Anal. Chem. 1998, 70, 4805-4816
Height Equivalent to a Theoretical Plate Theory for Rectangular GC Columns Glenn E. Spangler
Technispan LLC, 1133C Greenwood Road, Pikesville, Maryland 21208
The theory for the height equivalent to a theoretical plate (HETP) of a miniature rectangular gas chromatographic (GC) column is developed in analogy to Golay’s theory for an open tubular GC column. The HETP is similar to that for an open tubular column except for the nonequilibrium or mass-transfer term. Unlike prior theories, the nonequilibrium or mass-transfer term is complexly related to column geometry. The theory successfully predicts the performance of fast chromatography, in the form of both short microbore and micromachined columns. For a given column length, rectangular columns have lower HETPs than conventional capillary columns and higher volumetric flow rates. A good rule of thumb is that the resolution can be adjusted by selecting the column height (provided it is much less than the column width), and the volumetric flow of carrier gas can be adjusted by selecting the column width (or cross-sectional area). A satisfactory compromise is to use a low inlet pressure that also provides an HETP that remains nearly constant over a range of pressures. Since the introduction of open tubular column (OTC) technology by Golay in 1957 for gas chromatography (GC),1 there has been an interest in developing high-speed chromatography with retention times on the order of 10-30 s.2 However, the development efforts have been slow. One of the early problems was the unavailability of suitable inlet systems.3 To prevent adverse broadening of peaks, an inlet system must efficiently inject sample with a pulse width on the order of milliseconds. Today, mechanical valves with sample loops, sample-focusing devices, and pneumatic switches used in combination with vacuum sampling, exist to handle these problems.4-7 Both microbore (100-µm-diameter) capillary columns (1) Golay, M. J. E. In Gas Chromatography 1958; Desty, D. H., Ed.; Academic Press: New York, 1958; pp 36-55. (2) Sacks, R.; Smith, H.; Nowak, M. Anal. Chem. 1998, 70, 29A-37A. (3) Gasper, G.; Arpino, P.; Guiochon, G. J. Chromatogr. Sci. 1977, 17, 256261. (4) van Es, A.; Janssen, J.; Cramers, C.; Rijks, J. HRC&CC, J. High Resolut. Chromatogr. Chromatogr. 1988, 11, 852-857. (5) Klemp, M. A.; Akard, M. L.; Sacks, R. D. Anal. Chem. 1993, 65, 25162521 (see also U.S. Patent 5,141,532, August 25, 1992). (6) Liu, Z.; Phillips, J. B. J. Microcolumn Sep. 1989, 1, 249-256 (see also U.S. Patent 5,135,549, August 4, 1992). (7) Snyder, A. P.; Harden, C. S.; Brittain, A. H.; Kim, M.-G.; Arnold, N. S.; Meuzelaar, H. L. C. Anal. Chem. 1993, 65, 299-306 (see also U.S. Patent 4,970,905, November 20, 1990). 10.1021/ac980328z CCC: $15.00 Published on Web 10/10/1998
© 1998 American Chemical Society
have been used for high-speed chromatography purposes. Microbore columns produce a higher rate of theoretical plates, but resolution and zone capacity are less than that obtained from longer 0.25-mm-diameter columns.6,8-12 While these issues are mitigated somewhat by the introduction of multicapillary columns (a bundle of microbore columns), multicapillary columns are bulky and lethargic to temperature programming.13 Tightly wound columns used in combination with integrally woven heaters are allowingminiaturelow-powergaschromatographstobeassembled.14-16 Another approach to high-speed gas chromatography is to use a microfabricated GC column. This involves etching a groove on a silicon wafer and covering it with a cover plate using fabrication processes available to the microelectronics industry.17,18 While wet chemical etched columns have been assembled, their performance has been variable compared to more conventional GC technology.19-23 Much of this is due to the inability to provide a satisfactory bonded phase and to match the geometry of the column to the detector. With the advent of deep-etching capabilities and newer wafer-bonding techniques, this is changing.24-27 State-of-the-art reactive ion etchers (RIE) allow rectangular (8) Phillips, J.; Luu, D.; Lee, R. J. Chromatogr. Sci. 1986, 24, 396-399. (9) Liu, Z.; Zhang, M.; Phillips, J. J. Chromatogr. Sci. 1990, 28, 567-571. (10) Tijssen, R.; Van den Hoed, N.; Van Kreveld, M. E. Anal. Chem. 1987, 59, 1007-1015. (11) Gonnord, M. F.; Guiochon, G.; Onuska, F. Anal. Chem. 1983, 55, 21152120. (12) Akard, M.; Sacks, R. Anal. Chem. 1994, 66, 3036-3041. (13) Multicap Capillary Columns; Alltech Bulletin 356; Alltech Associates, Inc.: Deerfield, IL, 1997. (14) Overton, E. R.; Carney, K. R. Trends Anal. Chem. 1994, 13, 252-257. (15) Ehrmann, E. U.; Dharmasena, H. P.; Carney, K.; Overton, E. B. J. Chromatogr. 1996, 34, 533-539 (see also U.S. Patent 5,611,846, March 18, 1997). (16) Holland, P. M.; Mastacich, R. V.; Everson, J. F. Field Analysis Methods Hazardous Wastes and Toxic Compounds; VIP-71; Air & Waste Management Assoc.: Pittsburgh, PA, 1977; pp 615-623. (17) Sze, S. M. Physics of Semiconductor Devices; Wiley: New York, 1981. (18) Kovacs, G. T. A.; Petersen, K.; Albin, M. Anal. Chem. News Features 1996, 68, 407A-412A. (19) Terry, S. C. Ph.D. Dissertation, Stanford University, 1975. (20) Terry, S. C.; Jerman, J. H.; Angell, J. B. IEEE Trans. Electron. Dev. 1979, ED-26, 1880-1979. (21) Jerman, J. H.; Terry, S. C. Environ. Int. 1981, 5, 77-83 (see also U.S. Patents 4,471,647, September 18, 1984. and 4,474,889, October 2, 1984). (22) Angell, J. B.; Terry, S. B.; Barth, P. W. Sci. Am. 1983, 248 (4), 44-55. (23) Reston, R. R.; Kolesar, E. S., Jr. J. Microelectromech. Syst. 1994, 3 (4), 134146, 147-154. (24) Delapierre, G. Sens. Actuators 1989, 17, 123-138. (25) Williams, K. R.; Muller, R. S. J. Microelectromech. Syst. 1996, 5 (4), 256269. (26) Wallis, G.; Pomerantz, D. I. J. Appl. Phys. 1969, 40, 3946-3949 (see also U.S. Patent 3,397,278, August 13, 1969). (27) Sensor Technology and Devices; Artech House: Norwood, MA, 1994; Chapter 5.
