Optimization of two-dimensional
isocratic gas chromatography: a Pareto
approach
Abstract
A mathematical model has been constructed to calculate the two-dimensional peak capacity
in GCxGC, as a function of column geometries, mobile phase characteristics and system
configuration (pressures and modulation time). The parameters investigated are column
length, diameters, film thickness, pressure, and modulation time for both dimensions in
isocratic mode. The model is used to relate isocratic peak capacities with total analysis time,
and it is used to optimize the parameters above (column geometries and system configuration)
to maximize peak capacities and minimize analysis time. The way to approach this (multiobjective) optimization was by the means of the Pareto methodology. The inspection of the
Pareto front revealed that the optimal modulation times could be shorter than the usual. Also,
in order to optimize time, both columns are operating at sub-optimal conditions relative to the
minimum plate height in the plate-height equation.
Contents
1.Introduction ............................................................................................................................. 4
2.1. Description of the peak-capacity model for two-dimensional chromatography. ............ 6
2.2. The number of cuts per peak ......................................................................................... 10
3. Experimental ........................................................................................................................ 10
3.1. Parameters used ............................................................................................................ 10
3.2. Equations workflow ...................................................................................................... 11
3.3. Pareto optimization ....................................................................................................... 13
4. Results and discussion ......................................................................................................... 14
Appendix 1:.............................................................................................................................. 20
Appendix 2:.............................................................................................................................. 21
Appendix 3:.............................................................................. Error! Bookmark not defined.
Appendix 4:.............................................................................................................................. 26
1.Introduction
In separation science, the chromatographer is often confronted with the problem of the
optimization of the chromatographic system. This task is relatively straightforward in onedimensional chromatography. For example, simple rules can be applied to find the optimal
column geometries and flow regimes in one-dimensional column chromatography. This task
becomes more troublesome in the case of two-dimensional chromatography. The reason is the
fact that the different parameters affecting the system (e.g., column geometries, flow rates
and modulation times) are no longer independent. Hence, they have to be optimized together,
increasing exponentially the number of possible combinations of the parameters.
In the past, a general framework to optimize column geometries, flow rates and modulation
times for HPLCxHPLC was developed1. In this seminal paper, the problem of the exponential
growth of the number of combination of parameters was solved by using the Paretooptimization methodology. This technique reduces the number of possible combinations to
the ones that are actually optimal, in terms of the different objectives that are being
maximized (or minimized). In this way, the number of possible combinations to be examined
is reduced from several millions to a few hundred. In addition, the changes of the different
optimal parameters along the different total analysis times can be inspected, similarly to the
work of Poppe [citation Poppe plots].
In that paper, we made use of the Pareto optimization methodology to find those conditions
that maximize peak capacity, minimize time, and (eventually) minimize dilution in
HPLCxHPLC, in both isocratic and gradient regimes. The parameters optimized concerned
column geometries, flow rates and modulation times. In that paper, two effects had
demonstrated to be crucial. On one hand, the worsening of the peak capacity of the first
dimension was considered. This worsening (sometimes called “Tanaka factor”) is due to the
(inevitable) undersampling that occurs when fractions of the first dimension are injected in
the second. This effect, commented by many others [citations], was introduced in the Paretooptimization framework, finding similar results as those found in the bibliography. Another
key effect was the worsening of the second dimension peak capacity due to the injection band
broadening. Without paying attention to this effect, peak capacity of the second dimension is
overestimated.
