller contributions. For the apparent plate heights, much of the error again stems from the variances obtained in the retention times and the peak widths. These affect individual column plate heights and are passed on to the system plate height. Variances in the inlet and intermediate pressures affect both the,fcorrection factors for both columns
PREDICTION
OF PLATE HEIGHT
9
in series. The errors obtained are unequal, with the error in thefratio from inlet to junction being the smaller of the two. Part of the error seen in the apparent theoretical plate height is attributable to the unequal column diameters. This error is due only to the model and not to instrumentation. The use of eqn. 8 produces an error in the prediction of the junction pressure. A better prediction could be obtained by using an equation derived from the pneumatic resistance of the two columns. There are several equations for this?,ii. Overall, the more exactly the pressure readings are known, the better the prediction will be. Further, variability in the inlet and outlet pressure causes changes in the outlet gas velocity, which affects the solute retention time and its peak width. Experiments should therefore be run during a time span in which the atmospheric pressure is constant. CONCLUSION
In order to utilize fully the effects of the pressure correction factors, it seems that the columns should be dissimilar in length. This would change the intermediate pressure, thus affecting the compressibility factor for both columns to a larger extent. Dissimilar lengths would also change the contribution of each column to the apparent plate height. Overall, the predictive capability of the equation is good. The accuracy of the prediction can be maximized by minimizing errors in the pressure readings and pressure variability. The effectiveness of the equation in predicting apparent plate heights leads to its coupling with models predicting the apparent capacity factor. Together, both models can be used to optimize system parameters for the complete resolution of a mixture when this is unobtainable on a single column. Additional studies on factors affecting the apparent plate height could include the effect of stationary phase film thickness, the effect of using two columns of different diameter with a complete pneumatic resistance junction pressure equation and the effect of temperature12J3. Together, these studies should give a complete optimization scheme for resolving a mixture on serially coupled capillary columns. GLOSSARY OF TERMS
;
c, Cl
relative retention ratio; axial diffusion term in the Golay equation, equal to 2 D,,o; resistance to mass transfer term in gas phase in the Golay equation, equal to (1 + 6k’ + 1l/Y*) r;. 24(1 + /?‘)* D,,. ’ resistance to mass transfer term in liquid phase in the Golay equation, equal to
df D b%O
DI
k’d;
6(1 + k’)* D1 ’ stationary phase film thickness; gas diffusion coefficient at outlet velocity; liquid diffusion coefficient;
10
.I
G. GUIOCHON,
Giddings’ pressure 9(P - 1) (P2 - 1).
ffl H2 f-4
k’ L Ll L2 n
P P, pi PO & %I UO fJ
correction
factor
for
plate
J. E. N. GLJTIERREZ
height,
equal
to
8(P3 - l)Z ’ plate height of the first column; plate height of the second column; apparent plate height for coupled system; capacity factor; column length; length of the first column; length of the second column; number of theoretical plates; ratio of inlet to outlet pressure; junction pressure; inlet pressure; outlet pressure; resolution factor; gas velocity at the column junction (Fig. 1); outlet gas velocity; standard deviation of the band profile.
ACKNOWLEDGEMENTS
The authors thank IBM Instruments, use of their double-oven gas chromatograph.
T. Bradley
and Joseph
Velisek for the
REFERENCES 1 2 3 4
G. Guiochon, J. Krupcik and J. M. Schmitter, J. Chromatogr., 213 (1981) 1899201. J. H. Purnell, M. Rodriguez and P. S. Williams, J. Chromatogr., 358 (1986) 39-51. T. W. Smuts and T. S. Buys, J. High Resolut. Chromatogr. Chromutogr. Commun., 3 (1980) 461470. H. T. Mayfield and S. N. Chesler, J. High Resolut. Chromatogr. Chromatogr. Commun.. 8 (1985) 595-601.