Analytical Chemistry, Vol. 70, No. 22, November 15, 1998 4805
columns to be microfabricated with a rectangular cross section and provide an aspect ratio where the corners of the rectangle are sharply defined.28-34 Unfortunately, any attempt to microfabricate such a column is muddled by a lack of theoretical models to predict performance. Such a model is necessary if optimum design parameters are to be selected for what could be a very expensive development effort. In 1983, Giddings et al. studied the use of rectangular columns (which they termed “open parallel plate columns, OPPC”) for chromatography.35 This was an outgrowth of work on field-flow fractionation and electrophoresis.36-38 They found that for equal linear flow velocities and diffusivities, the ratio of plate heights for an OPPC column with height w to an OTC column with an internal diameter dc satisfies
HETPOPPC/HETPOTC ) κ(w2/dc2)
cesses: static diffusion that causes the solute molecules to migrate away from regions of high concentration (whether the carrier gas is moving) and dynamic diffusion that causes the solute molecules to be transported more efficiently in one part of the column (for example, in the center) compared to another (for example, near the wall). To ensure usefulness for optimization purposes, the theory must be grounded in fluid dynamics. It must address the flow profile for the carrier gas through the column, the solute transport imposed upon the sample by the flowing carrier gas, retention of the solute by the bonded liquid phase contained in the column, and finally nonequilibrium concentration gradients that develop as a result of the separation process.40 If the number density for the solute in the gas phase is ng (molecules/mL), Fick’s law states that the flux BJ due to diffusion is41
(1)
where κ is a constant that depends only on the retention index. The width of an OPPC column also provides additional degrees of freedom for adjusting GC performance not otherwise available from an OTC column. It is possible to individually select the HETP and carrier gas flow by independently adjusting the column height and width, respectively. This capability provides a mechanism whereby the effects of detector dead volumes can be minimized without affecting resolving power. Although the experimental work of Giddings et al. was with liquid chromatography, similar results are expected for gas chromatography. For this reason, an effort was initiated to further develop the theory for rectangular GC columns. The results are reported here. THEORETICAL OVERVIEW Transport Equation. Any good theory for gas chromatography must address the efficiency with which a sample is separated into component parts as it passes through the column. Ideally the separation process would provide infinite resolution, but band broadening occurs as the sample components disperse around characteristic retention times. Band broadening depends on the operating parameters selected for the GC, and these can be optimized for performance. The optimization can be accomplished either heuristically or with the aid of a suitable theory that adequately describes the separation processes occurring within a column. Two theoretical measures of performance that have been developed for this purpose are the number of theoretical plates (N) and the height equivalent to a theoretical plate (HETP).39 Since the HETP is inversely proportional to N, the two are related. Both describe the effects of two physical pro(28) Ephrath, L. M. J. Electrochem. Soc. 1982, 129 (3), 62C-66C. (29) Plasma Etching: An Introduction; Manos, D. M., Flamm, D. L., Eds.; Academic Press: New York, 1989. (30) Kim, J. M.; Carr, W. N.; Zeto, R. J.; Poli, L. J. Electrochem. Soc. 1992, 139 (6), 1700-1705. (31) Bertz, A.; Werner, T. J. Micromech. Microeng. 1994, 4, 23-27. (32) Grill, A. Cold Plasma in Materials Fabrication; IEEE Press: New York, 1994. (33) Esashi, M.; Takinami, M.; Wakabayashi, Y.; Minami, K. J. Micomech. Microeng. 1995, 5, 5-10. (34) Jansen, H.; Gardeniers, H.; deBoer, M.; Elwenspoek, M.; Fluitman, J. J. Micromech. Microeng. 1996, 6, 14-28. (35) Giddings, J. C.; Chang, J. P.; Myers, M. N.; Davis, J. M.; Caldwell, K. D. J. Chromatogr. 1983, 255, 359-379. (36) Giddings, J. C. J. Chem. Phys. 1968, 49, 81-85.
4806 Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
BJ ) -Dg∇ B ng
(2)
where Dg is the coefficient of diffusion. When this is combined with the mass transfer flux ngb ν, the total flow of solute though a unit volume dV of the column is
total flow ) I(ngb ν - Dg∇ B ng) dS B
(3)
where b ν is velocity of the carrier gas and dS B is the area of the surface elements bounding the unit volume. After equating the total flow to the rate of decrease of solute in the volume, one obtains
∂ n dV ) -I(ngb ν - Dg∇ B ng) dS B ∂t g
∫
(4)
or with the application of Green’s formula,
∫[ ∂t
]
(5)
∂ng ν - Dg∇ B ng) ) 0 +∇ B ‚(ngb ∂t
(6)
∂ng
ν - Dg∇ B ng) dS B )0 +∇ B ‚(ngb
Since eq 5 holds for any volume,
is the transport equation describing the motion of the solute. Because Dg is independent of position, the transport equation can also be written as
dng ∂ng ) +b ν ‚∇ B ng ) Dg∇2ng - ng∇ B ‚ν b dt ∂t
(7)
(37) Giddings, J. C.; Young, Y. H.; Caldwell, K. D.; Myers, M. N.; Hovingh, M. E. Sep. Sci. 1975, 10, 447-460. (38) Cifuentes, A.; Poppe H. Chromatographia 1994, 39 (7-8), 391-404 and references therein. (39) Ettre, L. S.; Hinshaw, J. V. Basic Relationships of Gas Chromatography; Advanstar: Cleveland, OH, 1993. (40) Skelland, A. H. P. Diffusional Mass Transfer; Wiley: New York, 1974. (41) Jost, W. Diffusion in Solids, Liquids and Gases; Academic Press: New York, 1960.
where the substantive derivative, dng/dt, is introduced. The first term on the right describes the effects of diffusion, and the second term describes the effects of convection on the linear flow velocity. Previous theoretical treatments of GC have generally neglected convection (∇ B ‚ν b) and this practice will be continued here. When the initial condition is a pulse of sample introduced into the carrier gas at the entrance of the column, several approaches can be taken to solving eq 7. The first is to apply straightforward mathematical techniques and develop an analytical expression for the solute concentration in the column. This approach is best represented by the work of Lapidus and Amundson and van Deemter et al.42,43 and generally requires the use of error functions to properly describe the concentration profile. The second approach is to perform a moment analysis on eq 7 to arrive at a simplified expression for the HETP. This approach is best represented by the work of Golay.1 The third approach is to kinetically analyze nonequilibria that develop at the gas-liquid interface within the column to again arrive at an expression for the HETP. This approach is best represented by the work of Giddings.44 Since the desire was to both analyze the fluid dynamics and assess the separation capabilities of a rectangular column, Golay’s approach was adopted for the present study. The characterization of chromatographic peaks by their statistical moments can be traced back to the fundamental mass balance relationships that theoretically describe the system.45-47 For example, the zeroth moment is the area under the peak, the first moment is the peak’s center of mass related to the retention time, the second moment is a measure of the peak width, and the higher central moments are further measures of peak shape. Golay’s incremental second moment (du) was
Figure 1. Coordinate system for modeling fluid flow through a rectangular column.
