Review Article | Open Access
Hubert M. Quinn, "A Reconciliation of Packed Column Permeability Data: Column Permeability as a Function of Particle Porosity", Journal of Materials, vol. 2014, Article ID 636507, 22 pages, 2014. https://doi.org/10.1155/2014/636507
A Reconciliation of Packed Column Permeability Data: Column Permeability as a Function of Particle Porosity
In his textbook teaching of packed bed permeability, Georges Guiochon uses mobile phase velocity as the fluid velocity term in his elaboration of the Darcy permeability equation. Although this velocity frame makes a lot of sense from a thermodynamic point of view, it is valid only with respect to permeability at a single theoretical boundary condition. In his more recent writings, however, Guiochon has departed from his long-standing mode of discussing permeability in terms of the Darcy equation and has embraced the well-known Kozeny-Blake equation. In this paper, his teaching pertaining to the constant in the Kozeny-Blake equation is examined and, as a result, a new correlation coefficient is identified and defined herein based on the velocity frame used in his teaching. This coefficient correlates pressure drop and fluid velocity as a function of particle porosity. We show that in their experimental protocols, Guiochon et al. have not adhered to a strict material balance of permeability which creates a mismatch of particle porosity and leads to erroneous conclusions regarding the value of the permeability coefficient in the Kozeny-Blake equation. By correcting the experimental data to properly reflect particle porosity we reconcile the experimental results of Guiochon and Giddings, resulting in a permeability reference chart which is presented here for the first time. This reference chart demonstrates that Guiochon’s experimental data, when properly normalized for particle porosity and other related discrepancies, corroborates the value of 267 for the constant in the Kozeny-Blake equation which was derived by Giddings in 1965.
The value of the constant of proportionality between pressure gradient and fluid flow velocity in a column packed with granular material and how it relates to column porosity have been controversial topics for some time . This is especially true in the field of chromatography because columns packed with porous particles have, at least for the last 50 years, been the main vehicle by which chromatographic separations have been carried out, and, thus, this feature has added a new dimension to the controversy. The complication arises, in part, because of the difficulty of differentiating between the free space between the particles and the free space within the particles in columns packed with porous particles; a problem that does not exist in the case of columns filled with nonporous particles.
More recently, because current chromatographic analytical and purification applications are being driven by the need for faster and faster separations, the relationship between pressure gradient and high rates of fluid velocity has taken on a special significance . For example, it is now common to see the use of very small particles (circa 2 micrometers) in combination with very large operating column pressures (circa 15,000 psi) . Because of the obvious impact of high operating pressure on the friability of chromatographic support particles, this development has created a new focus on the design criteria for packed columns, especially with regard to the fractional porosities within the columns. It will be appreciated that as the internal porosity of chromatographic support particles increases, the particles’ ability to withstand high pressures decreases. This physical reality would seem to dictate that chromatographic support particles suitable for this newly defined operational regime of high column pressures ought to have a lower particle porosity than those in common use for conventional column pressures. Paradoxically, however, one finds examples in the recent literature pertaining to fully porous particles where low values for the constant of proportionality in the Kozeny-Blake equation are being justified on the basis of particle porosities which purportedly run counter to this principle . Clearly, something is wrong.
The value of the permeability coefficient dictated by the necessity to balance the pressure gradient/fluid flow equation after all independent variables are accounted for (and which presumably has its origin in the fundamental mechanisms which underlie the generation of pressure in the first place) is by no means self-evident. From the standpoint of first principles, it would appear that its value ought to be constant regardless of the physical parameters comprising the fluid flow apparatus. However, no fully developed theoretical explanation for any constant of permeability has yet been published. On the other hand, the literature is replete with discussions of the value for this parameter which purportedly are justified on the basis of empirical data. The problem is that the authors of this literature do not agree with each other and sometimes the same author disagrees with himself. In the last few years, Georges Guiochon, in particular, has published conflicting values for this parameter, all of which are supposedly based upon measured values, yet he has made no serious effort to reconcile the conflicts or to provide any plausible underlying rationale for their differences.
