pIace for temperature programming in capillary column gas chromatography involving analysis of mixtures having a wide boiling range of constituents. LITERATURE CITED
(1) Ambrose, D., et al., “G?: Chromatog-
raphy-Edinburgh 1960, R. P. IT. Scott, ed., p. 429, Butterworths, London, 1960. (2) Destv, D. H., Goldup, A., “Gas Chrom”atography-Edinburgh 1960,” R. P. IT.Scott, ed., p. 162, Butterworths, London, 1960.
(3) Desty, D. H., Goldup, A., Swanton, ~ ~ r ~ ~ t ~ ~ ~ ~ 83, 1961,
(4) Fryer, J. F., Habgood, H. W.,Harris, W. E., ANAL.CHEW33, 1515 (1961). (5) Giddings, J. C., J . Chromatoq. 4, 11 (1960j.
(6) Giddings, J. C., ISA Proc.: 1961 Int. Gas Chromatog. Symposium, Preprints, p. 41, 1961. ( 7 ) Habgood, H. IT.,Harris, IT;. E., ASAL. CHEM.32. 450 11960). (8) Purnell,’ J. H., k a t u r e 184, 2009 (1969).
Failure of the Eddy Diffusion Concept Gas Chromatography
(9) Rowan, R., ANAL. CHEX 33, 510 y (1961). ~ ~ ~ ~ ~ u n ~ g (10) Scott, R. P. W., J . Inst. Petrol. 47, 284 (1961). (11) Teranishi, R., Nimmo, C. C., Corse, J., ANAL.CHEM.32, 1384 (1960). RECEIVEDfor review February 26, 1962. Accepted ,May 4, 1962. Joint contribution from the Department of Chemistry, University of Alberta, and No. 172 from the Research Council of Alberta. Presented to the 45th Annual conference of the Chemical Institute of Canada, Edmonton, May 1962.
Of
J. CALVIN GlDDlNGS and RICHARD A. ROBISON Department of Chemistry, University of Utah, Salt Lake City, Utah F A critical examination of the status of e d d y diffusion in gas chromatography i s made in terms of past experimental results and some new experimental work. Previous experimental anomalies, such as negative eddy diffusivities, are examined in the light of both old and new theoretical concepts, and possible explanations are given for the departure from the expected pattern. A summary of this work strongly indicates the failing of the classical e d d y diffusion concept. This i s confirmed b y further experimental results using an inert glass b e a d column in which the plate height i s sometimes less than the particle diameter. An alternative to the classical concept, the coupling theory of eddy diffusion, i s in satisfactory agreement with past and present results.
T
HE widely accepted role of eddy diffusion as one of the major sources of peak spreading in chromatography is subject to grave doubt. This vievi has, of course, been expressed before, especially in the light of numerous experimental results which have not agreed with the eddy diffusion concept. Whatever doubt has arisen, hoaever, has apparently failed t o stimulate a n active concern with alternatives to the eddy diffusion concept and explanations for the anomalous and sometimes contradictory experimental results. I n only a few cases have workers attempted t o explain the reasons for the rather severe departure of their results from those predicted by eddy diffusion theory. The object of this paper is t o summarize and put in perspective the explanations n hich have been made, and, with some new suggestions, t o explore the relation-
ship between previous experimental anomalies and the theory of eddy diffusion. I n addition, experimental n-ork which bypasses some of the previous experimental difficulties is presented. Eddy diffusion is a name given to the spreading of chromatographic peaks which is a direct and sole result of the interaction of mobile fluid with the solid support. The phenomenon depends entirely upon the geometrical arrangement of the support particles and is thus nonchemical in nature. It arises from the nonequivalence in velocity of various flow paths which a fluid and its contained solute follow in migrating through the porous support. The less tortuous, higher velocity paths lead some solute to a position in advance of the bulk of solute and vice versa. Because a random distribution of flow paths exists, the resulting band spreading is random, and the subsequent concentration profile (providing one starts with a narrow pulse) is gaussian. There is no exact theory of eddy diffusion for the simple reason that the enormous geometrical complications of porous materials defy mathematical tractability. This is not t o say that exact relationships have failed t o appear (an example, using similitude principles, relates eddy diffusion t o particle diameter without assuming simple geometries), but simply that a purely theoretical calculation of the value of eddy diffusivity is now out of the question. The fact that eddy diffusion is, under some circumstances, a real phenomenon is demonstrated b y the existence of a t least three approximate derivations which all agree as to the order of magnitude of the effect. These derivations employ a similitude principle (23), a mixing stage model ( 5 , 25) (very much
like a plate height model), and a random walk model (9). They all predict a plate height contribution, A, which is proportional to the particle size (for geometrically similar packing) and independent of all nongeometrical parameters such as flow velocity and temperature. I n addition, i t is generally assumed that A is equal or larger than a single particle diameter (5, 15). These two sentences summarize the results of what it e will call the classical eddy diffusion concept of chromatography. In mathematical form A A
= 2Xdp
(1)
> d,
(2)