toward the sample. Derivation of an HETP that is equivalent to eq 9 for a rectangular column is the subject of this paper. Velocity Profile for Fluid Flow through a Rectangular Column. Golay assumed Poisseuille flow for the gas flow through his OTC column and expressed the linear velocity in terms of an averaged velocity across the cross-sectional area of the column
νz ) 2νjz(1 - r2/Rc2)
where Rc is the internal radius. In Appendix A, a similar expression is derived for laminar flow through a microfabricated rectangular column. In terms of the column dimensions shown in Figure 1, the velocity profile is
+∞
d du ≡
∫nj z
2
g 1
dz1
-∞ +∞
∫nj
g
+∞
)d
∫hf z
2 1
dz1
(8)
-∞
where nj g is the solute concentration averaged over the crosssectional area of the column and z1 is the location of the species within the column relative to the motion of the center of mass for the solute. The HETP is related to this incremental second moment by dividing it by the length of the column. For an OTC column, the HETP is
HETP )
νz )
-∞
dz1
[
]
2 2 2 2 Kν(x,y) dP 1 (x - b )(y - d ) 1 dP )2 2 2 2 2 (x - b ) + (y - d ) η dz η dz
(11)
where x and y are the positional coordinates within the column, b and d are the half-width and half-height of the column, η is the viscosity of the fluid, dP/dz is the pressure gradient along the column, and Kv(x,y) is the column permeability. When averaged over the cross-sectional area, Ac, of the column, the linear velocity becomes
P2i - P2o
b d
νjz )
∫∫
1 K (x,y) dy dx 2Aclη -b -d ν
2Dg 2kh2νjz (1 + 6k + 11k2) d2cνjz + + (9) νjz 3(1 + k)2D 96(1 + k)2D 1
(10)
x
P2i
(12)
z - [P2i - P2o] l
g
where νjz is the average linear velocity for the carrier gas through a column with a diameter dc, and h is the thickness of the bonded liquid phase that is characterized by a diffusion coefficient Dl (42) Lapidus, L.; Amundson, N. R. J. Phys. Chem. 1952, 56, 984-988. (43) van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Chem. Eng. Sci. 1956, 5, 271-289. (44) Giddings, J. C. Dynamics of Chromatography: Part I. Principles and Theory; Marcel Dekker: New York, 1965. (45) Kucera, E. J. Chromatogr. 1965, 19, 237-248. (46) Grubner, O. Adv. Chromatogr. 1968, 6, 173-209. (47) Grushka, E. J. Phys. Chem. 1972, 76, 2586-2493.
where Pi and Po are the inlet and outlet pressures respectively, l is the length of the column, and the column exit is at z ) l. In Appendix A, it is further shown that the average linear velocity can also be written as
νjz )
KQ 2lηAc
P2i - P2o
x
P2i
(13)
z - [P2i - P2o] l
where KQ is the column permeability for volumetric flow. Due to the retentive property of the stationary liquid phase, the solute Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
4807
tions and can be expressed by a difference between the two concentrations ng,x)(b,y)(d and ngw divided by a time constant, T. When this mass-transfer rate is balanced against the rate with which the gas phase replenishes the concentration near the wall, the boundary condition becomes
( )
( )
( )
Dg ∂ng Dg ∂ng Dg ∂ng ∂ngw )+ + ∂t 2b ∂x x)b 2b ∂x x)-b 2d ∂y y)d Dg ∂ng k ) (n - ngw) (16) 2d ∂y y)-d T g,x)(bory)(d
k
Figure 2. Golay’s model for boundary conditions in an open-tubular column.
velocity is a fraction R of this average linear velocity, Rνjz, where the fraction R is given by 1/(1+k). k is the partition coefficient for the solute in the liquid phase. Center-of-Mass Transformation. To simplify analyses, it is desirable to analyze the solute concentration in a coordinate system moving with the center of mass of the solute. Since the location of the center of mass, z1, is given by
z1 ) z -
∫νj
z
dt (14)
1+k
eq 7 becomes
[
]
[
]
∂ng AcKν(x,y) ∂2ng ∂2ng ∂2ng 1 ∂ng ) Dg + 2 + + νjz ∂t KQ 1 + k ∂z1 ∂x2 ∂y ∂z12 (15) Boundary Conditions. Before a solution can be developed for eq 15, it is necessary to define boundary conditions. An important boundary condition is the rate with which solute transfers between the mobile gas and stationary liquid phases. This mass-transfer rate describes a concentration change per unit time.48 That is, the rate with which the solute concentration increases or decreases in the stationary liquid phase is equal to the rate with which it decreases or increases, respectively, in the mobile gas phase. A graphical representation of Golay’s boundary conditions is shown in Figure 2. Golay assumed that the concentrations of the solute in the stationary liquid phase, ngw, and in the mobile gas phase, ng, are in equilibrium. The equilibrium is described by the partition coefficient, k, which is the ratio of the concentration in the stationary phase to that in the mobile phase. Golay further noted that the solute concentration in the gas phase near the boundary wall, ng,x)(b,y)(d, may not be equal to the concentration in the center of the column, ng (Giddings also described this with his “equilibrium departure term m”).48 Golay reasoned that the mass-transfer rate between the stationary liquid phase and mobile gas phase is related to the difference in solute concentra-
( )
where first-order kinetics is assumed. Golay further argued that the time constant, T, is associated with the solute migrating into the liquid phase and depends on the thickness, h, of the liquid phase and the coefficient of diffusion, Dl, for the solute in the liquid phase.
1 h2 3 Dl
T)
Equation 17 is just the time constant associated with the temporal behavior of polymeric materials whenever one of their surfaces is exposed to a vapor concentration.41 Variational Analysis of the Transport Equation. To obtain a solution for the solute concentration, ng, a variational analysis must be performed on eqs 15 and 16. Following Golay, this is accomplished by mathematically allowing the solute concentration to deviate from an average value, nj g, by the amounts ∆ng and ∆ngw: b d
n jg ≡
∫∫ n
g
dy dx
-b -d b d
∫∫
4808 Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
b d
) dy dx
∫∫
1 n dy dx Ac -b -d g
-b -d
ng ≡ n j g + ∆ng; ∆ng , n jg ngw ≡ n j g + ∆ngw; ∆ngw , n jg b d
∫∫∆n
g
dy dx ≡ 0
(18)
-b -d
The significance of this approach is that all possible causes for variable concentrations (e.g., Taylor flow, eddy diffusion, etc.49) are lumped into one term, ∆ng. When the definitions of eq 18 are substituted into eqs 15 and 16, the equations become (see Appendix B)
∂n jg ∂2n jg ∂n jg ) A1 2 + A 2 ∂t ∂z ∂z 1∂t 1
(48) Giddings, J. C. Dynamics of Chromatography: Part I. Principles and Theory; Marcel Dekker: New York, 1965; Chapter 3. (49) Skelland, A. H. Diffusional Mass Transfer; Wiley: New York, 1974; Chapter 6.
