Nature of Gas Phase Mass Transfer in Gas Chromatography J. CALVIN GlDDlNGS Department o f Chemistry, University o f Utah, Salt lake City 7 2, Utah
,An attempt is made to evaluate the importance of gas phase mass transfer terms and to relate these to plate height. Theoretical estimates, made on the basis of the observed properties of randomly packed particles (without reference to chromatographic data) and the generalized nonequilibrium theory of chromatography, compare favorably with empirical values obtained by various authors. Comparison of theoretical and experimental results shows that the principal contribution to the plate height arises in the interaction of different flow channels which have unequal velocities due to a large variation in size. This, in fact, is the only contribution of sufficient magnitude to explain the experimental results. The diffusion processes within individual support particles can account for no more than 10% of the observed plate height, and the velocity variation effects within normal flow channels, often given credit for the entire plate height term, can account for no more than 2% of the observed value. The dependence of the plate height contribution upon the degree of retention is shown both theoretically and experimentally to be mild. The gas phase mass transfer effects show little resemblance to capillary columns where the retention dependence is, by comparison, very large. At least six distinct terms related to gas phase nonequilibrium contribute more or less to the over-all plate height of a normal chromatographic column.
T
HE contribution of gas phase nonequilibrium to the plate height in gas chromatography is significant. Emphasis on the importance of this term was apparently first focused by Jonin 1956 (MI g8). Since that time a number of authors have, using diverse experimental techniques] isolated a term which is undoubtedly a consequence of the failure to reach complete equilibrium in the gas phase (6, 8, 14, 28). Although the experimental evaluation of this term is subject to some uncertainties, the comparison of the results of difTerent investigators, made below, shows satisfactory agreement and clearly
1186
0
ANALYTICAL CHEMISTRY
establishes the order of magnitude of the effect. This agreement is fortunate, for it offers a good deal of conclusive evidence regarding the validity of various theoretical models. The theoretical problem of formulating the plate height contribution due to gas phase nonequilibrium is a formidable one. The immense complications of the interstitial geometry make it impossible to calculate exact expressions for plate height effects. However, these complications do not exclude the possibility of making satisfactory plate height approximations in terms of the observed structural characteristics of porous media. In addition] theory should serve in correlating the performance of different solid support materials. This work WM undertaken in the hope that theoretical considerations might explain some of the observed plate height characteristics and relate them to packing structure. In so far as chromatographic performance is concerned, a packed column is generally considered in terms of a bundle of noninteracting capillary tubes aligned in the flow direction. The open tube cross sections serve as models for the interparticle flow channels of the granular material. The plate height is then made up of contributions from longitudinal diffusion, nonequilibrium in the liquid phase, nonequilibrium in the gas within the particle] and nonequilibrium within the flowing gas between the particles. [The contribution of eddy diffusion is probably negligible a t ordinary velocities (14). This is the only phenomenon commonly related to the interaction of “separate” flow channels.] If one proceeds to calculate the total gas phase contribution (the last two terms) using reasonable geometrical parameters] the calculated value is an order of magnitude less than the observed values. The reason for the discrepancy becomes obvious if one observes the structure of randomly packed particles. Such an aggregate is composed of small regions where the particles are in close proximity, separated by gaps of fairly large volume. Since flow velocity increases with the square of the linear dimensions of a flow channel, the randomly occurring gaps must certainly carry a gas stream of
above average velocity. This differential flow established an interchannel nonequilibrium which exists over a region much larger than a single particle diameter or flow channel. This phenomenon] which is due to the interaction between nonequivalont flow channels, accounts for plate height values of the observed order of magnitude. STRUCTURE OF THE PACKING AGGREGATE
Any aggregate of unconsolidated particles will show a random variation in the size and location of the open gaps or channels running between them. This is true, as may be verified visually, even for completely uniform particles of spherical or cubical shape. Despite the high degree of randomness, however, the channels are observed to divide roughly into two groups: the flow channels which result from the intimate contact of particles in regions of high density and close packing and the larger channels between particles that do not fit into an intimate structure with each other, or alternatively] fit roughly into a low density packing. The varying character of packing structure as described above has been observed by workers concerned with fluid flow in porous media. Collins (‘7) summarizes this view with the statement that “sand packs of uniform grain size usually consist of small regions of more or less regular packing separated by regions of irregular packing in which ‘bridging’ has occurred.” There is probably a reasonably close correspondence between aggregates of solid particles and molecules in the liquid state. In his “hole” theory of liquids, Eyring has visualized regions of regular packing (crystalline structure) separated by holes of slightly less than molecular size (23). The volume increase attendant to melting (approximately 5 to 15%) equals the volume of the holes in the liquid. By comparison to this increase, a closest packed array of spheres has a volume increase of 15 to 30% upon randomization ( I , 3, 81).
