Anal. Chem. 1982, 5 4 , 1533-1540
significant difference in the extracted amounts of each ion, and probably more importantly, no uniform trend is evident in the data. For a much larger data set (on the order of 100 samples) most of the variability in the averages would presumably disappear. It can[certainly be expected that the range in the amounts of ionic contaminants actually present on surfaces among various locations will typically be much greater than any variations caused by pH effects. ACKNOWLEDGMENT The author is indebted to P. C. Milner, G. B. Munier, L. A. Psota-Kelty, C. A. Russell, and C. J. Weschler for very helpful discussions spanning several years. LITERATURE CITED (1) Hermance, H. W.; Russell, C. A.; Bauer, E. J.; Egan, T. F.; Wadlow, H. V. Envlron. Scl. Techno/. 1971, 5 , 781-785.
1533
Munier, G. B.; Psota, L. A.; Reagor, B. T.; Russiello, 6.; Sinclair, J. D. J . Electrochem. SOC. 1980, 2 . 265-272. Munler, G. 6.; Psota, L. A,; Sinclair, J. D. Paper 179, presented at The Electrochemical Society Meeting, Hollywood, FL, Oct 5-10, 1980. Munler, G. 6.; Psota, L. A.; Sinclair, J. D. Paper presented to the Corrosion Symposium, 28th Congress of the International Union of Pure and Applled Chemistry, Vancover, Brltish Columbia, Canada, Aug 16-22, 1961. Slnclair, J. D. J . Nectrochem. SOC. 1978, 725,734-742. Flegl, Fritz ”Spot Tests in Inorganlc Analysis”, 5th ed., translated by R. E. Oesper; Elsevier: New York, 1958. Moody, G. J.; Thomas, J. D. R. “Selective Ion Sensitive Electrodes” Merrow Publlshlng Co.: Watford Herts, England, 1971. Sawicki, E.; Mulik, J. D.; Wlttgenstein, E. “Ion Chromatographic Analysls of Environmental Pollutants”; Ann Arbor Science: Ann Arbor, MI, 1978. Blaedel, W. J.; Meloche, V. W. “Elementary Quantitative AnalyslsTheory and Practice”; Harper and Row: New York, 1963. Frledlander, S. K. “Smoke, Dust and Haze-Fundamentals of Aerosol Behavlor”; Wlley: New York, 1977.
RECEIVED for review December 2,1981. Accepted May 6,1982.
Influence of Retention on Band Broadening in Turbulent-Flow Liquid and Gas Chromatography Michel Martin and Gealrges Guiochon” Laboratoire de Chimie Anaiyflque Physique, Ecole Polytechnique, 9 1 128 Palaiseau, France
Solute peak broadening in1 caplliary chromatographiccolumns (GC and LC) operated under turbulent-flow conditions is calculated according to the, Aris general dispersion theory. I t Is shown that the plate height increases greatly wlth lncreasing solute capacity factor, mainly because of the strong k’ dependence of the contribution of mobile-phase masstransfer contrlbutlon, although, In GC the stationary-phase contribution Is not negligible. The experimental data of Doue and Gulochon compare favorably with theory. I t is proposed that the mobile-phase nonldeaiity Is malnly responslble for the 1 order of magnitude dlaicrepancy between the calculated plate-helght values and thw experimental data of Giddings et ai. The potential of turbulence appears to be limited to very fast analyses wlth columns of low or moderate efficiencies.