n here d, is the mean particle diameter and h is a geometrical constant of order unity. Only one theoretical treatment has been given with results in wide variance with Equations 1 and 2. This theory, which predicts A t o be a function of flow velocity and t o be capable of approaching zero a t low velocities, is of interest in view of the failure of experimental results to conform to Equations 1 and 2 . For lack of a better name we will call this the coupling theory of eddy diffusion (see later). I n approximate mathematical terms
A=
1 1
(3)
A =0
A = 2Xdp
(4)
v-+O
v + m
1
2Xd, + c,u
where v is the mean gas velocity and C,v is the nonequilibrium or mass transVOL 34, NO. 8, JULY 1962
* 885
~
r
e
fer term for diffusion in the interparticle g a s space. EXPERIMENTAL ANOMALIES
I n this section a summary will be made of the important cases of departure between experimental work and the classical eddy diffusion theory of Equations l and 2 . The presence of experimental departure from theory does not, of course, prove the theory to be incorrect. Such departure can arise from various sources as will be discussed. One primary goal of this paper is to separate. as far as possible, the significant from the superfluous departures, This matter will be discussed in the next section. The principal experimental anomalies reported to date fit in one of the following categories: Appearance of A Values Which Are Nearly Zero or Negative. This is perhaps the most 11-idely noticed forin of departure from classical theory. Many authors have recently rejected the necessity of a n A term for the description of their data (1,2, 4, 14, 22, 28). Others have, under certain circumstances, found the consistent appearance of negative A values (2, 16, 27, 291, Variation of with Particle Size. It is almost universally found t h a t A increscee as p:irticle size decreases (3, 7 , 8,i7.22,30). This has often been explained as resulting from the difficulty in obtaining uniforni pscking n-ith fine particles. It is difficult to imagine, hone[ er. that this effect would be sufficient t o cause nearly a direct inverse proportion betn een h and d, as observed experimentslly. Thus this result is considered as a true experimental anomaly. Khile the consequences will be discussed a t greater length later, most of these experimental results are consistent nith A values being entirely independent of particles size (1, 3, 7 , 17, do, 30). I n addition to the above result, Bohemen and Purnell have found a correlation between X and d, such that X becomes negative for particles qnialler than about 50 to 60 mesh. Variation of A with Flow Velocity. Glueckauf (1?), Bohemen and Purnell (21, and Purnell (29) have reported experimental evidence for a strong variation of A with velocity. The type of variation ia not, hom-ever, consistent; Glueckauf, using a liquid carrier and an inert g l a s bead support, found a rapid increase of -4 with velocity followed perhaps by a slight decrease. Purnell et al. (2, 29) found that the A term beet combined with the B (molecular diffusion) term and thus had an inverse dependence on flow velocity. Variation of A with Retention Volume. Kieselbach (21, 22) has 886
ANALYTICAL CHEMISTRY
reported t h a t light gases tend to have larger A values than more highly retained components. This effect has also been observed by Littlewood (26). Variation of A with Solute. The value of A has been observed to differ in going from one solute to another (3, 16). A few other anomalies, such as the variation of ,4 with percentage liquid ( 8 ) ,have been reported in the literature, but for the most part the data are not sufficient to justify a discussion. O R I G I N OF EXPERIMENTAL ERRORS
It would be very difficult to explain completely each of the foregoing anomalies without a great deal more knowledge about the individual apparatus and experimental method and about the basic chromatographic parameters. Considered altogether, however, certain patterns are evident. I n a very general way, experimental anomalies may arise either because the assumed theoretical expressions are incorrect, or because of errors in the collection of data (26). To begin n-ith the former, the following sources may be responsible for some of the departure between theory and experiment : 1. Erroneous Assumptions on Nature of A . The evaluation of t h e eddy diffusion effect has always proceeded under t h e assumptions of the classical theory, particularly t h e assumption t h a t A is constant. If t h e coupling theory of eddy diffusion is correct, or if sonie unknon-n velocity dependent processes occur, then A will actually be velocity dependent. According to the coupling theory the effects of eddy diffusion are negligible beloiv a certain velocity estimated as about 10 to 100 em. per second by Giddings (11) and about 100 per second by Jones (19). If the experiniental range is significantly below these velocities, then one expects the experimental result, A = 0. If the expeiimental range includes these velocities, then an uncertain result is obtaincd because one has forced the effect, realljvelocity dependent, into a constant. It is important to note, hon-ever, that for a large velocity range, the coupling theory combined with reasonable parameters indicates that the plate heightvelocity curve mill in many cases be slightly concave d o m at high velocities. [Some idea of the effect of using the van B v Deemter equation, H = il Cv, to fit points that really conform to the coupling theory can be gained in the follon-ing two examples. If one obtains points from the equation H = u l / v 1/(1 l/v) and makes a least squares fit of A , B, and C, the folloming results are obtained. Using 6 points in the interval v = 0.5 - 3, we have A = 0.58, B = 0.86, and C