When optimizing GCxGC systems, several differences exist compared to the liquid
chromatography case. On one hand, active modulation is applied between the first and the
second dimensions, so the injection band broadening effect in the second dimension can be
neglected (similar to the focusing effects that are desirable in the HPLCxHPLC case). On the
other hand, no extra mobile phase is injected in the interphase between the first and the
second dimension. This means that the chromatographer has less degrees of freedom to
manipulate the system, as the complete mass of mobile phase that abandons the firstdimension column (and no more than it) is injected in the second. This does not happen in the
HPLCxHPLC case, in which the valves-system in the interphase between the two columns
allows to add extra mobile phase in the second dimension, thus changing selectivity and
modifying the flow regimes. This restriction in the GCxGC systems forces the diameter of
the second dimension column to be smaller than the first, in order to make the
1
Comprehensive Study on the Optimization of Online Two-Dimensional Liquid Chromatographic Systems Considering Losses in
Theoretical Peak Capacity in First- and Second-Dimensions: A Pareto-Optimality Approach. G. Vivo´ -Truyols, Sj. van der Wal, and P. J.
Schoenmakers. Anal. Chem. 2010, 82, 8525–8536
chromatographic process of the second-dimension separation fast enough for a true GCxGC
separation. As has been demonstrated elsewhere, this in turn forces both (1st and 2nd
dimension) columns to work at suboptimal flow velocities according to the plate-height
equation.
In this paper, we extend the Pareto optimization concept to the GCxGC case. This allows to
examine several effects with a new insight. First, we examine the GCxGC system optimizing
the total peak capacity and the analysis time, finding a collection of optimal conditions
(opposed to a single optimal condition, as proposed by de Koning et al.). This allows to
examine how the column geometries change along the total analysis time (in a similar fashion
as the Poppe plots used in HPLC []). Second, the so-called “Tanaka factor” is introduced for
the first time in gas chromatography, and its consequences are examined in detail. Third, we
want to know the impact of the restrictions mentioned earlier (i.e. working at suboptimal
conditions in the plate-height equation) when a collection of optimal conditions is examined.
This allows us to inspect whether these restrictions are important at low or at high peak
capacities.
2. Theory
2.1. Peak-capacity model for isocratic two-dimensional gas chromatography.
In this section, the equations used to calculate the peak capacity in isocratic two-dimensional
gas chromatography are described. As the objective of this work is to optimize GCxGC
systems attending to (maximize) peak capacity and (minimize) total analysis time, a model
should be constructed that relates both objectives with the factors being optimized (including
column lengths, column diameters, film thickness, modulation time, and column pressures).
Schoenmakers et al.2 defined the conditions to be met for a two-dimensional separation to be
called comprehensive, i.e. (i) every bit of the sample is subjected to two different separations
and (ii) the resolution of the first-dimension separation is essentially maintained. In other
words, this means that the sample is separated following two different retention mechanisms
while the separation in both dimensions is maintained. If the two-dimensional
chromatography is performed in time, there should be a mechanism (i.e. modulation) that
collects small fractions of the first dimension and injects them into the second separation. For
the separation to be comprehensive, the first dimension separation should be (normally 100
times) slower than the second dimension.
Therefore, the total analysis time 2Dt is defined as the sum of the time taken for the lasteluting compound to elute from the first dimension (1t) and the time taken to run the second
dimension(2t)3:
(1)
We will follow through this paper the notation described elsewhere4 , in which the left-hand
superscript indicates the separation dimension (e.g. 1tr stands for the first-dimension retention
time, 2L stands for the second-dimension column length, etc.). In Eq. 1, the second term 2t is
equal to the modulation time 1tw. In Eq. (1) this term can be neglected, since (as mentioned)
retention times in the first dimension are around 100 times larger than the retention times in
the second.
The concept of total peak capacity can be defined as the maximum number of base-line
separated peaks that can be separated if the space is occupied with well-ordered, adjacent
peaks. The maximum total peak capacity (2Dn) that can be achieved by the system
corresponds to a situation in which the two separation mechanisms are completely orthogonal.
As we are interested only in maximizing this (theoretical) quantity, the total peak capacity is
defined as the product of the first- (1n) and second dimension (2n) peak capacities5
(2)
2
3
P. Schoenmakers et al., LC GC Eur. 16 (2003) 1
Anal. Chem. 2010, 82, 8525–8536, eq (9)
4
Philip J Marriott, Grace Wu Zeying and Peter Schoenmakers. Nomenclature and Conventions in Comprehensive Multidimensional
Chromatography – An Update LC-GC Europe, 25 (2012) 266-275.
