Anal. Chem. 2003, 75, 4462-4466
Transient Flow in Response to a Pressure Pulse in Gas Chromatography Tal M. Nahir*
Department of Chemistry, California State University, Chico, California 95929-0210
The analysis of flow through a gas chromatography column has traditionally assumed the presence of steady-state conditions. However, when rapid changes in inlet pressure are introduced, a significant transient period is observed, leading to a failure of the steady-state model. Through the introduction of a one-dimensional continuity equation into the basic set of equations, a nonlinear partial differential equation is derived to describe the evolution of pressure profiles in a capillary gas chromatography column. A numerical solution is used to solve the differential equation for the case of a pulse injection under isothermal conditions, and comparisons with experimental holdup and retention times show very good agreement. Conventional analysis of flow through a gas chromatography (GC) column assumes the existence of a steady state where the flux of molecules (and atoms) into any column section is equal to the flux out of that section. Under these conditions, the combination of Poiseuille’s law with the ideal gas law allows to predict key parameters such as flow rates and holdup times.1 The validity of this theoretical framework has been repeatedly confirmed for common applications.2 Consider now the modification of flow when some change in inlet pressure is introduced. In particular, let us focus on a column at steady state that is subjected to a sudden drop in inlet pressure. The effect of this perturbation will be initially experienced close to the inlet and then propagate through the column toward the outlet; eventually, a new steady state will be established. We might expect the analysis of the transition between the initial and final steady states to include a situation where the mass flow into any section of the column is not equal to the mass outflow. For example, soon after the inlet pressure abruptly drops, it should be possible to find a column section with a reduced inflow of molecules at the end closer to the inlet (because of the drop in pressure), while the outflow is still identical to its initial steadystate value (since the effects of the inlet perturbation are yet to reach the end closer to the outlet). Thus, because the mass fluxes are not equal, we cannot assume that the transition between the initial and final steady states proceeds through a series of steady states. However, until now, it appears that GC theory has * E-mail:
[email protected]. Fax: 530-898-5234. (1) (a) Littlewood, A. B. Gas Chromatography, 2nd ed.; Academic Press: New York, 1970; p 24. (b) Cazes, J.; Scott, R. P. W. Chromatography Theory; Marcel Dekker: New York, 2002; p 30. (2) Gonzalez, F. R.; Nardillo, A. M. J. Chromatogr., A 1997, 757, 97-107.
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neglected to probe the possibility of a time-dependent transition between two such steady states. Recently, we have suggested that a significant period of nonsteady-state flow was the reason for several otherwise unexplained measured retention times when rapid changes in inlet pressure were introduced.3 In this work, we consider the incorporation of a pressure pulse into a GC runsa technique that has been reported to improve the efficiency of sample transfer into the column4 and to enhance selectivity in series-coupled column ensembles5sand probe the effects of applying abrupt pressure changes in the course of an analysis. We propose a modification to traditional GC flow equations and experimentally examine the validity of the new theoretical model for a time-dependent evolution of pressure profiles in a capillary column during a pulse injection. The assumptions are similar to those proposed previously for transient phenomena in gas flow through a packed bed6 and a concrete wall7 and lead to a nonlinear partial differential equation for the pressure. This equation is solved numerically for a boundary condition where an abrupt drop in inlet pressure occurs, and the calculated flow through a column is compared with measurements from an actual system. The experimental results, which could not be explained using conventional steadystate assumptions, confirm the theoretical predictions from the time-dependent model. EXPERIMENTAL SECTION Most measurements of holdup times were carried out on a Hewlett-Packard HP 6890 Plus Series GC system equipped with a Hewlett-Packard HP 5973 mass-selective detector operating at near-vacuum conditions (GC/MS). Air samples of 1 µL were (3) Nahir, T. M.; Gerbec, J. A. J. Chromatogr., A 2001, 915, 265-270. (4) (a) Wnorowski, A.; Yaylayan, V. A. J. Anal. Appl. Pyrolysis 1999, 48, 7791. (b) Godula, M.; Hajsˇlova´, J.; Alterova´, K. J. High Resolut. Chromatogr. 