(17)
with
(19)
A1 )
After substituting eq 20 into eq 24, the HETP for a rectangular column becomes
kTνj2z Dg kνjz + + × 1 + k (1 + k)3 (1 + k)2
{ ∫[ d
4Acνjz 1 2dDg 0 K (b2 + y2) Q
dy dy dx dx
|
∫∫∫∫K (x,y)
-
x)b
[
4Acνjz
ν
]
{ ∫[
ν
]}
4Acνjz 1 (1 + k)Ac KQ(x2 + y2)
b
d
∫∫
4Acνjz 1 2dDg 0 K (b2 + y2) Q
-b -d
dy dy dx dx
-
y)d
(
ν
]
and
(1 + k) k2νjz
[
d
1 b2y2 1 dy + 2 2 2 d b 2(1 + k) Dg 0 b + y kνjz
b d
∫∫ 2(1 + k)D g -b -d
(
∫
2 2
∫ x x+d d b
0 2
2
]
Kν(x,y) 1 x2y2 dy dx (21) 2 KQ (1 + k)Ac x + y2
Golay noted that eq 19 describes static diffusion, with the inclusion of additional terms as required to account for dynamic transport phenomena. The additional terms of A1 effectively increase the diffusion coefficient. Departure from this simple concept lies in A2, which makes a negligible contribution to the second moment. Evaluation of the Second Moment. By virtue of eq 19, the second moment of eq 8 can be written as +∞
du ) dt
(
+∞
)
∂fh 2 ∂2hf ∂fh z12 dz1 z1 dz1 ) dt A1 2 + A2 ∂t ∂z ∂t ∂z 1 -∞ -∞ 1
∫
∫
(22)
which, in Appendix C, is shown to be equal to1
du )
2(1 + k)A1 dz1 νjz
(23)
In eq 22, the averaged solute concentration is replaced by its normalized analogue as defined by eq 8. Because the HETP is defined as the incremental second moment per unit length of the column, it becomes
HETP ≡
2(1 + k)A1 νjz
4Acνjz
]
b νjz b2y2 1 dy + 1 + k b2 + y2 2bDg 0
ν
] } {∫ ∫ (
4Acνjz
∫
∫∫∫∫K (x,y) dy dy dx dx|
νjz x2d2 1 dx 1 + k x2 + d 2 Dg
[
ν
b
d
-b -d
-
y)d
)
Kν(x,y) 1 × KQ (1 + k)Ac
∫∫∫∫K (x,y) dy dy dx dx -
KQ(x2 + y2)
ν
] }
νjz x2y2 dy dx 1 + k x2 + y 2
(25)
A comparison of eq 25 with eq 9 for an OTC column shows common terms arising from longitudinal diffusion and resistance to mass transfer in the liquid phase. The difference lies in the resistance to mass transfer in the gas phase, a difference that depends on channel geometry.
dx +
)[ ]
-
KQ(x2 + d2)
∫∫∫∫K (x,y) dy dy dx dx -
2
|
∫∫∫∫K (x,y)
x)b
[
Kν(x,y) KQ
νjz x2y2 dy dx (20) 2 1 + k x + y2
kTνjz
l
d
∫
∫∫∫∫K (x,y) dy dy dx dx|
νjz νjz x2d2 dx 2 2 1+kx +d 2(1 + k)Dg
A2 ) -
2kh2νjz 2Dg 2k + + × 2 νjz 1 +k 3(1 + k) D
b νjz 1 b2y2 dy + 1 + k b2 + y2 2bDg 0
KQ(x2 + d2)
)[
HETP )
(24)
DISCUSSION To establish the validity of eq 25 for the HETP, it is necessary to compare its predictions to fast GC results. The open-tubular data of Guiochon et al.50 and the microfabricated GC data of Terry will be used for this purpose.19 Advanced math computation software will assist in performing the complex computations.51-53 Next, a simpler expression for the HETP will be considered and evaluated in terms of the present theory. Then finally, the significance of the column width on the performance of a rectangular column will be addressed. Fast GC Data. Guiochon et al. studied the contributions of instrument parameters to the HETP of a microbore capillary column.50 Significant contributions were found for fast-eluting peaks but negligible contributions for slow-eluting peaks. Figure 4 of their paper shows HETP values for n-octane (a slow-eluting compound) and furnishes parameters necessary to fit the data to eq 9. Curve A in Figure 3 is a plot of eq 9 using the parameters provided by Guiochon et al. The inlet presssure was deduced from the linear velocity reported by Guiochon et al. for their hydrogen carrier gas by assuming Poisseuille flow. Since the linear velocity used in eq 9 was corrected for compressibility, curve A provides a better fit to Guiochon et al.’s data (also shown in Figure 3) than that provided by the authors themselves. Also shown in Figure 3 are two other plots. These plots are for square columns (57.6 µm × 57.6 µm) with cross-sectional areas (50) Gasper, G.; Annino, R.; Vidal-Madjar, C.; Guiochon, G. Anal. Chem. 1978, 50, 1512-1518. (51) Porter, M. L. Personal Eng. 1996, (June), pp 29-35. (52) The University Scientist. Sci. Comput. Autom. 1995, 12 (10), 29-39. (53) For a nominal fee, the author can provide the worksheet files used to perform the computations of this paper with MathCAD 7.0.
Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
4809
Figure 3. Comparison of theoretical results for high-speed GC data: (curve A) Golay’s theoretical results for an open-tubular column; (curve B) Golay’s theoretical results for a square column; (curve C) this work for a square column with equal sides. Diamonds are a replot of Guiochon et al.’s data. The column length is 85 cm, OTC column diameter is 65 µm, square column width is 57.6 µm, gas-phase diffusion coefficient is 0.260 cm2/s, liquid-phase diffusion coefficient is 9 × 10-7 cm2/s, film thickness is 0.25 µm, carrier gas viscosity is 200 µP, retention factor (k) is 3.0, and γ is 1.0.
Figure 4. Theoretical HETP values versus inlet pressure for various column heights (micrometers). Diamonds indicate Terry’s operating points. The column length is 50 cm, column width is 130 µm, gasphase diffusion coefficient is 0.135 cm2/s, film thickness squared/ liquid-phase diffusion coefficient (h2/Dl) is 0.34, carrier gas viscosity is 200 µP, retention factor (k) is 15.25, and γ is 1.0.