As we discuss below, there are several inherent flaws in Guiochon’s teaching of permeability which are at the root of his conflicting values for the coefficient. We suggest that these flaws originate in the fact that his teaching does not distinguish between the respective impacts on column permeability of mass transfer by convection versus mass transfer by diffusion and spherical particles versus particles having an irregular morphology. Accordingly, without substantial refinement and adjustment, Guiochon’s teaching of column permeability, at best, does little to overcome the shortcomings of Darcy’s law and, at worst, adds fuel to the fire of the aforementioned controversy.
2. Guiochon’s Textbook Teaching
In his influential textbook, Fundamentals of Preparative and Nonlinear Chromatography , Guiochon (together with his coauthors) bases his teaching of the pressure/flow relationship in streamline flow on the Darcy equation which he displays therein as his equation ; a rearrangement of which we repeat here: where , , , , , and Guiochon’s residual permeability coefficient.
All terms are self-consistent from a unit-of-measure perspective.
Note that Guiochon does not discriminate in his expression of the Darcy equation between spherical and irregular particles. In other words, in his statement of the equation, a spherical particle with a measured diameter of cm is equivalent to an irregularly shaped particle of the same measured diameter. Peculiarly, however, he goes on to say that “traditionally, a value of is used for irregular particles and for spherical ones” [5, p. 153].
Similarly, without including any porosity dependence term in the pressure/fluid flow relationship or specifying any particular relationship between column pressure gradient and column porosity for columns packed with either porous or nonporous particles, he states that varies “between and , depending upon the compactness of the packed bed, that is, on the external porosity” [5, p. 153].
In essence, then, Guiochon bundles the variables of particle morphology and column porosity into his residual permeability coefficient, . Indeed, this is why we call it his residual permeability coefficient.
In order to explore these implicit terms which are embedded in Guiochon’s teaching for his , we turn, for comparison purposes, to the well-known Kozeny-Blake equation: where , , , and .
As we noted above, Guiochon acknowledges in his teaching that particle irregularity does influence pressure drop. In a modification of the Kozeny-Blake equation proposed by Carman, this factor was expressly accounted for by adding a parameter to represent particle sphericity, which we denote with the symbol (Carman gave it the symbol in his original publication ). It will be appreciated that the value of is equal to 1 for spherical particles and is less than 1 for particles having an irregular morphology. Accordingly, the Kozeny-Blake equation (as modified by Carman)1 is expressed as where Carman’s particle adjustment factor for deviations from perfect sphericity and .
As for column porosity, note that the Kozeny-Blake equation includes a combination of porosity terms which, taken together, can be viewed as its porosity dependence parameter, which we notate with the symbol : where the volume of free space between the particles in the column/the volume of the empty column.
Comparing (1) and (3), the reader will observe that there still is another difference; the fluid velocity frames are different. Accordingly, in order to establish the relationship between the values of Kozeny-Blake’s and the value of Guiochon’s , one needs to delineate the impact of Guiochon’s velocity frame upon his permeability coefficient. Before attempting to do this, a review of the various velocity frames typically used to characterize flow in porous media may be helpful.
3. Fluid Velocity Frames
3.1. Superficial Velocity
The fluid velocity term in both the Darcy equation and the Kozeny-Blake equation is the volumetric flow rate of the fluid divided by the cross-sectional area of the empty column. This velocity is known as the “average fluid superficial linear velocity” or, simply, the “superficial velocity” and is defined as follows: where volumetric flow rate, cross-sectional area of the column, and superficial velocity.
Thus, superficial velocity is not influenced in any way by the existence or structure of the packed bed. Furthermore, it does not change depending upon whether the particles in the bed are porous or nonporous.
3.2. Interstitial Velocity
The convective fluid velocity between the particles of a packed bed is called the “average fluid interstitial linear velocity” or, simply, the “interstitial velocity.” Interstitial velocity is the superficial velocity adjusted for the cross-section of the actual fluid flow in a packed column. It is related to the superficial velocity through the external porosityof the column, , as follows: where .
If interstitial velocity is to be substituted for superficial velocity in the Kozeny-Blake equation, one must also include the external porosity term. This term is the compensation factor that maintains the integrity of the value of the permeability coefficient between pressure gradient and fluid superficial velocity. The equation then becomes
3.3. Mobile Phase Velocity
Guiochon does not use either the superficial velocity or the interstitial velocity in the equations in his textbook. Instead, he uses the “mobile phase velocity.” Mobile phase velocity is uniquely a chromatographic term. It is the velocity calculated as a result of injecting an unretained, fully permeating solute into the column and measuring the time ofits elution. It is defined as follows: where the time it takes an unretained, fully permeating solute to traverse the column and mobile phase velocity.