5 L. S. Ettre and J. V. Hinshaw, Chromatographia, 10 (1986) 561-572. 6 L. Rohrschneider, Fresenius Z. Anal. Chem., 170 (1959) 256263. 7 H. J. Maier and 0. C. Karpathy, J. Chromatogr., 8 (1962) 308-318. 8 C. N. Reilley and G. P. Hildebrand, 9 M. J. E. Golay,
10 11 12 13
Anal.
Chum., 36 (1964) 47-58.
in V. J. Coates, N. J. Noebels and I. S. Fagerson (Editors), Gas Chromatography, Academic Press, New York, 1958, p. I. J. C. Giddings, S. L. Seager, L. R. Stucki and G. H. Stewart, Anal. Chem., 32 (1960) 867-870. G. Guiochon, in M. Lederer (Editor), Chromatogruphy Reviews, Elsevier, Amsterdam, 1966, p. 6. T. S. Buys and T. W. Smuts,. J. High Resolut. Chromatogr. Chromatogr. Commun., 4 (1981) 102-108. T. S. Buys and T. W. Smuts, J. High Resolut. Chromatogr. Chromatogr. Commun., 4 (1981) 317-322.
Printed
1274
THEORETICAL PREDICTION OF COUPLED CAPILLARY COLUMNS
G. GUIOCHON*
in The Netherlands
PLATE
HEIGHT
IN
SERIALLY
and J. E. N. GUTIERREZ
Department of Chemistry, Georgetown University, Washington. DC 20057 (U.S.A.)
SUMMARY
An equation has been derived and tested for the prediction of the plate height on a serial combination of capillary columns in gas chromatography. The results show good agreement with experimental data. This equation, together with an equation for the apparent capacity factor, can be used to optimize the system parameters in resolving a mixture.
INTRODUCTION
Effective gas chromatographic (GC) separations depend on the column tivity (k’) and efficiency (N), as predicted by the resolution equation: R, = n+/4 [(a -
selec-
1)/a] [k’z/(l + k;)]
Proper selection of the stationary phase, column length and instrument operating parameters is therefore crucial in optimizing the separation. Serially coupled columns have experienced a recent revival owing to the lack of variety of commercially available stationary phases for capillary GClp5, i.e., serial coupling of two capillary columns with different stationary phases gives a new “single” column with an apparent intermediate phase. Previous work focused primarily on the prediction of apparent k’ values for several models1,2,4 displayed excellent solutes derived from coupled columns6-8; agreement with experimental data. Thus, solute selectivity can be varied and accurately predicted. Complete resolution, however, depends on both the capacity factor and’the plate number. To date, only Hildebrand and Reilley8 have attempted to predict apparent plate numbers. Their model, however, assumed the solutes’ plate number to be equal on both columns. The aim of this study was to extend the utility of serially coupled capillary GC by deriving and testing an equation which predicts the theoretical serial plate height from the individual column plate heights. THEORETICAL
We assume 0021-9673/87/$03.50
we have in series two open-tubular 0
1987 Elsevier Science Publishers
B.V.
columns
in the configuration
G. GUIOCHON,
4
P
P
P,
PO
0
Ll
L,+ L*
Y
u,
“0
%
”
Fig. 1. Pressure
and flow velocity
J. E. N. GUTIERREZ
along a column
series
shown in Fig. 1 with lengths Li and Lx. The columns are also assumed to have identical inner diameters, and therefore the same permeabilities. The peak variance stems from the integration of local variances associated with their finite distances: Jdo2
= 1 Hdx
(1)
where o2 is the peak variance, His the local plate height and I the abscisa along the column. The average theoretical plate height arises from this equation if we integrate it, assuming that H is independent of x and the variance of the injected band is zero:
(2)
cr2 =ji;L where L is the column length. length-average plate height:
For open-tubular three terms:
columns,
ln other words,
Golay
equated
the experimental
the column
plate height is the
plate height to the sum of
H = B/u + C,u + Clu
(4)
where H is the local plate height, u is the local velocity, and C, and Ci are the coefficient of resistance to mass transfer in the gas and liquid phase respectively. With thin-film coatings, the last term of the plate height due to resistance of mass transfer in the liquid phase can be neglected. The coefficients B and C are proportional to D, and l/D,, respectively, where D, is the solute diffusion coefficient in the gas phase. As D, is inversely proportional to the pressure and the product pu is constant along the column, the local plate height is constant along the column. It is important to emphasize here that the local plate height depends only on the mass flow velocity of carrier gas (mass flow rate divided by column cross-sectional area). Each elementary contribution do2 is produced at a certain pressure p and expands during elution. Due to carrier gas compressibility, Giddings’ correction factor1° must be included. The final plate height equation is
H = (Blue + C,uo)f
(54
where u. is the outlet flow velocity 9 (P f=
-
1) (P2 -
8 (P3 -
l)*
1)
and P is the inlet to outlet pressure
ratio.