equal to the 65-µm-diameter column of Guiochon et al. Curve B was generated using Golay’s HETP for a square column,
51k2 Dg 8h2kνjz 2 νjzAc HETP ) 2 + + νjz 105(1 + k)2 Dg 3Dl(1 + k)2 1 + 9k +
(26)
with a mean linear velocity,
νjz f 〈ν〉 )
jAc Pi2 - Po2 96ηl P0
(27)
where j is James and Martin’s gas compressibility factor defined in Appendix A. Curve C was generated using eq 25 and (see Appendix A)
νjz f 〈ν〉 )
jKQ Pi2 - Po2 2Acηl P0
(28)
Except for the differences in column dimensions, the GC parameters of Guiochon et al. were used in all cases. While the theory for a square column is not expected to accurately describe an open tubular column (see Cifuentes and Poppe for additional comments on this point),38 Figure 3 shows surprisingly good results. Golay’s theory provides HETP values that are consistently higher than those of Guiochon et al., while the present theory provides HETP values that are in essential agreement with the experimental data. The difference is probably related to the method used to calculate linear velocities in each case. Golay introduced an elliptical approximation for the linear velocity early in his theoretical development, while the present theory delays such an approximation until the boundary conditions are addressed. For the present theory, the exact expression for the linear velocity is evaluated using numerical integration whenever possible. Microfabricated GC Data. For ethanol, Terry observed plate heights of 0.76, 0.35, and 0.28 using inlet pressures of 15.5, 5.5, 4810 Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
Figure 5. Volumetric flow of carrier gas exiting rectangular columns with various channel heights (micrometers). Diamonds indicate Terry’s operating points. The column length is 50 cm, column width is 130 µm, gas-phase diffusion coefficient is 0.135 cm2/s, film thickness squared/liquid-phase diffusion coefficient (h2/Dl) is 0.34, carrier gas viscosity is 200 µP, retention factor (k) is 15.25, and γ is 1.0.
and 3.0 psig, respectively, for his microfabricated GC column.19 To obtain these results, Terry injected his samples into a helium carrier gas flowing through a 30 µm × 130-160 µm × 50 cm long column at room temperature. He assumed that the gas-phase diffusion coefficient was 0.135 cm2/s, and he measured the partition coefficient at 15.25. Using these parameters, eq 25 is compared to Terry’s data in Figure 4. Terry’s data correspond best to eq 25 when the ratio of the film thickness squared to the liquid-phase diffusion coefficient (h2/Dl) is 0.34 s. For this value, the calculated HETPs are 0.92, 0.35, and 0.20 for the three inlet pressures, respectively. For a liquid-phase diffusion coefficient of 9 × 10-7 cm2/s (the value chosen by Guiochon et al. for their n-octane data), the liquid film thickness is 5.6 µm. While Terry did not report a thickness for his liquid film (poly(ethylene glycol) 400), the reasonableness of 5.6 µm demonstrates the adequacy of the present theory. The volumetric flows observed by Terry for his three operating conditions are shown in Figure 5. Also shown in Figure 5 is a plot of the exit flow (see Appendix A)
Q˙ o ) -
1 ηPo
d
b
∫ ∫K (x,y) dx dy ν
-d -b
[
]
Po2 - Pi2 2l
(29)
versus inlet pressure (see Appendix A). Again there is a good
Figure 6. Comparison of the nonequilibrium constant CM calculated from this work (A) and by Giddings (B). The column length is 50 cm, column width is 130 µm, gas-phase diffusion coefficient is 0.135 cm2/ s, carrier gas viscosity is 200 µP, retention factor (k) is 15.25, and γ is 1.0.
agreement between Terry’s results and theory. The exit flow is found to increase as the inlet pressure and height of the rectangular column are increased. Comparison with Gidding’s HETP. In 1961, Giddings derived an expression for the HETP of a rectangular liquid chromatographic column.54 Unlike gas chromatography, where peak broadening is controlled by longitudinal diffusion in the mobile phase, nonequilibrium or mass transfer in the mobile phase, and slow diffusion in the stationary phase, peak broadening in liquid chromatography is controlled by nonequilibrium or mass transfer in the mobile phase. After applying nonequilibrium theory to the analysis of the variance associated with peak broadening, Giddings wrote the HETP as35,54,55
Hc ) Cmνjz )
51 d2 2 2 12 R - R+ νj 3 5 35 zDg
(
)
(30)
where laminar flow between two parallel faces of infinite extension is assumed. The difference between eq 30 and the similar term in eq 25, is that eq 30 is a simpler expression. Because the physical processes underlying the nonequilibrium or mass-transfer term are the same for both gas and liquid chromatography, eq 30 should also apply to gas chromatography. Figures 6 and 7 show plots of the nonequilibrium coefficient, CM, derived from Giddings’ and the present theories using the experimental parameters of Terry. In Figure 6, it is evident that Giddings’ results (eq 30) approach that of the present theory (eq 25) when the column height goes to zero. This is consistent with the fact that eq 31 was derived by assuming a large aspect ratio for the column width to column height. However, as the column height approaches the column width, eq 30 deviates from eq 25 until the difference is 60% for a square column. Figure 7 shows that the percentage difference is dependent on the partition coefficient. Although the difference decreases as the partition coefficient increases, there appears to be a minimum near 20%. Significance of the Column Width. Giddings et al. noted that the relevant dimension for determining resolution in a (54) Giddings, J. C. J. Chromatogr. 1961, 5, 46-60. (55) For a square column, this expression for the nonequilibrium term numerically equals that given by Golay. The only difference is that Golay uses “width x height” in place of Giddings “height squared”.
Figure 7. Comparison of the nonequilibrium constant CM calculated from this work (A) and by Giddings (B). The column length is 50 cm, column height is 30 µm, column width is 130 µm, gas-phase diffusion coefficient is 0.135 cm2/s, carrier gas viscosity is 200 µP, and γ is 1.0.
Figure 8. Dependence of HETP and exit flow on column width. Diamonds indicate an operating point used by Terry. The column length is 50 cm, column height is 30 µm, gas-phase diffusion coefficient is 0.135 cm2/s, film thickness squared/liquid-phase diffusion coefficient (h2/Dl) is 0.34, inlet pressure is 20 psia, carrier gas viscosity is 200 µP, retention factor (k) is 15.25, and γ is 1.0.
rectangular column is the height, while the relevant dimension for establishing volumetric flow is the cross-sectional area.35 Figure 8 shows the variation of HETP and column exit flow for a 30 µm-deep rectangular column versus column width using Terry’s operational parameters in eqs 25 and 29. As the column width increases for a constant inlet pressure, the HETP and column exit flow both initially increase, but as the column width exceeds a multiple of the column width, the HETP assumes a plateau value. Provided the width/height ratio is large, this allows the volumetric flow for the carrier gas to be adjusted by varying the column width. This is important when a rectangular column is interfaced to a detector. For example, Terry used a miniature thermal conductivity detector to prevent contributions of dead volumes to his GC results. To duplicate Terry’s results, other detectors will require larger flow rates. For a rectangular column, the increased flows can be provided by increasing the column width, and this can be accomplished without affecting resolution. CONCLUSIONS Golay’s theory for HETP has been modified for application to a rectangular GC column. The modified theory correctly predicts HETP data collected by Guiochon et al. on fast GC columns and HETP and flow data collected by Terry on a microfabricated column. Miniature rectangular columns have lower HETPs than conventional capillary columns and can be configured to deliver higher flows of carrier gas to a detector. A good rule of thumb Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
4811
∂2νz ∂x
2
+
∂2νz ∂y
2
)
1 dP η dz
(A3)
Equation A3 is solved using the boundary conditions,
Figure 9. Linear velocity profile for gas flow through a rectangular column.