In order to separate chromatographically a large number of components, one requires a large plate number. Such a search for high efficiencieri has led to the development of new technologies, including narrow diameter open or packed capillary columns in gas chromatography (GC) and in liquid chromatography (LC) and microbore columns or segmentedflow capillary columns in LC. All of these approaches are now under active study; however none has yet to be shown to be superior to the others in achieving extremely large plate counts. The utility of turbulent flow was advocated 15 years ago for very high-efficiency capillary chromatographic columns. This stems from two observations: first, in the turbulent-flow core, the velocity profile is largely flattened, thus decreasing flow inequalities (I), and, second, the effective diffusion coefficient of the solute is increased considerably by the formation of eddies which quickly relax the concentration gradients resulting from these flow inequalities ( 2 ) . As a consequence, peak broadening arising in the mobile phase as a result of the velocity profile (at times called Taylor dis-
persion) is largely expected to be reduced. This has in fact been observed in chemical engineering studies of gas as well as liquid flow in pipes. The use of chromatographic columns under turbulent-flow conditions was, therefore, considered as an attractive means of achieving high efficiencies. Such an advance would be especially welcomed in LC, as theory predicts (in agreement with experimental results) that due to the slow rate at which molecular diffusion relaxes concentration gradients generated in a Poiseuille velocity profile, only columns with a diameter of less than a few micrometers are competitive with conventional packed columns using 3-5 pm particles. Broader columns would be much easier to prepare and operate if, through the use of turbulent flow, plate heights of retained compounds smaller than the column diameter could be achieved. In order to evaluate the potential of turbulent-flow chromatography, one needs a quantitative theory of peak broadening under such conditions. Obtaining such a description is obviously more complicated than that for laminar flow because of the extreme complexity of the flow profile in the former case. Still,the degree of complexity is higher in packed columns than in open-tube capillary columns for which the geometry is rigorously known. The determination of peak broadening in capillary chromatographic tubes operated in turbulent flow conditions has been described by Pretorius and Smuts (3) in the two cases corresponding to capacity factors, k’, of 0 and 1,respectively, using the framework of the Aris general dispersion theory ( 4 ) which allows for a nonparabolic flow profile and a variable diffusion coefficient. They found higher plate height values for k’ = 1than for k’ = 0. The few experiments made with gas chromatographic capillary columns under conditions of turbulent flow have also shown a significant increase in plate heights with increasing capacity factors ( 5 , 6 ) . Such an increase is also observed in capillary columns operated in laminar flow conditions and is accounted for by the Golay theory (7). However, this theory, which assumes a steady parabolic flow profile, cannot be applied to
0003-2700/82/0354-1533$01.25/00 1982 Amerlcan Chemical Society
1534
ANALYTICAL CHEMISTRY, VOL. 54, NO. 9, AUGUST 1982
turbulent flow. In the following, the Aris dispersion theory is used for the determination of the influence of the capacity factor on the plate height in gas chromatography (GC) as well as in liquid chromatography (LC). Because the theoretical calculations are more easily carried out when the stationary phase is a liquid, it is assumed in what follows that the situation corresponds to one of fluid-liquid chromatography. Moreover, only open-tubular columns are considered since the pressure drop necessary to obtain turbulent flow in packed columns is too large for practical consideration.
THEORY According to the theory of peak broadening in cylindrical open-tubular columns, fiisbgiven by Golay (7)for laminar flow and then extended by Arb to any type of flow (4),the apparent plate height is the sum of three terms: mobile-phase longitudinal diffusion, mass-transfer resistance in the mobile phase, and mass-transfer resistance in the stationary phase. This treatment thus neglects contributions from axial diffusion in the stationary phase (usually very small), and from interfacial resistance which is commonly supposed to be small and which is almost impossible to take into account in the present state of development of the field (8,9).With the reduced parameters h and v defined by h = H/d, (1) the apparent plate height, L(ut/t# can be written