The velocity experienced by the flowing fluid in a column, under steady state conditions, is not altered by the existence of “blind” pores in the particles of that column. Blind pores are pores within the particles which have only one fluid opening; that is they are not through-pores. Accordingly, under steady state conditions, the pores are already filled with identical stagnant fluid and there exists no driving force to exchange this stagnant fluid with the fluid flowing in the interstices of the particles. It will be appreciated that column permeability data is usually taken under steady state conditions in order to take advantage of a constant value for fluid viscosity. Although a solute which is dissolved in the flowing fluid will penetrate the stagnant fluid in the pores (assuming it is not strictly excluded by either physical dimension or chemical interaction) because of molecular diffusion, the driving force for this solute migration is the concentration gradient of the solute between the fluid outside the particles and the fluid inside the particles. This intraparticle migration of the solute does not contribute to friction and thus must not be included in velocity considerations related to pressure gradient. Mobile phase velocity, therefore, is not strictly a fluid velocity term; instead, it is a misnomer and it would better be termed the “unretained fully permeating solute velocity.”
Although Guiochon defines the relationship between fluid flow and pressure gradient in the context of a residual permeability coefficient, , and acknowledges that column porosity plays a role in that relationship, his use of mobile phase velocity fails to recognize that molecular diffusion in blind pores of the particle fraction of a column packed with porous particles does not influence pressure gradient.
Mobile phase velocity is related to fluid superficial velocity, however, through the total porosity of the column. “Total column porosity,” , is a term peculiar to the chromatographic literature which relates to the use of columns packed with porous particles. It is the fraction of the column volume taken up by the free space between the particles plus the fraction of the column volume taken up by the free space in the internal pores of the particles. Thus, it is the sum of the external and the internal column porosities: where total column porosity and internal column porosity.
Mobile phase velocity is related to fluid superficial velocity through the total porosity of the column as follows: Thus, if mobile phase velocity is used in the Kozeny-Blake equation, one must insert the compensation factor into the equation in order to maintain the integrity of the value of the permeability coefficient, , between pressure gradient and fluid superficial velocity, and the equation then becomes And, now reconciling (1) and (3), we get
4. The Flow Resistance Parameter
Before proceeding any further, we digress to define another term which will become important to our understanding of Guiochon’s work. The chromatographic literature contains a term called the “flow resistance parameter” . The fluid velocity underlying this parameter is based upon Giddings’ definition of “mean fluid velocity through a cross-section” and is found in his 1965 text book at page 207 2. It is defined as follows: where .
The flow resistance parameter, therefore, is equivalent to the reciprocal of Guiochon’s term :
5. Refining the Terms For Column Porosity
In order to continue with our comparison of Guiochon’s teaching to that embodied in the Kozeny-Blake equation and, in particular, to accommodate columns packed with porous particles, we need to refine the porosity dependence term in the Kozeny-Blake equation. To accomplish this, we adopt the definition of column porosity as taught by J. Calvin Giddings ; namely, where the porosity dependence term in Kozeny-Blake (porous particles) and where the ratio between the external and total column porosities.
Using this methodology in the most general case involving porous particles to express total column porosity (where total column porosity is not the same as external column porosity) as opposed to in the special case involving nonporous particles (where total column porosity and external column porosity are the same), the Kozeny-Blake equation (using mobile phase velocity as in (10)) may now be reexpressed as follows: It will be appreciated that in the special case of columns packed with nonporous particles, (17) is identical to (3) because here = and, thus, = = and .
6. The Relationship between and
We can now connect the teaching of Guiochon to that embodied in the Kozeny-Blake equation as follows: or As illustrated by (19), the term in the Kozeny-Blake equation is the flow resistance parameter normalized for the column porosity dependence term.
Note that if is a true constant, the two parameters upon which it depends, and , will vary in direct proportion to each other. This is illustrated by the data in Table 1 herein, which is a reference chart where particle porosity, , is varied from nonporous at one extreme to highly porous at the other, where volume of the free space within the particles in a packed column/the total volume of all the particles in the column.