is that the resolution can be adjusted by selecting the column height (provided it is much less than the column width), and the volumetric flow of carrier gas can be adjusted by selecting the column width (or cross-sectional area). A satisfactory compromise for selecting operating parameters is to use a low inlet pressure that also provides an HETP that remains nearly constant over an extended range of inlet pressures. While prior uses of miniature rectangular columns have been limited, this will change as the reliability of deep-etching capabilities improves and is applied to microfabricating rectangular GC columns. ACKNOWLEDGMENT I thank Dr. John Carrico of SRI International for alerting me to the problems associated with microfabricating a GC column. I also thank Professor Edward Kolesar of Texas Christian University for encouraging me during the effort and for the flow data he kindly provided for Appendix A. I am also indebted to an unknown reviewer who lead me to the pertinent work of Giddings et al. The research was sponsored by the Defense Sciences Office of the Defense Advanced Research Projects Agency (DARPA) issued by the U.S. Army Missile Command as Contracts DAAH01-96-CR156 and DAAH01-97-C-R129.
dP/dx ) dP/dy ) 0
(A4)
b ν ) 0 at x ) (b, y ) (d
(A5)
and the solution is56
[
]
Kν(x,y) dP 1 (x2 - b2)(y2 - d2) 1 dP νz ) ) η dz 2 (x2 - b2) + (y2 - d2) η dz
(A6)
where Kν(x,y) is the column permeability. Since Boyle’s law (i.e., the product of the pressure times volumetric flow rate is constant) requires that the pressure distribution along the column is58
Pi2 - P2 z ) 2 l P -P2 i
(A7)
o
where l is the length of the column, Pi is the inlet pressure, and Po is the outlet pressure; the first derivative of the pressure becomes
dP ) dz
Po2 - Pi2 2l
x
(A8)
z Pi2 - (Pi2 - Po2) l
Equation A6 then becomes APPENDIX A. GAS VELOCITY/FLOW PROFILE Linear Velocity. The linear velocity profile for a gas flowing through a rectangular column is obtained from the Navier-Stokes equation56
F
[∂ν∂tb + (νb‚∇B)νb] ) -∇PB + η∆νb
(A1)
where F and b ν are the localized density and velocity of the gas, P is the driving pressure, and η is the coefficient of viscosity. Referring to the model in Figure 1, the gas is assumed to flow in the +z direction through a channel with width 2b and height 2d. From the continuity equation,
∂F/∂t ) -∇ B ‚(Fν b) ) 0
(A2)
it follows that ∇ B ‚ν b ) 0, ∂νi/∂xi ) 0 (parallel flow!), (ν b‚∇ B )ν b ) 0, and ∂2νz/∂z2 ) 0. For steady-state (i.e., ∂ν b/∂t ) 0) conditions, equation A1 becomes:57 (56) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Pergamon Press: Reading, MA, 1959; p 49.
4812 Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
Kν(x,y) νz ) 2lη
Po2 - Pi2
x
(A9)
z Pi2 - (Pi2 - Po2) l
The general shape of the velocity profile described by eq A9 is shown in Figure 9 where Kν(x,y) is plotted (vertical axis) across the cross section of the column (horizontal plane). The heightto-width ratio (b/d) for the column was assumed to be 10. At z ) l, the linear velocity, νo, of the carrier gas exiting the column is
|
Kν(x,y) dP Kν(x,y) Pi2 - Po2 νo ) ) η dz z)l η 2lPo
(A10)
Elliptical Approximation for the Linear Velocity. Whenever there is a need to integrate the linear velocity of eq A9 over x and y, difficulties are encountered as the function Kν(x,y) becomes discontinuous within the range of integration. (57) Schlichting, H. Boundary-Layer Theory; McGraw-Hill: New York, 1968; p 78-80. (58) Littlewood, A. B. Gas Chromatography: Principles, Techniques, and Applications; Academic Press: New York, 1970; p 26, eq 2.27.
Kν(x,y) ) -
2 2 2 2 1 (x - b )(y - d ) 2 2 2 2 (x - b ) + (y - d2)
(A11)
Two approaches to circumventing this problem are to approximate eq A11 with another expression that can be more easily integrated or to avoid the regions of discontinuity by numerically integrating around the discontinuity. A reasonable approximation for eq A11 is the elliptical approximation:59
Kν(x,y) )
[
]
b2d2 x2 y2 γ- 2- 2 2 2 2(b + d ) b d
[
b d
)-
]
b2d2 1 x2 y2 dP 1 - 2 - 2 dy dx 2 2 4bdη -b -d2(b + d ) dz b d
∫∫
b2d2 dP 1 6η (b2 + d2) dz
(A13)
Terry
Kolesar
calcns
5 10 15 20 25 30 35 40 45 50
1.1 2.5 4.0 6.1 8.6 11 14 17 20 24
1.5 3.5 4.0 6.0 7.0 10 13 16 18 23
1.0 2.6 4.0 6.0 8.0 11 13 16 20 24
or from eq A8
∫∫
1 Q˙ o ) ηPo
[
2
]
Q˙ )
b +d νz ) 6νjz Kν(x,y) b2d2
(A14)
The elliptical approximation also allows the indefinite quadruple integrals
∫∫∫∫K (x,y) dy dy dx dx ) ν
[
]
x2y2 y2x4 x2y4 b2d2 γ (A15) 2 2 2 2(b + d ) 4 24b 24d2 to be evaluated that are otherwise not easily solved using eq A11. Volumetric Flow. The volumetric flow of gas passing through a column, Q˙ , is the linear velocity νz integrated over the crosssectional area of the column.
Q˙ )
∫∫ν
z
dx dy )
PoQ˙ o P
(A16)
Boyle’s law is again applied to equate Q˙ with the volumetric flow of gas exiting the column, Q˙ o, at z ) l. When this expression for Q˙ is combined with eq A6, one obtains
Q˙ o ) -
∫∫K (x,y) dx dy P [dP dz ]
1 ηPo
ν
(A17)
(59) Landau, L. C.; Lifshitz, E. M. Fluid Mechanics; Addison-Wesley: Reading, MA, 1959; p 58.
[
]
Po2 - Pi2 Kν(x,y) dx dy 2l
(A18)
Table 1 compares values obtained from eq A18 with data collected by Terry60 and Kolesar61 using microfabricated rectangular columns in silicon. There is essential agreement between theory and experiment. The volumetric flow rate, Q˙ , of eq A16 can also be written as
so that the linear velocity corresponding to eq A6 becomes 2
output flow rate (µL/s)
inlet pressure (psig)
(A12)
where γ is a variable parameter with a value between 1 and 2 (γ ) 1 for an ellipse). When γ is 1.0, the integrals of eqs A11 and A12 over the cross-sectional area are equal, and eq A12 amounts to a series expansion of eq A11. Using this approximation, the average linear velocity can be written as
νjz) -
Table 1. Volumetric Flow at the Outlet of a Miniature Column versus Inlet Pressure
PoQ˙ o 1 )P η
≡∫∫K (x,y) dx dy dP dz ν
KQ dP η dz
(A19)
which defines the column permeability for volumetric flow. Relationship between Linear Velocity and Volumetric Flow. When PQ˙ ) PoQ˙ o is substituted into eq A7 2 2 (Pi/Po)2 - (νjo/νjz)2 z (Pi/Po) - (Q˙ o/Q˙ ) ) (A20) ) l (Pi/Po)2 - 1 (Pi/Po)2 - 1
where νjo/νjz is the ratio of the linear gas velocities obtained by dividing the volumetric flows by the cross-sectional area of the column. After rearranging terms and eliminating νjo by using eq A10, the linear velocity averaged over the cross-sectional area of the column becomes
νjz )
KQ 2lηAc
Pi2 - Po2
x
(A21)
z Pi - [Pi2 - Po2] l 2
where Ac is the cross-sectional area. Unlike the approximation of eq A13, this is an exact expression for the averaged linear velocity. Through eqs A6 and A19, νz is related to νjz by AcKν (x,y)/KQ and νz can be written as (60) Terry, S. C. Ph.D. Dissertation, Stanford University, 1975. (61) Kolesar, E. S., Jr., private communication, 1996.
Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
4813
νz )
AcKν(x,y) Kν(x,y) νjz ) KQ 2lη
Pi2 - Po2
xP
2
i
(A22)
z - [Pi2 - Po2] l
The values of νz and νjz at the column exit can be evaluated by setting z ) l. Mean Linear Velocity. It is generally accepted in gas chromatography that the mean linear velocity 〈νz〉 characterizing the flow of carrier gas through a column is related to the average linear velocity at the column exit (i.e., eq A21 evaluated at x ) l) through a compressibility factor j 62-66
〈ν〉 ) jνjo ) jQ˙ o/Ac
time compared to the time required for establishment of equilibb d ∫-d dy dx is rium in the column. When the operator 1/Ac∫-b applied to all members of eq B1, the equation becomes
kνjz ∂n jg ∂n jg νjz + + ∂t 1 + k ∂z1 KQ b
νjz (1 + k)Ac
d
∂∆ng
∫ ∫ ∂z -b -d
b
d
∂∆ng dy dx ∂z1
∫ ∫K (x,y) ν
-b -d
dy dx )
1
[ |
Dg ∂∆ng 2d ∂y
[ | | ]
Dg ∂∆ng 2b ∂x
-
y)d
∂∆ng ∂y
-
x)b
y)-d
| ]
∂∆ng ∂x
+ Dg
+
x)-b
∂2n jg
(B3)
∂z12
(A23) where the definitions of eq B2 have been applied. Now, if we similarly define ngw as
where j is
ngw ≡ n j g + ∆ngw
3 P2 - 1 j) 2 P3 - 1
(A24) The boundary condition of eq 16 (see main text) may be written as
and P is the relative pressure (Pi/Po) as initially proposed by James and Martin.67 APPENDIX B. VARIATIONAL ANALYSIS OF TRANSPORT EQUATION The HETP is derived by first performing a variational analysis on the modified transport equation:
[
]
[
2
2
2
]
∂ng AcKν(x,y) ∂ ng ∂ ng ∂ ng 1 ∂ng ) Dg + 2 + + νjz ∂t KQ 1 + k ∂z1 ∂x2 ∂y ∂z12 (B1) To treat mass transfer perpendicular to gas flow, two new variables, nj g and ∆ng, must be defined. The new variables are b
( ) ( )
∂(n j g + ∆ngw) Dg ∂∆ng kνjz ∂(n j g + ∆ngw) )∂t 1+k ∂z1 2b ∂x
k
( )
Dg ∂∆ng 2b ∂x
x)-b
-
( )
Dg ∂∆ng 2d ∂y
+
y)d
Dg ∂∆ng 2d ∂y
-b -d b d
b
)
∫ ∫ dy dx
g
∂n jg kνjz ∂∆ngw + ∂t 1 + k ∂z1
νjz b KQ -b
d
∫ ∫K (x,y) ν
-d
b ∂∆ng νjz dy dx ∂z1 (1 + k)Ac -b
d
∂∆ng
∫ ∫ ∂z -d
d
g
dy dx ≡ 0
dy dx )
1
∂2n jg ∂z12
(B6)
dy dx
-d
ng ≡ n j g + ∆ng; ∆ng , n jg
∫ ∫∆n
(B5)
(1 + k)
d
∫ ∫n
1 Ac -b
-b -d
b
y)-d
Dg
∫ ∫n
+
x)b
in the moving coordinate system. Adding eq B3 to eq B5, member by member, yields
d
g dy dx
n jg ≡
(B4)
(B2)
-b -d
Following Golay, it can be argued that only the averaged concentration nj g in the z1 direction is of physical importance, but only after the diffusive process has operated for a long period of (62) Giddings, J. C.; Seager, S. L.; Stucki, L. R.; Stewart, G. H. Anal. Chem. 1960, 32, 867-870. (63) Giddings, J. C. Anal. Chem. 1964, 36, 741-744. (64) Hawkes, S. J. J. Chem. Educ. 1983, 60, 393-398. (65) Ettre, L. S.; Hinshaw, J. V. Basic Relationships of Gas Chromatography; Advanstar Communications: Cleveland, OH, 1993; Chapter 5. (66) Mittlefehldt, E. J&W Sep. Times 1994, 8 (1), 10-14. (67) James, A. T.; Martin, A. J. P. Biochem. J. 1952, 50, 679-690.