where f and j are dimensionless compressibility parameters [taken to be equal to 1 in LC for which the mobile phase is supposed incompressible (10) and as given by the classical expressions in GC ( I I ) ] . It can be shown that, as long the gas phase is ideal, these laminar-flow expressions are valid whatever the type of flow (12). In laminar flow, D, is simply the molecular diffusion coefficient D," at the column outlet pressure. In turbulent flow, the instantaneous values of the velocity fluctuate wildly about their mean value (13). Therefore, u and D, in this case represent not only crosssectional averages but also time-smoothed values of the mobile-phase velocity and the solute diffusion coefficient in the mobile phase, respectively, at the column outlet. In laminar flow, C, is given by the Golay theory on the basis of a parabolic flow profile as llkI2 c, = 1 9 66k' (4) 0 + kq2
+
+
According to the Aris dispersion theory ( 4 ) , C, is given in turbulent flow as
cm-Dmo = Dm
and
A,
+ Alk' + A2kl2 (1 + k12
(5)
1
I,=
x3
dx
J21c/o
@ ( x ) = S,'2x'r$(x') dx'
(12)
In these expressions, x is the dimensionless radial coordinate defined by x = 2r/d,
(13)
where r is the distance from the column axis. 4 ( x ) and + ( x ) are the dimensionless radial flow and diffusion profiles, respectively. In order to evaluate the coefficients A for turbulent flow, it is necessary to have expressions for r$(x) and +(x). Pretorius and Smuts (3) used the flow profiles determined empirically by Tichacek et al. (14) at a few values of the Reynolds number in the turbulent regime to calculate numerically @ ( x ) in eq 12. Then, assuming that the Reynolds analogy is valid, i.e., that the transfer of matter, heat, and momentum by the turbulence are exactly analogous (and, therefore, that the turbulent diffusion coefficient is equal to the turbulent kinematic viscosity), they derived an expression for the diffusion-coefficientprofiie in terms of the Moody friction fador, which is equal to four times the classical Fanning friction factor. This expression also depends on the velocity gradient profile d$(x)/dx. Because the exact shape of the velocity profiie is not rigorously known in turbulent flow and since the velocity gradient profile is strongly dependent on the precise shape of the flow profile (especiallyin the vicinity of the wall), determination of the diffusion-coefficient profile necessary to compute the integrals I is prone to quite significant errors. Moreover, with the data of Tichacek et al., these computations can only be made at Reynolds numbers for which the flow profiles have been determined. For these two reasons, it is preferable to have analytical expressions for the radial flow and diffusion profiles. For turbulent flow, it is generally assumed that a good approximation of the velocity profile is given by the 1/7 power law for Reynolds numbers lower than lo5 (13) 117
which gives
and, from eq 12 @ ( x ) = 1-
c
-x
+ 1)(1-
x)8/7
As noted previously, radial transport of the solute in the mobile phase is greatly increased by eddies in turbulent flow. However, in that case, molecular diffusion takes place within and between the eddies so that the total transport is the result both of molecular motion and of turbulent mixing. Therefore, it is commonly assumed that the total diffusion coefficient, D, is the s u m of a molecular part, D,', and of a turbulent part, DT (15) D = D," + DT (17) The molecular diffusion coefficient is constant and does not depend on the radial coordinate. The turbulent diffusion coefficient is usually much larger than Dm0and does depend on the radial coordinate. According to the Reynolds analogy, the radial profiles of the turbulent diffusion coefficient, the turbulent (or eddy) viscosity, and the turbulent (or eddy) thermal conductivity should be identical. Even if the absolute
ANALYTICAL CHEMISTRY, VOL. 54, NO. 9, AUGUST 1982
values of these coefficients (especially the first two) are not exactly identical (15),the similarity of the radial profiles has been established from experimental data obtained for tubes as well as for flat plate channels (15-21). These profiles are characterized by a minimum at the center of the tube, a maximum around x = 0.51, and a zero value at the point of contact with the wall. In order to formulate an analytical expression for the turbullent diffusion-coefficient profile, it was decided to use the simplest polynomial expression which gives the boundary conditions: zero at x = 1; zero derivative at x = 0 and at x = 0.5. This derivative is therefore proportional to x (1 - 2x) aind the diffusion coefficient is proportional to DT
m
1 + 3 x 2 - 4x3
It is currently supposed tlhat turbulence starts to set in at Re > 2100. Linear regression analysis of the turbulent diffusion data obtained in tubes (ref 20,21) for x = 0.5 at 11 different values of Re gives
- 1.906
(20)
Therefore, combination of eq 18 and 20 for x = 0.5 gives the turbulent diffusion coefficient profile as
+ 3x2 - 4x3)
DT/pmo = 0.01Re0.84(1
(21)
It is remarkable that this dependence of DTon Re is close to the expected theoretical relationship (DT 0: Re7/8) and is identical in one instance with that measured experimentally (22).The relative diffusion-coefficient profile +(x), which is required for evaluation of the integrals, I, is then obtained by combination of eq 17, 18 and 21
- +
+
+(x) = D(r) -- 1
0.01Fle0.84Sc(l 3 x 2 - 4x3) Dmo where Sc is the Schmidt number, defined as
SC= pmo/Dmo = qmo/ (pDmo)
(22)
(23)
Typical values of Sc range from 0.5 to 5 for gases and from 250 to 2500 for liquids. The average cross-sectional value of the diffusion coefficient is next obtained after integration of eq 22
D,/Dmo =I 1 + 0.009Re0%c
(24)
Noting that the mobilephase reduced velocity u is related to the Reynolds and Schmidt numbers according to eq 2,19, and 23 through u = Re Sc (25) it can be said, according to eq 3 and 24, that the axial diffusion contribution, h ~to,the plate height can be split into two parts, one due to molecular diffiusion and the other to turbulent diffusion
hA =
(--- + 2 IZeSc
Table I. Variations of the A Coefficients with Re for Sc = 2.6 (GC)
4000 10000 20000
19.53 x 9.23 x 5.20 x
lo-$
26.5 27.1 27.5
326.7 380.9 422.7
88.6 102.3 112.8
Table 11. Variations of the A Coefficients with Re for Sc = 500 (LC)
(18)
Although only an approximation, the expression gives a value of 1.25 for the ratio DT(x = 0.5)/D~(x= 0) which is in good agreement with experiment. Further, while the equation gives only a relative indication of the profile, the absolute value of DT is observed to be strongly dependent on the Reynolds number, Re, defined as
log (DT/pmo) := 0.842 log Re
1535
-)f 0.018 Re0.16
It appears, therefore, that for turbulent flow in a capillary chromatographic tube the reduced plate height depends only
4000 10000 20000
9.81 X 4.55 X 2.54 X
lo-* loe8
28.2 28.2 28.2
739.1 796.6 842.6
192.1 206.5 218.0
on the Reynolds and Schmidt numbers, the capacity factor k’, and the term (D,o/D,o)(df/d,)2 (assuming that the compressibility factors f and j are close to 1 as is the case in that follows).
RESULTS The computations have been carried out in two steps. First, eq 6-11, 16, and 22 were used to evaluate numerically the coefficients A for each value of Re and Sc; the plate height was then calculated according to eq 3 and 5-8. Calculations were made for 10 values of Re between 2000 to 20000 and 11 values of Sc between 1 and 2500. While they could have been made for higher values of Re, it is unlikely that such values could be reached in practice because of the associated high pressures. Indeed, under turbulent-flow conditions, the pressure drop, -dP/dz, is proportional to Re7f4while, in laminar flow, it increases only as Re (13). While the formation of eddies is expected to start when the Reynolds number reaches about 2100, the calculations involving expressionsvalid in the turbulent flow regime have been extended down to Re = 2000. It is interesting to note first the values of the coefficients A in eq 5. We report in Table I and I1 for Schmidt numbers of 2.5 and 500 (typical of GC and LC, respectively) and three Reynolds numbers, the values of Ao, the relative values of AI and A2 to Ao, and the ratios of the term CmDm0/Dmat k’= 1 to the corresponding term at k’ = 0. The values in Tables I and I1 can be compared with those obtained in laminar flow conditions, for which A. = 1/96 = 1.04 X A1/Ao = 6, A2/Ao = 11, and Cm(k’= l)/C,(k’ = 0) = 4.5. It can be seen that the values of the coefficient A0 in turbulent flow are, in GC, about 3 orders of magnitude and, in LC, 5 to 6 orders of magnitude smaller than the corresponding values in laminar flow. This is a consequence of the large increase in radial mixing induced by eddies as well as by flattening of the velocity profile. Moreover, A. decreases with increasing Re; however the values of AoRe increase slightly with increasing Re. Therefore, the contribution of mobile-phase mass-transfer resistance to the plate height will increase slightly with increasing Re. This may appear surprising as the turbulent plate height in retentionless pipes generally is expected to decrease slightly or to remain constant with increasing Re (3,6,14, 23-27). These discrepancies probably arise because the plate height is strongly dependent on the shape of the flow profile near the wall and this profile is not known with the required precision. In the present calculations, it was found that whatever the value of Sc, the plate height term H, does not vary by more than 35% with Re over in the range of Re values considered here. The slight change in H, with Re is in any event far less significant than that with k .’
1536
ANALYTICAL CHEMISTRY, VOL. 54, NO. 9, AUGUST 1982
Table 111. Plate Height and Relative Contribution to Plate Height of the Resistance to Mass Transfer in the Stationary Phase for Sc = 2.5 (GC) Re = 2000 Re = 10000 Re = 20000 k’
4000
8000
12000
16000
Re
Flgure 1. Variations of the reduced plate height with the Reynolds number in GC (Sc = 2.5) at k ’ = 0.1, 0.5, 1, 5, and 10.