The “holes” or large channels within a porous aggregate are irregular in shape. Often a long dimension will exist which may be as much as several
particle diameters in length. These channels, when directed along the flow axis, prohahly play a major role in the development of gas phase nonequilihrium. The direct observation of spherical and cubical aggregates shows that such channels occur a t intervals separated by approximately two full particle diameters. This, in fact, will he the assumption used in developing the chromatographic theory. The validity of such an assumption is, of course, unprovable without a rigorous definition of what will and what will not he classified as a suitable channel. This approximation was made, however, in view of the observed properties of particle aggregates, and was in no way influenced by chromatographic evidence. The cross-sectional area of the channels is of obvious importance in determining the extent of nonequilihnum. Although these channels show a good deal of variation in cross-seetional shape and area, they often seem related, in spherical arrays, to particles uhich have formed (usually not perfectly) the nucleus of a simple cubic lattice. The usual small channel, on the other hand, is of a size dictated by one of the close packing structures. The difference in the size of such channels is illustrated in Figure 1, where a single plane from each lattice is shown. The largest circle (corresponding to the area where the hulk of flow occurs) which can be inscribed in the simple cubic interstice has a dianeter 41% as large as the constituent spheres. In the close-packed structure the inscribed circle has a diameter only 15% as large as the spheres. Such a difference in size, which observation shows to be of the correct order, will clearly lead to signiiicant velocity differences from one region of the packing to another. A variation in channel size is also observed with the common chromatcgraphic support materials. Figure 2 is an enlarged photograph of 30/60mesh Chromosorh W. A numher of channels can he seen with crosssectional areas significantly larger than n
The development of accurate plate height expressions for ChromGography depends upon clearing two basic theoretical obstacles. First, one must be able to obtain models which faithfully reflect the structural and flow profile characteristics of the column material. Second, the plate height must he accurately formulated in r terms of the structnral models, no matter how complicated. The first obstacle, involving the acquisition of realistic
Figure 1. Approximate size of interparticle f o w channels in close-packed and simple cubic structures Roughly corresponds to size of small ond large Row channels, rerpectively, in chromologrophic posting
Inodels,
is essentially a nonchromatoits existence and I:raphie problem-i.e., .nl..+:nn A, . U"
S".U"A""
+-.,
"YY
A*.",l "Fp""U
..X^"
Up,"
-.l+-1'"
^^
"V
currence of the chromatographic process. The second obstacle, the development of chromatographic theory, is intimately connected with the fundsi mental nature of chromatography. Of the two theoretical areas, the chromatographic formulation involves the least difficulty. The honeqnilihrium theory (If, f3, 16, 18, 801, developed by the author, can he used for the exact calcnlation of plate height in terms of known or assumed flow velocity and structural characteristics of the packing. If the physical characteristics of the model are too complex, a numerical solution must he sought, but in all cases a solution can he obtained. 'The current bottleneck in the theoretical description of chromatography lies principally in the nonchromatographic area. Thus it is a relatively simple matter to obtain the liquid phase mass transfer term once the configum tion of the liauid on the solid sunnort is known. The' liquid configurathi prohlem has, however, only recently been
Figure 2. Surface structure of 30160mesh Chramosorb W
attacked (13). A similar difficulty exists with the problem under discussion. I t may he presumed that really accurate models for the support structure will require a long period of development. In view of this difficulty, the present theoretical treatment is justified as follon-s. Even simplified models should yield approximate plate height expressions which are in error a t most hy a factor of 2 or 3. As shown later, this is entirely adequate to rule out some of the sources previously postulated for the gas phase nonequilibrium term. Hence the major chromatographic effect can be correlated well with certain structural characteristics of the support. This, in turn, leads to the only clues a t present availahle for the improvement of this particular aspect of chromatography. Thus the following development should have Dractical as well as theoretical significance. The earliest theories of noneauilihrium effects m the mobile phasi centered around a diffusion film which presumahly encircled individual beads in an ion exchange column (6). In these simplfied theories no effort was made to account for the interaction of adjacent particles or for the dsnsion between noneqnivalent flow channels as discussed here. The first quantitative discussion of mass transfer in the mobile phase of gas chromatography was given by van Deemter, Zuiderweg, and Klinkenherg in 1956 (9). These authors assumed that gas phase mass transfer occurs in a pore or channel which is the order of one-fifth the particle diameter. The contribution of such a small dsnsion distance was assumed negligible. In contrast to this, Jones concluded that the gas phase contribution was the dominant one (86). Jones snggested that this contribution arose from two terms, one describing the role of diffusion in the moving gas and the other accounting for the stagnant gas presumably found within the support An approximate particles (85, 28). chromatographic theory was developed to describe these effects. The theory of gas phase mass transfer in capillary columna was given by Golay in 1958 (844). Subsequently many authors have attempted to transfer the concepts and equations pertaining to the capillary column over to packed columns (5, 10, 18). This theory, a t best, is applicahle only to flow in the single channels within the support which are comparable to open capillary tubes. The experimental values show that this theory contributes 8 term from 10 to 500 times too small, assuming any reasonahle model. Consequently a difierent mechanism must he used to explain the experimental results. VOL 34, NO. 10, SEPTEMBER 1962
1187