(5b)
PREDICTION
OF PLATE
HEIGHT
5
As the processes that cause band broadening in the two columns are independent of each other, the total variance of the coupled columns is the sum of the variances due to each column:
a: = of + 0: Insertion
(6)
of eqn. 2 into eqn. 6 yields
H,L, = HlLl
+ HzLz
(7)
where HI, Hz and H, are the measured plate heights of the first column the second column and the apparent plate height for the coupled system, respectively. HI, Hz and H, are measured at the same column outlet velocity. Rearrangement of eqn. 7 and introduction of the pressure correction terms give the following equation: H
t
=
HlLl[fl(Pi, PJ/fi(Pi, PO)] + H2bdh(f’m Po)/f2(f’i,PO)] L1
+
L2
(8)
where fi and f2 are the pressure correction factors (eqn. 5b) for the first and the second column respectively, and Pa is the junction pressure (see Fig. I), given by’ P,” = (PFL2 + PGL,)/(L,
+ L,)
(9)
for two columns having the same internal diameter and operated at the same temperature. Then the gas has the same viscosity in both columns and the two columns have the same permeability. Pi and PO are the inlet and outlet pressures. Pi is chosen so that the outlet velocity at which H, is measured for the column couple is the same as the one at which HI and H2 have been measured for the first and second columns. The plate-height contribution of each column in the coupled system can be related to the individual plate heights of the uncoupled columns through the pressure correction factors. Owing to the inverse proportionality of pressure to D,, the terms B/u and C,u in the plate-height equation are constant along the column. Eqn. 6 permits the prediction of the apparent plate height of the column series from the measured plate height of each column at the same mass flow velocity. In order to use it we must relate the plate height of column 1 at the same mass flow rate with an outlet pressure Pa and with an outlet pressure atmospheric. Let HI,, be the plate height of the first column in the coupled system (inlet pressure PO, outlet pressure Pa) and HI,,, be the plate height of this column alone (same mass velocity, inlet pressure Pi, outlet pressure PO). Then
HI,, = (B/us + CgUa)fl(Pi,Pa)
(104
H l,al =
(lob)
(B/u0
+
Cguo).h(Pi,
PO)
Note that since D, is inversely proportional to pressure, since pu is constant along a column and u. is the same for the two experiments, the terms B/u, and B/u0 are equal, and so are the terms CIu, and C,u,. Dividing eqn. 10a by eqn. lob gives
H1.S= Ha,adfl(Pi,f’a)Lfl!f,Pi, PO)]
(104
G. GUIOCHON,
6
J. E. N. GUTTERREZ
This is valid because the mass-flow velocity in the column is the same in the two experiments. The final equation relates the apparent serial plate height to the individual column plate heights
H
=
{ffl,al[fl(Pi,
Pz~)if;(~i> pO)l}Ll + {H2,aifZ(Pa~ pO)l/lf2(pi~ pO)ll L2
(11)
t
-L
+
L2
Eqn. 11 is valid provided the measurements are made at the same outlet flow-rate. This requires the use of different inlet pressures for column 1 (Pi), column 2 (P,) and the column series (Pi). Otherwise, if efficiency measurements are made at different gas velocities, interpolation of the data on a Van Deemter curve must be carried out. EXPERIMENTAL
Data analysis Data analysis was carried out using an IBM (Danbury, CT, U.S.A.) CS 9000 computer. BASIC programs were written for the calculation of the apparent theoretical plate height. Retention times were obtained by fitting between 15 and 120 points over 90% of the peak height to a parabola and calculating the coordinates of the maximum of the parabola. Similarly, peak widths at half-height were obtained by fitting points between 40% and 60% of the peak height to a straight line. The constants obtained from fitting were used to calculate the desired parameter. Plots of In(signa1) vs. (t - t# were made to check whether the peaks were approximately Gaussian. The plots give validity to the peak fitting and the calculated values.