1999, 22, 395-402. (c) Siegmund, B.; Leitner, E.; Pfannhauser, W. J. Chromatogr., A 1999, 840, 249-260. (d) De Boer, J. J. Chromatogr., A 1999, 843, 179-198. (e) Oehme, M.; Baycan-Keller, R. Chemosphere 2000, 41, 461-465. (f) De Boer, J.; Allchin, C.; Law, R.; Zegers, B.; Boon, J. P. Trends Anal. Chem. 2001, 20, 591-599. (g) Gennaro, M. C.; Marengo, E.; Gianotti, V.; Maurino, V. J. Chromatogr., A 2001, 910, 79-86. (h) Martin, J. W.; Muir, D. C. G.; Moody, C. A.; Ellis, D. A.; Kwan, W. C.; Solomon, K. R.; Mabury, S. A. Anal. Chem. 2002, 74, 584-590. (i) Covaci, A.; de Boer, J.; Ryan, J. J.; Voorspoels, S.; Schepens, P. Anal. Chem. 2002, 74, 790798. (5) (a) Leonard, C.; Sacks, R. Anal. Chem. 1999, 71, 5501-5507. (b) Veriotti, T.; McGuigan, M.; Sacks, R. Anal. Chem. 2001, 73, 279-285. (c) McGuigan, M.; Sacks, R. Anal. Chem. 2001, 73, 3112-3118. (6) Addison, P. A.; Buffham, B. A.; Mason, G.; Yadav, G. D. Chem. Eng. Sci. 1994, 49, 561-572. (7) Verdier, J.; Carcasse`s M.; Ollivier, J. P. Cem. Concr. Res. 2002, 32, 13311340. 10.1021/ac030103a CCC: $25.00
© 2003 American Chemical Society Published on Web 07/22/2003
injected using an Agilent 7683 series injector into a HP-5MS column (30 m × 0.250 mm × 0.25 µm nominal dimensions). Mass spectra were taken in the m/z range from 15 to 115 at 11.90 Hz. Holdup times were recorded at the apex of air peaks. Using a measured holdup time of 0.942 min at 40 °C and an inlet pressure of 200.2 kPa (vs vacuum), and assuming an actual column radius of 125 µm, a column length of 28.37 m was calculated from
tm )
32L2 2
3rc
3
3
pin - pout η 3 3 (pin - pout )2
(1)
where tm is the holdup time, L is the length of the column, rc is the radius, η is the viscosity of the carrier gas (η ) 0.373T0.698 µPa‚s for helium8), pin is the pressure at the inlet, and pout is the pressure at the outlet (∼0 here). As it turns out, the only quantity needed for the digital simulations is (L/rc)2. The analysis of n-nonane, n-decane, and n-undecane (from Fluka) was performed on a Hewlett-Packard HP 6890 Series GC system with a Hewlett-Packard flame ionization detector (FID) operating at ambient pressure. Approximately 1 µL of vapor from above a mixture containing the three n-alkanes was injected manually into a HP-5 column (30 m × 0.320 mm × 0.25 µm nominal dimensions). The holdup time was determined by assuming a simple exponential relationship between the adjusted retention times and the number of carbon atoms for the three n-alkanes and extrapolating to calculate the retention time for methane; this value was also consistent with the observed time for the first peak (trace contamination) on the chromatogram. Measurements at 100 °C and inlet pressure of 200.0 kPa (vs 100.3 kPa ambient pressure) yielded a value for tm of 0.540 min; assuming an actual column radius of 160 µm, a column length of 28.68 m was calculated from eq 1. Analysis conditions were programmed using the instrument’s software in the pulsed split mode with a 10:1 split ratio. When the Agilent 7683 series injector was used (GC/MS), the actual inlet pressure was adjusted (by the instrument) to the preset pulse magnitude and the sample was injected within 10 s after the displayed inlet pressure had stopped changing. Prior to injection, a noticeable perturbation was seen as the pressure was stabilizing at its pulse value (Figure 1). Manual injections (GC with FID) followed a similar pattern. Ambient atmospheric pressures were ∼100 kPa for all measurements. Numerical solutions were obtained using Microsoft Visual Basic 6.0 (part of Microsoft Excel 2000). The calculations did not include corrections for the leakage of carrier gas (helium) through the capillary walls.9 RESULTS AND DISCUSSION Derivation of a Partial Differential Equation. To account for the possibility of an imbalance between the flow in to and out of a thin isothermal column section, we need to expand the basic set of relationships normally associated with motion inside GC columns. For a fixed volume, any imbalance in mass flow must correspond to a change in gas density. The rate of this change is (8) Hinshaw, J. V.; Ettre, L. S. J. High Resolut. Chromatogr. 1997, 20, 471481. (9) Cahill, J. E.; Tracy, D. H. J. High Resolut. Chromatogr. 1998, 21, 531-539.
Figure 1. Time dependence of inlet pressure during a pulse injection. During tpre (before an injection is made), a gradual rise is observed until the pressure stabilizes at ppulse; this pressure is maintained for a period of tpulse after the injection at time 0 and then drops sharply to prun. The small perturbation before injection was noticed in all experiments.
described by a one-dimensional continuity equation:
∂c/∂t ) - ∂(cu)/∂z
(2)
where c is the density (or concentration, in moles per volume), t is the time, u is the carrier gas velocity, and z is the distance from the inlet.10 If we assume that a steady state exists at all times, the left side of this equation will be equal to zero. The other relevant relationships are directly from traditional analysis of flow in GC columns.11 For a capillary column, we assume that the velocity of the carrier gas is proportional to the pressure gradient according to Poiseuille’s law:12 2
u ) -(rc /8η)(∂p/∂z)
(3)
and that the ideal gas law applies:
p ) cRT
(4)
where p is the pressure, R is the universal gas constant, and T is the temperature. From eq 4, we can substitute c with p on both sides of eq 2; in addition, u on the right side of eq 2 can be replaced with the corresponding expression in eq 3. After rearrangement, the result is a nonlinear partial differential equation for the pressure inside the column: 2
rc ∂2p2 ∂p )∂t 16η ∂z2
(5)
This expression may also be obtained in a manner similar to the derivation of the time-dependent diffusion equation.13 (10) (a) Hess, S. L. Introduction to Theoretical Meteorology; Krieger: Huntington, NY, 1979; pp 212-213. (b) Davis, H. F.; Snider, A. D. Introduction to Vector Analysis; Allyn and Bacon: Boston, 1979; pp 104-105. (11) Keulemans, A. I. M. Gas Chromatography; Reinhold: New York, 1957; pp 131-132. (12) Castellan, G. W. Physical Chemistry; Addison-Wesley: Reading, MA, 1964; p 574. (13) Atkins, P. Physical Chemistry, 6th ed.; W. H. Freeman: New York, 1998; p 751.
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It is worth noting that eq 5 reduces to the steady-state relationship when the time-dependent term on the left side of the equation is equated to zero:
d2p2/dz2 ) 0
(6)
Solving this equation with constant pressures at the inlet and the outlet as boundary conditions yields the well-known steady-state expression11
[ ( )] 2
p ) pin 1 -
pout z 1L p 2
1/2
(7)
in
Numerical Solution and Digital Simulation. Equation 5 can be numerically solved by the explicit method, where a new value is computed according to a propagation through a two-dimensional grid of space-time.14 If the length of the column is divided into j equal-sized segments, and the system is allowed to evolve at small time increments, δt, then the new magnitude of the pressure in the ith segment, p′i, is computed from three earlier (by δt) pressures: pi-1 at a point closer to the inlet (-L/j from i), pi at i, and pi+1 at a point closer to the outlet (+L/j from i) according to 2
δt rc p′i ) pi + (pi-12 - 2pi2 + pi+12) (L/j)2 16η
(8)
To avoid oscillations in the calculated p′i values, a proper time increment must be chosen. For optimal results, the magnitude of δt was set to one-third of the largest allowed value, and a heuristic approach was applied to calculate a stability condition corresponding to eq 5.15 As a result, the time increment for all computations in this work is
δt )
1 4η L2 j2 3pmax rc2
(9)
where pmax is the largest inlet pressure. Substituting δt into eq 8 and dividing both sides by pmax yields a simpler form for predicting a new value for the pressure in the ith segment of the column:
p′i pmax
)
pi pmax
+
[( ) ( ) ( ) ]
1 pi-1 12 pmax
2
-2
pi
pmax
2
+
pi-1 pmax
2
(10)
Note that the pressures can be expressed in dimensionless forms. A new pressure profile is computed by repeating the calculation in eq 10 for i from 1 to j - 1 (the values at i ) 0 and i ) j are the programmed inlet pressure and the constant outlet pressure, respectively). Once a complete pressure profile is known, we can proceed to simulate the incremental motion of a component through the column. The velocity of the carrier gas is calculated according to eq 3, where the pressure gradient is approximated by the ratio of (14) Britz, D. Digital Simulation in Electrochemistry, 2nd ed.; Springer-Verlag: Berlin, 1988; p 31. (15) Richtmeyer, R. D. Difference Methods for Initial-Value Problems; Interscience: New York, 1957; pp 92 and 108.
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Figure 2. Time-dependent evolution of pressure profiles after the inlet pressure is abruptly dropped to half its initial magnitude (outlet pressure near vacuum). Numerical solutions of eq 5 according to eq 10 are shown in lines b-e for two sizes of column segments: Lines correspond to j ) 200 (n ) 100, 1000, 10 000, and 100 000 iterations) and circles correspond to j ) 20 (n ) 1, 10, 100, and 1000 iterations). The initial and final steady-state profiles, lines a and f, respectively, were plotted according to eq 7.
the pressure difference to the distance between two locations:
∂p p′i+1 - p′i ≈ ∂z L/j
(11)
The only calculation required for the simulation is at the location where the component of interest is currently found, i.e., the corresponding column section. Multiplying the carrier gas velocity by the time increment from eq 9 yields a distance increment, l, which the compound of interest travels during δt. An expression for a dimensionless parameter is obtained after division by the total column length:
l/L ) u δt/L ) -
1 1 1 p′i+1 - p′i j 6 k + 1 pmax
(12)
where k is the retention factor. Note that the pressures are again reduced to dimensionless quantities. The retention time for a compound is calculated by counting the number of iterations, n (i.e., the number of new pressure profiles that must be generated), until the sum of the distance increments, ∑(l/L), is equal to 1, and multiplying n by δt. Pressure Pulse Experiments. The time dependence of inlet pressure during a pulse injection is shown schematically in Figure 1. The general pattern suggests an establishment of a steady state corresponding to ppulse before an injection is made at time 0; the relatively high-pressure lasts for a period of tpulse until it is reduced to prun. Instead of assuming an instantaneous change at tpulse from one steady state with ppulse to another steady state with prun, the analysis of the gradual change in pressure profiles is now performed according to eq 5. Using the numerical solution method, the time evolution of pressure profiles after the pulse ends is revealed in Figure 2. To simplify the presentation, we have considered a drop in inlet pressure to half its pulse value and a constant outlet pressure near zero (a common GC/MS configuration where the
detector operates at near-vacuum conditions). Clearly, the introduction of a new inlet pressure results in a noticeable period when the pressure profiles adjust from an initial steady state (line a) to a final steady state (line f). Note that the results for significantly different sizes of column segments (j ) 20 vs j ) 200) are very similar, with only one noticeable exception in the very first calculation with j ) 20 after the pressure drops (trace b at z/L ) 0.05). An estimate of the time required for the transition between the two steady states in Figure 2 is provided by calculating the time increment according to eq 9. For a 30.0-m column with a 125-µm inner radius at 40.0 °C and using a pulse pressure of 250 kPa versus vacuum (pmax ) ppulse ) 250 kPa), the calculated δt is 0.158 ms when the column is divided into j ) 200 segments. A comparison between line e (n ) 100 000) and line f suggests that the system at 15.8 s after the end of the pulse is still significantly different from the eventual steady state. It seems reasonable to expect that the nature of the evolution of pressure profiles in response to a pressure pulse would affect retention times. The effects should be most evident for relatively mobile compounds that have low retention factors but minimal for those with high retention factors. For example, in programmedtemperature GC, a pulse injection in conjunction with initial low temperatures is expected to have relatively little effect on strongly retained and practically immobile moleculesstheir motion is probably negligible until the temperature is substantially higher, when a new steady state (with prun) might have already been established. Therefore, to illustrate the effects of a pressure pulse on the flow in the column, a series of measurements of the holdup time (where k ) 0) and retention times of weakly retained compounds (relatively small k) were conducted; the results from these experiments are reported and discussed next. The most noticeable feature in Figure 2 is the temporary positive pressure gradient near the inlet, which persists for a significant period of time after the inlet pressure drops. From eq 3, the corresponding flow is predicted to be negative, and any mobile substance in this region will be moving toward the inlet. This suggests the possibility that a compound might exit the column through the inlet if it has not traveled far enough when the pulse is applied. Figure 3 shows experimental results from a series of runs with identical ppulse and prun and a range of pulse lengths that confirm this predictionsthe marked decrease in the size of the air peak at the shortest pulse (tpulse ) 0.10 min) correlates with the loss of most of the sample through the column inlet. Figure 4 shows the overall effect of a pulse injection on holdup time and demonstrates the good agreement between the experimental results and the theoretical model developed in this work. Two reference holdup times at constant inlet pressures (no pulse) are included for comparison: a relatively short time corresponding to a pressure pulse extending longer than tm and a relatively long time for a run without a pressure pulse; for the constant values of 200.2 and 125.2 kPa (vs vacuum); the holdup times are 0.942 and 1.493 min, respectively. When the pulse is short (e0.30 min), the compounds are still relatively close to the inlet and their overall motion is slowed by the change in pressure gradient. Therefore, although the inlet pressure never falls below 125.2 kPa, the holdup time is longer than that predicted at a constant inlet pressure of
Figure 3. Superposition of experimental chromatograms showing air peaks at a range of pulse lengths. The peaks are labeled with the magnitude of tpulse, ranging from 0.10 to 0.90 min. Initial pressure was ppulse ) 200.2 kPa, and final pressure was prun ) 125.2 kPa (vs vacuum). These results were obtained on a GC/MS at 40 °C.
Figure 4. Dependence of theoretical (solid line) and experimental (circles) holdup times on the length of pulse period. The dashed lines represent holdup times at 200.2 and 125.2 kPa (vs vacuum). Experimental conditions were the same as for Figure 3.
that magnitude. In contrast, compounds relatively close to the outlet will move faster than predicted by the instantaneous establishment of steady-state conditions with prun. For instance, if a molecule is at z/L ) 0.95 when the pulse ends, Figure 2 shows that the pressure gradients during the relaxation period (lines b-e) are steeper than what is predicted at steady state (line f). A careful examination of Figure 4 when tpulse is similar to or longer than tm reveals that the experimental values are somewhat shorter than the theoretical predictions. This result could not be related to the abrupt drop in inlet pressure, since the effects of the pulse period are experienced after the air peak has already eluted out of the column. The reason for the discrepancy appears to be the lack of establishment of a steady state during the relatively short period between the time the inlet pressure reaches ppulse and the time of injection (see Figure 1). From the general discussion of the evolution of the pressure profiles in response to an abrupt pressure change, we can predict that when the pressure rises from prun to ppulse, a relatively large negative pressure gradient will be observed near the inlet, and that gradient will gradually decrease as the system approaches steady state with inlet pressure of ppulse. If air is injected too early, it will experience a pressure gradient steeper than what would be found at the corresponding steady state, resulting in shorter holdup times. Analytical Chemistry, Vol. 75, No. 17, September 1, 2003
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Figure 5. Comparison of calculated band trajectories between steady-state (lines with circles) and transient (lines) analyses. The pulse ends after 0.50 min, when ppulse drops to prun (from 200.0 to 100.0 vs 100.3 kPa ambient pressure, respectively). A chromatogram representing results obtained on a GC with FID at 100 °C is shown on top.
Indeed, in a series of experiments where the pulse length was longer than the holdup time, a convergence to theoretical steadystate predictions was observed when the magnitude of prun was
4466 Analytical Chemistry, Vol. 75, No. 17, September 1, 2003
set closer to ppulse. In addition, note that an injection when the pressure gradient near the inlet is steeper than predicted by steady state also explains why the observed holdup times in Figure 4 are consistently shorter than theoretical predictions (which assume an initial steady state). A summary of the main effects of the transient response during a pulse injection on several components is shown in Figure 5. Here, a chromatogram and band trajectories are shown for four components: an unretained substance, n-nonane, n-decane, and n-undecane. The response of each component to the abrupt pressure drop at the end of the pulse period depends on its location inside the column. Thus, n-undecane, which moves through the column more slowly than the other components (its k is larger), is held back longer than predicted by steady-state calculations, because it is subjected to a positive pressure gradient at tpulse (0.50 min); note the temporary decrease in the corresponding value for z/L at that time, which shows that this compound reverses its direction of motion. In contrast, the holdup time is somewhat shorter than predicted by steady-state calculations, since the unretained component is closer to the outlet when the pulse period ends, and it is subjected to a steeper pressure gradient. Received for review March 14, 2003. Accepted June 7, 2003. AC030103A