Subsequent theories of gas phase nonequilibrium have dealt with diffusion processes both inside and outside the individual support particles (4, 11, $5). A rigorous expression for diffusion within the porous particles, assumed spherical, was derived by the present author using the nonequilibrium theory (11). A comparable term was derived by Jones (85). Unfortunately, the latter is not functionally correct, since the plate height contribution is erroneously predicted to go to zero for nonpartitioning vapors. Recent theories of diffusion in the channels adjacent to the support particles show no material improvement over previous single-channel models. Jones (26)has developed a highly technical random walk model for the evaluation of plate height terms. Such models ( I @ , based on probability considerations, have not yet been developed to the point where they reflect moderate changes in channel structure, velocity profile, etc. Precise plate height values cannot be computed for any but the simplest physical models for the support aggregate. While it may be argued that any theory is approximate in the complex situation existing in practice, it is extremely important to determine whether departures from theory are due to shortcomings in the model assumed for the packing structure or in the calculations relating the plate height to this model. If this distinction cannot be made, it is impossible to judge the propriety of different models for the solid support and to correlate chromatographic performance with support structure. Thus the Jones theory cannot be used, for example, to choose between single-channel flow and the interacting flow of multiple channels as suggested here. I n addition, several parameters must be fixed by reference to chromatographic (rather than structural) data, and a correlation parameter exists which is admittedly intractable. The first step in applying the nonequilibrium theory to the flow processes in the support aggregate consists of choosing a physical model based on direct observations of the aggregate. The model chosen here is simple and direct, although it is anticipated that later models might be more elaborate. Based on the observation that large flow channels occur with a spacing of several particle diameters, a repeating unit of flow is chosen which consists of an inner core of circular cross section where the gas flow is rapid, and an outer annulus in which a moderate flow rate occurs through the interstices between the particles. For the present time, the outer annulus is assumed to contain a structure sufficiently fine to be treated as a homogeneous medium. The geometrical characteristics of the model are shown in Figure 3. Each such flow 1188
0
ANALYTICAL CHEMISTRY
unit is only one or two particle diameters in length. The effects encountered when the flow leaves one such unit and enters another (end effects) are Support
particles
I
where Lii is the mean velocity of the solute zone and E is the mean concentration (in moles per cubic centimeter of the column volume) of solute. -4pplying this equation first to the outer annulus, in which case eo is held constant, the solution for el becomes
where go and g, are integration constants and r is the distance from the center of the system. The three boundary conditions needed to fix the values of go,gl, and eo are el = eo
3ei/br
=
at
T
= r0
0 at r =
(3) T~
r" 1-
(4)
Figure 3. Model used to approximate repeating flow unit in chromatographic packing
closely related to eddy diffusion and are discussed in the concluding section. Except for these effects, the units may be considered essentially infinite in length. The various physical parameters pertaining to the model are designated by a 0 or 1 subscript, depending on whether they apply to the inner core or outer annulus. Thus the velocity of the gas stream is uo in the core and u1 in the annulus. Other such parameters are the velocity of the solute zone, ui; the diffusion coefficient, Dd; the outer radius, rt; the concentration, cb- and the nonequilibrium departure term, et (i = 0 or 1). Other terms are introduced in the derivation. Because of the high symmetry of the model, the plate height contributions of the core and annulus are assumed additive (18). This assumption makes it possible to treat the two regions individually, as can be seen from the mathematical condition (18) for additivity, H , (DO,D1)= H , (Do,a ) H , ( Q), D1).Thus the plate height term, a function of both diffusion coefficients, is the sum of plate height contributions obtained with only one diffusion process controlling the equilibration a t a time. Each individual (DO,a), for examcontribution-H, ple-is not difficult to calculate because D1 = 03 implies that there are no gradients in region 1 and consequently e1 = constant. In the following paragraphs the two terms, H , (a,Dl) and alternately H, (DO, m), with EO and assumed constant, are derived. Following earlier developments of the nonequilibrium theory (18) the differential equation describing the variation of E* in region i is
where co* and c1* are the respective solute concentrations in the inner core and outer annulus, assuming complete lateral equilibrium. These equations, used in conjunction with Equation 2, yield
where Q = co*r$/Er12 is the fraction of solute in the inner core (region 0) a t equilibrium. The plate height contribution of the outer annulus can be established in the standard fashion as
When EO is substituted from Equstion 6, this gives
+
where E
= r 0 1 ' and ~
Some values of f($ are shown in Table I. With the outer annulus disposed of, attention is next focused on the inner core. We note that in this region the zone velocity, ~ 0 equals . the average gas velocity, go, since there is no retention in free gas space. If the flow in this central region is amumed t o be parabolic, we have 2'0
= 200 (1
- TZ/To2)
(10)
When this is substituted in Equation 1 and the latter integrated,
go
+ 91
In r
at r
(12)
= t-0
deo/dr = 0 at r
=
0
(13)
These equations lead ,to (15)
go = 0
[11c0
(11)
The integration constants and el are fixed by the conditions so = c1
as products of one another, may be discarded. This gives
+
[-a
e l = - -Qro2B
Do
Q(:
In E 3 dz (6
I-)!! -
;)
(16)
(I7)
The parabolic (as opposed to uniform) flow assumption is responsible for a more complicated plate height expression than was needed for the outer annulus, Equation 7. -2 = .cia In
+ Qoo)]
=
ro2/r12(1- P,) = t2/(1 - F,)
-
X
Since el, eo, and i j ~are specified by the previous equations, this expression can be evaluated in a straightforward manner to yield Qro2
242zDo
(Oo(1lBo - Sa)
+
- (SDo
- 6a) [ Q o ~ (1 - & b i l l (19)
This reduces to capillary column theory under the special set of conditions in which u1 = 0, .ii = &BO, and Q = R, as can be verified by comparison with Equation 44 of a previous article (18). In the case under consideration here, Equation 19 can be simplified without a great deal of error. The flow in the inner core is rapid and thus o0 may be considered large compared to u1, fi, and &&I. Hence, all of the latter terms which appear to the second power, or
Table 1.
