Evidence of Coupled Eddy Diffusion in Gas Chromatography JUDSON M. HARPER’ and EARL G. H A M M O N D Department o f Dairy and Food Industry, Iowa State University o f Science and Technology; Ames, Iowa
b A method was developed to calculate the eddy diffusion contribution to plate height in gas liquid chromatography at various gas velocities. The presence of a coupled eddy diffusion term, Ac’, was demonstrated. The maximum value of A,’ as u’ -+ was shown to b e equal to the conventionally determined constant, A = 2 X d, with the average value of X = 1 .O. The mass transfer coefficient, Cc’, a measure of resistance to diffusion in the interparticle gas space in coupled eddy diffusion, was shown to b e inversely proportional to the diffusivity of the solute in the gas phase. The magnitude of C,’ is somewhat larger than had been theoretically predicted for the resistance to lateral diffusion processes connecting unequal velocity channels within the packing.
T
exact nature of eddy diffusion, that contribution to band spreading in porous media resulting from nonuniform flow paths, has been the subject of considerable discussion recently. The pioneering work of van Deemter, Zuiderweg, and Klinkenberg (5) showed an eddy diffusion contribution to plate height, -4= 2 X d,, where X is dependent on the packing geometry and d, is the average particle diameter. From the theory of the plate height model, A is expected to have a value > d, (5) a n d to be independent of the flow velocity, u. Similar conclusions regarding A were reached using a mixing stage model (4, 19). Giddings (8-10), using a random walk model, predicted that the eddy diffusion contribution to plate height is velocity-dependent. His theory gave a coupled eddy diffusion contribution, il,, which can be written HE
1
A c = E
-2 +X,d,-
Neither of these two positions has been proved or disproved conclusively using graphical or numerical curve-fitting methods on gas chromatographic data. In recent carefully performed work (2, 21), constant A values were obtained which agreed with van Deemter’s theory. Giddings and Robison (11) have reviewed reported experimental results which indicate that A may be velocity-dependent, but their experimental work to establish coupled eddy diffusion was inconclusive. Knox and McLaren (18),studying the spreading of air peaks, found little evidence of eddy diffusion or coupled eddy diffusion. Also using data on air peaks, Kieselbach (17‘) concluded that a significant A term was absent when mixing volumes within the system, which skewed the air peaks, were eliminated. Using Kieselbach’s data, Giddings (8) showed that they lend support to his coupled theory of eddy diffusion, although the A terms were small. I n the same paper he also provided conclusive evidence against the classical theory by using chromatographic data obtained with liquid carriers. ,41so studying liquid column chromatography, Gordon et al. (16) found evidence consistent with the coupled eddy diffusion theory, but they felt that with gas carriers this effect would be noticed only at velocities much higher than those found in gas chromatography columns. Work at this laboratory on the effect of solid support characteristics on plate height in gas liquid chromatography revealed inconsistencies which led us to a careful examination of coupled eddy diffusion. The first inconsistency appeared when the equation
N
1 -
where
CJL 1
1
-+2Xd,
1
(1)
C,u
where C, is the mass transfer term for diffusion in the interparticle gas space. This term shows A , = 0 as u --t 0 and A , = 2 X d , a s u - - t 03. 486
ANALYTICAL CHEMISTRY
and (4)
was used to evaluate C,’ and C,. In
this equation, P is the ratio of inlet to outlet pressures, p,/po, u‘ is the effective linear flow velocity or the ratio of the volume flow rate a t p , = 1 atm. to the total gas volume per unit of column length, and u’ f 2 / p , can be considered the average effective velocity, fi. B’, Co’, and Ci are the contributions of longitudinal diffusion, resistance to gas phase mass transfer and resistance to liquid phase mass transfer to the plate height, H . The pressure correction factor, f i , proposed by Giddings and Robison (11) was essentially 1.0 for the experimental conditions used and so was eliminated in the calculation procedure. I n this procedure, values of A and B’ are evaluated from the intercept and slope, respectively, of an H os. l/u’ plot a t low values of u’, where the C,‘ and C I contributions are negligible. Writing Equation 2 in the form of
H
-A
- B’/u’ U’
=
C,‘
+ C1f2/p0= c (5)
shows that a plot of C us. f i / p , should be linear with a slope of Ci and intercept of Cot. Our data gave values of C which became very small at f 2 / p 0 E 1 and resulted in a curve with a pronounced downward hook rather than a straight line. Similar trends can be noted at a fixed p , in the plots of DeFord, Loyd, and Ayers (6),who used a similar technique to evaluate C,’ and Ci, although they did not gather data at very low flow velocities. The eddy diffusion term, A , plays a major role in the evaluation of C as u’ + 0. A coupled eddy diffusion term which approached zero as u’ decreased would tend to make the C us. f 2 / p o plot more nearly linear. Also, when Equation 2 was plotted by using values of the constants evaluated by the two gas method of Bohemen and Purnell (3) and Perrett and Purnell (go), values of H were obtained which were larger than the experimental values at low values of u’ (see Figure 1). Again a coupled eddy diffusion term would help to correct this discrepancy. 1 Present address, James Ford Bell Central Research Laboratories, General Mills, Inc., Minneapolis, Minn.
