LITERATURE CITED
(1) Deemter, J. J., van Zuiderweg, F. J., Klinkenberg, A*, Chem. Eng. sei+ s f 271 (1956). (2) Giddings, J. c.,A x - a . CHEx 34,1186 (1962). (3) Ibid., 35, 1338 (1963).
(4) Giddings, J. C., Robison, R. A,, Ibid., 34, 885 (1962). (5) Golay, M. J. E., in “Gas Chromatography--1958,” D. H. Desty, ed., p. 36, Butterworths, London 1958. (6) Jones, W. L,, ANAL,CHEM,33, 829 ( i m i’I \LY”I,.
(7) Kieselbach, R., Ibid., 35, 1342 (1963). (8) Klinkenberg, A., Sjenitzer, F., Chem. Eng. Sci. 5, 258 (1956).
(9) Taylor, G. I., Proc. Rou. Soc. (London), Ser. A 223, 446 (1954).
A. KLINKENBERG
Bataafse Internationale Petroleum Maatschappij N.V. (Royal Dutch/Shell Group) The Hague, The Netherlands
The Nature of Eddy Diffusion SIR: In the preceding article (5) Klinkenberg has raised certain questions about the coupled theory of eddy diffusion and has proposed a simple classical equation to describe the eddy diffusion phenomenon. For immediate clarification of the issue, the “classical equation,” as I have used the term, is
H
=
B u
+ A f C,U + CLU
(1)
and is seen to contain the classical eddy along with longitudinal diffusion term 9, diffusion, gas phase mass transfer and liquid phase mass transfer. I n reduced coordinates (reduced plate height h and velocity Y ) this equation is h
= 2Y/Y
+ 2h + + WY
(2)
QY
The concepts and terms behind these equations can be traced back as much as a quarter century, but van Deemter, Zuiderweg, and Klinkenberg (1) first made the equations explicit. hluch to their credit they included the C, term in their general formulation (see their equation 49 and the following equation) although it was subsequently dropped on numerical (not conceptual) grounds. The concept of the coupled theory (3) leads to a different (non-additive) combination of the same general terms. I n a simplified form the equation is h r -2 Y Y
1
+ 1/2x
+
l/WS
+
Qv
(3)
and the “new” simplified classical equation just discussed by Klinkenberg (5) h = 27/u
+ 2X
(simplified classical
(6)
- inert)
Very briefly, one way to clearly distinguish between the three equations above is by means of the following welldefined properties subject to experimental test. Equation 6 always gives a plate height-velocity curve with a negative slope, dh/dv < 0. Equation 5 (and 6) always yields a curve that is concave up, d2h/d? > 0. Only equation 4, the coupled form, is consistent with a plate height curve which is concave down in certain regions. Unfortunately, using these simple criteria, the data in the paper (.2) criticized by Klinkenberg (5) are not very conclusive. More recent results covering a broader
12
10
a
(coupled equation) Since publication of the article (2) criticizedf by Klinkenberg much new evidence has accumulated which not only supports this equation in its broad aspects but, more importantly, makes classical theory and its modifications untenable. With inert (non-sorbing) peaks the last term of equations 2 and 3 goes to zero. We will therefore compare the equations 1
27
h=-+ Y
’/zX
(coupled h = 27/Y
l/WY
(4)
- inert)
+ 2x +
(classical 490
+
WY
- inert)
ANALYTICAL CHEMISTRY
h e 4
2 0.
40
20
60
range, from several workers, are decisive on this point however. In Figure 1 is an example of results obtained in our laboratory (8) using methane in a Chromosorb P column a t inlet pressures up to 160 atmospheres. The reality of the concave down region (or “flattening”) is unmistakable. The same clear result has been obtained for glass beads. In the most extensive experimental work yet performed, Knox has shown (6) that concave “downness” is always present in liquid systems, thus proving that the results are not a spurious consequence of turbulence. In addition to the data from purely inert peaks and columns, several workers (4, 7 , 9 ) have recently obtained results in which the C I term has been subtracted out from retained peaks. I n each of these cases the curves become concave down a t moderate velocities. In view of the above evidence it is very difficult to find any merit in equation 6. It fails to describe even qualitatively all the interesting theoretical and practical features of plate height curves (although our discussion pertains to inert peaks, this is the foundation for retained peaks as well). First it misses the usually distinct minimum in plate height, then it misses the flattening off (concave “downness”) a t higher velocities. Although these phenomena occur only in the transition regions between velocity extremes, this is the region where nearly all practical gas chromatography is executed. Even if equation 6 had any practical uses, it would still be unsatisfactory as a theoretical equation because it tells nothing at all about the important role of gas phase nonequilibrium. Returning to the specific paper criticized by Klinkenberg ( 5 ) , it should be noted that the experimental evidence presented in that paper ( 2 ) is already outdated by the very considerable advances made by Knox and in this laboratory. It merits only enough attention to explain the issue raised. It is pointed out by Klinkenberg ( 5 ) that the equation
z/ (5)
Figure 1. Experimental curve for reduced plate height vs. reduced velocity
h =a
+ b / u + + du2 CY
(7)
has several questionable aspects. This author agrees completely; the limitations were obvious from the outset, but perhaps not sufficiently emphasized. This can be made clear by explaining the origin of the equation. Because of the inherent difficulty in making use of least squares on the “summed” form of the coupling expression [equation 6 of that paper (@], equation 7 was used as a simple but effective criterion of coupling. Nowhere was it stated that equation 7 would yield physically meaningful parameters. In other words the parameter d was used as a criterion of “flattening,” an asymmetric property of coupled curves. It serves this role admirably well for d will be negative for any data obeying coupling and average to zero for any data obeying classical theory. It is of no consequence whatsoever that a negative d will lead h back through zero since there is no suggestion that the equation
does, or should, fit points beyond the immediate experimental range. We see then that d is only a qualitative criterion, but nonetheless a decisive one, in choosing between mechanisms. In the final analysis equation 7 is meaningless only because of the inherent shortcomings of classical theory-if classical theory were adequate one could indeed attach physical meaning to all parameters except d and this, as an average, would equal zero. Finally, to put the now dubious reputation of equation 7 in perspective, the clear failure of classical theory makes the universally used classical equations just as erroneous as equation 7 . If the use of equation 7 is to be condemned, one might argue with like conviction that the use of all classical equations is equally objectionable-or even more so because the parameters of classical theory are invariably given a physical interpretation of doubtful significance.
LITERATURE CITED
(1) Deemter, J. J. van, Zuiderweg, F. J., Klinkenberg, A., Chem. Eng. Sci. 5, 271 (1986). (2) Giddings, J. C., ANAL. CHEM. 35, 1338 (1963). (3) Gidd&s,’J. C., “Dynamics of Chro-
matography. Part 1. Principles and Theory,” Marcel Dekker, Inc., New York, 1965. (4) Harper, J. Si., Hammond, E. G., ANAL.CHEM.37, 486 (1965). ( 5 ) Klinkenberg, A., Ibid., 38, 489 (1966). (6) K,nox, J. H., 3rd International Symposium on Advances in Gas Chrornatography, Houston, October 1965. ( 7 ) .Littlewood, A. B., private communication. Newcastle-upon-Tyne, England (1965). (8) Myers, M.N., Giddings, J. C., ANAL. CHEM.37, 1453 (1965). (9) Saha, N. C., Giddings, J. C., Zbid., 37, 830 (1965). J. CALVIN GIDDINGS Department of Chemistry University of Utah Salt Lake City, Utah WORK supported by Public Health Service Research Grant GM 10851-08 from the National Institute of Health.
The Nature of Eddy Diffusion in Gas Chromatography SIR: In previous correspondence Klinkenberg (8) and Giddings (6) have exchanged views on the merits and limitations of the original van Deemter, Zuiderweg, and Klinkenberg (1)theory of band-widening in gas-liquid chromatography as compared to Giddings’ coupling theory. It is felt that for a better understanding of the problems a t stake a general statement on the present position, especially with regard to the concept of eddy diffusion, would be useful. Such a statement is accordingly presented. It is, in part, based on some observations made recently (9) in a general paper on residence time distributions in chemical engineering. The classical equation by van Deemter et al. ( I ) was based on the concept of a uniform forward gas velocity (“plug flow”) with some superimposed spreading mechanisms-viz., axial diffusion, axial eddy diffusion, spreading by transfer to/from stationary fluid, with resistance to transfer in both phases. This accounts for a 4-term equation of the general form of equations 1 and 2 (6) but with both linear terms vanishing for a nonsorbing solute. Plug flow, however, is only a first approximation for the real flow pattern. I n each pore the velocity ranges from zero to a maximum value. In any real packing, not being a closest packing of equally sized spheres, there are moreover wider and narrower pores in parallel and there may even be packing
irregularities extending over larger regions. I n the van Deemter et al. concept the total convective spreading by all these mechanisms was called “eddy diffusion.” A consequence of the flow not being plug flow is that, upon the passage of a chromatographic band, there are radial concentration gradients. These give rise to lateral diffusion, which reduces the axial spreading and ultimately leads to an apparent axial diffusivity (“Taylor diffusion”). Accordingly “eddy diffusion” is reduced if there is appreciable lateral diffusion. A statement that this is to be expected in gases for heat transfer was made by Klinkenberg and Sjenitzer (IO), who have, however, not examined its consequences for GLC. Giddings dealt with this subject in many publications ( 2 , S , 6 , 7 ) . Such lateral diffusion occurs most easily in the case of the single pore. It is less important in the cases of the pores in parallel and of the packing irregularities, where the distances to be covered are greater and the particles are moreover obstructing the diffusion. This means that eddy diffusion should be split up into its component parts before the effect of lateral diffusion on it can be studied. Giddings accordingly considers the sum of “coupling” terms between “various eddy diffusion and mass transfer effects” ( 9 , s ) . Lateral diffusion becomes appreciable if the time of diffusion r 2 / D across a pore
radius r is comparable to the time of passage d,/v through a pore. Since r and d, are proportional, this leads to the conclusion that the concept of eddy diffusion as a purely convective process must be modified if D/vd, = 1 / u surpasses a \ alue in the neighborhood of unity. See Giddings ( 7 ) . The chromatographer is interested in working a t minimum h-Le., for h = 2y/u 2~ C Y , at h,,, = 2d2ic
+ +
I-
+
If the three parameters in this equation are all of the order of unity, as they seem t o be, Y 5: 1 is the preferred working parameter. Thus it is concluded that the eddy diffusion concept is going to fail when it is most neededLe., in a region where its classical contribution 2X is relatively most important in respect to the other terms, the sum of which is to be minimized. Giddings ( 7 ) arrived a t this same conclusion by comparing the velocity a t which the influence of radial diffusion becomes important to gas velocities used in practice. Comparing the classical prediction h = 2y/u 2X for a nonsorbing solute to the Kieselbach data, descending branch reaching h = 2 a t v = 1, horizontal branch h = 2 for Y > 1, (4) we see indeed that the limiting cases for low and high Y are well represented but for the region Y ‘v 1 this is not the case. The value of h approaches its limiting
+
VOL. 38, NO. 3, MARCH 1966
491