Figure 1(a) is a plot of the data contained in Table 1 as versus . Proceeding from the lower left hand corner to the upper right hand corner of the plot, particle porosity increases. It is obvious from the plot that all the data in Table 1 falls on a single line with a slope of 267, which we say is the constant value of . As is illustrated by the plot, in the range of porosity covered by the data, the values for vary from about 500 to about 1900 and, correspondingly, the values of vary from approximately 2 to approximately 7. Note also that the upper boundary value of for packed columns is about 5.5 with larger values related to capillaries.
Finally, we point out that once the value of is correctly identified, a measured value for the flow resistance parameter may then be used to calculate the value of the external porosity of the column, , when the latter’s value has not or cannot be measured.
7. Guiochon’s Modified Permeability Coefficient,
As reflected in some of his recent publications, which we will explore in more depth later, Guiochon inadvertently, apparently, confuses the term with another parameter, , which we identify here and define for the first time. is not the same as because it is based upon Guiochon’s unique (from a permeability perspective) teaching for fluid velocity. It is defined as follows: where Guiochon’s modified permeability coefficient.
It will be appreciated that Guiochon’s , if used in the Kozeny-Blake equation in place of , accommodates his use of the mobile phase velocity in a way that is different from, but equivalent to, a direct conversion of mobile phase velocity to superficial velocity.
In Table 2, a cross-reference is established between Guiochon’s teaching for columns packed with both porous and nonporous particles, the Kozeny-Blake equation (as modified by Carman), and, for validation purposes, Giddings’ teaching based upon actual permeability measurements from his Table in his 1965 text .4
In his worked example, Guiochon’s fluid velocity, , is found in his text as his equation (2.56) , at page 61, and equates to mobile phase, not superficial, velocity. Accordingly, when applied to columns packed with porous particles, his worked example combines the contribution to fluid velocity of mass transfer by molecular diffusion in the free space within the particles with mass transfer by convection in the free space between the particles. As discussed above, if one is attempting to compare Guiochon’s teaching for the value of the permeability coefficient with that of the Kozeny-Blake equation, the conversion factor is the total porosity of the column.
In addition, because Guiochon’s particle size parameter has an embedded assumption due to particle shape/roughness, his worked example also commingles the contribution to spherical particle diameter equivalent of particle shape/roughness with other nonparticle variables. Unfortunately, his worked example is not based upon completely spherical particles where this would not be an issue. We know this because he states that in his worked example equals 1 × 10−3, which he says is the value of the permeability coefficient for columns packed with irregular particles.
In order to move forward, one must simplify. If this were to be done experimentally, it would best be accomplished by experiments with nonporous particles, in which case the variable of molecular diffusion in the pores of the particles would be eliminated and by using only smooth, spherical particles, such as glass beads, in which case the variable of particle shape/roughness would be eliminated. If both of these things were done, one would be able to directly and unambiguously determine the true value of .5
Accordingly, since his textbook contains a general teaching without any expressed restrictions with respect to particle porosity, let us first assume that the column in Guiochon’s worked example is packed with nonporous particles. Secondly, let us assume that the value for the external porosity of the column (which is the same as the total porosity in a column packed with nonporous particles) is the value traditionally assumed for a typical, well-packed column, that is, 0.4.
As illustrated by example A in Table 2, this would lead to an apparent value for of 444, which is obviously much too high.6 Using the adjusted particle size of 8 micrometers (rounded), which is calculated based on a value of 0.78 for , we derive values of 267 for and 107 for (the relationship between these two values is, of course, equal to the total porosity of the column, , as is dictated by (20)).
Similarly, as illustrated by example B in Table 2, when the column of Guiochon’s worked example is adapted to accommodate spherical, nonporous particles, Guiochon’s teaching of the value for of 1.2 × 10−3 results in an apparent value of 369 for 7, again, too high [10, 11]. On the other hand, if we correct the value of , we get a column external porosity of 0.3620,which is too low for a typical well-packed column () . Moreover, in the case of both corrections, Guiochon’s teaching overstates the pressure gradient (187 psi compared to the 135 psi which Giddings’ experiments would generate). Accordingly, it is only when we correct for the pressure drop that we achieve appropriate values of 0.4 for , 267 for , and 107 for .