EXPERIMENTAL CP
SUFTORT
L=l5%
J
0
I
1
I
IO
20
10
Y
1
40
’ (em. sec-1)
Figure 1 . Fit of experimental data with curves calculated considering constant and coupled eddy diffusion 2-Octanone eluted with helium at 103’ C.
CALCULATING Ac’
7 can be written eliminating A,’ and
Values of A,’ were determined at various values of u’ by using the equation
Cl.
A,‘ = H - BO‘/u’- C,‘U‘ - Cia (6) and experimental values of H with independently calculated values of Bc’, Cg’, and C1found in the following manner: B c f ,the longitudinal diffusion coefficient may be evaluated from a plot of H us. l/u’, However, Bo’ is not the slope of the plot at low values of u’ as is the usual case. For if eddy diffusion depends on u’, the plot should not be linear but curved upward, and B,’ should be the slope of the asymptote to the H us. l/u’ plot passing through the origin-that is, B,’ should not make a contribution to H at infinite velocity and should account completely for it at zero velocity. Independent values of C,’ can be calculated by using a method similar to that employed by Bohemen and Purnell (S), who obtained H and u’ data from a column with two different eluting gases. The difference in plate height with different gases and velocities is given by :
(2 2)+ -
(CP’lUf1-
C,’ZU‘Z)
+
where subscripts 1 and 2 represent the different gases. I n theory Ci is independent of the kind of gas and Act1 = Act2 at high values of u’, so that, when iil = ii2 and u‘ is large, Equation
~C,’lU’~
- CgfzUfz)
(8)
From theory, Bc’l/Bc‘2 = C,’2/CQ’1 = D,’1/D1’2, the ratio of the diffusivities of the solute in the gas at p , = 1 atm. This ratio may be obtained experimentally, calculated from empirical equations, or obtained from the ratio of the Bo‘ values evaluated as above. C,’ can then be calculated by using Equation 8 when AH is measured a t .iLl = a2. The value of C,’ as calculated in this manner can be made nearly independent of B,‘ by measuring AH at high values of u’ where B,‘/u’ becomes small. Equation 7 can also be used to evaluate Cl as described by Perrett and Purnell (20). If u’ values are large and chosen so that the ratio u f 1 / u f 2= D,’l/Do’~,B,’ and C,’ terms will cancel and since the A,‘ terms are equal a t high values of u’, AH = Hi
- Hz
= Ci(G1
-
fiz)
(9)
Values of Ci are most reliable when A H , AC, and u’ are large. If D,’ is known, CI may be obtained independently of Ao’,B c f ,and Cg‘. Because A,’ is evaluated as a difference, fluctuations in the results must be expected because of the random errors in H . From A,’ us. u’ curves, 2Xd, can be estimated as the plateau value of A,’ at high u’and C,’ can be estimated at the slope of the curve as u’ 0.
-
Twelve chromatographic columns were used in this study-three amounts of stationary phase on four different solid supports. One support was JohnsManville Chromosorb P (CP). The other three were supports of varying density prepared by lyophilizing solutions of sodium sulfate, sodium hexametaphosphate, and sodium silicate (16). All suppork were -40+60mesh, with Carbowax 20M as a liquid phase. The amount of liquid phase per packed volume was standardized on all columns to the amount that was required on C P at levels of 5 , 15, and 25 weight %. The gas chromatographic apparatus was constructed in our laboratory, with every effort being made t,o reduce the dead volume. The thermal conductivity detector was a Gow-Mac Model 9285 having low dead volume. The sample heater was designed so the vaporized sample was immediat,ely swept into the column by the carrier gas. The column was connected to the sample heater and detector by short pieces of stainless steel capillary tubing. Special care was taken to minimize the dead volume a t the ends of the packed columns. With these precautions, the total dead volume was 1. I n c Calculated using Equation 12. support .-L at 5 and 15y0 liquid load there was breakup of t’he support during the packing of the columns. This led to modest increases in 2Xd,, but model is proportional to the distance a of w are not completely consistent with the value of C,’ increased considerably. the small values of A that were found. velocity path persists. Thus, packing inhomogeneity may lead The velocity at which A,’ is half its to some coupled eddy diffusion processes maximum ( I O ) is: LITERATURE CITED having a value of 2Xd,, which is large 4 AD,‘ (1)Berge, P. G. van, Pretorius, V., compared to its associated Cc’, resulting U’l,, = -~ (10) ANAL.CHEM.36,693(1964). in a large C,‘ contribution added to Cv‘. P2d, (2)Bohemen, J., Purnell, J. H., J. Chem. The value of Cc’, like all constants Soc. 1961. 