5
Leonid M. Blumberg, Matthew S. Klee. Quantitative comparison of performance of isothermal and
temperature-programmed gas chromatography. Journal of Chromatography A, 933 (2001) 1–11
For Gaussian-shaped peaks it is accepted that the peak width at the base is 4σ. Therefore, if
base-line separation is required to construct this “well-ordered” peak arrangement, the peak
capacity for any dimension becomes
(3)
where t0 is the dead time and t is defined above.
.
The value of  for the first dimension, 1, depends on the band broadening contributions of
the 1D chromatographic process (1σpeak) and the band broadening contribution due to the
relatively low-frequency modulation time (tw):
(4)
Where δdet2 is a constant with values between 12 [6] and 4,76 [7]. In this model δdet2=12 which
is chosen to be sufficiently high in order not to contribute to the total peak width and is a
representative value for practical situations. The value δdet2=4,76 has been derived
experimentally with statistical overlap theory applied to two dimensional separations, but it
will be not used here. The chromatographic band broadening 1peak is a function of the plate
height (1H), the first dimension retention time (1tr) and the column length (1L):
(5)
The plate-height 1H can be modeled using the Golay equation8:
(6)
Where 1CE is the column efficiency, 1Dm,o is the diffusion coefficient of the analyte in the
mobile phase (see eq. 20), 1Ds,o is the diffusion coefficient of the analyte in the stationary
phase (see Eq. 21), 1uout is the outlet linear velocity, f(1k) and g(1k) are correction factors (see
below), 1dc is column diameter, 1df is the film thickness and 1f1 and 1f2 are pressure
correction factors. Superscript 1 indicates that the values are calculated for the first dimension,
but an equivalent expression holds for the second dimension. f(1k) is defined as
6
7
Comparison of one-dimensional and comprehensive two-dimensional separations by gas chromatography. Leonid M. Blumberg,
FrankDavid, Matthew S. Klee, Pat Sandra. Journal of Chromatography A, 1188 (2008) p.p. 8 -9
Optimization of temperature-programmed gas chromatographic separations I. Prediction of retention times and peak widths from
retention indices. Henri Snijders, Hans-Gerd Janssen, Carel Cramers. Journal of Chromatography A, 718 (1995) 339-355
8
Jan Beens a, Hans-Gerd Janssen , Mohamed Adahchour , Udo A. Th. Brinkman. Flow regime at ambient outlet pressure and its influence
in comprehensive two-dimensional gas chromatography. Journal of Chromatography A, 1086 (2005) 143.
(7)
Where
1
k is the retention factor. Similarly, Function of retention factor g(1k) is
(8)
We will make use of the relationship between the retention factor (1k) and the retention time
(1tr) via the column deadtime 1t0
(9)
In Eq. (6), the pressure correction factors are:
(10)
and
(11)
In Eq. (10) and (11), 1p0 is the (dimensionless) ratio between the inlet(1pin) and outlet
pressures (1pout).
(12)
The values of 1pin and 1pout are related also through the volumetric flow rate (1F) at the outlet
of the column9
(13)
Where  is the dynamic viscosity of the carrier gas. We have been using the empirical
equation suggested by Etre et al.10 to calculate this viscosity. For hydrogen (the carrier gas
used in all the computations), the viscosity at a certain temperature (Ti) is obtained as follows:
(14)
Similar relationships can be found for other carrier gases.
9
Jan Beens a, Hans-Gerd Janssen , Mohamed Adahchour , Udo A. Th. Brinkman. Flow regime at ambient outlet pressure and its influence
in comprehensive two-dimensional gas chromatography. Journal of Chromatography A, 1086 (2005) 141–150.
10
L. S. Ettre. Viscosity of gases used as the mobile phase in gas chromatography. Chromatographia Vol. 18, No. 5. Mai 1984.