The chemicals used for the experiment were purchased from Sigma. The compounds were dissolved in HPLC-grade methylene chloride obtained from J. T. Baker (Phillipsburg, NJ, U.S.A.). Initial stock concentrations of 10 mg/ml were made. These were further diluted to 0.10 mg/ml. Instrumentation The instrument used was a double-oven IBM 9630 gas chromatograph equipped with dual flame ionization detectors. The instrument was operated in the split mode with a splitting ratio of 1:50. The detector gas flow-rate settings were set for optimal sensitivity according to the instrument manual. Inlet pressures were measured using a 4.5in. test gauge obtained from USG Industries. The column outlet pressure was assumed to be atmospheric and read from a mercury barometer. The columns were purchased from Quadrex (New Haven, CT, U.S.A.). Both had a l-pm film coating. The 10% phenylmethylsilicone column dimensions were 24.8 m x 0.34 mm I.D. and the Carbowax 20M had the column dimensions 19.1 m x 0.37 mm I.D. The two columns were connected via a zero dead volume connector purchased from Valco Instrument (Houston, TX, U.S.A.). Samples of 4 ~1 were injected using a IO-p1 Hamilton (Reno, NV, U.S.A.) syringe. In each injection, a small amount of natural gas was included to provide the unretained methane peak. All samples were fatty acid methyl esters.
PREDICTION
OF PLATE
HEIGHT
I
Measurements were made succesively on each column and on the two possible serial combinations. As far as possible, we tried to achieve the same gas flow-rate in all the experiments. RESULTS
AND DISCUSSION
Experimental and theoretical plate heights and the capacity factors are reported in Tables I-III, together with the relative differences. There seems to be a systematic difference between the experimental and predicted plate heights. This is attributable to a certain lack of consistency of the mass flow regulator in the system. Hence the experimental values will be directionally lower or higher than the predicted values. The capacity factors indicate good agreement with a previously proposed model’. The relative difference, ranging from 3 to 9%, are well within experimental error. Variability of the apparent capacity factors due to column order is barely seen. This stems from the similar lengths of the two columns, which affect equally the intermediate pressure Pa in eqn. 9. The reversal of the elution order of oleic and stearic acids on the two column series, (1 + 2j and (2 + l), is worth noting, however. TABLE
I
CHARACTERISTICS Column
OF THE COLUMNS
1 = 10% phenylmethylsilicone;
USED
column
2 = Carbowax
20M.
Colllmn
Pi(kPa)
PO(kPa)
rofmm)
L(m)
d,(v)
1 2 1+2 2+1
132.2 114.8 194.0 194.0
101.2 101.0 100.8 100.6
0.17 0.19
24.8 19.1
1 1
TABLE
II
COLUMN
CAPACITY
Cf., ref. 1: k&, = Compound
Caproate Caprylate Caprate Laurate Myristate Palmitate Stearate Oleate
FACTORS
k; (P: - P,“) + k; (P,3 -
k;
0.229 0.519 1.138 2.443 5.182 10.88 22.66 20.76
P;)
(P: - Pa)
k;
0.197 0.374 0.714 1.370 2.625 5.012 9.532 10.39
Column 2 -+ column I
Column I + column 2
0.232 0.487 1.02 2.12 4.39 9.04 18.5 17.7
0.217 0.467 0.985 2.06 4.26 8.77 17.9 17.0
Difference (%)
k&
6.47 4.11 3.43 2.83 2.96 2.99 3.24 3.95
0.224 0.469 0.975 2.02 4.18 8.58 16.9 17.6
kbred.