360. Using the approximation found in this dealing with gas phase mass transfer, (3)Ibid., p.’ 2630. (4)Carberry, J. J., Bretton, R. H., should be inversely proportional to Do’, work that 2 A = p, U’I = ”’. This A.I.Ch.E.J.4,367(1958). so the reliability of C,’ can be confirmed Ad, ( 5 ) Deemter, J. J. van, Zuiderweg, F. J., by using the relationship, Cc’pICc‘l = for the test equation predicts that Klinkenberg, A., Chem. Eng. Sci. 5, 271 solute eluted with helium = 12 and D,’l/Dg‘p. The ratio of experimentally (1956). for nitrogen = 4 cm. per second; these (6) DeFord, I). D., Loyd, R. J., rlyers, determined values of C,’ is given in B. O., ANAL.CHEM.35,426 (1963). are very close to the actual values (see Table I and should be 3.0, the ratio (7)Giddings, J. C.,Ibid., 34, 1186 (1962). Figure 2). of the diffusivities found using the (8)Ibid., 35, 1338 (1963). The magnitude of C,’ should depend Gilliland equation. ;ill experimental (9)Giddings, J. C.,J . Chromatog. 5, 61 on the lateral mass transfer by diffusion ratios were < 3, with more than half > (1961). (10) Giddings, J. C.., Suture 184, 357 and column-wide effects due to column 2 and only one < 1. Because of fluctu(1959). coiling and nonuniformity of particle ations in the experimental data, which (11) Giddings, J. C., Robison, R. A., (Y), considering the size. Giddings strongly influence :Ic’ determined by ANAL.CHEY.34,885 (1962). column parameters, predicted difference, and t’he uncertainty of (12)Giddings, J. C., Seager, S. L., Stucki, L. R., Stewart, 6 . H., Ibid., fitting a curve to the experimental data, 32,867(1960). dP2 the values of C,’ are subject to rather (13)Gilliland, E. R., I n d . Eng. Chem. cc’= D,’ large errors in some cases. Consider26,681(1934). ing the errors involved, these experi(14)Glueckanf, E.,“T’apor-Phase Chroand w = 0.70 (1 - 0.15R) (12) matography,” D. H. Desty, ed., p. 29, mental data seem to confirm that C,’ a Academic Press, New York, 1957. Dv‘-l. The ratio Bc’l/Bc’zshould also where R is the fraction of solute in the (15)Gordon, J. AI., Krige, G. J., Haarbe 3.0 theoret’ically, and experimental gas phase. The data in Table I1 show hoff, P. C., Pretorius, lr.,ANAL.CHEM. values were 2.74 to 3.00. that w calculated by using Equation 11 35, 1537 (1963). (16)Harper, J. AI., Hammond, E . G., Giddings (IO) predicted from thewas in all cases larger than those using Zbid., 37,490 (1965). oretical considerations that C,’ = Equation 12. Recause of fluctuations (17)Kieselbach, R., Ibid., 35, 1342 in the data, a relationship between R (1963). -~ ” d p $ ) , where p d, is the distance of a and w could not be confirmed. The (18)Knox, J. H., McLaren, L., Ibid., 200 35,449 (1963). experimental values for CP are in agreecomplete diffusion step in the random (19)Kramers, H.,Alberda, G., Chem. ment with the proposal, considering walk model with /3 in the order of unity. Eng. Sci. 2, 173 (1953). that Giddings expected a two- or Values of Cell = 0.0018 and CCt2= (20)Perrett. R. H.. Purnell. J. H.. AXAL. threefold error in the numbers used. CHEM.34,’1336(1962). ’ 0.0054second were calculated by using p (21)Ibid., 35, 430 (1963). It appears, however, that the resistance = 1,d, = 300 microns, and Dot1 = 0.255 to lateral diffusion processes connecting and Dolz = 0.0834 sq. cm. per second. RECEIVED for review Augrist 14, 1964. unequal velocity channels within the These values are in good agreement with Accepted January 25, 1965. Work subpacking is greater than theoretically data for CP, considering the error in mitted by J. 11. Harper in partial fulfillexpected, especially for the lyophilized measuring Cc‘. Because C,’ was larger ment of the requirements for the degree of supports. This suggests again that the doctor of philosophy to the faculty of Iowa than the lyophilized supports, (3 ranged State University of Science and Techvelocity irregularities exist over a between 1.5 and 3.0 with some evidence nology, Ames, Sovember 1963. Joiirnal longer distance in the lyophilized than that p = 2 A. This indicates that the Paper No. 5-4923,Iowa Agricriltiiral and in the C P support, thus making C,’ distance a molecule must diffuse to Home Economics Experiment Station, larger. However, these large values Ames, Project KO. 1517. complete a step in the random walk ~
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1
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(3
VOL. 37, NO. 4, APRIL 1965
489