In Eq. (15), the linear velocity at the outlet of the first-dimension column (1uout) can be
related to the volumetric flow rate (1F) at the end of the column and the column diameter
(1dc)11 as described by Poiseuille12.
(15)
Finally, the average linear velocity in the first dimension (1ū) is a function of the outlet linear
velocity (1uout) and the second pressure correction factor in the first dimension (1f2).
(16)
In the following equations, we have to make use of the relationship between the retention
time of the un-retained compound and the column length:
(17)
In two-dimensional chromatography, the mass-flow exiting the first-dimension column is the
same as the mass flow entering the second dimension column. Applying Poiseuille equation
on this equality yields
(18)
Where the superscripts “1” and “2” refer to the first- or second-dimension separations, as
described earlier.
To calculate the diffusion coefficient in Eq. (6), we make use of the empirical computation of
the mobile phase diffusion coefficient for binary gas mixtures13, which is calculated from the
molar masses of the components (Mm and Mo), the temperature T (K), the pressure p (Pa) and
their diffusion volumes , (m3). In this paper, we have been using hydrogen as carrier gas
(m), and we used C12H26 as a model molecule (o).
(19)
From Eq. (19), a relationship between the diffusion coefficient in the first and the second
dimensions can be established (considering that the temperature in the first-dimension
column is the same as the temperature in the second dimension):
(20)
11
Jan Beens a, Hans-Gerd Janssen , Mohamed Adahchour , Udo A. Th. Brinkman. Flow regime at ambient outlet pressure and its influence
in comprehensive two-dimensional gas chromatography. Journal of Chromatography A, 1086 (2005) 143.
12
http://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation. Last visit to the website: 13/March/2013.
13
Journal of Chromatography A, 1037 (2004) 147–189
To calculate the diffusion coefficient in the stationary phase, a good approximation is to
consider that Ds is 50.000 times smaller than Dm,o14:
(21)
By combining eq. 3 -11, the peak capacity (for any separation dimension) becomes
(22)
When this equation is used to calculate the peak capacity in the first dimension, ti2 is the
modulation time and t is the retention time of the last eluting compound in the first
dimension. When the equation is applied to calculate the peak capacity in the second
dimension, ti2 becomes the response time of the detector (the time between two adjacent data
points) and t is the modulation time. The solution of this integral can be found in Appendix1, eq. (23).
2.2. The number of cuts per peak
It is important to get an estimation of the average number of second-dimension injections
during the elution of a first-dimension peak (i.e. the number of “cuts per peak”). This is
because at very low modulation rates, the number of cuts per peak could be so low that Eq. 4
is no longer applicable. The number of cuts per peak is 41peak/2tw . As 1is not constant, the
average of the number of cuts per peak is calculated by integrating 41peak /2tw over the firstdimension t and dividing the result by the time spanned in the integration limits:
(27)
To simplify the notation, in Eq. 27 all t values correspond to 1t, except
the one for the
term. Solution of this integral can be found in
appendix 2, eq. (28). 3. Experimental
3.1. Parameters used
One should note that the equations described in section 2 are valid within (reasonable)
parameter ranges. In this section, some of these parameter ranges are discussed.
As for column diameters, the model is basically valid with column diameters 0.50 mm – 0.03
mm. With diameters below this range the equations are no longer valid due to phenomenon
14
Jan Beens a, Hans-Gerd Janssen , Mohamed Adahchour , Udo A. Th. Brinkman. Flow regime at ambient outlet pressure and its influence
in comprehensive two-dimensional gas chromatography. Journal of Chromatography A, 1086 (2005) 143.
called “slip flow” (in which the gas interactions with the capillary wall start to become
significant15). As for the film thickness, the equations describe the processes accurately for
thin film columns. Increasing the film thickness at constant temperature affects the capacity
ratio16.
Table 1 presents the values of the parameters used to optimize the system. Parameters
including a range of values are optimized. Parameters with only a single value are fixed..
