(2) Gluckauf, E., Nature, London 156, 748 (1945). (3) Goodwin, L. F., J . Am. Chem. SOC. 42, 39 (1920). (4) Keller, R. A., Stewart, G. H., ANAL. CHEM.3 4 , 1834 (1962). (5) Martin, R. L., Zbid., 33, 347 (1961); 35, 116 (1963). (6) Martire, D. E., Ibid., 33, 1143 (1961). ( 7 ) Martire, D. E., Pecsok, R. L., Purnell, J. H., Nature 203, 1279 (1964). (8) Messinger, Ber. 29, 3336 (1888).
(9) Parcher, J. F., Urone, P., J . Gas C h r m t o g . 2, 184 (1964). (10) Pecsok, R. L., de Yllana, A,, AbdulKarim, A., ANAL.CHEM.36,452 (1964). ( 1 1 ) Purnell, J. H., “Gas Chromatography,” p. 13, Wiley, New York, 1962. (12) Sawyer, D. T., Barr, J. K., ANAL. CHEM.3 4 , 1518 (1962). (13) Scholz, R. G., Brandt, W. W;:
“Symposium on Gas Chromatography, No. 3, Instrument SOC.of Am., Mich. State Univ., preprints, p. 9, June 1961. (14) Urone, P., Parcher, J. F., J . Gas Chromatog. 3 , 35 (1965).
(15) Urone, P., Pecsok, R. L., ANAL. CHEM.3 5 , 837 (1963).
RECEIVEDfor review August 23, 1965. Accepted December 7, 1965. Investigation made possible by National Science Foundation Grant No. GP-1000. The gas chromatographic instrument used was obtained through a separate National Science Foundation Grant No. GP-2117. Presented at the Third International Symposium on Advances in Gas Chromatography, Houston, Texas, October 1965.
Turbulent Flow Chromatography: A New Approach to Faster Analysis VICTOR PRETORIUS and
T. W.
SMUTS
Department of Physical and Theoretical Chemistry, University o f Pretoria, Pretoria, Republic of South Africa
b Expressions have been derived for plate height and minimum analysis time in open tubular columns in which laminar and turbulent flow of the mobile phase may be employed. Using these equations the role of various column parameters in fast analysis has been studied in the laminar and turbulent flow regions of gas and liquid chromatography. In the gas chromatography turbulent flow can lead to analysis times about one tenth of those obtained under comparable circumstances by using laminar flow. In liquid chromatography turbulence can shorten analysis times by as much as a factor of lo4. In general turbulent flow chromatography entails the use of slightly longer column lengths and much larger pressure drops than would be needed under similar conditions in laminar chromatography.
E
attempts to decrease analysis time in chromatography have been balked by the existence of an optimum mobile phase velocity, and consequently of an apparently fixed analysis time for a given chromatographic system. It was, however, soon pointed out (12, 13) that the analysis time could be shortened by increasing the mobile phase velocity and increasing the column length to compensate for the accompanying deterioration in the column efficiency. This improvement can be effected as long as the longitudinal diffusion term contributes significantly to the plate height (12, 13, 15). Several expressions for the minimum time needed to resolve a given pair of solutes, to a given extent, have been derived. According to the most convenient of these (12,13) ARLY
274
ANALYTICAL CHEMISTRY
where
The effect of various column parameters on this minimum time has been investigated. Under favorable circumstances analysis in gas chromatography may be carried out in a few seconds (14). Although experimental results are largely lacking, the minimum analysis time in liquid chromatography would appear to be longer than that in gas chromatography by a factor equal to the ratio of gaseous to liquid diffusion lO4-if similar sepacoefficients-Le., rations are compared. Analysis times of several hours would seem to be consistent with practical experience. -411 these studies have been solely concerned with laminar flow of the mobile phase and the conclusions reached are essentially correct for this situation. The present paper shows that, if chromatography is carried out with turbulent flow in the mobile phase, the minimum analysis time can be shorter than that with laminar flow. The treatment is confined to open tubular columns. The improvement is most significant in liquid chromatography. GENERAL EXPRESSION FOR PLATE HEIGHT OPEN TUBULAR COLUMNS
IN
The theory of chromatography in open columns is well developed. The best known work is that of Golay (9), who assumed a parabolic radial velocity profile and mass transfer by molecular diffusion-assumptions which are essentially correct for laminar flow (6, 6, 1 7 ) . Under turbulent conditions the
Golay expression breaks down, since the radial velocity profile is no longer parabolic but becomes velocity-dependent (3) and radial mass transfer is enhanced by convection. Of the several studies of band dispersion under turbulent conditions (1, 2, 18, 19), that of Aris is the most convenient for the present development (2) and from it may be written:
where
(9’1‘
1 l+k =
111
and 9
10
-2112
(T)
(4)
is of the form:
$(r) is a function which relates the dispersion coefficient, D ( r ) , at a radial position, r , to a reference value, D,by the expression D(r)
=
D+(r)
(6)
+(r) is a similar function defined by 47) =
Wiid
(7)
where the mean velocity is chosen as reference, The term “dispersion coefficient” is used in a general sense to include eddy diffusion and molecular diffusion.