Some Numerical Values of f(f)
€ 0.00 0.10 0.20 0.25 0.50 1*00
(21)
@ = VllBO (22) where the fraction of free gas space which is within the inner core is a. This is related to the fraction, F,, of the column which is occupied by solid material. The significance of p is obvious. The parameter, Q, which may be written as co* [ 2 / E , can, in view of Equation 21, be written as cO*a(l F,)/E. Since the ratio of the average zone v e h i t y , fi, to the average gas F,)/E, we velocity, 0 , is R = co*(l have
Q = CYR
=
(30)
The total plate height contribution due to the interaction of nonequivalent channels is given by the sum of Equations 8 and 20. The next step involves the reduction of these equations to the numerical values suggested by the structure of the support. I n view of the approximate nature of this study, several approximations are made which entail errors of the order of 10% each. There are two parameters which are useful in the following analysis: CY
980 +
- S(u1 + 6
f(€) m
6.33 3.75 2.98 0.95
0.00
(23)
Other expressions which are obvious in view of the preceding work are uo = BO and ti = Roo. I n addition, u1 = Rlul --
[email protected] only a small part of the free gas is in the inner core, the R value of the outer annulus, R1, is approximately equal to R. Thus u1 Rp8o. The mean core velocity, go, is related to the over-all mean velocity, B, by 8 = 00 a ~ l (l a) = ijo(a P - ab) G (a B)t?o. The difference term, u,, - u1 = oo - Rlpoo E go (1 BR). The difference term, .ii - u1 = ~ ( -1Q) uo Q - UI = ( U O - uJQ = aROo(1 - BR) . Since diffusion within the inner core is simply free gas diffusion, Do = Do, where the latter is the gaseous diffusion coefficient of the solute. The diffusion coefficient for the outer annulus is D1 = R1Do/T2 G RD,/T2. The Rl term appears because only the fraction, R1, of solute is free to diffuse assuming negligible diffusion in the liquid. The tortuosity, T, appears in order to account for the complex diffusion paths in and around the support particles. In terms of the operational parameters just described, Equations 8 and 20 become
+ +
+
+
)
annulus
(Outer
(24)
Xumerical values for the foregoing parameters are assigned as follows: Porous Solids € = 0.20
rl/dp
=
1.25
T 2 = 1.50
@ = 0.05
Nonporous Solids = 0.20 rl/dp = 1.25
T 2= 1.50
I9 =
F a = 0.15 CY = 0.05
0.10
Fa = 0.60 CY = 0.10
The porous solids include diatomaceous earth and related materials and nonporous solids include solid beads, crushed glass, etc. The parameter, a, has been computed from Equation 21, using the tabulated values for 4 and F,. The value, rddV = 1.25, is based on the previous observation that approsimately two particles were found between the larger flow channels. When half of this distance is added to the approximate core radius, d v / 4 , the above value is obtained. The value of B = o l / ~ o is difficult to estimate with any degree of precision. The large channels appear to have a diameter the order of three times the small ones. Under normal circumstances flow velocity increases with the square of the diameter, and thus j3 s 0.1. Two errors inherent to this assumption will probably compensate one another. I n nonporous solids, a large amount of gas occupies the “thinning” regions of the flow channels (in the outer annulus) where the particles are in close proximity and the velocity is near zero. This will be offset by the restrictions which prevent the free movement of gas into and out of the large channels. Thus both velocities will be reduced by a significant amount. In porous solids the velocity, ul, is reduced by another factor of 2 because of the stagnant gas in the particles interior (the gas velocities used here are average values taken over all free gas space). If the specific values listed above for porous solids are substituted into Equations 24 and 25, the following expressions are obtained
where w =
0.550 (1
- 0.05R)a +
G O.TO (1 -
0.143 (1 0.1523)
- 0.145R)
In the case of nonporous solids w = 0.550 (1 - O.lR)’+ 0.072 (1 - 0.29R) S 0.62 (1 - 0.3R)
(27)
(28)
An important conclusion related to these equations is that the R or k [ k = (1 VOL. 34, NO. 10, SEPTEMBER 1962
1189
R)/R] depcndence of the plate height term is rather moderate. Thus H, is reduced by about 15 to 3oyo on going from a highly retained solute to a nonpartitioning solute. I n contrast, the plate height of a capillary column is reduced by a factor of 11 for the same change in solute character. In addition to the plate height expressions found above, a number of other contributions due to mass transfer effects in the gas phase will be found: (1) mass transfer within individual porous particles, (2) nonequilibrium in the small flow channels, (3) lateral interaction between separate flow units in a given cross section, the average one of which was considered here, (4) nonequilibrium across the entire tube due to tube bending, and (5) nonequilibrium across the tube due to the nonuniformity of particle sizes between inside and outside regions (21). Most of these effects are probably negligible. No. 4 has been shown to be ordinarily small through theoretical calculations (17). No. 5 is probably small, since the effect is noticed only in preparative scale columns. No. 3 may be considerable, but a t present there is m simple way to evaluate its contribution. Nos. 1 and 2 are discussed in the following paragraphs. The plate height contribution due to intraparticle nonequilibrium has been obtained rigorously for spherical particles with interior pores small compared to the diameter (11): Although most support particles are not spherical some of the widely used supports such as Chromosorb have well rounded surfaces (see Figure 2) and are probably described in an adequate fashion by the theory if an average diameter is used. The theoretical expression is 2(1 - *R)zT* dpzV = 60(1 - 0)