Table 1: parameters to be varied:
Variable
Variable range (units)
Constants
Value (units)
1L
50- 150 (m)
1D
m,o
2L
1- 10 (m)
2D
m,o
3,32631E-05 (m2 /s)
1d
c
0,2 (mm)
Ds
6,65262E-10 (m2 /s)
2d
c
0,1 (mm)
ɳH2
0.000103285 (Pa .s)
1p
in
80- 400 (kPa)
Temperature
373,15 (K)
1t

1440- 7200 (s)
2p
out
100 (kPa)
2t

1- 10 (s)
1d
f
0,25 (μm)
2d
f
0,25 (μm)
=
(m2 /s)
3.2. Equations workflow
Figure 1 represents schematically the way the total time and peak capacity are calculated
from a collection of values.
15
James A. McGill. LOW-DENSITY GAS DYNAMIC FACILITY. California Institute of Technology. Pasadena, California. September
22, 1966.
16 T.H.M. Noij. Trace analysis by capillary gas chromatography. Technische Universiteit Eindhoven. januari 1988. p.p. 49.
Figure 1: Equation scheme for the calculation of isocratic peak capacities
Input
Range of variables: 1L, 2L, 1dc, 2dc, 1tr, 2tr=tw, 1pin, 2pout
Locked variables: 1df, 2df
Constants: T, 1CE, 2CE,
Calculated constants: ɳ eq. (14), Ds,o eq. (21)
Calculate for each variable from top to bottom
1
pout =2pin , (eq. 18)
1
p0, 2p0 Eq. (12)
1
f1, 2f1 Eq. (10)
1
f2, 2f2 Eq. (11)
1 2
F, F Eq. (13)
1
uout, 2uout Eq. (15)
1
ū, 2 ū Eq. (16)
1
t0, 2t0 Eq. (17)
1
Dm,o Eq. (19)
Solving peak capacity Eq. (23) from appendix I
Calculate the total analysis time and the total peak capacity 2Dtw
(Eq. 1)
ɳ (Eq. 2)
2D
Calculate the average number of cuts per peak eq. (28, appendix
II), and do not consider any particular combination of parameters
when the number of cuts per peak is below 1.5.
Search for Pareto front for (maximum) peak capacities and
(minimum) time variables
The computation starts by considering the values of 1L, 2L, 1dc, 2dc, 1tw, 2tw, 1pin, 2pout, 1df and
2
df (normally the parameters of the model), along with some of the constants (T, 1CE, 2CE,
det). In this step the values of viscosity and diffusion coefficients given at the input were
considered constant. These values were calculated using Eqs. (14), (19) and (21) for
dodecane /C12H26 on a polysiloxane column.
The second step consists of calculating 1pout, via Eq. 18, which is equal to 2pin. Next, Eq.(10)
is used to calculate 1p0 and 2p0, and Eq. (10) and (11) are applied to calculate the pressure
correction factors 1f1 and 1f2. Next, Eq. (13) is applied to calculate the flow rate for both
dimensions, 1F and 2F. The value of the flow is used in Eq. (15) to calculate the value of the
gas velocity at the outlet, u0. The average linear velocity is then calculated according to Eq.
(16), and the result is used in Eq. 17 to calculate the value of the dead time.
The parameters calculated in the previous equations are used to calculate the peak capacity in
the first and in the second dimension via Eq. 27 The total analysis time is simply calculated
with Eq. (1) and the total peak capacity is calculated with Eq. (2).

3.3. Pareto optimization
To calculate the optimal solution, multiple combinations of parameters (with the ranges
specified in Table 1) are used. The protocol explained above is applied for every single
combination of parameters to calculate the total peak capacity and the total analysis time.
This results in millions of peak-capacity/time pairs, each one associated to a combination of
the parameters described in Table 1. Prior to apply the Pareto optimization, the combination
of parameters yielding a number of cuts per peak below 1.5 is discarded. In this paper the
value of 1.5 cuts per peak has been selected as a threshold, in accordance with the optimal
value observed elsewhere [citation Tanaka, Gabriel].