From Equations 4 and 5 it follows that the integrals I l l , Z12, and 113 can be written as:
files can be used. This may be done as follows: Radial mass transport in the turbulent region may be considered to consist of a radial convection contribution superimposed on the usual molecular diffusion-Le.,
+
D ( r ) = D M ~ D.d(r) ro2
(9)
(17)
It may be assumed (4, 10, 19) that the
=I*
I
I
total radial mass transport is governed by an equation of the same form as Fick's first law of diffusion-Le.,
For final evaluation of these integrals it is more convenient to transform the radial coordinate, r , into a dimensionless variable
Similarly the radial transport of momentum is determined by
I
IO^
r rD
=
=
I
I
lob
I$
RE
r,
Equations 8,9, and 10 now become Z,l
.
lo3
It would appear reasonable (4, 19) to assume that mass and momentum are transported by the same mechanism in the turbulent region. Consequently
2
v,d(r) = D d ( r ) (20) F can be expressed in terms of the Moody friction factor-viz.
and Equation 3 may be written as
(19)
Figure 1. Calculated and experimental values of reduced plate height at various Reynolds numbers in open tubes k = O a, b. Limits of experimental data for liquids and gases c. Calculated from Equation 1 5 for liquids d. Calculated from Equation 15 for gases e. Theoretical values derived b y Taylor
D(r) may now be expressed in terms of
M,Rero '-[I6
+1
]
Sc (26)
and
j~ and js are functions correcting for the compressibility of the mobile phase. The expressions used for f~ and fs are those derived by Giddings et al. (8, 16) for laminar flow, since it may be shown that they also hold for turbulent flow. Equation 15 may be regarded as a general form of the Golay expression, in that it is valid for both laminar and turbulent flow. The equation readily reduces to the Golay expression, since for laminar flow d4.D)
=
2 [l
- (TD)21
measurable quantities by using Equations 17 to 21:
(22)
b7
At this stage it is convenient to express Equation 22 in terms of dimensionless groups :
D(rD) =
(16)
and
D(rD) = D M ~ By using Equations 12, 13, and 14 it h a y be shown that 111 = 1 1 / 4 8 , 1 1 2 = and 113 = 6 / ~ . Substitution of these values in Equation 15 leads to the familiar Golay expression. To evaluate Equation 15 for turbulent flow, +(r') and $(r') must be known. A purely theoretical evaluation of these functions is extremely complicated and leads to inaccurate results even for the simple case where no retention is considered. A more satisfactory solution to this problem may be found by casting +(r') and $(r') in a form in which empirically measured velocity flow pro-
where Sc = Schmidt number =
V M ~ / D M (24) " Re = Reynoldsnumber =
and rD
=
r ro
It now follows from Equations 6 and 23 that
D = DMM Equation 26 enables one to evaluate $(rD) from experimentally measured velocity profiles and, in turn, to evaluate Equation 15 for various values of the column parameters. An indication of the validity of this approach may be obtained by comparing the plate height calculated by these means with experimental data for the simple case where the solute is not retarded-i.e., k = 0. The comparison is most conveniently made by plotting (from Equation 15) the reduced plate height, h M = HM/r,,, against the Reynolds number which, from Equation 24, is proportional to flow velocity for fixed values of r, and v M m . D p has been assumed to be lo-' sq. cm. per second for gases and 10-6 sq. cm. per second for liquids. The experimental curve shown in Figure 1 is a compilation representing results obtained by a number of investigators (11). The calculated curve has been obtained from the equations derived here by using the experimental velocity profile data reported by Tichacek et al. (19), which are also a compilation from a variety of sources. The extent to which these results agree is emphasized by curve e in Figure 1 , VOL. 38, NO. 2, FEBRUARY 1966
275
which represents the values obtained by Taylor on the basis of a theoretical study of the radial dependence of the dispersion coefficient and of the flow velocity. I n chromatography one would, however, be more concerned with the situation where k > 0. I n Figure 2 the reduced plate height has been plotted against the Reynolds number for k = 1, which is the most important value from a practical point of view. As before, the data have been calculated from Equation 15. The previously mentioned values of D . p have been used; Dsm has been taken to be 10-~ sq. cm. per second, and d, = 10-5 cm. The curves for the laminar region can be assumed to be essentially correct. The reduced plate height in this region differs by a factor of 4, a t each value of the Reynolds number for k = 0 and k = 1. Jf the k-dependence of the reduced plate height were the same in both the laminar and turbulent regions, one would expect the curves for k = 0 and k = 1 in the turbulent region also to differ by a factor of 4. The most plausible explanation which we can put forward at the moment for the anomalous behavior of the liquid curve, b, in the turbulent region, is that the evaluation of the integrals is extremely sensitive to the precise shape of the empirical velocity profile in the vicinity of the column walls. Measurements made in this region are magnified in the case of liquids, since gaseous diffusion coefficients are times those of liquid ones. On the basis of these arguments we have concluded that curve b is a gross overestimation of the plate height and that a more realistic indication of these values may be obtained by assuming that the reduced plate heights for turbulently flowing liquids when k = 1 are 4 times those for k = 0. The main features of