+
+
+ +
+
= 1-08. Using 8 points in the interval v = 0.25 - 2, we have A = 0.39, B = 0.94, and C = 1.16. The actual plots show a close adherence to the theoretical points so that in an actual case i t would be difficult to throw out the results because of the failure to obtain a good fit (14). Despite the fact that the above examples are a very special (and in some Yays exaggerated) case, i t is probably true that van Deemter parameters obtained from data based on a coupling mechanism would show A values less than 2Xd, and greater than zero, B values slightly less than they should be, and C values slightly greater than the Ce term but less than the sum of C t and C, terms. ] 2. Failure to Correct Plate Height Expression for Column Pressure Drop. The effect of using pressure correction terms for t h e plate expression is best illustrated in the paper by Giddings, Seager, Stucki, and Stewart (14). The usual van Deemter equation, employed by using outlet velocities rather than pressure corrected velocities, showed Consistent h values of about 1.0. Using pressure correction terms on the same equation, and employing the same data, the eddy diffusion term n-as consistently negative m-ith h G - 0.5. The constants n-ere obtained by a least squares fit to ensure complete objectivity. -4 result similar to this has been reported by Littlenood for glass bead columns (26, 27). The use of pressure correction terms changed the apparent eddy diffusion contribution from about 0.5 to slightly negative values. These results show definitely that experimental A values are strongly influenced by pressure correction terms whenever pressure gradients are noticeable. 3. Failure to Include M a s s Transfer in the Gas Phase. I n the absence of noticeable pressure gradients, t h e analysis of plate height-velocity d a t a to yield A is not influenced by the presence or absence of i n a s transfer in the gas phase. Othernise it becomes critical, insofar as A values are concerned, to include gas phase mass transfer even though it does not make a major contribution t o plate height. This is illustrated in the pre\iously quoted work (f4)nhere the same data n ere fitted by least squares to a number of possible plate height equation?. Khereas the van Deemter equation modified by pressure correction terms jiPlds -0.5, the same equation modified by both pressure connpction terms and m a s transfer in the gas phase yields X g 0.1, a value very close t o zero. This change occurs despite the fact that the gas phase mass transfer term is only 15% as large as the liquid phase mass transfer term. The results of 1, 2, and 3, above, shoK that the values obtained for A depend very critically upon the as-
sumed plate height expression. This conclusion should apply to any reliable method of data interpretation since i t applies to one, the least squares method. I n the following cases, reference will be made to the occurrence of experimental anomalies which result from errors in the collection of data. 4. Inherent Uncertainty of Data. Both Littlewood (66, 27) and Bohemen and Purnell (3) have emphasized the inherent difficulty in obtaining reliable eddy diffusion constants. The effect is masked to a large extent by other simultaneous phenomena and is subject to ordinary experimental errors. 5. Errors of Graphical Methods. While all data have not been interpreted by graphical methods, those which have are generally considered to be suspect as far as the A term is concerned. Ayers, Loyd, and DeFord ( 1 ) have illustrated this point in connection with attempts to equate A to the intercept of the limiting tangent in a plate height-flow velocity plot. The intercepts are substantial even Bowhen eddy diffusion is absent. hemen and Purnell ( 3 ) have also reported uncertainties in the graphical method. 6. Equipment Errors. It is very probable t h a t a n apparent eddy diffusion term originates within the equipment, either in the sample introduction system or the detector. This has been verified in some careful work by Kieselbach (22), who was apparently able to eliminate eddy diffusion with proper equipment design. Littlewood (26) has speculated that the introducer will contribute a velocity independent term to the band width and thus to the plate height. He followed this suggestion through with a calculation showing that the entire A term for a C 0 2 sample could be accounted for by the 0.5-ml. introducer volume of his apparatus. The effect of finite sample size is essentially the same as this, and may be included in the same category. The broad implication of the foregoing analysis of anomalies and sources of errors is that the classical eddy diffusion theory as described by Equations 1 and 2 has failed to account for experimental results. This is based on, first, the fact that three of the important sources of anomalies (Nos. 2, 5, and 6) are responsible for A values that are too large as opposed to one (So. 3) which indicates values too small, and second, a great deal of experimental evidence exists n hich, despite this trend, indicates that A values are near zero. Significantly, the morerecent results (1,g6), where a great deal of care has been taken to eliminate experimental error, shorn the strongest indication of negligible A values.