Pareto optimization is applied then. Pareto optimization has been described elsewhere
[Massart et al., part A], and only a brief definition is going to be given here. In short, Pareto
optimization consists basically in the calculation of what is called the “Pareto front”, i.e.
those experiments for which it is impossible to improve one objective without worsening one
of the others. In other words, the Pareto front in our calculations would be those combination
of parameters (column lengths, pressures, etc.) for which it is impossible to obtain higher
peak capacities without increasing the total analysis time. The Pareto optimization has been
implemented writing home-made routines in Matlab (The Mathworks, Natick, MA, USA).
4. Results and discussion
4.1. Minimum film thickness and minimum column diameters.
In the current model, the optimization is performed exclusively in terms of (maximum)
peak capacity and (minimum) total analysis time. In other words, the model selects any
combination of parameters that yield the maximum peak capacity and yield the minimum
time. From this consideration, it is obvious that the model was selecting the minimum film
thickness and the minimum column diameters available. This is because both parameters
have an impact in the plate height. However, a practical threshold was set up for film
thickness and column diameters, in order for the system to be able to hold a reasonable
column loadability. As column loadability was not considered in the model, we had to set up
this thresholds manually. The values of film thickness and column diameters presented in
table 1 are already the minimum logical values attending experimental practice in terms of
column loadability.
4.2. Total peak capacity vs. analysis times.
Fig. (2) depicts the Pareto front for the (maximization of) total peak capacity vs. the
(minimization of) total analysis time. A similar effect described in 4.1. occurred with the
column length of the first dimension. The Pareto optimization was selecting all the time the
longest column in the first dimension available. In fact there is no theoretical limit for the
column length in the first dimension (the longer, the better). We had to set up again a practical
threshold, limiting the column length of the first dimension to 150 m. In this case, however,
we studied the Pareto fronts when the column length was fixed to different values, as
depicted in Fig. (2).
The Pareto fronts depicted in Fig. (2) indicate that for an analysis time of around one hour a
range of peak capacities between 18.000 and 25.000 can be achieved with a 50m and a 150m
column, respectively. Logically, the peak capacity increases as the column length increases.
Figure 2: Retention time vs. Peak capacity for different column lengths.
Fig. (3) depicts the values of optimal factors along the Pareto front, together with the H/Hmin
ratio (the actual value of the plate height to the optimum (minimum) plate height) for the 100
m column. It can be observed that, for longer analysis times (and more complicated samples),
the second dimension column length increases. The increase in the second dimension
modulation time as the total peak capacity increases is a logical result from the increase in the
second dimension column length. it can be observed also that both 1H/1Hmin and 2H/2Hmin
become closer to 1 (optimal values) as the peak capacity increases. As time is available, the
chromatographic process is slower, making the flow rate velocity closer to its optimum as, at
all times, the system operates at the mass-transfer domain of the plate-height equation (see
section 4.3).
Figure 3: Pareto optimized curves, plotted for different parameters, for a 100m column.
4.3. Plate height equations for a Pareto optimal case
Figure 4 depicts the plate height equation for the Pareto point depicted in Fig. 3. Using the
column geometries and modulation times found at this optimal point, different inlet pressures
were used to generate different values of outlet velocity, to construct the plate-height equation.
Table 2 depicts the values obtained at this optimal point. As can be seen, the system works at
the mass-transfer range of the plate-height equation for both dimensions. However, this is
more evident for the second dimension, as flow rates are higher to make the two-dimensional
chromatography possible. This is the reason why the velocity approaches the optimal velocity
when the total analysis time is increased.
Table 2: First- and second dimension H-values and velocities for a selected optimal peak
capacity.
1u
1u
1
1
TPC
H
Hmin
0
0opt
23435
0,00024239
2
0,00020675
Hmin
0,00010733
2
H
0,00021314
0,8911
2u
0
5,5130
0,4994
2u
0opt
1,3921
Figure 4: van Deemter curves plotted of the first and second dimension.