4814 Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
where the k(∂∆ngw/∂t) term has been dropped by virtue of ∆ngw , nj g, and it is negligible compared to ∂nj g/∂t. The ∂∆ng/∂z1 terms, which could be dropped when compared to ∂nj g/∂z1 within the same order of differentiation, are not negligible when compared to ∂2nj g/∂z12, which is a higher order of differentiation. To obtain an expression for ∆ng, a relationship is needed between ∆ng and the differentials of nj g. This is accomplished by substituting nj g + ∆ng into eq B1 and dropping terms in ∆ng when there are similar terms in nj g:
[
]
∂n jg AcKν(x,y) jg 1 ∂n + νjz ) ∂t KQ 1 + k ∂z1
[
Dg
∂2∆ng ∂x2
+
∂2∆ng ∂y2
+
jg ∂2n ∂z12
]
(B7)
Subtracting eq B6 from eq B7 and again dropping the appropriate ∆ng terms, the result is
[
]
[
]
AcKν(x,y) jg ∂2∆ng ∂2∆ng ∂n jg 1 ∂n + νjz ) Dg + ∂t KQ 1 + k ∂z1 ∂x2 ∂y2 (B8)
-k
Solving for ∆ng and recalling that ∆ng is finite at x ) y ) 0, eq B8 becomes
{[
4Acνjz 2
KQ(x
∫∫∫∫K (x,y) dy dy dx dx +y)
]
jg jg νjz x2y2 ∂n x2y2 ∂n -k 2 2 2 2 1 + k x + y ∂z1 x + y ∂t
}
(B9)
[ | [ |
∂ng ∂n j g Dg ∂∆ng k + νj ) ∂t 1 + k z ∂z1 2b ∂x
-
x)b
Dg ∂∆ng 2d ∂y
| ] | ]
∂∆ng ∂x
ν
KQ(b2 + y2)
]
(B10)
ν
]}
)[
kTνjz (1 + k)
[
]
νjz ∂n jg ∂n jg + ∆ng,x)(bory)(d (B11) ngw ) -T ∂t 1 + k ∂z1
Because ∆ng is a function of x and y along the boundary, ∆ng,x)(bory)(d must be averaged
{ ∫[
d 4Acνjz 1 1 ∆ng,b ) ∆ngo + 2Dg d 0 K (b2 + y2) Q
b
kνjz
∫∫ 2(1 + k)D g -b -d
[
∫
ν
[
k2νjz
]
d
∫
]
1 x 2d 2 dx + 2 b 0 x + d2
∫
)[ ]
+∞
d
]
ν
]
νjz x2y2 dy dx (B14) 1 + k x2 + y2
1 b2y2 dy + 2(1 + k)2Dg d 0 b2 + y2
(
ν
∫∫∫∫K (x,y) dy dy dx dx|
Kν(x,y) KQ
APPENDIX C. DERIVATION OF HETP Using Golay’s approach, plate theory is derived by analyzing the second moment of the transport equation.68 The incremental second moment is given by
νjz ∂n jg b2y2 dy + dy dy dx dx 2 2 x)b 1+kb +y ∂z1
b 4Acνjz 1 b 0 K (x2 + d2) Q
(
Kν(x,y) 1 x2y2 dy dx (B15) 2 KQ (1 + k)Ac x + y2
∫∫∫∫K (x,y)
|
-d
-
y)d
∫∫∫∫K (x,y) dy dy dx dx -
2
d
d
∫∫
b
Equation B10 in combination with the boundary conditions of eq 16 (see main text), requires
0
∫∫∫∫K (x,y) dy dy dx dx|
4Acνjz
+
y)-d
b
∫
b νjz νjz x2d2 dx 1 + k x2 + d 2 2(1 + k)Dg -b
A2 ) -
-
x)b
νjz 1 b2y2 dy + 1 + k b2 + y 2 2bDg
z)-b
∂∆ng y)d ∂y
∫
∫∫∫∫K (x,y) dy dy dx dx|
4Acνjz
4Acνjz 1 (1 + k)Ac KQ(x2 + y2)
When eq B8 is integrated only once, it becomes
-k
[
KQ(x2 + d2)
ν
2
{
d kTνjz2 Dg kνjz 1 + + 1 + k (1 + k)3 (1 + k)2 2dDg 0
[
∆ng ) ∆ngo + 1 2Dg
A1 )
du ≡
∫nj z
2
g 1
dz1
-∞ +∞
∫nj
g
+∞
)d
∫hf z
2 1
dz1
(C1)
-∞
dz1
-∞
-
y)d
[ ] ∫[ ] }
where hf is the normalized solute concentration. By virtue of eq B13, eq C1 can be written as
νjz ∂n j g k d b2y2 ∂n jg x2d2 dx dy 2 2 2 2 1+kx +d ∂z1 d 0 b + y ∂t
∫
b ∂n jg x2d2 k dx b 0 x2 + d 2 ∂t
+∞
(B12)
du ) dt
(
+∞
)
∂fh 2 ∂2hf ∂fh z1 dz1 ) dt A1 2 + A2 z12 dz1 ∂t ∂z ∂t ∂z 1 -∞ -∞ 1
∫
∫
(C2) to provide a weighted contribution to ∆ngw. When the expressions for ∆ng in eq B9 and ∆ngw in eq B11 (using ∆ng,b for ∆ng,x)(bory)(d) are substituted into eq B6, the transport equation becomes
∂ng ∂2n jg ∂n jg ) A1 + A2 2 ∂t ∂z ∂z 1∂t
(B13)
In eq C2, the A1 term describes the effects of pure diffusion. Since Golay argued that the A2 cross-term makes a negligible contribution to the incremental second moment, the incremental second moment can be written as69
1
where
(68) Golay, Press: (69) Golay, Press:
M. J. New M. J. New
E. In Gas Chromatography 1958; Desty, D. H., Ed.; Academic York, 1958; pp 36-55. E. In Gas Chromatography 1958; Desty, D. H., Ed.; Academic York, 1958; pp 40-41.
Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
4815
( ) |
+∞
du) dt
∫A
-∞
1
∂2hf 2 z1 dz1 ∂z12 +∞
HETP )
[
+∞
∂fh ∂fh ) A1 dt z1 -2 z1 dz1 ∂z1 -∞ ∂z -∞ 1 2
|
+∞
∫
) A1 dt[-2fhz1 -∞ + 2 ) 2A1 dt
l
4Acνjz
∫hf dz ] 1
-∞
[
ν
where terms have been set to zero whenever the essential positive behavior of hf allows, and the definitions for u and hf have been applied. In the moving coordinate system (where the time required for a component with a given k to travel an elementary distance dz1 is [(1 + k)/νjz] dz1), eq C3 becomes eq C4.
du ) (2(1 + k)A1/νjz) dz1
(C4)
HETP ≡ 2(1 + k)A1/νjz
(C5)
Because the height equivalent to a theoretical plate is defined as this incremental second moment divided by the length of the column, we get eq C5. When eqs B14 and 17 (see main text) are substituted into eq C5, the HETP becomes eq C6. Because νjz varies along the length of the column, it is replaced by the mean linear velocity as defined in Appendix A to evaluate
4816 Analytical Chemistry, Vol. 70, No. 22, November 15, 1998
-
x)b
]
b νjz 1 b2y2 dy + 1 + k b2 + y2 2bDg 0
4Acνjv
∫
∫∫∫∫K (x,y) dy dy dx dx|
KQ(x2 + d2)
(C3)
∫
∫∫∫∫K (x,y) dy dy dx dx|
KQ(b2 + y2)
+∞
{
d 2kh2νjz 2Dg 1 2k + + νjz 1 + k 2dDg 0 3(1 + k)2D
ν
]} { (
νjz x 2d 2 1 dx 2 1 + k x + d2 Dg
)[
4Acνjz
1 (1 + k)Ac KQ(x2 + y2)
b
d
∫∫ -b -d
-
y)d
Kν(x,y) KQ
∫∫∫∫K (x,y) dy dy dx dx ν
] }
νjz x2y2 dy dx 1 + k x2 + y2
(C6)
the HETP. Furthermore, greater accuracy is obtained if KQ is evaluated by numerically integrating Kν(x,y) over the crosssectional area of the column. The evaluation of the indefinite quadruple integrals of Kν(x,y), however, is more difficult. For these integrals, Kν(x,y) must be linearized using an elliptical approximation before performing the integrations. See eq A15 of Appendix A.
Received for review March 20, 1998. Accepted August 25, 1998. AC980328Z