Difference I%)
0.1213 0.445 0.923 1.90 3.88 7.90 15.5 16.0
5.03 5.1 5.33 5.98 7.18 1.93 8.28 9.09
8
G. GUIOCHON,
TABLE
III
PLATE
HEIGHTS
Compound
H (I) H (2) (x lo--“) (x IfIF)
J. E. N. GUTIERREZ
Column 2 + column I
Column 1 + column 2
Difference HexP.
Caproate Caprylate Caprate Ldurate Myristate Palmitate Stearate Oleate
8.0 7.1 5.9 5.0 4.2 3.1 3.1 3.1
11 IO 11 11 11 9.3 8.6 8.2
12.3 10.6 9.9 9.1 8.2 7.8 7.5 7.6
9.5 8.6 8.1 7.7 6.9 6.2 5.8 5.7
22.8 18.9 18.2 15.4 15.9 20.5 22.0 24.5
6.2 6.3 6.7 7.0 7.0 6.9 6.8 6.7
9.5 8.6 8.1 7.7 6.9 6.2 5.8 5.7
53.2 36.5 20.8 10.0 1.40 10.1 14.7 14.9
The plate heights show relative differences ranging from about 10 to 25%. The error between the predicted and the experimental values may be deemed high. HOWever, typical plate height experiments on a column usually yield a lo-15% variability. Hence the predicted values from the derived equation show good agreement with the experimental data. The predicted plate heights for either column combination are the same. This appears anomalous. However, this should be expected as the contributions of column A or B to the apparent system plate height are affected more by the length ratios [i.e., L,/(La + Lb) than by the pressure correction factor ratios. The pressure corection factor varies from 1.O to 1.125. Therefore, the pressure correction ratios are effective only when a high inlet to outlet pressure ratio is used. Otherwise, at pressure ratios of less than 2, the effect of the pressure correction factor on the plate height seems negligible. In an effort to decrease the error in the predicted plate height, a study of the propagated errors was carried out. Much of the error obtained in the predicted values was attributed to the accuracy of the inlet and outlet pressure readings. For the apparent capacity factors, variances in compound retention times affected the individual column capacity factors. Retention time variances ranging from 0.15 s for the first eluted to 16 s for last eluted compounds show their cumulative effects on the a.ccuracy of the apparent capacity factors given in the tables. Further, the mobile phase holdup time, t,, contributes markedly when broken down into its theoretical components. For later eluting compounds the intermediate pressure P, also begins to decrease the accuracy of k&,,, as it weights the P, error by a factor proportional to ki - k;. The intermediate-pressure term is also subject to errors arising from both the inlet and outlet pressure readings. Other variables such as column length, viscosity and column diameter produce smaller contributions. For the apparent plate heights, much of the error again stems from the variances obtained in the retention times and the peak widths. These affect individual column plate heights and are passed on to the system plate height. Variances in the inlet and intermediate pressures affect both the,fcorrection factors for both columns
PREDICTION
OF PLATE HEIGHT
9
in series. The errors obtained are unequal, with the error in thefratio from inlet to junction being the smaller of the two. Part of the error seen in the apparent theoretical plate height is attributable to the unequal column diameters. This error is due only to the model and not to instrumentation. The use of eqn. 8 produces an error in the prediction of the junction pressure. A better prediction could be obtained by using an equation derived from the pneumatic resistance of the two columns. There are several equations for this?,ii. Overall, the more exactly the pressure readings are known, the better the prediction will be. Further, variability in the inlet and outlet pressure causes changes in the outlet gas velocity, which affects the solute retention time and its peak width. Experiments should therefore be run during a time span in which the atmospheric pressure is constant. CONCLUSION