4.4. Number of cuts per peak.
Fig. (xxx) depicts the values of the number of cuts per peak along the Pareto front. One
should note that the number of cuts per peak depends on the retention time, since in isocratic
elution the band broadening is not constant. As can be seen, the number of cuts per peak is
extremely low at times around the dead time. For example, for 100 m columns the number of
cuts per peak is around 0.2, which means that a modulation time spans for the elution of 5
peaks. In other words, 5 separated peaks are mixed and analyzed in a single injection.
Obviously, this jeopardizes the separation in the first dimension. However, this is properly
accounted in Eq. (4), in which the value of 2tw=5x(41peak= 20x1peak. Following Eq. (), this
means that
. In other words, this means that the actual
band broadening used for the calculation of the peak capacity is 5.86 larger than the
chromatographic band broadening. This makes sense, since, as it is mentioned earlier, 5 peaks
are embedded into the same second-dimension injection. On the other hand, the number of
cuts per peak is at the end of the chromatogram is around 10 (although it varies more along
the total peak capacity). At these retention times,
, making the contribution
to the band broadening due to undersampling almost negligible. As can be seen, the average
number of cuts per peak reaches the imposed limit (1.5) in almost all Pareto points. The
number of cuts per peak at t0 decrease with increasing peak capacity, whereas the number of
cuts per peak increases at tw. This is because the difference in band broadening between the
beginning and the end of the chromatogram grows as the total analysis time (and column
length) increases, which is a logical effect in isocratic chromatography.
4.5. Comparison with other column lengths
When comparing a 100 meter column to a 50 and 150 meter columns, it can be observed that
the general optimal pressure increases with increasing column length (Figure 5). This
relationship can be related to the increase in flow rate to compensate for the increase in
column length. This influence can be observed from eqs.(13) and (5). As can be seen, the
H/Hmin ratio is almost the same for all column lengths. This is due to the fact that an increase
in column length is compensated by an increase of pressure, making the situation respect to
the plate-height equations similar for all lengths.
Figure 5: The average overpressure given per column length.
5. Conclusions
Typical 2D configurations may consist of a first dimension (60 m length, 0,25 mm i.d., 0,25
μm film thickness) column together with a second (1,5 m length, 0,25 mm i.d., 0,1 μm film
thickness) column17. The total peak capacity can be found for a 50 m column in the first
dimension and a 1,5 m column in the secondis around 10000 peaks. Furthermore it can be
observed that this peak capacity can be increased significantly by increasing the second
dimension column length. If this is done, the optimal modulation time increases. In practical
applications the second dimension column lengths might be smaller than those provided in
the literature. Also, modulation times tend to be shorter. This is because we allowed a lower
number of cuts per peak in the system (normally 1.5) opposed to the typical 4 cuts per peak
proposed in the literature. The value of 1.5 is the optimal value of number of cuts per peak
when the worsening of the first dimension separation due to undersampling is considered
properly.
From the van Deemter plots in section 4.3. it can be concluded that both dimensions are
operating at higher than optimal flow conditions – the mass transfer region, irrespective to the
time needed for the seaparation. Again, this is a departure from previous studies since the
average number of cuts per peak is smaller than the reported cuts per peak in older studies. As
the number of cuts per peak is smaller, the first dimension can be faster, allowing to operate
in the mass-transfer range of the Van Deemter equation. Apparently it is beneficial, time wise,
to undersample the first dimension separation to get faster first- and second-dimesnsion
analysis. Finally it can be observed that the pressure (logically) increases with increasing
column length and speed, while other other parameters remain the same.
17
http://www.restek.com/Technical-Resources/Technical-Library/Petroleum-Petrochemical/petro_PCAN1789-UNV
Appendix 1:
The solution to the integral in equation 22 is:
(23)
Where the terms are:
(24)
(25)
(26)
Appendix 2:
Solution for Eq. (27) is:
(28)Where the terms are:
(29)
(30)
(31)
(32)
(33)
(34)
Appendix 3:
Appendix 4: