Evidence on the Nature of Eddy Diffusion in Gas Chromatography From Inert (Nonsorbing) Column Data J. CALVIN GlDDlNGS Department o f Chemisfry, University of Utah, Salf lake City 72, Utah blnformation on the basic mechanisms of gas chromatography can advantageously be obtained from inert columns where the absence of sorption simplifies the results. A critical interpretation of some inert column data recently brought to light has thus been made. Both gas and liquid mobile phases have been used. These systems complement one another nicely, and permit the study of an enormous range of effective velocities. Of particular importance to the gas studies are the recent results obtained by Kieselbach on air peaks. The over-all results are in serious conflict with the classical theory of eddy diffusion, but lend positive support to the author’s coupling theory.
0
still controversial hubjects in the field of gas chromatography is that related to the effect of the microscopic flow pattern on plate height, or the so-called eddy diffusion effect (8, 14). The early theoretical discussions of this phenomenon (8,9 ) indicated that the plate height contribution was an additive quantity and constant for a given column-i.e., XE OF THE
H (eddy diff.)
= 2Xd,
(1)
nhere
x 2 1/2 This result will be called the classical equation for eddy diffusion (8). The term d, is the average particle diameter and X is a constant depending only on the geometry of the packing and thus independent of the carrier gas, temperature, flow velocity, etc. In 1959 this author proposed the coupling theory of eddy diffusion ( 7 ) . This theory indicates that the 2A d, term is coupled with mass transfer in the gas phase (the C,v term) to yield a term which is less than either 2Xd, or C,v alone (for a discussion of this theory, see the references). 111 its most recent form, still approximate, the coupling equation is written as a sum of terms (5) H (eddy dif1.j =
1338
1 - ~ _ _ ~ -
x 1 / 2 hid,
+ 1/C,,v
ANALYTICAL CHEMISTRY
~
For brevity we have called this H (eddy diff.), although it does incorporate the gas phase mass transfer term along with eddy diffusion. The mass transfer term must be added separately to Equation 1. The original equation involved the simpler but less realistic case in which only a single term appeared
H (eddy diff.) = 1/2 d, 1 1/C,v (3) The C, term in these expressions does
+
not include the contribution of stagnant (nonmoving) gas, since the latter contributes a separate, additive term to the plate height ( 5 ) . The bulk of experimental evidence bearing on the eddy diffusion controversy has been obtained from functioning gas chromatographic columns [this e1 idence has been summarized in a recent paper (S)]. While most of the recent data are in distinct disagreement with the classical theory of eddy diffusion, the results are not entirely conclusive. Perrett and Purnell (14, in fact, have concluded from their own data that the classical theory is valid. The contradictory nature of such data stems in part from the contribution of the partitioning process (the Cl term) which sometimes swamps out the other contributions and always obscures them to some extent. It has been clear for some time that data acquired from nonsorbing column-solute systems, where interference is at a minimum, would considerably advance our knowledge of the elementary flow processes leading to eddy diffusion. Despite certain experimental difficulties, such data have now been obtained by several workers (2, 8, 10, 12. IS). When combined with results from nonchromatographic studies, some important conclusions can be reached. The evidence presented in this paper falls into two categories: that obtained from the gas chromatography of air peaks, and that obtained from the study of dispersion in liquid systems. The latter has been almost entirely ignored by the proponents of gas chromatography. The laws of fluid dynamics are quite definite, however, in showirig tliet the flow pattern iz generally the same in gas and liquid flow systems at low Reynolds numbers.
One system can be scaled into another so that the combined effect of %ow and diffusion is the same. In this scaling process, one can reach very high effective gas flow velocities by using slowly flowing liquid systems because of the enormous difference between diffusion coefficients in the two systems. Essentially, then, the liquid data takes us into the very high flow regime where gas chromatography data is not available (the flow would be turbulent for gases at these velocities), and where the distinction between the classical and coupling theories is most pronounced. In the high velocity limit, the classical theory predicts that the C,u term, assumed to be additive with ZXd,, is predominant, and that the plate height is consequently proportional to flow velocity. The coupling theory, Equations 2 or 3, predicts that the plate height gradually approaches a constant value. The liquid data (17) is conclusive on this point; the effective diffusivity a t high flows is nearly proportional to flow rate, which means that the plate height is essentially independent of flow velocity. This not only indicates that the classical theory of eddy diffusion is incorrect, but it also provides definite support for the coupling theory. Some effects are still not understood completely, however,-i.e., some X values appear to be larger than those normally expected. The details of these liquid studies will be presented in a later section. In addition to the liquid data, both Kieselbach (11) and Knox (12) have recently acquired gas chromatographic air-peak data. These data (in contrast to the liquid data) are especially suited to the examination of inert peaks (or columns) in the vicinity of the plate height minimum. Knox has discussed his data in terms of the alternate theoretical approaches, and not much need be added to this discussion. He has found that use of the classical theory requires a A well below the minimum of 1/2 usually associated with that theory, and that use of the coupling theory requires an unusually large A value. The coupling effect, if present, was not readily discernible because of the la1ge A. The data acquired by Kieselbach, made available to the present author
before publication ( I I ) , show a definite indication of the coupling effect. Evidence for this resides in the consistent “flattening” of the Flate height-velocity curves a t high velocities. This evidence u-ill be discussed shortly. In addition to the above authorb, Sorem ( I S ) has also studied air peaks in packed columns. 3ince his data are not nearly as extensilre as Kieselbach’s (only two columns hiving been used), only slight reference will be made to this work. The implications of low velocity measurements have been studied insofar as eddy diffusion is concerned. These results are summarized elsen here (8). In comparing data from liquid and gaqeous systems, it is uqeful to use the rc,ducrd or dimensionless variables. h = H / d p (reduced plate height) and Y = vd,/D, (reduced velocity). The advantage of these variables iq that a plot of h vs. Y qhoiild remain essentially thr yam? even though the particle m e i5 varied or if one ures a diffcwnt gas or even a 1ic uid carrier. The plots nil1 vary t o some extent because of trans-column effwts and packing difference, but to a fir it approximation, if care is used, such as plot of data should yield a universal curve nearly applicable to all system>. This fact is illustrated by n riting the plate height expression, assuming coupling, for an inert column
Table 1.
Values of A, B, C, and E Obtained from Different Columns by Assuming Different Plate Height Equations Eqn. 11 Column Eqn. 9 Eqn. 10 A x 103 4.81 1.33 6.79 I, dry Chromosorb B X 10 8.30 8.51 8.26 c x 104 0.68 1.78 0.66 E x 107 ... -7.26 ... 11, dry A x 103 -3.94 -22.6 -2.20 Chromosorb B X 10 8.63 9.53 8.61 c x 104 2.40 8.38 2.48 E x 107 ... -38.0 ... 111, Siliclad B x 103 10.4 5.13 12.5 Chromosorb B X 10 7.73 8.00 7 .ti!) c x 104 1.02 2.57 1.07 E x 107 ... -8.64 ... IV, dry A x 103 2.84 2.96 4.36 Microbeads B x 10 S.26 8.25 8.24 c x 104 0.42 0.39 0.41 E x 107 ... 0.08 ... T, Siliclad .i x 103 I0.U 6.47 12.1 Chromosorb H x 10 7.0s 7 .:E 7.0.; c x 104 0.87 1.79 0. $141 E x 107 ... -3.87 ... 6.76 5.03 1.23 -4 x 103 VI, Siliclad 7. 73 7.93 7.71 Chromosorb B x 10 0.99 0.95 1.98 c x 104 ... -5.20 E X 107 VII, Siliclad A x 103 3.74 0.66 Chromosorb I3 x 10 8.42 8. 6(i 8.41 c x 104 1.23 2.44 l.% E X 10‘ ... -6.44 ..
+
+
+
- _-
+
r ) . d i
t o the classical esprccaion. Proceeding as abow, it is easy t o show that h
2X
+ 2r/v + wv
(8)
KIESELBACH’S DATA
(4)
nhert, the C,, of Equition 2 has been The mean rcplaccd 1)y ~,d,2/L’,. fli)w vvlocity (lcxngtli/tiinc) is u and the iricilrc*i~lardiffusion cncfficient iq D,,. \I7hvn thi? expre4or i j divided by d p , tlie follon ing is obtxined
Replacing the rcduced d p/Do,by Y , we have
velocity,
:LJI(I licwc~tlic valuc of h a t a
Y given ihiiild d q m d only on the conptants 7 , A,, and a i , and not on the variables rl,, and D ,. Thece cor stants 1%ill varv slightly from one sys em to another, but the change should not ordinarily be large. Trans-coluran effects, important in large-scale gas chromatography and perhaps in some liquid systems, have not been d u d e d in thiq equation. If they u w e included, h would lose its reduced form and become :I function of d, and nu. The coneq)t of redulxxd plate lieiylit mid reduccd velocity c a i also be applird
The results obtained by Kieselbach (11) were taken from straight, 1-meter glass columns with flow distributors. ~ The The sample was 0.05 ~ m of . air. time constant of the detector was compensated to 0.01 second. Inlet preysures up to 6.5 ntnioipheres led to outlet velocities a5 high 3s. 247 em./ second. One glass bead rolumn and
sis Chromosorb columns (some of the Chromosorb was treated with Siliclad) were run using 100,’120mesh materials. The plate height values obtained by Kieselbach were measured with rather high inlet-to-outlet pressure ratios. P. Denoting the appafent (or measured) plate height as H , the true (or local) plate hcLght may then be obtained as H = H/fl, where fl i. the prmsure correction factor (6, 16), 0 ( P 4 - 1) (P2 - 1)/’8(P3 [Recent experimental evidence on the validity of this correction factor has been obtained by DeFord, Loyd, and dyers ( d ) ] . The H values so obtained have been fitted by least squares to the two equations (see later for a discussion of these equations in relationship to those i i w l by Kieselbaah) , H
=
A
+ B / t o + CZ’,
(9)
H =A
+ B/v, + Ct’, + Ev.*
(10)
where v o is the outlet velocity. The parameters, A , B, C, and E, obtained from this fit are shown in Table I. In addition, the data have been fitted to the equation
fi
=
A
+ Blv, + CV,
(11)
In all caqes A , B, C, and I3 are givcii in cgs unit\-i.e., em., ern.?, second, w c onds, and second?/cin. The number of significant figures does not indicate the precision of the data, but enables one to see the precise effect of fitting different equations to the same data. The main feature in Tablt I bearing on the eddy diffusion problem is the value of E obtained by fitting the data to Equation 10. The E term may be regarded as an asymmetry parameter which measures the departure of T I from the classically-expected valiicx, Equation 9, a t high vclocitiek. T h t ~ fact that E is ntgativc. in all biit on(. case indicates that tlic platc height i i 1cqs than evpcctPd at high vclocitiw, a result consiqtent with the coupling theory. There is a good deal of variation from column to column uhich prohably results from rxperimental error and which might e a d v account for the single positive E value. The general trend of E toward negative values is too consistent to ascribe to random errors , however. inother fact demonstrated by Table I is that the values of A decrease, first, as one uses the pressure correction term VOL 35,
NO. 10, SEPTEMBER 1963
1339
(or 27) = 1.17, c (or U ) = 0.886 and d = -0.133. The parameters, a, B, c, and d are obtained from the reduced form of Equation 10
h
lo
h
-1b I
h 5
-
ts;
COLUMN . .
n
m
$1
P
A
I*
II
0 -
I
0
I
I
20
IO
I
I
30
40
50
r/ Figure 1.
+ b / v + cv
ANALYTICAL CHEMISTRY
Terry, I3lackwel1, aiid Rayne (17) have compiled results obtained by a number of investigators on inert columns using liquid carriers. Their data can be translated directly into a plot involving h and Y . Figure 2 shows these results with lo;: h plotted agaimt log v. These reaults we particularly interesting because the reduced flow velocity, Y, is in some instances l o 4 times higher than that a t which the minimum plate height is expected to occur. The difference between classical and couuline throrics is narticu-
___ Ot
COUPLING THEORY W = I , -f-067 1
-3
-2
-I
(12)
LIQUID DATA
h e e n the individual columns. To make the combined results most ineaningful, it is necessary to disregard the isolated point connected with column V occurring at v > 4 (because it is associated with a column showing consistently high h values), and to disregard the results of columns I and I1 (because li. values are consistently lox and high, respectively, and the range of v values is less than that for the other columns). A least squares plot of this modified data yielded the following parametcrs: n (or 2 A) = 0.164, b
-I
dv2
The value of d, of particular importaim as a criterion of curve flattening rclated to the B of Table I by the ex5.3 X d pression E = dg3d,’D This gives a n E value of -i.1 x 10-7 seconds*/cm., in general accord with the values in Table 1. The magnitude of d makes this term quite significant since a t v = 3.3, the absolute value of d v 2 is 50% of the nonequilibriuiii term, cv. Values similar to these are obtained if all the Chromosorb data are used in place of the modified data-i.c., a = 0.10, b = 1.2. c = 0.80, and d = -0.10. Thus Iiieselbach’s data ~ h o i van unmistakable trend toward the kind of curve flattening predicted by the coupling tlieory.
Kieselbach’s d a t a for Chromosorb columns
jl (proceeding from Equation 11 to Equation 9), and second, as an additional parameter, E, is included to allow for high velocity deoartures (from Equation 9 to Equation 10). Thus the average h values from about 0.3 for Equation 11 to the neighborhood of zero for Equation 10. Both values are below the classical value, h > 0.5. Precise values of A or h are, as pointed out by IGesclbach ( I I ) , difficult to determine because the random error of plate-height mciasurement is the same order of magnitude as A . Kieselbach has found rxccllent justification for considering A a$ entirely negligibile and discarding it from his data analysis. The above equations, with A included, do not imply any disagreement with Kieselhach; instead they are used to avoid the possible criticism that A and the classical theory were discarded prematurely. Even though A may be essentially negligible, its incorporation in the equations does not hinder the study of the asymmetry properties characterized by E. Because of the rather large experimental uncwtainty obtained from individual columns, it was felt that more significant results might be obtained by combining the data from all Chromosorb columns. These data are shown in Figure 1, plotted in terms of the reduced variables, h = H / d p and v = vd,/D.. The value of Do n-as assumed to be 0.695 cm.2/secomd [based on measured D, values from both N P and 0 2 (15)],and d, was assigned the value 0.0137 cm. It is clear that the separate columns give very similar values, but it is also clear that consistent (although slight) differences exist be1340
a
=
I
a 0
=
0
I
2
3
4
. . I _ _
5
6
L O G 3 =LOG* DG
Figure 2. theories
Comparison of liquid d a t a with predictions of classical and coupling
With orily one exception, each series of result5 shown in Figure 2 is essentially independent of velocity. This is predicted by the coupling expression, Equation 6, which shows ihat h approaches 2 2 h , = 2X, a constant, a t high velocities. These results r i n contrary to the classical theory, Equation 8, in which h approacheq W V , a term proportional to velocity. In the original paper the authors conclude, in fact, that the slope of the line reppesenting the top three sets of data is 0.17, a value far closer to the coupling value of zero than the classical value of unity. A number of interesting points are raised when the magnitude, as opposed to the slope, of the results is considered. One rather definite result is that the magnitude of the plate height a t high velocities is from 10 to 100 times less than can be explained on the basis of classical theory. Figure 2 shows the range of h values predicted by classical theory. This range is based on the assumption that w values are never less than 0.1 (1 > w > 0.1). All known experimental and theoretical evidence confirms that w is above! perhaps well above, 0.1. (In the cirigmal paper the authors compared the experimental results with those predicted in terms of flow through small interstitial capillaries. The latter yields an w value of the order of loe too small, and does not therefore form the klasis for a valid comparison.) Thus 1 he experimental results appear to be irreconcilable with the classical theory. The unusual featuie of the experimental results is thzk they seem to divide into two di: tinct categories. The higher values of h form a plateau in which H is approximately 50 particle diameters ( h , = 50, X = 25). The lower results group in a manner indicating that H is ~pproaching only about 2.5 particle diameters (ha = 2.5, X = 1.25). The difference between the two can hardly be ascribed to normal packing variations. It would be useful if there were 6,ome a priori indication of what h ought to be. Preliminary calculations in this laboratory indicate that some A, terms are approximately unity in magnitude, and that other values may approach infinity (these values would bil associated with single interstitial capillaries which, at the velocities being considered, would still be contributing a small term proportional to velocity). Fhile the matter is quite complicated, it seems clear that X should be a t least unity and perhaps well above. ?Sonethelm, the different
series of results should show more consistency. In considering the over-all relationship between the classical and coupling theories and the experimental results, it should be kept clearly in mind that extraneous effects may easily lead to an increase in apparent plate height, but (normally) a decrease will not occur. Thus dead volume a t the column inlet and outlet may give rise to H values which are larger than the true column value. In addition, trans-column effects, the kind that detract from preparative-scale columns, may also lead to a spurious contribution. This, suprisingly, may be constant with velocity changes (although trans-column effects ordinarily yield a term proportional to velocity, the lateral diffusivity is also proportional to velocity a t the high Y values encountered here and the velocity dependent terms should thus cancel). It is consequently possible that the larger values of h are not very meaningful, being merely maximum possible values with the actual values considerably less. Even though additional contributions to the plate height may exist in these cases, the experimental values are stiil signScantly less than the range of classical prediction. Thus these considerations make the classical theory even more untenable, but do not detract in any way from the coupling hypothesis. CONCLUSIONS
While the liquid studies provide nearly conclusive evidence against the classical theory, a small A term has been obtained quite consistently in the recent literature (1, 11, 12, 14). The magnitude of this A term is nearly always less than the minimum classical value, A = d,. The origin of this constant term can be explained quite readily by considering the coupling expression, Equation 2. This equation is a summation of terms, each of which will behave differently in the range normally encountered in gas chromatography. One such term originates in the flow and diffusion processes occurring in single interstitial channels. The C,, contribution in this case is small and the plate height term thus approaches H = C,, v over most of the experimental range. In contrast to this, there is apparently a term due to the interaction between unequal flow regions which are well separated from one another in the column. This long-range interaction yields a big Cod term because the equilibration distance is very large (perhaps
from 3 to 30 or so particle diameters), and the plate height term consequently yields H = 2X,d,,, a constant, over most of the experimental range. Terms such as this may well explain the appearance of a constant term in the plate height expression. In general, terms will appear ranging the entire spectrum between the C,,v and the 2h,d, type. The coupling of these terms is not always apparent because the experimental range may not be sufficiently great to discern it. Thus experiments over a limited range may well fit the approximate equation
An equation of this form was suggested earlier by Knox (12). The use of this equation, judging by the evidence presented here, should be limited to a relatively narrow range of velocities, and the h and C, values should be treated as incomplete quantities which are expected to vary with the particular velocity range under consideration. LITERATURE CITED
(1) Berge, P. C. van, Haarhoff, P. C., Pertorim, V., Trans. FaTaday Soc. 58, 2272 (1962). (2) Bohemen, J., Purnell, J. H., J'. Chem. SOC.1961, 360. (3) Deemter, J. J. van, Zuiderweg, F., Klinkenberg, A., Chem. Eng. Sci. 5, 271 (1956). (4) DeFord, D. D., Loyd, R. J., Ayers, 35, 426 (1963). B. O., ANAL.CEIEM. (5) Giddings, J. C., Ibid., 34, 1186 (1962). (6) Ibjd., 35, 353 (1963). (7) Giddings, J. C., Nature 184, 357 (1959). (8) Giddings, J. C., Robison, R. A., ANAL.CHEM.34, 885 (1962). (9) . , Glueckauf, E., Ann. N . Y . Acad. Sci. 72, 614 (1959). . (10) Kieselbach., R.., ANAL. CHEM. 33, ' 806 (1961). (11) Ibid., 35,1342(1963). (12) Knox, J. H., McLaren, L., Ibid., p. 449. (13) Norem, S. D., Ibid., 34, 40 (1962). (14) Perrett, R. H., Purnell, J. H., Ibid., 35,430 (1963). (15). Seager, S. L., Geertson, 1,. R., Giddings, J. C., J. Chem. Eng. Data 8, 168 (1963). (16) Stewart, G. H., Seager, S. L., Giddings, J. C., ANAL.CHEM.31, 1738 (1959) (17) Thiy, W. M., Blackwell, R. J., Rayne J. R., Meeting of the Society of Petroleum Engineers, Houston, Texas, October 1958.
RECEIVEDfor review April 10, 1963. Accepted July 12, 1963. This work ~ 8 . 8 supported by the U. S. Atomic Energy Commission under contract AT-( 11-1)748.
VOL 35, NO, 10, SEPTEMBER 1963
1341