As is evident from Table 2, when the porosity of columns packed with nonporous particles varies, the value of also varies because it is based upon the measurement of mobile phase velocity, which, in turn, changes with the porosity of the column. Indeed, for typical columns packed with nonporous particles having external porosities close to 0.4 (0.38–0.42) , Table 2 reveals that the value of varies by as much as 6 points, while the value of remains constant at 267.
Having established a range of values for based upon columns packed with nonporous particles, one can now proceed to evaluate the range of values for based upon columns packed with porous particles, which is to say that we can now evaluate the impact of particle porosity on permeability. Among other things, Table 2 shows that when it comes to porous particles, Guiochon’s is even more variable (and, thus, even less useful) because the value of the coefficient can differ on a column-to-column basis by as much as 90 points, depending on the combination of the internal andexternal column porosities at play.
Figures 1(a) and 1(b) are a plot of the data found in Table 1. In this plot, Guiochon’s modified permeability coefficient, , is plotted against column total porosity, . Once again, it will be appreciated that (a) all data falls on a line with a slope of 267 which represents the value of and (b) the line has no intercept, which means that when is zero, is also zero. As shown in the plot, values of the parameter are highest at low values of and lowest at high values of . Further, as is evidenced by the respective definitions for the terms and contained herein, the only time that the value of Guiochon’s does not differ from the value of (267) is when the value of is 1, its value in a theoretical packed column devoid of particles, that is, acapillary.
Finally, Figure 1(c) is another plot of the data in Table 1, but this time the values for are plotted against particle porosity, . The equation of this line does have an intercept, which represents the value of when the particles are nonporous. It is also obvious from the equation of this line that it is only when particle porosity is unity, that is, in a capillary, that the value of is 267.
It will be appreciated that since is the sum of the internal and external porosities of a packed column, any given value for involving a column packed with porous particles can represent many different combinations of these parameters. Vice versa, any given value for can generate many different values for depending on the amount of particle porosity which is present. On the other hand, as is pointed out earlier, column internal porosity and/or particle porosity play no role in determining the permeability of a packed column; the only porosity parameter which is relevant for such purposes is column external porosity. In other words, as a permeability coefficient, is not a particularly informative concept because it still fails to provide any meaningful tie to the underlying physical phenomena that govern the pressure drop/fluid flow relationship.
In the case of permeability measurements, one quantifies only total pressure drop. Pressure drop, however, is a commingling of the contributions of the several different parameters embodied in the pressure/flow relationship. Therefore, one must accurately represent the contribution of each factor before adding them together to equal the pressure drop. Obviously, if one mistakes the contribution of any given element, for instance particle size, this will skew the contribution of some other element when summed, for instance, column porosity.
Our frame of reference has been calibrated with respect to the interrelationships between the values of , , , and based on the data reported in Giddings’ table in his 1965 textbook. Since his data base was derived from experiments using nonporous particles, the permeability results of which were extended to columns packed with fully porous particles by using his parameter, the validity of which, in turn, was established by independent means, there is no uncertainty with respect to the critical variable of external porosity, , inherent in our frame of reference. It is for this reason, therefore, that our frame of reference trumps any reported data which uses any other technique, including the technique of inverse size exclusion chromatography by itself to establish values for . In addition, our frame of reference is internally consistent due to the grounding of the parameters at the boundary condition of a theoretical capillary, at which point the values of and are unity and the values of and are equal. In essence, these interrelationships exemplify the conservation laws of nature which are the ultimate standard of measure when considering partial fractions of a single entity.
Consequently, as Figures 1(b) and 1(c) illustrate, if one knows based upon other independent means, the value of a column’s total porosity or its particles’ porosity, respectively, one can at least verify whether results of permeability experiments expressed in terms of are reasonable. Or, to put it another way, if the results reported do not conform to what these plots tell us one should expect from columns and particles based upon independently derived porosity values for the particles under study, we know that something is wrong. Accordingly, this is the way in which we use below to analyze and evaluate Guiochon’s experimental work of the last decade.