chromatography in laminar and turbulent regions are essentially determined by the behavior of the curves in Figure 2. In laminar chromatography the plate height initially decreases as the flow velocity increases as a result of a diminishing contribution of molecular diffusion to band spreading. A minimum plate height is reached when longitudinal molecular diffusion becomes comparable to the resistance to mass transfer. Hereafter the plate height increases linearly as a result of the last-mentioned bandspreading processes. At relatively high flow rates turbulent flow begins to set in and this both flattens the velocity profile and increases the radial mass transfer by superimposing convective transport on the normal molecular diffusion. Consequently, the contribution of resistance to mass transfer in the mobile phase, in the turbulent flow region, becomes progressively less than that in the laminar region. If now the
276
ANALYTICAL CHEMISTRY
lo5
I
I lo4
I 0’ fi
t
/4-+ TURBULENT
LAMlNIlR
IO2
IO1
IO0 IO*
IO‘
loo
lo2
10’
10’
lo4
los
lo6
RE Figure 2. Calculated values of reduced plate height at various Reynolds numbers in open tubes k = 1, K = 1000 for gases, K = 100 for liquids, and ro = 0.01 e, d. a, b, e.
a, b, d, e. C.
cm. Gases Liquids Calculated from Equation 15 with Dm = lo-’ and 10-6 rq. cm./rec. for gases and liquids, rerpectidy Values corrected from data in Figure 1
initial resistance to mass transfer-i.e., in the laminar region-is greater than or even comparable to that in the stationary phase, the effect of turbulence will be to decrease the plate height. The greater the turbulence the more the resistance to mass transfer in the mobile phase is decreased. The plate height will be correspondingly decreased until the band spreading is largely caused by resistance to mass transfer in the stationary phase. From this point on an increase in the flow velocity will have little beneficial effect on the plate height and will, in fact, ultimately increase it. From Equations 1 and 15 and Figure 2 it may be seen in general terms that the analysis time, T, in the laminar region can be decreased by increasing the flow velocity to a point where the plate height is dependent only on resistance to mass transfer-Le., where it increases linearly with flow velocity. This, of course, will be true only if, a t the same time, the column length is increased so that the total number of plates remains constant. If the flow rate is increased beyond the previously mentioned point, the “rate of production of plates” remains approximately constant, since the now linear increase in the plate height necessitates a proportional increase in the column length, which effectively offsets the advantages of the higher flow velocity. In the turbulent region, however, not only can the increase of plate height with flow velocity be lessened but, under certain circumstances, the plate height itself can actually be decreased.
The extent to which the analysis time may be decreased in going from the laminar to the turbulent flow regions is essentially dependent on the relative sizes of the contributions of the resistance to mass transfer in the mobile and stationary phases to the plate height, as shown below. In gas chromatography, which is not a particularly favorable situation, analysis time may be reduced by as much as a factor of 10 by employing turbulent rather than laminar flow. In liquid chromatography, the gain is far greater; in general, analysis times may be reduced by about a factor of lo4 on the basis of the calculated results and probably even more in actual practice. GAS-LIQUID CHROMATOGRAPHY
The implications of the equations derived above for turbulent flow chromatography are most conveniently illustrated by considering gas-liquid chromatography. The purpose here has not been to carry out an exhaustive study of the effect of all the column parameters but rather to delineate a few of the more important trends. Mass Distribution Coefficient, k. The precise effect of the mass distribution coefficient on the analysis time is not immediately apparent from Equations 1 and 15. A fairly detailed study by Purnell (12, IS) for the laminar region has shown that, for a wide range of experimental situations, the optimum value of k lies between 0.1 and 6. It would appear that a value of
k
Figure 3. Effect of mass distribution coefficient on analysis time a, d.
b, c, e. a, c. b. d, e.
liquid chromatography; DP = 10 -5 sq. cm./sec. Gas chromatography; D,; = lo-' sq. cm./sec. Laminar flow laminar flow Turbulent flow
k = 2 may in general be chosen for the optimum, since the advantage of being able to fix a unique value of k outweighs the errors thereby introduced. In the turbulent region the role of k is more difficult to assess. We have found it convenient to rewrite Equation 1 in terms of dimensionless groups in order to limit the range of parametric values which otherwise would have to be considered. This leads to
The left side of Equation 27 is plotted against k for several values of the Reynolds number and the group &/To in Figure 3. It is evident that in the laminar region the optimum value of k lies between 0.6 and 2 , which is in agreement with the conclusion reached by Purnell et al. In the turbulent region the optimum value of k does not differ much from that in the laminar region and lies between 0.6 and 1. Since large values of k are relatively difficult to obtain in open tubular columns, the smallest value of k within the optimum range would be used in practice. For the purposes of further discussion we fix kept 'V 1 for both the laminar and turbulent regions, although smaller values could, if necessary, be used.
