6
1
a t a helium velocity of 16 em. per second. Finally, the influence of particle size on gas phase effects is shown in Figure 8. These results are rather surprising, in that the curves tend to meet at about Y 1.3 and reverse order completely. This would indicate perhaps that the finer supports have more favorable short range equilibration effects while the large particles are better off with respect to long range (or trans-column) effects, since the latter do not couple out so rapidly. Since the recent paper of Sternberg and Poulson (IO) it has been generally believed that the chromatographic process is intrinsically more efficient for relatively large particle-tocolumn diameter ratios. The present results show that this conclusion must be re-examined, preferably with more experimental work. The effect of particle size on various parameters is summarized in Table V.
-
'L*
00
R
Figure 10. Effect of temperature on minimum plate for different systems with DNP coated 60- to 80-mesh support
(hO)*m,n]. If R was the predominant factor controlling w , the three curves would form one continuous curve. If liquid load predominated, the curves would be almost horizontal-Le., very little R dependence-stratified one above another. The actual case is more complicated than either simple extreme. The 10% curve tends to lie above the 20% curve of Gas Chrom S, indicating that, a t constant R , w decreases somewhat for increases in load. The effect of temperature on various plate height terms is summarized in Table IV. Reference is again made to the relative liquid and gas contributions
CONCLUSIONS
Of the four supports studied, Chromosorb P is clearly the best in terms of column efficiency parameters. However, other considerations-e.g., adsorption- may strongly influence the choice of a support for particular applications.) A large part of this is due to the very fine pore structure of Chromosorb P (about half of its free volume is in pores < 1-micron diameter). I n view of this fact, it is somewhat
surprising that Chromosorb P is not even more efficient, a t least in connection with liquid phase mass transfer. A number of effects have been noted here which should be studied more thoroughly. Of particular interest is the effect of particle size on both gas and liquid phase terms. The entire problem of finding a consistent pattern in the variation of the gas phase plate height needs a great deal more exploration. Such studies would undoubtedly lead to better methods for the fabrication of gas chromatographic columns. LITERATURE CITED
(1) Blandenet, G., Robin, J. P., J . Gas Chromatog. 2, 225 (1964). (2) Dal Nogare, S., Chiu, J., ANAL. CHEM.34, 890 (1962).
(3) Giddings, J. C., Zbid., 34, 1186 (1962). (4) Ibid., 36, 741 (1964). (5) Giddings, J. C., J . Chromafog. 13,301 (1964). (6) Giddings, J. C., Schettler, P. D., ANAL.CHEM.36, 1483 (1964). (7) Houghton, G., Kesten, A. S., Funk, J. E., Coull, J., J . Phys. Chem. 65, 649 (1961). (8) Perrett, R. H., Purnell, J. H., ANAL. CHEM.35, 430 (1963). (9) Saha, N. C., Giddings, J. C., Zbid., 37, 822 (1965). (10) Sternberg, J. C., Poulson, R. E., Zbid., 36, 1492 (1964). RECEIVEDfor review January 12, 1965. Accepted March 29, 1965. Investigation supported by Public Health Service Research Grant GAI 10851-08 from the National Institutes of Health.
Pressure Profiles, Fluctuations, and Pulses in Gas Chromatographic Columns PAUL D. SCHETTLER and J. C. GlDDlNGS Department o f Chemistry, University o f Utah, Salt lake City, Utah Fluctuations in column pressure, due to inadequate pressure regulation, flow startup, or other flow changes, hinder the use of gas chromatography for qualitative and quantitative analysis and research purposes. The time-dependent pressure profiles resulting from these fluctuations have been studied. These profiles should b e used whenever there is a departure from steady-state flow and thus from the classical pressure profiles of James and Martin. Nonsteady profiles are obtained here as a numerical solution to the partial differential equation of gas flow in porous media. Cases have been solved which correspond to sudden pressure jumps at the inlet and to injected gas peaks. The time of pressure relaxation is in most cases about 0.4 reduced time unit, roughly
comparable with the retention time of an inert peak. A sensitive thermistor anemometer has been constructed for measuring the flow changes arising out of pressure fluctuations. Ordinary tank regulators are subject to small but rapid pressure fluctuations. A comparison of theoretical and experimental transient flow rates shows good agreement.