This implication may be expanded somewhat by considering the individual experimental anomalies. First, of course, is the appearance of nearly zero or negative A values. As mentioned above, the recent trend with carefully constructed equipment and advanced technique is in this direction. It is difficult to imagine errors of sufficient magnitude to make these data conform to the classical eddy diffusion picture. The only certain source of error which would ivork in this direction is the failure to include mass transfer in the gas phase, No. 3. As indicated, this requires the presence of large pressure gradients, and even then is not a major effect. In addition, several investigators (1.4, 21) have reported A values beloiv theoretical even while using a gas phase mass transfer term. I t is, of course, conceivable that some basic omission from the theoretical plate height expressions would lead to A values far below their actual value. This is highly unlikely because of the basic agreement of theory and experiment in most areas. Kegative A values, which are physically meaningless, may arise in the failure to consider gaseous nonequilibrium when a high pressure drop exists. It was mentioned under S o . 3, earlier, that the addition of a gaseous mass transfer term a t high inlet pressures changed A from -0.5 to -0.1, This concept is consistent with the results that negative A values are most prominent when fine particles ( 2 ) and long columns (14) are used, both having relatively large pressure gradients. The second anomaly concerns the inverse variation of A with d, leaving A nearly independent of particle size. This cannot readily be explained in a manner consistent with the classical concept by any of the sources of experimental errors discussed here. The anomaly is clearly consistent with KO. 6, above, in which the equipment contributes a constant apparent eddy diffusivity. Because of the failure of A to vary with d, it would seem that no part of the measured values truly originate in the column and that as far as the column is concerned, L4 0. The third anomaly, the variation of A with flow velocity, is difficult to detect and has thus not been TI-idely looked for nor observed. The results reported by Glueckauf (17) are in broad agreement n ith the coupling theory of eddy diffusion ( 1 1 ) but are not consistent nith the classical theory. The observed variation of A with the inverse of velocity ( 2 ) is difficult to explain in terms of column parameters and may have its origin n-ithin the equipment. The fourth anomaly, the inverse variation of A with retention volume, is also difficult to explain in terms of the classical theory. -4s pointed out by N
-
Kieselbach (62),this result is consistent with (and strong evidence for) an A term originating within the equipment but not in the column. Kieselbach demonstrated this origin of A by adapting his equipment in such a manner as to eliminate it. The A term apparently originated in a skewing of peaks caused by precolumn mixing volume. This is one of the best experimental indications that A is negligible within a typical column. The fifth anomaly, the variation of A with solute, is rather erratic, and probably originates as inherent uncertainty in the data, No. 4. The foregoing results are strong evidence for the failure of the classical eddy diffusion concept of chromatography, The experimental section of this paper gives additional proof for this conclusion. THEORY
I n contrast to experimental results, the concept of eddy diffusion has remained surprisingly fixed in chromatographic literature, The only major exception is found in the recent coupling theory of eddy diffusion (10, 11). The relationship between these theories and the physical processes occurring within the colunin will now be discussed. The coupling theory of eddy diffusion emphasizes the interaction between the purely geometrical parameter, A, and various nongeometrical quantities including flow velocity and diffusion rate. The similarity of the processes occurring in packed and capillary columns illustrates the coupling interaction. In capillary columns there are nonequivalent (nith respect to velocity) paths, rapid near the column center and sluggish a t the outside. The individual solute molecules do not travel for long a t a fixed velocity because they rapidly diffuse into regions mith some other velocity. This continual interchange of velocities by a solute molecule is responsible for keeping peak Fpreading (plate height) reasonably small. In a packed column a similar distribution of velocities exists. The variation is wider, however, since a velocity bias may exist across one or more particles (trans-particle) as well as within a single channel (trans-channel). The eniallnese of the plate height n-ill be a measure of how rapidly molecules exchange their velocities. The unique thing about packed columns is that the velocity exchange can occur by two mechanisms. I t can occur by diffusion (as in capillary columns) from one flow path to another, or it can occur without diffusion (unlike capillary columns) by simply following a given laminar flow path which randomly acquires all possible velocity states in its passage through the column. The classical VOL. 34, NO. 8, JULY 1962
e
887