As has been shown, k 'V 1 and it is evident from the foregoing discussion that d, should be made as small as possible for high speed analysis for both the laminar and turbulent regions. A t the same time it is convenient, particularly for practical reasons such as the pressure drop across the column, to be able to increase r,. The main function of K , in the present context, is to ensure that, in the light of the restriction on d, and ro, a value of k 'V 1 is maintained. A concise illustration of this point is shown in Figure 4. It is evident that the analysis time, in both the laminar and turbulent regions, can be improved by choosing a situation in which K is large. In the laminar region K cannot profitably be increased beyond K 500. In the turbulent region, however, much larger values of K are still advantageous, because the iimiting value of K depends ultimately on the extent to which the stationary phase resistance to mass transfer, and particularly d,, can be reduced in relation to the mobile phase resistance to mass transfer. Since in turbulent flow the mobile phase contribution is progressively decreased, the stationary phase contribution can also be decreased a t a constant k by further increasing K . For example, for K = 100, ro = 0.05 cm., D M = ~ 0.1 sq. cm. per second, D s ~= l o p 5 sq. cm. per second, k 'U 1,and CS = 1.04 X second for film distribution of the stationary phase. The value of CM in the laminar region is 4.7 x 10-3 second, whereas a typical value in the turbulent second. region (Re = lo4) is 3.7 >( If now K = 1000 is chosen, C, = 1.04 x 10-5 second, for the same values of the other parameters and remains unchanged for both laminar and turbulent flow. For both values of K , C.M 2 CS for laminar flow; for turbulent flow CM Z Cs for K = 100 and becomes much less than Cs for K = 1000. The effect of turbulent flow can be sufficiently great to allow comparable analysis time with laminar flow for values of K which are as little as one tenth of those needed in the laminar region. Flow Velocity. The role of flow velocity has been discussed in general terms. An important distinction between the laminar and turbulent flow
-
Table I.
uo, To,
u-1000
4 d
10-2
I
".,
t
" I '01
I
I
to1
IO'
lo3
d
lo5
lo6
( 0 '
RE
Figure 4. Effect of concentration distribution coefficient on analysis time at various Reynolds numbers for column radius ro = 0.05 cm. in gas chromatography Valuer calculated from Equations 1 and 15 with DS = 10-6 rq. cm./sec. and DT = rq. cm./rec.
regions is that it does not pay, in terms of analysis time, t o increase the flow velocity in the former beyond a certain point, whereas in the latter the flow velocity cannot be decreased below a certain point. An increase in the flow velocity beyond this point, within the limits discussed, will lead to faster analysis times. An indication of the magnitudes of the velocities involved is given in Table I, for k = 1, K = 1000, D,um= 10-I sq. cm. per second, and D S m = 10-5 sq. cm. per second. The flow velocities tabulated for the turbulent region relate to Re = 6 x lo4 and are therefore larger than the minimum but less than the maximum permissible flow velocities. They represent the sort of flow velocities which might be considered in practice. Column Radius. It follows from Equations 1 and 15 and Table I that for both laminar and turbulent flow the analysis time decreases with a decrease in the column radius. The limiting factor for laminar chromatography will, in practice, probably center around the problem of successfully manufacturing very fine uniform-bore capillaries, since relatively
Typical Flow Rates and Values of Analysis Time Parameter in Laminar and Turbulent Regions for Various Values of Column Radius
Concentration Distribution Coefficient, K . The essential function of
the concentration distribution coefficient is most clearly brought out by the relationship
IO0
cm.
10-2 5 x 10-2 8 x 1 x 10-1 2 x 10-1
Laminar region
cm./sec. 400 60 40 20 15
T,
sec.
1.51 x 10-3 3.0 x 9 . 7 x 10-2 1.54 X 10-l 6.05 X 10-1
Turbulent region 7 , sec.