In order to utilize fully the effects of the pressure correction factors, it seems that the columns should be dissimilar in length. This would change the intermediate pressure, thus affecting the compressibility factor for both columns to a larger extent. Dissimilar lengths would also change the contribution of each column to the apparent plate height. Overall, the predictive capability of the equation is good. The accuracy of the prediction can be maximized by minimizing errors in the pressure readings and pressure variability. The effectiveness of the equation in predicting apparent plate heights leads to its coupling with models predicting the apparent capacity factor. Together, both models can be used to optimize system parameters for the complete resolution of a mixture when this is unobtainable on a single column. Additional studies on factors affecting the apparent plate height could include the effect of stationary phase film thickness, the effect of using two columns of different diameter with a complete pneumatic resistance junction pressure equation and the effect of temperature12J3. Together, these studies should give a complete optimization scheme for resolving a mixture on serially coupled capillary columns. GLOSSARY OF TERMS
;
c, Cl
relative retention ratio; axial diffusion term in the Golay equation, equal to 2 D,,o; resistance to mass transfer term in gas phase in the Golay equation, equal to (1 + 6k’ + 1l/Y*) r;. 24(1 + /?‘)* D,,. ’ resistance to mass transfer term in liquid phase in the Golay equation, equal to
df D b%O
DI
k’d;
6(1 + k’)* D1 ’ stationary phase film thickness; gas diffusion coefficient at outlet velocity; liquid diffusion coefficient;
10
.I
G. GUIOCHON,
Giddings’ pressure 9(P - 1) (P2 - 1).
ffl H2 f-4
k’ L Ll L2 n
P P, pi PO & %I UO fJ
correction
factor
for
plate
J. E. N. GLJTIERREZ
height,
equal
to
8(P3 - l)Z ’ plate height of the first column; plate height of the second column; apparent plate height for coupled system; capacity factor; column length; length of the first column; length of the second column; number of theoretical plates; ratio of inlet to outlet pressure; junction pressure; inlet pressure; outlet pressure; resolution factor; gas velocity at the column junction (Fig. 1); outlet gas velocity; standard deviation of the band profile.
ACKNOWLEDGEMENTS
The authors thank IBM Instruments, use of their double-oven gas chromatograph.
T. Bradley
and Joseph
Velisek for the
REFERENCES 1 2 3 4
G. Guiochon, J. Krupcik and J. M. Schmitter, J. Chromatogr., 213 (1981) 1899201. J. H. Purnell, M. Rodriguez and P. S. Williams, J. Chromatogr., 358 (1986) 39-51. T. W. Smuts and T. S. Buys, J. High Resolut. Chromatogr. Chromutogr. Commun., 3 (1980) 461470. H. T. Mayfield and S. N. Chesler, J. High Resolut. Chromatogr. Chromatogr. Commun.. 8 (1985) 595-601.
5 L. S. Ettre and J. V. Hinshaw, Chromatographia, 10 (1986) 561-572. 6 L. Rohrschneider, Fresenius Z. Anal. Chem., 170 (1959) 256263. 7 H. J. Maier and 0. C. Karpathy, J. Chromatogr., 8 (1962) 308-318. 8 C. N. Reilley and G. P. Hildebrand, 9 M. J. E. Golay,
10 11 12 13
Anal.
Chum., 36 (1964) 47-58.
in V. J. Coates, N. J. Noebels and I. S. Fagerson (Editors), Gas Chromatography, Academic Press, New York, 1958, p. I. J. C. Giddings, S. L. Seager, L. R. Stucki and G. H. Stewart, Anal. Chem., 32 (1960) 867-870. G. Guiochon, in M. Lederer (Editor), Chromatogruphy Reviews, Elsevier, Amsterdam, 1966, p. 6. T. S. Buys and T. W. Smuts,. J. High Resolut. Chromatogr. Chromatogr. Commun., 4 (1981) 102-108. T. S. Buys and T. W. Smuts, J. High Resolut. Chromatogr. Chromatogr. Commun., 4 (1981) 317-322.