T
theory and practice of gas chromatography are based largely on the assumption that steady-state flow has been achieved in the gas chromatographic column. Vnder these circumstances the pressure profile will be given by the classical James-Martin expression (6). Any perturbation of or departure from steady-state flow will alter HE
the pressure profile and make it timedependent. This will seriously hinder qualitative analysis, since retention times will not be reproducible, and it will hinder quantitative analysis, particularly with thermal conductivity cells, because peak area is flow-sensitive. The use of gas chromatography to obtain accurate research data will also be affected. To understand further the importance of nonsteady flow and pressure profiles, the present work gives the theory of flow and amplifies it with numerical examples obtained by computer. Thus the James-Martin treatment of steady flow has been extended to the next logical step with the removal of the steady-state hypothesis. Time-dependent pressure profiles may originate in several ways. Whenever VOL. 37, NO. 7, JUNE 1965
835
Figure 1 . Schematic diagram of elevated outlet pressure control Tank regulator adjusted to desired pressure. Needle valve opened until flow from regulator is much larger than flow from column. Outlet pressure held constant, independent of column flow rate, within limits o f ordinary pressure gauge
column flow is started, stopped, or changed In any fashion, a certain relaxation time will be required to reach another steady state. The continuous changes in flow encountered with temperature and pressure (velocity) programming ill likewise cause a departure from the usual assumption of a steady state. Pressure fluctuations at the inlet or outlet, always present to some extent, will also perturb the steady state. The treatment of nonsteady flow is much more difficult than that for the steady state. Each pressure profile must be obtained numerically as a solution to a nonlinear partial differential equation. The form of this differential equation has long been known in connection n i t h gas floa in
porous materials ( I ) . Nonsteady flow in gas chromatography has been treated independently in recent unpublished work (4). In connection with the use of gas chromatography for research on the nature of plate height terms, of particular interest in this laboratory ( 3 ) , all shortcomings in instrumentation and methodology, whether related to pressure control or not, must be carefully eliminated. The defects in instrumentation are generally of two types. First are those effects which cause a uniform distortion for all peaks obtained under the same experimental conditions. Dead volumes and electronic time const’ants are examples of this, which are under considerable study a t the present time (5, 8). Second are those effects which produce random fluctuations in peak shape, width, and retention times. One such error is caused by flow fluctuations. Flow is ordinarily controlled by the controlled pressure of a two-stat’e tank regulator ( 2 , ?). Such regulation is inadequate for exact,ing theoretical interpretations which require the measurement of second moments (plate height), particularly with high pressure outlet and low pressure drop. I t may also fail to give accurate and reproducible results in qualitative and quantitat ive analysis. Consider a system using an elevated outlet pressure, p,, with an uncertainty
-1
1 Figure 2. Figure 1
I
(p,
* Ap,)’ - ( p o
(piz -
PO’)
f
Ap,)’
* 2 ( p , A ~ tf poApo)
(1)
If ~ , ~ - - p ,is~ small, as in cases of columns of low permeability and/or slow flow, the second term will be significant or even dominant in the expression for the flow rate. I n addition to the steady-state change in flow with pressure fluctuations, there will be a transient flow rate as “rarefaction” or “compression” waves move along the column. In an effort to study pressure fluctuations in a column we have constructed a highly sensitive thermistor anemometer for measuring flow fluctuations; employed a system which exhibits improved flow rate stability by virtue of decreasing the Ap’s, and solved numerically the appropriate differential equation describing pressure transients in packed beds for the case of a discontinuous pressure jump and for the pressure transient caused by an injected sample. EXPERIMENTAL
An anemometer was constructed from a Cow Mac (922581) thermistor bead (commonly used in gas chromatography) mounted in a special block such that the bead was directly in the flow stream. Unlike a hot-wire anemometer, in which the sensing element is heated by an adjacent hot wire, this thermistor was used (following chromatographic practice) as one aym in a Wheatstone bridge and heated directly by the passage of electric current. However, no balancing thermistor was used, and this necessitated a 500- to 3000-ohm balance resistor capable of adjustment to 0.1 ohm or better. The bridge output was connected through an attenuator to a 1-mv. recorder. The anemometer was used in place of the detector to sense the flow variations at the end of the gas chromatographic column. The column was held above atmospheric pressure, as shown in Figure 1. A typical recorder trace is shown in Figure 2. The chromato-
~
COLUW PRE-
TANK CUUMN ETECTOR
NEEKE
REGULATOR
-
Flow fluctuations caused by outlet system of
Ordinate. Flow fluctuations resulting in pressure variations o f to 0.1 p.5.i. Places where bridge hod to b e readjusted noted
836
of Apo and an inlet pressure of p , f Ap,. The steady-state flow rate (mass of carrier gas per second) will be proportional to
ANALYTICAL CHEMISTRY
f 0.05
Figure 3.