(29)
where cp is the fraction of gas in interparticle space. This quantity is approximately 0.5. Using T 2 = 1.5, the value of w (the coefficient of dp2C/Dp) is
replace the usual R (or k) value, since in the derivation R' is taken as the fraction of the total solute actually within the gas space of the flow tubes. I n capillary columns, of course, this is all of the solute vapor, but in columns packed with porous materials it is one half or less of the total. Thus R' R/2. Assuming further that 2ro = 0.3dp (which gives a flow channel with twice the diameter of an inscribed circle in the plane of a close-packed layer) and 0, = 2 0 (6),we have w =
0.0019 (1.5RZ - 8R
+ 11)
(32)
This value ranges between 0.021 (R = 0, k = a )and 0.0085 (R = 1, k = 0) for porous supports. For nonporous supports the corresponding range is 0.010 to 0.00094. These values are noticeably smaller than the three contributions already treated. While values of the specific parameters may be argued one way or another, it would be difficult to imagine changes of a magnitude which would lead to any different conclusions. Thus the treatment of a packed column as a bundle of noninteracting capillaries is clearly wrong, a conclusion which will be further substantiated by experimental evidence. The total calculated contribution of gas phase mass transfer in columns with a porous support may be summarized approximately by o G
0.82
- 0.20R
(33)
This expression comes from the addition of Equations 27, 30, and 32. For nonporous supports w
E0.63
- 0.20R
There are several other contributions
to gas phase mass transfer effects which would increase these values to some extent. COMPARISON WITH EMPIRICAL RESULTS
(18)
where 0, is the average velocity within the flow channel and ro the effective channel radius, An R' (or k') must
where E (or C,) and B are the gas phase mass transfer and longitudinal diffusion coefficients in the plate height-velocitypressure equation and y is the tortuosity
O.lO(1
- 0.5R)Z
(30)
The contribution of flow pattern and diffusion within the normal sized channels between particles can be calculated by comparison to capillary column theory. This particular contribution appears to be the only one bearing any real relationship to capillary column processes as such. The expression for a capillary (or open tubular) column is
o =
1190
ANALYTICAL CHEMISTRY
EB/2ydpa
w =
+
7+ 1
1 [A k2Dd Ed2 dp2 (1 k ) * (1 k )
+
(36)
Using the second column, which is intermediate in both liquid load and mesh size, and butane as the solute, w is 0.88. If the theoretical equations are used, a rather wide variation in w from about 0.63 to 3.95 is indicated. On the whole, however, the values are somewhere in the vicinity of unity. Both high and low k values were employed. Bohemen and Purnell have evaluated gas phase nonequilibrium effects using 20% columns of polyethylene glycol on Sil-0-Cel (6). Acetone and benzene were the principal solutes. The results were written in terms of the theory of capillary columns. In terms of these results, w is
(34)
The first reported attempt to isolate quantitatively a gas phase mass transfer term from experimental data was apparently made by the author and his coworkers (22). Using a 20% load of silicone grease on firebrick, the gas phase term for p-xylene was found to be only about 15% as large as the liquid phase term. Nonetheless, reasonably consistent values were obtained for the gas phase term on seven columns of three different lengths. These results are shown in Table I (22) under both Equations I11 and IV. Using the nomenclature and equations of that paper, it can be shown that the w term is
w =
or labyrinth factor associated with B. Assuming that y = 0.7, dp = 0.02i cm. (50/60 mesh), and the best experimental values, B = 0.066 and E = 0.007, the value of w is determined as 0.45. This result is applicable to the low R ( R slightly less than 0.1) or high k range. Kieselbach has used columns with different mesh sizes of Chromosorb, different liquid loadings of silicone oil and different solutes in order to isolate a gas phase mass transfer term (27, 28). While some doubt was expressed concerning the detailed equation used in this analysis, the over-all gas phase contribution can berdbtained from .Table I1 (27, 28). Using Kieselbach's nomenclature and the tabulated values of Dd,2 and Ed,2, the value of w is found to be
(35)