eddy diffusion concept is based solely on the latter mechanism of velocity exchange. The theory of capillary columns is based entirely on the former. Most authors erroneously assume that the two effects are additive in their contribution t o the plate height of a packed column. This is obviously in error because both exchange mechanisms together can lead to a more rapid exchange of velocities than either alone, and consequently (29) to a smaller plate height than either mechanism by itself would provide. The approximate theory of the coupling of these two mechanisms, leading to Equation 3, is given elsewhere (10, 11). I n understanding this theory, the difference between trans-channel and trans-partide mass transfer is one of degree and not kind. I n the three-dimensional packing structure there is no such thing as a n isolated channel, and each flow region is connected to each other by numerous lateral diffusion paths. This continuous variation in the type of mass transfer probably smooths out the variation of A with average flow velocity v , perhaps requiring a sum of terms similar to the right-hand side of Equation 3. The theory just discussed has been criticized by Klinkenberg and Sjenitzer (24). Their arguments are based on their assumption of the additive nature of independent exchange processes to the plate height. This assumption can be proved incorrect through a number of rigorous calculations on simple systems (12). Klinkenberg and Sjenitzer (25) originally stated that (‘the axial spreading caused by the parabolic velocity distribution and the effect thereon of radial molecular diffusion are, however, interdependent.” This idea was neither explained nor quantitatively developed, and has thus had no further significance in the evolution of chromatographic theory. One further aspect of the eddy diffusion concept has been discussed by Golay (18). He emphasizes that the phenomenon does not involve true eddies as implied by the name since the Reynolds number is the order of unity. This has never been a point of true controversy in chromatography since the term was so named only because of certain formal analogies with true eddy diffusion (23). The authors, unlike Golay, have not encountered the opinion that true eddies are involved i n the phenomenon, and hence feel that the name has not been particularly misleading. A note of caution should be interjected here, however, in regard to dismissing universally all true eddy defects. While a Reynolds number of unity is three orders of magnitude below turbulence in open tubes, i t is apparently right on the threshold of turbulence in packed media, Collins ( 6 ) , for instance, states that “In sands and 888
ANALYTICAL CHEMISTRY
sandstones, the transition from laminar to turbulent flow occurs rather gradually in the range of Reynolds number from one to ten.” The onset of anomalous friction factor effects has long been known to occur at abnormally low Reynolds numbers in porous materials (SI). While i t is probable that this effect can ordinarly be neglected in chromatographic columns, caution is needed in the interpretation of plate height effects when the Reynolds number exceeds unity. EXPERIMENTAL
Since it was considered advantageous to work with the simplest possible system, inert glass beads were used for the solid support. To avoid some of the previous experimental pitfalls and thereby obtain conclusive experimental results, some departures were made from previous procedures. First, large uniform beads of average diameter d, = 0.117 em., were used to make the eddy diffusion effect, if present, large enough t o be clearly discernable. The large bead size employed was slso successful in avoiding the complicating effects of high inlet,/outlet pressure ratios. T o minimize spurious results caused by the equipment, a long column (12.03 meters) rms employed. I n addition a short (25 em.) correction column was used to subtract out end effects, including those of detector and injection system. Finally, a least squares procedure mas employed t o ensure against unnecessary errors in the interpretation of the data. Data were collected on a PerkinElmer Model 154C using a 0.25-inch 0.d. copper column. The borosilicate glass beads, presumably n-ith d , = 0.15 cm., nere obtained from Mine and Smelter Supply Co., Salt Lake City, Utah. The measurement of 33 beads by a micrometer yielded a n average d p = 0.117 em. with a standard deviation in diameter of 0.010 cm. Their inertness was confirmed by the simultaneous elution of all gases and vapors, including helium and p-xylene. Gas flow was controlled by capillary tubing in combination with a Cartesian diver manostat. and measured with a soap bubble meter. I n the CO9-X9case. 0.05-cc. samdes were withdrakn &om a cylinder of cbmmercial C 0 2 and injected with a standard glass hypodermic syringe calibrated in units of 0.01 cc. All air was carefully purged from the syringe with a C 0 2 stream. Measurements were made at room temperature (-25’ C.) and atmospheric pressure (-640 mm. of Hg). The temperature was recorded before and after each run, and the data The \\-ere all corrected to 25’ C. width-at-half-height method was used for obtaining plate height. The p-xylene-K2 and p-xylene-He systems were treated similarly. The temperature was maintained a t 150” C. and the xylene sample size was 0.4 p1. The experimental plate height was obtained using a short correction tube