U,,cm./sec. 45,000 9,000 5,625 4,500 2,250
9.12 X 2.44 x 10-3 6.67 x 10-3 1.07 X 4.68 x
VOL. 38, NO. 2, FEBRUARY 1966
277
*
IO’
I
\
K=10,100,1000
I I
IV
1
I
I
I
I K=lOO
I
low flow rates are involved. In turbulent flow chromatography, on the other hand, the main problem will almost certainly be concerned with the pressure drops required to produce the very high flow rates involved. Column Length. The column length necessary to separate two components with a separation factor, CY, to the extent of R = 1.5 is given by
L
=
36
(5)’ (k) l + k
H (29)
Any differences between column lengths, needed for a given separation, in laminar and turbulent chromatography would be reflected by differences in the plate height. From Figure 2 it is apparent that, in principle, any plate height in the laminar region can be duplicated in the turbulent region. This would imply that for any particular separation a similar column length could be used in the two systems. In practice, however, the enormous velocities which are necessary to reduce the plate height in the turbulent region to very low values would limit one to plate heights, in general, larger than those in the laminar region, so that longer column lengths could be expected for turbulent chromatography. However, the difference between the length needed for seDarations by the two forms of chromatography need not necessarily be excessive. For instance, for the values of the parameters pertaining to gas chromatography stated in Figure 2, which are not untypical of a practical situation, it may be
278
ANALYTICAL CHEMISTRY
1.01
I
1.1
2
K
Figure 5. Effect of concentration distribution coefficient on analysis time at various Reynolds numbers for column radius r o = 0.05 cm. in liquid chromatography Valuer calculated from Equations 1 and 15 with DE = cm./rec. and Da = 10 rq. cm./rec.
I
I
1*001
10001
Figure 6. Minimum analysis times required at various values of a a-d.
e-h. a, b, e, g. c, f, d, h. b, d, g, h.
sq.
0,
e, e, f.
Liquid chromatography Gas chromatography Laminar flow Turbulent Row r o = 0.01 cm. ro = 0.1 cm.
seen from Figure 4 that the minimum analysis time will be obtained for laminar chromatography a t Re = 20, which corresponds to a reduced plate height (see Figure 2) of h % 3. Reynolds numbers of lo4 and even higher can be obtained in practice, so that reduced plate heights of h ‘V 6 are feasible in the turbulent region. Column lengths for turbulent chromatography need not, therefore, be radically longer than those needed for laminar chromatography under similar conditions. Pressure Drop across Column. It is abundantly clear that the gain in analysis time which can be brought about by employing turbulent flow is effected a t the cost of an increase in pressure drop across the column. Several examples of the pressure drops which could be expected in practice are shown in Figure 7. These have been calculated for the necessary column lengths and various radii by using the Poiseuille equation for the laminar r e gion and (5)
would appear to be no difficulty on this point even for relatively wide tubes with relatively smooth surfaces (20). Secondly, what are the relative sizes of the stationary phase C-terms? Fast analysis times are favored by relatively small C-stationary terms for both laminar and turbulent flow chromatography-particularly for the latter. In gas-liquid chromatography Cs can be varied over wide limits to values as small as 10” second ( 7 ) , which would roughly correspond to the situation for K = 1000 in Figure 4. In gas-solid chromatography similarly small Cs-terms appear to be obtainable, in fact, more readily than in gas-liquid chromatography, for fast analysis, would depend on what we have just stated. The relative merits of laminar and turbulent chromatography for gas-solid systems would be similar to those in gas-liquid chromatography, for comparably small values of CS.
AP=
The role of the various column parameters in liquid chromatography broadly follows what has been discussed in connection with gas chromatography; although the trends are the same, the actual values can be radically different. Mass Distribution Coe-fficient, k. From the data in Figure 3, obtained by the method disc&sed, it is clear that the optimum value of k for liquid chromatography lies between 0.5 and 2 for the turbulent and laminar
for the turbulent region. GAS-SOLID
CHROMATOGRAPHY
The essential difference between gassolid and gas-liquid chromatography is twofold. First, can k ‘V 1 be attained in gas-solid Chromatography? There
LIQUID-LIQUID CHROMATOGRAPHY
IO
8
7 IO IO
6
IO
5
I o4
z
IO
3
a IO1
0
IO
Io-' lo-'
I
I
1.001
1~0001
I
I
\I
1'01
1.1
I
I
IO' 1~0001
2
1~001
oc
I
I
1.1
1.01
2.0
o(
Figure 8.
Figure 7. Required pressure drops at various values of a a-d.
e-h. c, d, g, h. e, a, b, f. e, a, c, g.
b, f, d, h.
a, c, f, g. b, c, d, e. c, e, g.
Gas chromatography Liquid chromatography Laminar flow Turbulent flow ra = 0.01 em. ro = 0.1 cm.
regions, and that a value of k = 1 can be employed. Concentration Distribution Coefficient, K . I n Figure 5 the analysis
time parameter, 7 , has been plotted against the Reynolds number for various values of K and the following values of the relevant parameters: T , = 0.05 cm., DMm=Dsm=10-6 sq. cm. per second. As in the case of gases, short analysis times are favored, in both the laminar and turbulent regions, by increasing K . However, in contrast to gas chromatography, little is gained in liquid chromatography by increasing K beyond about 100. This difference in behavior is caused by the overriding contribution, to the plate height, of the mobile phase resistance to mass transfer, as discussed under the column radius. Flow Velocity. An indication of the flow velocities which could be expected in practice is given in Table I1 for a number of column radii, and for K = 100. The flow velocities which must be considered in the turbulent region are about lo* times higher than those in the laminar region; the analysis time is decreased by approximately the same factor. Column Radius. As with gases, the analysis time is decreased by decreasing the column radius. It will probably be of great practical significance that comparable analysis times can be obtained for the two cases by using a column in the turbu-
Required column lengths at various values
of a
a, b, d, f. a-c. d-g.