(37) where x is an empirical parameter, f ( k ) is a function of k applicable to capillary columns, u is the mean interparticle gas velocity, and 0, as before, is the mean gas velocity averaged over both interparticle and intraparticle space. The ratio u/0 is near 2 for porous packings and unity for glass beads and other nonporous materials. If the average value, x = 1.35, is taken along with f ( k ) = 10 (corresponding to k = 10 to 30) and u/ci = 2, the value of w is 1.12. These values vary slightly with solute, As carrier gas, and particle size. pointed out by the authors, these variations are not sufficient to lead to any definitive conclusions. Norem has made an experimental study of air peaks in both glass bead and Chromosorb columns (SO). Although he did not quantitatively interpret his data in terms of gas phase masa transfer, this can be easily done, aa suggested by Dal Nogare and Chiu (8), by observing the minimum value acquired by the plate height as a function of velocity, Assuming eddy diffusion
to be negligible (14), the plate height of an air peak can be written as
Table II. w
Through differentiation of this expression it can easily be shown that w and Hminare related by
Assuming that d p = 0.032 cm. for the 40/60-mesh glass beads and d p = 0.037 cm. for the 30/60-mesh Chromosorb, and observing that the minimum value of H was near 0.05 cm. in both cases, it is found that w = 0.44 for the glass beads and 0.33 for Chromosorb. If allowance is made for the difference in y found by Norem and others to exist between the two columns, the Chromosorb column acquires a higher value ( w G 0.4) than the glass bead column ( w % 0.3), a trend which agrees with theory. The difference, however, cannot yet be regarded as a significant one. DaI Nogare and Chiu have evaluated a large amount of data from which a gas phase mass transfer term has been isolated (8). Solutes were varied from those with low volatility to those with high volatility, in order to eliminate the liquid phase mass transfer term a t the two extremes of R or k (this method is similar to that of Kieselbach). These authors used a wide range of particle sizes, and employed both Chromosorb and glass beads. For Chromosorb columns, the nonpartitioning solutes, k = 0 or R = 1, showed an average Hmin,/dp= 2.1 and thus w = 0.79. I n the limit of k = ~0 or R = 0, H,,, / d p = 3.0, thus giving w = 1.62. The results were essentially the same for glass beads at low k values, but no results were reported for high k’s. Empirical w values are summarized in Table I1 for porous supports. The agreement among these results is satisfactory, considering the very wide range of experimental systems and methods of interpretation. At the present time, the best empirical w value can be estimated as 1.20 for R = 0 and 0.60 for R = 1. These are average values with a high weighting given to the results of Dal Nogare and Chiu, since the latter are perhaps the most extensive and consistent of all. The increase of w with increased retention is about twofold between the extremes. The experimental results are not yet consistent enough to show a reliable difference between Chromosorb and glass bead supports. The same may be said for the effects of particle diameter, tube diameter, carrier gas, and many of the other variables of gas chromatography, One additional experimental study which merits discussion was made in the author’s laboratory using very large
0.45 -1.0 1.12
0.33 0.79 1.62
Empirical w Values for Porous Supports k support 0.09 10 Firebrick
R
0-1 10 0 0 W
( d p G 0.12 cm.) uncoated glass beads in a column with an inside tube diameter-bead. diameter ratio of slightly more than 4. The plate height was found to acquire values less than dp within a certain range of flow velocities. A true minimum was not located, but the use of Equation 39 shows that w must be less than 0.18 for any plate height less in magnitude than dp. The reason for the abnormally small w value is not understood. It is possible that the influence of the wall in reducing the diameter of the exterior flow units was dominant, or that the bulk of the plate height term originates in the “long range” interaction of separate flow units, a phenomenon that would be ineffective with only four particles in one tube diameter. The possibility also exists that a uniform packing, often found for even smaller tube diameterparticle diameter ratios, reduces the plate height. A marble-jar model does not support this hypothesis, however. I t is hoped that additional work will be done to ascertain if the experimental results are truly reproducible, and to investigate the role of small tubeparticle diameter ratios. CONCLUSIONS
The agreement of the theoretically determined w values for porous supports (Equation 33) with the empirical values in Table I1 is very good, considering the nature of the approximations made. At this initial stage of theoretical investigation a two- or threefold error is expected and any closer agreement is probably accidental. A large uncertainty is also expected for the experimental values, although the results of Dal Nogare and Chiu show a high degree of consistency from one chromatographic system to another. In view of the probable theoretical and experimental errors, it is interesting to see what conclusions can be drawn and to discuss the chromatographic implications. One conclusion to be drawn from the theoretical and experimental work is that mass transfer effects in packed columns are not related in any substantial way to the effects in capillary columns. The best estimate (Equation 32) of the flow effects in the average channel leads to a plate height contribution which is less than 2% of the best