8. The Origin of Guiochon’s Value for Column Permeability
In 1966, Guiochon wrote a review paper entitled “Problems Raised by the Operation of Gas Chromatographic Columns” . In that paper, he reviewed the development of the permeability equation from Darcy to the then present. Among other things, he reported that “a mean ε of 0.38 is found for a number of packings obtained from powdered products of various origins, consisting of spherical or irregular particles. Almost all the measured values are between 0.35 and 0.45” (p. 14). He then went on to discuss what he called the “semiempirical Blake-Kozeny equation,” which he reported as [13, p. 15. Equation ]. What sticks out, of course, is the fact that Guiochon states here that equals 150, the value for the permeability coefficient championed by Ergun, not the value posited by Carman. To be fair, we should point out that Guiochon gave warning that he was uncomfortable with any particular value for the permeability coefficient. For even though he recited the Kozeny-Blake equation with a value of 150 for its permeability coefficient, he stated that “the values of given by (this) equation are not very accurate, and the deviations can be as large as 50%” [13, p. 15]. Guiochon attributed most of this uncertainty to the inability of practitioners of that era to effectively measure the value of a column’s external porosity. Since, as he pointed out, “(the Kozeny-Blake) equation is very sensitive to variations in ,” he concluded that the equation is “of little use” and that it was better practice to rely upon grosser measurements of permeability [13, p. 15]. Thus, Guiochon at this timeframe made a conscious decision to stay within the relatively safe, but admittedly more obscure, boundaries of Darcy’s teaching on column permeability.
In 2006, however, Guiochon emerged from under the shadow of Darcy permeability when he published a review paper which purported to be a comprehensive summary of the current state of the art in chromatography . In light of the fact that Guiochon and his coauthors had virtually contemporaneously published a new version of their textbook, one would have expected that Guiochon would take this opportunity to reaffirm his textbook teaching. So, it should come as no surprise that he would open his discussion of the pressure drop/fluid flow relationship with the following assertion: “the permeability of packed beds used in liquid chromatography is in the order of 1 × 10−3” [14, p. 9]. As one would expect, the authority which Guiochon cites for this seemingly familiar proposition is the 2006 edition of his own textbook. For the first time, however, he also provides some citations to third party sources. In particular, he cites a 1997 chromatography handbook written by Uwe Neue, Chief Scientist for Waters Corporation [15, p. 9]. As one can see, Neue’s equation—like Guiochon’s—does indeed postulate a permeability coefficient with a value of 1 × 10−3.
At this point, however, Guiochon abandons his traditional caution about the value of the permeability coefficient and marries his permeability equation to the Kozeny-Blake equation, thereby endorsing not only the porosity dependence term in that equation but also a specific value for . Referring to his value of 1 × 10−3, he states the following: “the specific column permeability is related to the column porosity through the Kozeny-Carman equation: ” [14, p. 9]. Not surprisingly, Neue makes the same connection: “the specific permeability depends on the particle size and the interstitial porosity of the packed bed. The relationship is known as the Kozeny-Carman equation: ” [15, p. 30].
Neue’s permeability equation and Guiochon’s textbook permeability equation, however, differ in that Neue uses the superficial, not the mobile phase, velocity. Accordingly, since all other features of their equations are the same, one would expect them to get different values for their permeability coefficients. Neue himself recognizes this when he says in his handbook “this value is also in good agreement with our experience, but values as low as 500 have been reported in the literature.8 However, there are several reasons why different values reported by different workers do not represent true differences in the resistance parameter. First, lower values are obtained for porous packings if the specific permeability is measured on the basis of the breakthrough time of an unretained peak instead of using the flow rate”; see [15, p. 32] (when he distinguishes the “breakthrough time of an unretained peak” from the “flow rate,” Neue is, of course, referring to the mobile phase velocity and the superficial velocity, resp.). Consequently, if one takes Neue’s point and turns it on its head, one can see that if he and Guiochon were to get the same value for their permeability coefficients , but one (Guiochon) is using mobile phase velocity to derive it and the other (Neue) uses superficial velocity to derive it; this would indeed represent a “true difference” in their equations.
So, how can Guiochon cite both Neue and himself for the proposition in his review paper that “the permeability of the packed beds used in liquid chromatography is in the order of 1 × 10−3”? The answer is that, at least for purposes of his review paper, he adopts Kozeny-Blake’s fluid velocity convention in place of the velocity frame he uses in his textbook. Here is what he says: “this approximation is valid only if the superficial velocity is used” [14, p. 9] (emphasis supplied).