of 25-cm. length packed in the same manner as the main 1203-cm. column, The use of this tube presumably makes i t possible to subtract out all end effects including that of detection and injection systems. This technique has been used by Giddings and Seager (15) and by Kieselbach (21). The procedure should be valid except a t rapid flow rates where the standard deviation in the peak elution time, 7, is the same order of magnitude as the time constant of the recorder, approximately 1 second. This perhaps led to some error since a few measurements were made in which the 7 of the correction tube approached 1 second. The error was probably small because i t occurred over only a limited velocity range and then only in the correction tube where a 10 to 20% error would be insignificant (the great length of the main tube almost does an-ay nith the necessity for a correction tube). The equation used to determine plate height using the correction twhniqur, is
11here
the subscript, 1, refers to the correction column. The column length is L and the observed retention time is t . This equation can be applied a t each flow velocity, but in practice i t is simpler to construct a plot of rl us. velocity so that the necessary data are readily available. An alternative to the preceding correction method is the extrapolation of plate height to infinite length columns, Le., to 1/L = 0. The extrapolation was difficult to make, and the method was considered unsatisfactory. The collection and interpretation of data by the above methods was adjusted to a range of flow velocities which would yield the best criterion of the eddy diffusion effect. These velocities were made low to correspond to the region where axial molecular diffusion predominates the plate height. Ignoring the C,v term as a first approximation, the classical and coupling theories of eddy diffusion yield the following equations = 2hd,
+ B/v (classical theory)
(6)
H = B/v
(coupling
(7)
H
theory)
4 plot of H t s . I/v should yield a straight line of intercept 2x4, if the classical theory is correct and zero if the coupling theory is correct. [This is the same plot used by Glueckauf (16) under slightly different experimental conditions.] I n actual fact the C,v term, although small, will slightly disturb these relationships and a slightly curved line is expected. It can easily be sholvn that a tangent to the curve a t a
Figure 1. Experimental plate height d a t a and least squares plots showing intercept near zero and occasion plate height value less than d,
symmetry was good for both tubes. Twenty peaks from the main tube showed a n average ratio of 1.03 for the half width of the leading compared t o the trailing edge. This result was rather difficult to determine precisely because of the difficulty in locating the peak center. For the above reasons the existence of plate height values less than d, appears to be real. Each of the plots shown in Figure 1 v a s obtained by a least squares fit to the data. The flow velocity (in centimeters per second) can be obtained by multiplying the flow rate by a factor of 12.3. Each experimental point is shown for p-xylene in helium. The points shown for the p-xylene-xz and C02-N2 systems are the average of approximately nine experimental determinations. The intercepts with the H axis are -0.009 i 0.003, 0.027 =k 0.002, and +0.033 f 0.007, respectively. The slopes are 0.0778 2: 0.0013, 0.0202 =t0.0003, and 0.0223 i 0.0007, respectively. The uncertainty caused by purely random errors is indicated by the standard deviation shown after each value. The deviation from zero intercept is not due to random fluctuations of experimental data. It is not entirely clear $That the source of this deviation is, but there are a number of factors connected v, ith the instrument deal volume and various time constants which are not completely accounted for. A slightly positive intercept is, of course, expected as a consequence of the C,v term discussed earlier. The reeults are conclubive in showing t h a t the intercept is near zero and thus the primary objective of these euperiments, the evaluation oi the magnitude of the eddy diffusion effect, has been fulfilled. The almost inescapable conclusion of the work reported in this paper is t h a t the cla-sical eddy diffusion concept cannot be applied successfully to most gas chromatographic columns. It may be argued that derivations of the classical equation do not necessarily carry with them the result A 2 d , (Equation 2). Hoirever, the early work nearly always suggested that A was larger than d, and the mixing stage model quite definitely implies -4 d , a t high Reynolds numbers and A > d, a t low Reynolds numbers ( 5 ) . Values of A less than, 5ay, d,/2, as found here, are highly inconsistent with the 1-lassical concept (16). It may further be argued that glass beads, especially the large ones uEed here, pack n-ith great uniformity because of their regularity, and thus A is smaller than normal. If, hoir ever, one packs marbles in a jar such that the ratio of diameters corresponds to the bead/tube diameter ratio, Ridespread bridging occurs which leaves many large and erratic channels as anticipated by classical theory.