Reference
Chromosorb Sil-0-Cel Chromosorb Chromosorb Chromosorb
(W (87) (6)
(30) (8) (8)
empirical value a t the very greatest. In going from nonsorbing vapon to completely sorbed vapors the gas phase contribution of a capillary column increases elevenfold, whereas the empirical evidence for packed columns shows a rather mild twofold increase. The almost unexpectedly large empirical w values show the necessity for considering the interaction of separate flow channels (the word “channel” is used here in a restricted sense, since all flow paths are laterally connected to all others in the actual three-dimensional network). No other reasonable mechanism accounts for w values of this magnitude. As mentioned earlier, it is entirely possible that an even longer range interaction (between separate “units” of flow) may be needed to explain the full effects of maw transfer in the gas phase. The theoretically predicted variation of H,with R seems to be less than that indicated by the empirical evidence. On going from a nonsorbed to a completely sorbed vapor, the predicted increase in H, is 1.32 whereas the empirical increase is about 2. The theory is successful in explaining the moderate variations observed. The quantitative results are probably as good as expected a t this stage of development. In previous work the present author proposed a coupling between the classical eddy diffusion effects and lateral mass transfer by diffusion in such a manner as to yield a plate height term less than that provided by either alone (14, 19). This so-called coupling theory of eddy diffusion fits into the present treatment as follows: The basic source of peak spreading in chromatography is the existence of regions of different downstream flow velocities. This velocity is zero within the stationary liquid, very high within the central core of the flow unit considered here, and intermediate a t other places. It is obvious that solute molecules being carried in either the fast or slow stream paths will tend to get ahead of or behind the peak center, and a peak spreading will thus occur. If the solute molecules change from one region to another rapidly, they will not remain in any given region for a sufficient time to get very much in front of or behind the average. Thus a rapid exchange of solute between regions of different velocities is critical for minimizing peak VOL 34, NO. 10, SEPTEMBER 1962
1 191
spreading. This is why H , is always inversely dependent upon Do. There are, however, two mechanisms leading to the exchange of solute molecules. They can change velocities either by diffusing out of a region or by being carried out along the stream path with the carrier gas. Since each stream path goes through every different kind of channel in its passage through a column, the latter mechanism by itself, without diffusion, would provide a fairly rapid exchange. In either case the new velocity acquired by the solute molecule after a short time is a random one due to the random nature of the packing. Since the second mechanism is actually increasing the rate of exchange, it gives a term which essentially increases D, in the plate height expression. To a good approximation, then, the terms become additive in the denominator rather than in the numerator as usually assumed. The stagnant or nonmoving gas provides an exception to this process. There is no stream path to carry the gas away and so its only mechanism of exchange is by diffusion. An exception is also found in the predictable column-wide effects such as found in coiled columns, except a t extremely high velocities. If the total calculated plate height contribution due to the random variation in channel size and flow velocity is Z CQr.and that due to diffusion in the stagnant gas is C,, than allowance for eddy diffusion gives a contribution (excluding column-wide effects) of the approximate form
where A i is the eddy diffusion term for the i t h process. Many of the results and equations obtained here for gas chromatography are applicable to other forms of chromatography as well. The film diffusion concept of ion exchange chromatography (Sa) does not account for the truly dominant effects of interchannel interaction. The applicability of shallow bed kinetic studies to deep bed performance may also be questioned. Since the mobile phase contribution to plate height is larger than generally supposed, diffusion in the mobile phase may often be the rate-controlling mechanism. This conclusion may also be applicable to paper chromatography, where diffusion within the fibers has been considered as rate-controlling (29). The physical model used here for the packing structure and flow profile should be greatly refined for more precise calculations. The repeating flow unit should have a geometry (hexagonal rather than circular) that continuously fills all space. A more exact analysis should be made of the effects of flow constrictions which inhibit the free flow 1 192