As a matter of pure mathematics, Guiochon’s permeability coefficient of 1 × 10−3 can only be equal to if the value of is 0.4, the typical value for the interstitial fraction of a well-packed column. Guiochon confirms in his review paper that this is indeed what he assumes in his statement of the Kozeny-Blake equation [14, p. 9]. Accordingly, his adoption of the value of 180 for solely depends on the validity of his assertion that the value of the permeability coefficient is 1 × 10−3.
We already know from our discussion of Guiochon’s textbook teaching that when this value is associated with a permeability equation based on mobile phase velocity, it is wrong. As reflected in our Table 2, when Guiochon’s worked example involving this figure is corrected to account for the embedded factor due to the irregularity of the particles which his worked example assumes, the value of the permeability coefficient generated by his equation is not 1 × 10−3; it is 1.66 × 10−3.
So, where did this value of 1 × 10−3 come from? Although Guiochon in his review paper cites Neue’s handbook for this value, Neue himself offers absolutely no authority for it. On the other hand, Guiochon does provide one other citation in his review paper for this value. The citation is to an article by Halász et al. published in 1975 .
If one then goes to that article, one sees that Halász et al. do indeed proffer the same equation as Neue (albeit in a somewhat different format) and, yes, it does contain the value of 1 × 10−3. See [16, Equation ]. Nevertheless, the authors fail to identify from whence they obtained this value. On the other hand, they do make reference to a paper that Endele et al. wrote a year earlier, in 1974, with Klaus Unger entitled “Influence of the Particle Size (5–35 μ) of Spherical Silica on Column Efficiencies in High-Pressure Liquid Chromatography” . It appears that this is where the body is buried; for it is in this article that Halász reports having actually derived a value of 1 × 10−3 for the permeability coefficient.
In summarizing the results of their experiments, which, incidentally, are based on measurements of superficial, not mobile phase velocity (see their Equation ), Halász et al. calculate an average value for the permeability coefficient (which they notate as ) of . Accordingly, they state the following: “t is proposed to define an average particle size, , for columns packed with spherical silica using the balanced density packing method by the equation ” [17, p. 382 their Equation ].
To recapitulate, it thus appears that Halász was the source of the value of 1 × 10−3 for the permeability coefficient which Guiochon adopts in his review paper, Halász’ value was based upon measurements which involved spherical particles, and Halász’ value was properly derived from an equation which utilized the superficial velocity as the fluid velocity.
However, we still have a problem: if we are going to accept the value of 1 × 10−3 as a basis for calculating the constant in Kozeny-Blake, we still need to know the value of Halász’ columns’ external porosity. For example, in the absence of measured values for the external porosity, Guiochon’s assertion in his review paper that the value of the constant is 180 is without foundation.
Unfortunately, however, Halász bases his methodology on “assumed” versus “measured” values for external porosity. Nevertheless, he does provide some clues that may help us peel back this onion. First, he says that his value for the permeability coefficient applies to “columns using the balanced density packing method.” Secondly, after stating his proposed value for the coefficient, we find the following telltale sentence: “the permeability of columns packed by the conventional “dry” method are worse by a factor of about 2 than those packed by the balanced density method” [17, p. 382]. For this proposition, Halász cites yet another of his own articles, this time a paper written by himself and M. Naefe in 1972 entitled “Influence of Column Parameters on Peak Broadening in High-Pressure Liquid Chromatography” .
When one follows this thread by going to Halász’s 1972 paper, one finds a possible solution to the puzzle. For this is where Halász describes his calculations of the permeability coefficient from experiments with a number of conventional, dry-packed columns containing spherical silica particles. Not surprisingly, the resulting value for the permeability coefficient for such columns was not 1 × 10−3. Instead, Halász says that his experiments with such columns resulted in the following value for the permeability coefficient: “~” [18, p. 78]. In other words, consistent with what Halász says in his 1974 article, the value for the coefficient for these dry-packed columns was indeed “worse by a factor of about 2” compared to its value for columns prepared with the balanced density method.