0 0.01 0.05 0.1 0.25 0.5 1 2.5 5 10 25
h
hsl h(%)
h
hsl h(%)
h
hsl h(%)
0.174 0.544 1.97 3.66 8.10 13.9 21.7 32.7 39.2 43.5 46.6
0 60.5 76.6 75.3 65.9 53.3 38.4 20.8 11.8 6.3 2.6
0.235 1.93 8.25 15.2 31.3 48.3 65.3 80.1 85.0 86.9 87.6
0 84.6 91.6 90.6 85.1 76.7 63.8 42.4 27.2 15.8 7.0
0.264 3.60 15.9 29.3 59.0 88.0 112 126 124 119 114
0 90.7 94.9 94.1 90.3 84.2 73.9 54.1 37.4 23.1 10.8
-
Table IV. Plate Height and Relative Contribution to Plate Height of the Resistance to Mass Transfer in the Stationary Phase for Sc = 500 (LC) Re = 2000 Re = 10000 Re = 20000 k’
4000
8000
12000
16000
Re
Flgure 2. Variations of the reduced plate height with the Reynolds number in LC (Sc = 500) at k‘ = 0.1, 0.5, 1, 5, and 10.
Noteworthy in Tables I and I1 is that the relative coefficient Al/Ao is approximately independent of the values of Re and Sc and is equal to 27-28. This is more than four times larger than the corresponding value in laminar flow, which implies that the capacity factor will have a large influence on H in turbulent flow. This is reinforced by the A2/Ao values in the Tables, which increase slightly with increasing Re and which depend markedly on Sc. Significantly,it is about 30-80 times larger than its value in laminar flow. In turbulent flow, therefore, as long as resistance to mass transfer in the mobile phase is the predominant contribution to peak broadening, the plate height is expected to depend strongly on the capacity factor. The variation of h with Re is shown plotted in Figures 1 and 2,those corresponding to Schmidt numbers of 2.5 (GC) and 500 (LC), respectively, a t five different values of k’ (0.1, 0.5, 1, 5,and 10). For these qalculations (as in those which follow), in order to compute the contribution of stationaryphaae resistance to mass transfer, the ratio dJd, of stationary film thickness to tube diameter has been taken equal to 0.001 (typical of present-day work in capillary columns) and the diffusion coefficient ratio D,“/D,O equal to loo00 in GC and 5 in LC. For all curves in Figures 1 and 2,the plate height increases with increasing Re or u. This reflects the fact that the contribution of axial diffusion to the plate height, which decreases with increasing Re, is small. Indeed, in the case of k’ = 0, when this contribution is the largest, it amounts only to about 3% in GC and in LC at Re = 2000. In this case (k’ = 0), the plate height increases slightly with increasing Re, as a consequenceof the variation of with Re, but remains nevertheless remarkably small. Indeed, over the range of Re of 2000 to 20000,h varies from 0.18 to 0.26 for both values of Sc, which is quite similar to (while slightly lower than) the result of the theoretical analysis of Taylor (23,27). While experimental values of h for turbulent flow in pipes have been found to be somewhat larger (27),they are neveEtheless low
0 0.01 0.05 0.1 0.25 0.5
1 2.5 5 10 25
h
hsl h(%)
h
hsl h(%)
0.181 0 0.231 0 0.271 12.3 0.471 35.0 0.815 18.6 1.67’ 45.4 1.84 15.0 3.59 38.3 8.5 11.1 6.31 24.1 15.5 4.8 25.4 14.6 32.5 2.6 51.1 8.2 1.1 97.1 63.8 3.5 85.7 0.5 129 1.8 101 0.3 152 0.9 113 0.1 168 0.4
h
hsl h(%)
0.256 0.670 2.56 5.33 15.2 32.9 63.7 117 154 180 199
0 49.0 59.2 51.7 35.1 22.5 13.1 5.8 3.0 1.5 0.6