lent region which is 50 times wider than that which must be used for laminar chromatography. (In gases this difference amounted only to about a factor of 4.) By employing turbulent flow, analysis times can be obtained in liquid chromatography similar to those pertaining to gas chromatography under the same circumstances. This is to be expected, since the effect of turbulence is to make the radial mass transport in the mobile phase largely convective and independent of the nature of the mobile phase. If the analysis times in the turbulent region are compared for gas and liquid chromatography, at the same Reynolds number, for columns of the same diameter, the ratio of the analysis times should equal that of the kinematic viscosities-i.e., 15. For example a t Re = 6 X lo4 and ro = 0.05 cm. the ratio of the analysis time parameters is 29.
Gas chromatography Liquid chromatography laminar flow Turbulent flow r o = 0.1 em. ro = 0.01 em.
This discrepancy may be ascribed to the fact that the calculated plate heights used have been obtained from the calculated data in Figure 2 (curve b) which is approximately a factor of 3 higher than the corrected value (curve c ) at Re = 6 X 10'. A consequence of the difference between the kinematic viscosities of gases and liquids-i.e. a factor of 15- is that C , for liquid chromatography will always be greater than CM for gas chromatography by the same factor. As a result Cs in liquid chromatography can be made about 15 times larger than CS in gas chromatography, if the same relative contribution of the C M and CSterms to the plate height is maintained. For this reason it may be easier in liquid chromatography to devise column coatings, which fulfill the condition k = 1, than in gas chromatography. It follows from the data in Figure 2 (curves a and
Table II. Typical Flow Rates and Values of Analysis Time Parameter for Liquid Chromatography in Laminar and Turbulent Regions at Various Values of Column Radius
Laminar
/
ro, cm.
U,,cm./sec.
n ni
0.06
0.1
O.OO(
0.2
0.00' 0.002 0.001
0.3 0.5
Turbulent
sec. 1.51 X 10' 3.78 X 10' I,
U,,cm./sec. 3,000 600 375
T,
sec.
2.80 X lo-* 7.00 x 10-2 1.80 X 10-l
VOL. 38, NO. 2, FEBRUARY 1966
279
and the discussion on the role of the column length in gas chromatography, that, for a similar separation under similar conditions, the column length in liquid chromatography will be approximately 10 times longer in the turbulent region than in the laminar region. Pressure Drop across Column. Typical values of the pressure drop across the column which could be expected in practice are shown in Figure 7 . They have been calculated by the methods outlined. c),
LIQUID-SOLID CHROMATOGRAPHY
In liquid chromatography the role of the stationary phase resistance to mass transfer terms is relatively minor for high speed analysis, since it would, in practice, be difficult to reduce the mobile phase resistance to mass transfer term to values approaching that of the stationary phase. Any of the advantages which may be obtained by using gassolid in preference to gas-liquid chromatography would fall away for liquid chromatography. GENERAL DISCUSSION
At this stage it is interesting to look a t the relationship between the separation factor, CY, of a two-component mixture (in a complex mixture CY would refer to the two components most difficult to separate) and the analysis time parameter, column length, column radius, and pressure drop. This information is shown in Figures 6, 7 , and 8 for = 1, K = 100 for liquids, and K = 1000 for gases, since these would be the most interesting values. Although some of the values are only of academic interest, they have been included to illustrate what is possible and impossible in practice, and to compare laminar and turbulent flow chromatography over a wide range. As an example let us assume that two substances, which can be handled by both gas and liquid chromatography, are to be separated and that their separation factor in both systems is 1.5. In practice AP,T, L, or even r, could be the limiting factor; let us assume that AP is limited to 102 atm. From Figure 7 it may be seen that the chromatographic systems represented by the curves which intersect dashed line A B may be considered. Since the pressure drop is proportional to r,2 and the analysis time to ro, turbulent flow gas chromatography may be brought into region A B by very slightly increasing the column radius from 0.01 to 0.011 cm. With these values and the data in Figure 6, a t CY = 1.5, it follows that the fastest analysis time (3 X lo-* second) would be obtained by employing turbulent flow gas chromatography with ro = 0.011 cm. The necessary column length from Figure 8 is 90 cm. The next shortest 280
ANALYTICAL CHEMISTRY