+
/ '
2:0
'
4:O
'
6:O '
8:O
' 10.0 ' 12:O
Reciprocal flow rate
'
14.0 ' I
(mi'sec)
given 21 \\auld intercept the axis a t 2Xd, 2C,v following the classical theor! and 2C,v follon-ing the coupling theory. '1'0 choose effectively between these theories, the flov velocity should be adjusted such t h a t 2Xd, >> 2C,v, or approximatelj, d, >> C,v. One is forced here to estimate thc magnitude of the C, term. The data of Sorem (28) indicate that this may be a: large as dPz/5D,, nhere D , is the gaseous diffusion coefficient. (This result indicates t h a t gas phase nonequilibrium spans a distance of sweral particle diameters, and is probably insignificant within single flow channels.) The term is probably smaller here cince a parabolic profile over the entire tube cross section ~ o u l dnot give a value this large. Assuming t h a t the maximum velocity permissible corrcspontls to d, = X , v , the aim(' eupression for C, yields a minimum D,/v s B \ d u e of approximately d,. I n this range of velocities Equation 7 shows that €I = B/v, and thus a velocity significantly higher than d, should not cwrresponding to H be usrtl. This would seem to have a wide latitude, hoivevcr, since H is g~nerallyconsidered to be grcater than d,, under all circumstanres. This condition is not applicable here, however.
+
11
RESULTS A N D DISCUSSIONS
A plot of plate height t s . reciprocal flow rate for the three experimental systems is shown in Figure 1. Some
data are shown which were taken a t a velocity above that corresponding to H = d,. Since no etrong curvature is introduced by these points, and since they obviously do not affect the location of the straight line fit very much, they were included in the over-all analysis. Two important observations can be made from these plots. First, the intercepts are near zero and are distinctly and conclusively less than d,. Second, experimental plate height values have been obtained which are themselves less than d,. The latter result has apparently not been observed before. These results contradict those reported by Glueckauf (26) \There the same plot using glass beads of nearly the same size yielded intercepts greater than d,. Glueckauf's work n-as done with frontal analysis on a &meter column, but little other detailed information is given which might help to uncover the discrepancy in the two methods. T o check the validity of the result H < d,, measurements and data interpretation were independently made by another operator. The results were essentially the same. This is apparently not a spurious result caused by the correction procedure since the uncorrected data also yielded H < d , above a certain velocity. The result cannot be a consequence of the recorder time constant since the peak width for the main column (uncorrected data) was never less than 6 seconds. It is apparently not a result of peak asymmetry since the
VOL 34, NO. 8, JULY 1962
889
The work reported here is coneistent with, b u t not conclusive proof for, the coupling theory of eddy diffusion. The proof of this theory would be greatly advanced if i t were possible to shorn the actual variation of A with flow velocity. This would be enormously difficult using the ordinary gas chromatographic methods. Glueckauf (17’) has shown this kind of variation in a liquid system but i t is difficult to judge the reliability of his data. The fact that the classical eddy diffusion picture seems t o work in chemical engineering studies, many in the proper range of Reynolds numbers, indicates that the predicted transistion mith velocity occurs a t some point. The authors are not familiar with any evperimental work which is in significant disagreement with the coupling theory. LITERATURE CITED
(1) Ayers, B. O., Loyd, R. J., DeFord, D . D., A N ~ LCHEM. . 33, 987 (1961).
(2) Bohemen, J., Purnell, J. H., . “Gas Chromatography, 1958,” p. 6., D. H. Desty, ed., -4cademic Press, New York, 1958. (3) Bohemen, J., Purnell, J. H.. J . Chem. Soc. 1961, 360.