enough to stimulate interest in terms of chromatographic applications. The curves for k’ = 0 are not plotted in Figures 1 and 2 because they are nearly coincident with the abscissa. The rate of variation of h with Re is higher for all curves of Figure 1 than for the corresponding curves of Figure 2. This is related to the fact that for a given k’, the relative contribution of stationary-phasemass-transferresistance to the plate height is higher in GC than in LC, essentially because the values of (D,O/D,O)Sc (see eq 3 and 25) are larger in the former case. For both Sc values this relative contribution increases as Re increases and decreases with increasing k’for k ’larger than about 0.05. These relative contributions, h,/h, as well as the values of h as functions of Re and k ’for Sc = 2.5 and Sc = 500, respectively, are reported in Tables I11 and IV. It can be seen from these data that the plate-height values at k’= 1 are of the same order of magnitude as those obtained theoretically by Pretorius and Smuts (3). Figures 1 and 2 (as well as Tables I11 and IV) show a large increase of h with increasing k‘. This is a consequence of the large values of the ratios of the coefficients A , / A , and, especially, A2/Ao,as well as of the important contribution of stationary-phase mass transfer to peak broadening (which is more pronounced in GC than in LC). The variation of h with k ’is plotted in Figure 3 at Re = 10 000 for the two values of Sc, 2.5 and 500. The curves increase continuously with increasing k’. In LC, the rate of increase of h with k’reaches a maximum, dhldk’ = 57.7,at k’= 0.4. The GC curve tends to plateau at h N 80 for k’ larger than about 2. While not apparent in the range of k ’of Figure 3,the h vs. k ’curve in GC has a maximum which, at Re = 20000, is close to k’ = 2.5 as can be seen from the data of Table 111. This explains why, in Figure 1,the curve for k ’ = 5 intersects the curve for k ’ = 10 around Re = 13000. Such a maximum in h can be ex-
ANALYTICAL CHEMISTRY, VOL. 54, NO. 9, AUGUST 1982
Table V. Comparison of Experimental Data of Figures 6 and 7 of Fi$ure 6 O1 n -pentane n -hexane SC 1.25 1.38 k‘ 1.18 2.08 2.87 x 104 2.74 x 104 Dom/Dos 33.7 (hrn + h k l f 21.8 3.4 3.9 h,j hcalcd 25.2 37.6 hexptl 43.9 50.1 hexptl/’Clcalcd 1.74 1.33 a
Re = 6300;f = 1.092:,j=0.314.
1537
Reference 6 with Calculated Plate Height Values Figure 7 _n-heptane n-pentane n-hexane n-heptane 1.53 1.26 1.40 1.51 0.31 0.80 3.81 0.12 2.73 x 104 46.6 1.05 3.31 10.2 3.2 49.8 1.05 3.31 10.2 95.1 3.4 8.7 30.5 1.91 3.24 2.63 2.99 _ I .
Re = 2010;f = 1.097;j = 0.285.
I -
160 120
h 80
40
2
6
4
8
k’ Figure 3. VarlatllDns of the reduced plate height with the capacity factor at Re = 10000 for Sc = 2.5 (GC)and Sc = 500 (LC).
plained by the fact that, in GC, the contribution to the plate height due to stationary-phase mass transfer is important at high Reynolds numbers and that this contribution (according to eq 3) reaches a maximum at k’ = 1. No maximum of h vs. k’is observed in1 LC. In that case, the major contribution to peak broadening comes from mobile-phase mass transfer, which is larger in LC than in GC for high k’values. Indeed, the A2/Aoterm is about twice as large in LC for which the smaller value of A. is comlpensated by the higher value of Sc (cf. Tables I and 11). Ultimately, in GC as well as in LC, h, reaches a limiting value, A,ReSc, for large k’values which is equal to 55 in GC and 181 in LC a t Re = 10000. These values of the reduced plate height are very large compared to those currently achieved with open-tubular columns in laminar flow (01.3-1.5); use of such columns under turbulent-flow conditions does not, therefore, offer much in the way of improvement in system efficiency. COMPARISON WITH PUBLISHED EXPERIMENTAL DATA It is interesting to compare the plate heights calculated as above with experimental data. To our knowledge, only two papers dealing with capilLary gas chromatographic columns operated under turbulent-flow conditions have been published (5, 6) and none on the analogous situation in LC. Contrary to widespread opinion, turbulent flow is not necessarily associated with high column pressure drop. One can easily show (The derivations of eq 27 and subsequent equations are av