Flow system using Cartesian manostats Arrows show direction of gas flow
-010
0
$
30 kEC
I5 SEC
4 t-W
Figure 4. Flow fluctuations caused by outlet system of Figure 3
i 1
-005
z graphic system was subsequently changed to include a No. 8 Cartesian manostat (Manostat Corp., New York, N. Y.) as shown in the outlet section of Figure 3. The resulting recorder trace is shown in Figure 4.
It is immediately clear that the inclusion of a manostat results in an improvement of 1 to 2 orders of magnitude in column outlet flow regulation. The remaining slow drift was possibly due to i n a d q u a t e inlet regulation or temperature fluctuations. Thus as far as short-term transients are concerned a Cartesian manostat gives about one order of magnitude improvement over a two-stage tank regulator. This agrees fairly well with manufacturers' claims. THEORY
If the pressure a t either end of a column is changed suddenly to a new value, a transient effect on the flow rate will exist as well as a change in the steady-state flow. Consider a column for which the outlet pressure is discontinuously lowered by a small amount. For a very short time the column will maintain its original pressure distribution except a t the outlet, where d p / d z will be almost infinite. This implies a very high outlet flow rate fed by gas expansion near the end. A rarefaction wave will then travel up the column and the high outlet flow rate of the transient will be expected to decrease to a new steady-state value. If a chromatographic peak were to be emerging from the column as the pressure discontinuity occurred, the peak would be distorted by the high flow rates of the transient. Simple Approximation. T h e order of magnitude of this effect may be calculated as follows: The total free volume between the inlet and a peak a t distance L from the inlet in a column is given by
ai
03
Figure 5. Reduced time plotted against reduced outlet velocity ( v = v , ) and reduced inlet velocity (v = vi) = 1 A p = 0.1 P O
Initial inlet flow rote zero
nR T p=AfL
(3)
we have (4)
For a column of length L and pressure p , a change in pressure, A p , will displace a peak near the end of the column by AL. If we assume p = 20 p.s.i. and A p = 0.1 p.s.i., along with L = 100 cm., then AL = 0.50 cm. An inert peak with 0.1-cm. plate height in such a column would be displaced 16% of its standard deviation (6.2 cm.) by such a pressure discontinuity due to the transient alone. This can involve as much as 6% of the area of a peak. This is probably one limitation on quantitative accuracy, since ordinary pressure regulation is little better than suggested by this example. This would not be compensated by an expansion of the peak, since this effect would amount to only 0.5% of the standard deviation. This approach is rigorously applicable only when the drop in pressure occurs uniformly throughout the column, a case that occurs exactly only for columns with no flow. Further, it does not give any detailed information on the time required for a column to re-establish steady-state flow, the pressure distribution in the column up to this time, nor the transient inlet and outlet flow rates. Rigorous Approach. Gas flow in packed columns is described by the basic differential equation (1)
v = AfL where A is the cross-sectional area and f the porosity. If the mean column pressure is p
0 05
04
subject to the appropriate boundary
conditions. I n this equation K O is the specific permeability, z is distance, and t is time. This is identical to Fick's second law of diffusion (pressure replacing concentration) in the special case where the diffusion coefficient, K , p / f , is proportional to concentration. A nonlinear diffusion problem of this type has been discussed by Wagner (9). If we make a transformation to the following reduced variables, each of which is dimensionless,
p / p o (reduced pressure)
(6)
z = z / L (reduced distance)
(7)
r = K , p , t/fL2 (reduced time)
(8)
p =
Equation 3 reduces to (9)
Since this does not contain the specific parameters L , f, and K O ,and the outlet pressure, p,, it is directly applicable to all columns and conditions. I n the limiting steady-state case, bp/br = 0, treated first in another manner by James and Martin (6) we have b(p bp/bx)/bx = 0. A first integration gives p bp/bx = constant. A second integration gives the familiar quadratic form, p 2 = constant 2, which reduces to the James-Martin steady-state case upon inclusion of the proper boundary conditions. A simple and important case of nonsteady flow occurs when the inlet pressure suddenly jumps to a new value. The boundary conditions for this case are p = 1
at z
=
1 for all
'T
VOL. 37, NO. 7,JUNE 1965
(10)
837
2.0
1.8
p
1.6
LO+
-
J.4
LO2
-
12
LO X
X
Figure 6. Reduced distance plotted pressure for various reduced times
against
Figure 8. pressure
reduced
p = po
at x
=
0 for
7