The obvious implication of this is that when Halász got a value of 1 × 10−3 for the permeability coefficient in 1974, he must have been evaluating columns that were packed to an interstitial fraction considerably higher than the external porosity of the columns that were the subject of his 1972 experiments. Our problem, however, is still not resolved because Halász did not accurately determine the external porosity of the columns under study in either the 1972 or 1974 columns.
We surmise that, in their 1972 paper, Halász and his coauthors, made use of the teaching of Giddings outlined in his 1965 textbook, which at the time of their 1972 paper was seven years old. Here is what they say: “experience has shown that in a regular packed column, with the sieve fractions usual for chromatography, is independent of the particle size if the particles are more or less spherical. In an acceptable approximation, is equal to 0.4. In those columns packed with nonporous supports and are identical. Using porous supports (i.e., silica gel, alumina, chromosorb, Porasil, and so forth), , where is the total porosity and indicates the pore volume of the support.” It will be appreciated that stands for interstitial velocity (our ) and stands for mobile phase velocity (our ). The “experience” which Halász and his coauthors are referring to concerning “columns packed with nonporous supports, alumina, and chromosorb” is that recorded by Giddings in Table in his 1965 text. Continuing to establish the interrelationships inherent in the various entities involved in chromatographic columns with respect to both interstitial and mobile phase velocity, the authors conclude: “formally can be calculated for a porous support, assuming is equal to 0.4.” (emphasis supplied). The authors then announce the methodology underlying their frame of reference with regard to column permeability as follows: “we determined in all of our columns the , ratio to be 0.56 with a differential refractometer. Consequently, the total porosity was equal to 0.714.” It will be appreciated that their ratio of is one and the same as Giddings’ parameter, as utilized by him in his 1965 textbook and as we define hereinabove.
We can now identify the origin of the discrepancy between Halász and Giddings and, by extension, between Halász and Guiochon (and Neue). For even though Halász makes use of the results in Giddings’ Table , he fails to appreciate the teaching inherent therein regarding the importance of an accurate value for column external porosity when using porous particles. In establishing his values for , Giddings was careful to ground his values for the external porosity of columns packed with porous particles in a comparison of their measured pressure drops with measured pressure drops in columns packed with nonporous particles, where the values for (external) porosity were accurate because they are equal to the columns’ total porosity.
In their 1972 paper, Halász et al. used a refractometer to detect the inert solute, -pentane, in the mobile phase of -heptane. They injected the -pentane into the flowing mobile phase, the flow rate of which they presumably measured with a volumetric cylinder and stop watch. Since the exit time of the -pentane peak was identified by the refractometer, their measurement of holdup volume was accurate and, accordingly, so was their value of 0.714 for (there is no uncertainty related to column total porosity when using inert solutes). This, in turn, led to an accurate value for their term, mobile phase velocity. However, since they did not measure the interstitial velocity, , but rather used an estimate based upon the measured flow rate and an “assumed” value of 0.4 for external porosity, their calculated/estimated value for was arbitrary. This, in turn, means that the value of 0.56 for their ratio was likewise arbitrary.
In Table 3 herein, we show the results for column number 1 in the 1972 study which we designate as Column A1. Note that the authors’ raw values correspond to a value of 500 for , a value which is obviously far too high. As is plainly evident in our Figures 2 and 3, this column data set does not conform in any reasonable fashion to the norms in our reference charts. Accordingly, the data is badly flawed. In our corrected data for Column A1 in Table 3 herein, we adjust the column external porosity to the lowest value permitted in our frame of reference (0.38). This correction results in a revised value for Giddings’ parameter of 0.532. In addition, we are compelled to adjust the particle downward (to 11 micrometers, approx.), as is also dictated by our frame of reference. We justify our low corrected value for external porosity and our downward adjustment for particle size on the aggressive “dry-packing” process described by the authors; “ the best results were achieved with the following method: the column (i.d. = 2 mm), 25 cm long, was packed with 25 equal portions of the “brush.” After adding each portion, the column was vibrated and afterward tapped on the floor and also vigorously tapped with a rod from the side.” Based upon a column packing experience of the principal author herein spanning more than 30 years, we believe that these packing conditions generated a packed column with a low external porosity and also one in which the particles experienced significant breakage.