analysis time (7.9 X 10-l second) could be obtained by using laminar flow gas chromatography in a column ro = 0.01 em. and L = 80 cm. If, for the hypothetical pair of substances we are considering, the value CY = 1.5, which we have assumed for gas chromatography, can be increased by employing liquid chromatography, it is possible to obtain a situation where turbulent flow liquid chromatography would lead to approximately the same analysis time as in turbulent flow gas chromatography. In the example used here this would occur for CY = 4 and ro = 0.1 cm. The pressure drop required would be less than 1 atm. and the length less than 100 cm. If thermally labile compounds must be separated, as is often the case with large organic molecules, gas chromatography can be unsuitable. Assume again that the CY value for a solute pair of this type is 1.5. If a pressure drop of about 12 atm. can be attained, it may be seen from Figures 7 and 8 that turbulent flow liquid chromatography in a tube r, = 0.1 cm. and L = 2000 cm. can be employed, and that an analysis time as short as 100 seconds can be achieved. For the same separation by laminar liquid chromatography in a tube of the same dimensions the pressure drop is negligible, the column length is about 1000 cm., and the analysis time is 9 X lo5seconds = 600 hours = 24 days. In all the preceding discussion the relatively stringent condition R = 1.5 has been employed. In practice a value of R = 0.75 can still be analytically useful. Under such conditions the calculated analysis times, column lengths, and pressure drops would be about a factor of 4 less than those pertaining to R = 1.5. CONCLUSIONS
Turbulent flow chromatography in open tubes can, in general, lead to lower analysis times than can be attained by laminar flow. I n the case of gases the improvement would rarely exceed a factor of 10; in liquids the decrease in analysis time can be so large-i.e., by a factor of l O L t h a t it could mean the difference between feasible and impractical separations from the point of view of analysis time. Fast analysis by turbulent chromatography is achieved a t a cost, which, in terms of column length, is not excessive, since it amounts to a factor of 2 in comparison with laminar gas chromatography; in turbulent liquid chromatography columns would be about 10 times longer than in laminar chromatography. The cost in terms of pressure can be relatively high. In general, pressure drops of several atmospheres have to be faced for simpler separations, and more difficult ones could imply pressure drops of more than 100 atm. At first sight such pressures might appear frightening to chromatog-
raphers, but they are by no means excessive by modern standards. Fortunately, several possibilities, despite the high pressures sometimes involved, make us optimistic about the future of high speed analysis by turbulent flow chromatography. High pressures must be used in turbulent flow chromatography because the flow rates must exceed a minimum value in order to effect turbulence. The beneficial result accruing from turbulence is primarily to increase radial mass transport and, to a lesser extent, to flatten the velocity profile. If, however, the velocity profile can be flattened by other means, turbulence, and consequently high pressure drops, become unnecessary. Several possible ways of doing this are being explored. NOMENCLATURE
C
mass transfer resistance concentration of solute in mobile phase D = reference dispersion coefficient; may, in general, include both eddy and molecular diffusion = radial average of D(r) b = dispersion coefficient for mass D" transport by molecular mechanism D ( r ) = dispersion coefficient a t r D"r) = dispersion coefficient for mass transport by turbulent convection a t r df = thickness of stationary liquid layer I" = shear stress fs,f,w = functions for pressure correction H = plate height h = reduced plate height I,, I,,, Z12, Z13 = integrals defined by Equations 4, 12 to 14 K = concentration distribution coefficient k = mass distribution coefficient L = column length -1.1, = Moody friction factor ( M I = 4 x Fanning friction factor) = pressure drop across column AP = peak resolution of solute pair R c
= =
Re = 3 '!Jm = Reynolds number r = radial coordinate ro = radius of open tubular column r D = r/ro = dimensionless radial coordinate Sc = y m / D m = Schmidt number T = t,ime necessary for a given separation t = time, sec. Q = radial averag- of u(r) aa = fszi = longitudinal average of u X(r) = flow velocity a t r V = retention volume w = break width (w = 4 X standard deviation)
GREEKLETTERS a = separation factor p = kinematic viscosity for mo-
mentum transfer by molecular mechanism vf = kinematic viscosity for momentum transfer by turbulent convective mechanism p = density 7 = analysis time parameter +(r) = function defined by Equation 7 #(r) = function defined by Equation 7
SUBSCRIPTS ill = mobile phase S = stationary phase I = first eluted solute I1 = second eluted solute LITERATURE CITED
(1) Aris, R., Proc. Roy. SOC.A235, 67 (1956). (2) Zbid., A252, 538 (1959). (3) Bird, R. B., Steward, W. E., Light-
foot, E. N., “Transport Phenomena,” Chap. 5, 6, Wiley, New York, 1960. (4) Zbid., p. 629. (5) Desty, D. H., Goldup, A., “Gas Chromatography 1960,” p. 162, Butterworths, London, 1960. (6) Desty, D. H., Goldup, D. G. F., Whyman, J., J . Znst. Petrol. 45, 287 (1959). (7) Giddings, J. C., ANAL.CHEM.35, 439 (1963). (8) Giddings, J. C., Seager, S. L., Stucki, L. R., Stewart, G. H., Zbid., 32, 867 (1960). (9) Golay, M. J. E., “Gas Chromatography,” D. H. Desty, ed., p. 36, Butterworths, London, 1958. (10) Hofmann, H., Chem. Eng. Sci. 14, 193 (1961). (11) Levenspiel, O., Ibid., 17, 575 (1962). (12) Purnell, J. H., Ann. N . Y . Acad. SCZ. 12, 592 (1959). (13) Purnell, J. H., “Gas Chromatog-.