(4) Brennan, D., Kemball, e., J. Inst. Petrol. 44, 14 (1958). (5) Carberry, J. J., Bretton, R. H., A.I.Ch.E. Journal 4, 367 (1958). (6) Collins, R. E., “Flow of Fluids through Porous Materials,” p. 51, Reinhold, New York, 1961. ( 7 ) Deemter, van, J. J., Zuiderweg, F. J., Klinkenberg, A., Chem. Eng. Sei. 5 , 271 (1956). (8) Desty, D. H., Godfrey, F. M., Harbourn, C. L. A., “Gas Chromatography, 1958,” p. 200, D. H. Desty, ed., Academic Press, New York, 1958. (9) Giddings, J. C., J . Chem. Educ. 35, 588 (1958). (10) Giddings, J. C., J . Chromatog. 5 , 61 (1961). (11) Giddings, J. C., Nature 184, 357 (19.59) \ _ _ _ _
(12) Ib[d., 187, 1023 (1960). (13) Giddings, J. C., Seager. S. L., J . Chem. Phis. 33, 1579 (1c60). (14) Giddings, J. C., Seager, S. L., Stucki, L. R., Stewart, G. H., ANAL. CHEM.32, 867 (1960). (15) Glueckauf, E., -4nn. AT. Y . Acad. Sei. 72, 614 (1959). (16) Glueckauf, E., “Gas Chromatography, 1958,” p. 33, D. H. Desty, ed., Bcademic Press, New York, 1958. (17) Glueckauf, E., “Vapor Phase Chromatography,” p. 29, D. H. Desty, ed., Academic Press, New York, 1957. (18) Golay, M. J. E., “Gas Chromatog-
raphy, 1960,” R. P. W. Scott, ed., Butterworth, Washington, 1960. (19) Jones, W. L., ANAL. CHEM.33, 829 (1961). (20) Keulemans, A. I., Kwantes,,, A., “Vapor Phase Chromatography, p. 15, D. H. Desty, ed., Academic Press, New York, 1957. (21) Kieselbach, R., ANAL. CHEM. 33, 23 (1961). (22) Ibid., p. 806. (23) Klinkenberg, A., Sjenitzer, F., Chem. Eng. Scz. 5 , 258 (1956). (24) Klinkenberg, A , , Sjenitzer, F., Nature 187, 1023 (1960). (25) Kramers, H., Alberda, G.; Chem. Eng. Sei. 2 , 173 (1953). (26) Littlewood, A. B., “Gas Chromatography, 1958,” p. 23, D. H. Desty, ed., Academic Press, Xew York, 1958. (27) Ibid., p. 35. (28) Korem. S. D., AYAL. CHEY. 34, 40 (1962): (29) Purnell. J. H.. Ann. il;. Y . Acad. Sci. 72, 592 (1959). ’ (30) Rijnders, G. K. A,, “Gas Chromatography, 1958,” p. 18, D. H. Desty, ed., Academic Press, Kew York, 1958. (31) Scheidegger, A. E., “The Physics of Flow Through Porous Media,” p. 137, Macmillan, Yew York, 1957. RECEIVEDfor review January 18, 1962. Accepted April 23, 1962. Work supported by the U. S.Atomic Energy Commission under Contract AT-(11-1)-748. ’
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A Study of the Performance of Packed Gas Chromatog r a phy Columns S. DAL NOGARE and JEN CHlU Plastics Deparfrnent, E . 1. du Pont de Nemours and Co., Wilrningfon, Del. The van Deemter-Jones plate height equation was used to evaluate the performance of small packed columns. Three well defined limiting efficiencies a t optimum carrier gas velocity can b e derived for specific values of the partition ratio, k . The plate height i s shown to b e proportional to the support particle diameter. The existence of the postulated gas phase mass transfer and velocity distribution terms i s established, and an estimate of the gas phase correlation term and gas diffusion distance in terms of particle diameter i s made. The relation between fl (the ratio of free gas space to liquid phase volume in the packing) and resolution is derived. A criterion i s established for p consistent with best resolution and efficiency.
I
is generally accepted that the efficiency of chromatographic columns expressed as H.E.T.P. or number of theoretical plates is a good measure of column performance r i t h respect to T
890
ANALYTICAL CHEMISTRY
a particular set of operating parameters. Efficiency is derived from dimensions taken from a single chromatographic peak, TI hereas resolution, or separation, involves two peaks. Consequently, the parameters of retention and relative retention, as n-ell as the manner in v hich these are affected by temperature, liquid phase loading, support particle size, and carrier gas velocity, must be considered in applying gas-liquid chromatography to difficult problems. I n this study, the efficiency of small diameter packed columns was evaluated as a function of particle diameter and the column characteristic, p , related to liquid phase loading, for a series of nparaffins. A linear relationship was observed between efficiency and particle diameter and limiting ratios ryere established. The significance of these results and the implications with regard to resolution are discussed. THEORETICAL
Jones ( I S ) has recently reported an expanded H.E.T.P. rate equation for
packed columns n-hich contains, in addition to the familiar van Deemter terms, several terms describing peak spreading in the gas phase. This equation is formally related to Golay’s capillary column treatment ( I O ) and may be considered an equivalent expression applied to the more complex geometry of packed columns I n this Jyork m-e will base our discussion of experimental results on this equation. If care is taken to eliminate experimental contributions to the d term, Jones’s equation is H = B/u
+ Ciu + (Cl
+ cz + C3)u
(1)
where B and C1 are the van Deemter molecular diffusion and liquid phase mass transfer terms. The C1 term represents resistance to mass transfer in the gas phase, C, is the velocity distribution term, and C3is a correlation term accounting for interaction between C1 and CI. Because the correlation term is extremely difficult to evaluate, i t will not be considered in this discussion. This introduces the reservation