8
ANALYTICAL CHEMISTRY
of gas through the large channrls. These effects could be highly significant if the flow velocity is greatly reduced. In addition, the flow of gas into and out of the large channels will have a lateral component which will affect the plate height to some extent. One of the major difficulties in the analysis of gas phase mass transfer effects originates in the long-range interaction of unequal flow channels. This effect could perhaps be calculated using statistically observed channel sizes over the cross section of a column. Experimentally such effects might be determined by observing the effects of changes in the tube to particle diameter ratio. I t would be rather difficult to predict a t this point the effects of variation in particle shape and character on chromatographic performance. A correlation between the two should be sought, since a practical gain might possibly result. The real need, however, is to fabricate a uniform packing structure in which interchannel interactions are eliminated, This would reduce the gas phase mass transfer effects by a factor of about 10. If mass transfer effects in the liquid could be greatly reduced, and there is every reason to believe they can be through the control of pore structure ( I d ) , the factor of 10 would lead to about three times the number of plates normally found a t the optimum velocity. The speed of analysis could be increased anywhere up to four or five times with vacuum outlet ( I S ) and up to ten times with normal atmospheric outlet (a). Even a moderate increase in the uniformity of packing structure would be well worth the effort expended in its development, NOMENCLATURE
Subscript 0 or 1 appearing with any parameter except the integration constants refers to the parameter’s value in regions 0 and I, respectively. A symbol with a bar above it indicates the average over the region to which the symbol refers. concentration concentration a t equilibrium D solute diffusion coefficient D, solute, diffusion coefficient in carrier gas d, = average support particle diameter F, = fraction of column volume occupied by solid go, gl = integration constants H , = plate height contribution of nonequilibrium k = ratio of solute in liquid and gas, respectively Q = fraction of solute in inner core R = fraction of solute in gas phase T = distance from center of structural unit T = tortuosity c
c*
= = = =
u v
z a
B y e
4
= zone or peak velocity =
gas velocity
= distance in flow direction = fraction of free gas space
in
inner core = ratio of annulus to core gas velocity = labyrinth factor for axial diffusion = equilibrium departure term
@
= ro/n = fraction of gas in interparticle
w
= coefficient
space indicating magnitude of H., Equation 26 LITERATURE CITED
(1) Alder, B. J., J. Chem. Phys. 23, 263 (1955). (21 Ayers, B. O., Loyd, R. J., DeFord, D. D., ANAL.CHEM.33, 986 (1961). (3) Bernard, R. A., Wilhelm, R. H., Chem. Eng. Progr. 46, 223 (1950). (4) Bethea, R. M., Adams, F. S., Jr., ANAL.CHEM.33, 832 (1961). (5) Bohemen, J., Purnell, J. H., J . Chem. SOC.1961, 2630. (6) Boyd, G. E., Adamson, A. W., Myers, L. S., J . A m . Chem. SOC. 69, 2849 (1947). (7) Collins, R. E., “Flow of Fluids through Porous Materials,” pp. 4-5, Reinhold, New York, 1961. (8) Dal Nogare, S., Chiu, J., ANAL.CHEM. 34. 890 (1962). (9, Deemier, J.’J. van, Zuiderweg, F. J., Klinkenberg, A., Chem. Eng. Sci. 5, 271 (1956). (10) Desty, D. H., Goldup, A., Whyman, B. H. F., J. I n s t . Petrol. 45, 287 (1959). (11) Giddings, J. C., ANAL. CHEM.33, 962 (1961). (12j Ibid., 34, 314 (1962). (13) Ibid., p. 458. (14) Ibjd., p. 885. (15) Giddings, J. C., “Chromatography,” E. Heftmann, ed., Chap. 3, Reinhold, New York, 1961. (16) Giddings, J. C., J. Chem. Phys. 31, 1462 (1959). (17) Giddings, J. C., J . Chromatog. 3, 520 (1960). (18) Ibid., 5, 46 (1961). (19) Giddings, J. C., Nature 184, 357 (1959). (20) Ibid., 188, 847 (1960). (21) Giddings, J. C., Fuller, E. N., J. Chromatog. 7,255 (1962). (22) Giddings, J. C., Seager, S. L., Stucki, L. R.. Stewart, G. H.. ANAL.CHEM.32. 867 (1960). ’ (23) Glasstone, S., Laidler, K. J., Eyring, H., “Theory of Rate Processes,” McGraw-Hill, Yew York, 1941. (24) Golay, M. J. E., “Gas Chromatography, 1958,” D. H. Desty, ed., p. 36, Academic Press, New York, 1958. (25) Jones. W. L.. ANAL.CHEM.33. 829 (i96i). (26) Jones. W. L.. Southwide Chemical I - -
- - I
Confereke. A.C.S.-I.S.A.. Memphis, Tenn., Dei. 6, 1956. ’ (27) Kieselbach, R., ANAL. CHEM.32, ~
880 (1960). (28) Ibid., 33, 23 (1961). (29) Mallik. K. L.. Giddinm. - , J. C.. Ibid.. ‘ 34, 760 (1962). ’ (30) Yorem, S. D., Ibid., 34, 40 (1962). (31) Rice, 0. K., J . Chem. Phys. 12, 1 (1944). (32) Walton, H. F., “Chromatography,” E. Heftmann, ed., Chap. 11, Reinhold, New York, 1961. RECEIVEDfor review April 9, 1962. Accepted July 13, 1962. Investigation supported by a research grant from the
National Science Foundation.