raphy,” p. 152, Wiley, New York, London, 1962. (14) Purnell, J. H., J . Chem. SOC.1960, 1268. (1:) Scott, R. P. W., Hazeldean, G. S. F., Gas Chromatography 1960,” p. 144, Butterworths, London, 1960. (16) Stewart, G. H., Seager, S. L., Giddings, J. C., ANAL. CHEM.31, 1738 (1959). (17) Taylor, Geoffrey, Proc. Roy. SOC. 219A, 186 (1953). (18) Zbid., 233A, 446 (1954). (19) Tichacek, L. J., Barkelew, C. H., Baron, T., A.Z.Ch.E. J . 3, 439 (1957). (20) Zlatkis, A., Walker, J. Q., ANAL. CHEM.35, 1359 (1963).
-
I
RECEIVEDfor review July 8, 1965. Accepted November 29, 1965. Third International Symposium on Advances in Gas Chromatography, Houston, Texas, October 1965.
Aerogel Columns in Gas Chromatography ISTVA” HALASZ and HANS-OTTO GERLACH lnstitut fh Physikalische Chemie der Universitat, Frankfurt am Main, Germany
b Aerogel columns were produced by drawing out to 0.4-mm. i.d. glass tubes loosely packed with highly dispersed silica having a particle size of about 1 micrometer. Although the particles appear to be closely packed, the packing is highly dispersed and only 2 to 470 of the total volume is occupied by the stationary phase. The gas permeability of the aerogel columns was somewhat lower than that of the conventional packed columns, but minimum values of 0.01 2 cm. for h and 0.035 cm. for H were achieved with high carrier gas velocities. Fast analyses within seconds were carried out. The aerogel columns are compared with other column types and some analytical applications for the separation of C1 to Ce paraffins and olefins are demonstrated.
P
broadening in gas chromatography is usually characterized by the h or H values, where h = L w2/16 1 2 and H = L w2/16 ( t t R ) 2 . Previous experiments have shown that in very rough approximation the relative peak broadening ( h or H ) in packed columns decreases linearly with decreasing particle size, d,, of the support (1, 7 , 8, 11,18-20). But greater efficiency of the column with decreasing particle size must be paid for by increasing pressure drop of the carrier gas. The Kozeny-Carman equation (2, 5 ) shows that specific gas permeability, K ( l d ) , is proportional to d,Z: EAK
where E is the interparticle porosityLe., the fraction of column cross section available to moving gases :
w
e = l - L
VCP
(2)
where T1’, is the total weight of the support in the column, V , is the total volume of the column, and p is the apparent density of the support measured by mercury displacement. Equation 1 shows and experiments on regularly packed gas chromatographic columns have proved ($1) that permeability is independent of the inner diameter of the column. In classical packed columns the interparticle porosity is practically independent ( E = 0.40 to 0.41) of the mesh size (163 to 545) of the support, although the total porosity increases from 0.76 to 0.80 (6) *
Since in classical packed columns the interparticle porosity, E , is practically constant, the permeability, K , is proportional to the square of the particle size, the pressure drop, A p , is limited to 4 atm. on account of technical reasons, and the relative peak broadening decreases with decreasing particle size, compromise must be made in choosing the particle size. Values of d, N 0.10 to 0.15 mm. have proved to be most favorable. Equation 1 shows that a smaller particle size might be chosen without lowering the permeability, if the decrease in d, could be compensated for by increasing the interparticle porosity, E . Column efficiency could be further improved by increasing the velocity of
obtaining equilibrium between the moving and the stationary phases. Equilibrium can be the more rapidly attained, the closer the contact between the stationary and gas phase- ke., the better the stationary phase is “dispersed” in the gas phase. ITsually particles as spherical as possible are employed in the preparation of packed columns, in order to achieve uniform packing and gas flow through the column. But spherical particles-especially if they are poreless-are unfavorable on account of their minimum specific surface area a t maximum packing. On the other hand (15, 16) under certain circumstances irregularly packed columns*.g., packed capillary columns-are more efficient than regularly packed columns. In view of these facts we tried to find a fibrous material which combines maximum surface area with minimum packing volume. EXPERIMENTAL
Chromatographic Apparatus. The gas chromatographic apparatus was similar to that described (3). The flame ionization detector, FID, was combined with a recorder (full scale time 0.3 second) or with a high speed ultraviolet galvanometer recorder. It was possible to record undistorted peaks as narrow as 0.2 second. The outlet pressure, p,, of the column was atmospheric in all experiments. The gas holdup values, to, were determined with methane. Methane, instead of helium, could be used for this purpose, sipce by means of a microthermal conductivity cell it was found that methane at 25’ C. behaves VOL. 38, NO. 2, FEBRUARY 1966
281
