Perspective pubs.acs.org/ac
Perspectives on the Evolution of the Column Efficiency in Liquid Chromatography Fabrice Gritti and Georges Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, United States ABSTRACT: When analyses of mixtures of small molecules are carried out at mobile phase velocities close to (for isocratic runs) or somewhat above (for gradient runs) the optimum velocity, the eddy diffusion term contributes to at least 75% of the band broadening. Future improvements in column performance may come only from a reduction of the eddy diffusion term. The classical models of axial dispersion of Gunn and Giddings are revisited and their predictions compared to recently reported eddy dispersion data obtained by solving numerically the Navier−Stokes equations and simulating advective-diffusive transport in the bulk region and in confined geometries of reconstructed and computer-generated random sphere packings. The Gunn model fails to describe these data. In contrast, the Giddings model succeeds, provided that his original guesses regarding the values of two parameters of his model are adjusted. Accurate measurements of real eddy dispersion data in modern high-pressure liquid chromatography (HPLC) columns were performed by applying a well established experimental protocol. Their results demonstrate that the other contribution to band broadening, sample dispersion in the homogeneous bulk region of these packed beds, accounts for less than 30% of the total eddy dispersion at velocities larger than the optimum velocity. This shows that the resolution power of modern HPLC columns is essentially controlled by wall and/or border layer trans-column eddy dispersion effects, depending on whether the column is radially equilibrated or not. Under a preasymptotic dispersion regime, the performance of short and wide HPLC columns is controlled by the border effects. As the bed aspect ratio (D/dp) increases, the column performance tends toward that of the infinite diameter column. Further improvement appears possible using radial segmentation of the outlet flow. Under an asymptotic dispersion regime, the reduced column plate height of long and thin HPLC columns is controlled by the wall effects and can be optimized only by improving the packing procedures, keeping as low as possible the bed aspect ratio and maximizing the transverse dispersion coefficient.
T
packing methods and to reduce important back-mixing effects that limit column efficiency. This report explains how the recent trends in the evolution of column properties and performance have led to this situation, why these trends cannot be pursued any longer, and what might be attempted to keep improving column performance. Since the early developments of high-performance liquid chromatography, the characteristic size of the stationary phases used in high-pressure liquid chromatography (HPLC) were shrunk by 2 orders of magnitude. In the late 1970s, when HPLC began as an analytical method, it used coarse, irregular, ∼100 μm size particles.3−7 By successive steps, these particles became sub-2 μm, spherical ones.8 This evolution had two consequences. First, the column length was reduced from a few meters to a few centimeters while the average column efficiency increased and the analysis time was considerably reduced; second, the inlet pressure at which HPLC instruments were operated had to be increased, from a few bars to over 1200 bar. After nearly 50 years of research and development, the typical column is now 5 cm long, it is packed with sub-2 μm particles,
he primary quality of a column, its efficiency, is proportional to its length, L, and to the reciprocal of a characteristic scale or repetitive distance, d, of the chromatographic bed. This applies whether the column bed is made of densely packed particles (with d = dp, the average particle diameter1) or of a monolithic material (with d the average domain size2). The last 10 years have seen the column characteristics evolving after conflicting trends that might have been difficult to understand for many users. In their constant efforts to increase analytical throughputs, column manufacturers have offered shorter, narrower columns packed with finer particles. Their progress in designing, manufacturing, and packing particles has resulted in the standard column of today that provide higher efficiency and better sample resolution than those of 10 years ago, in a far shorter time. We could be tempted to extrapolate these trends and wonder how much farther manufacturers could improve column performance. Unfortunately, it seems that we have now reached physical limits and that new, serious problems will be extremely difficult to solve. These obstacles are so serious that it seems that the further progress that we may see in the near future should be expected to come from a different line of work. Instead of striving to reduce the particle size and shorten the columns that are packed with them, we should work to develop better © 2013 American Chemical Society
Received: November 16, 2012 Accepted: February 15, 2013 Published: February 15, 2013 3017
dx.doi.org/10.1021/ac3033307 | Anal. Chem. 2013, 85, 3017−3035
Analytical Chemistry
Perspective
that accounts for the fact that axial diffusion in a bed cannot take place along a straight line. The new dimensionless parameters, h and ν, are the reduced plate height and reduced velocities. The plot h = f(ν) is called the van Deemter or reduced HETP curve. It has two asymptotes, a vertical one for ν → 0 and a steep one for ν → ∞. This curve has a minimum, the coordinates of which indicate how well the column is packed and how fast analyses can be made. Equation 2 fits relatively well to most experimental HETP curves, but this apparently good agreement is strongly misleading. To explain this difficulty, we need to relate the parameters of the van Deemter equation, γ and C, to the column characteristics. For this purpose, we present recent progress made in our understanding of diffusion in complex media, beds of packed particles, monoliths, and structure of the porous networks in particles and monoliths. This requires the use of sophisticated models of effective diffusion coefficients in particles and in column beds.20−25 For instance, the impact of intraparticle surface diffusion in reverse-phase liquid chromatography (RPLC) packing materials was long neglected or underestimated. Although surface diffusion is inconsistent with the Langmuir adsorption model, it does take place in different adsorbents. It is particularly important with those used in RPLC.26,27 It was recently shown that surface diffusion actually contributes to at least 80% of the effective particle diffusivity.28−33 This observation requires a revision of the expressions providing the coefficients γ and c. The coefficient a has long been the source of theoretical perplexity. After initial investigations, Giddings suggested that it was not a constant as written by van Deemter et al.17 but a function of the mobile phase velocity.1 Later, an empirical study34 confirmed the coupling between a flow-induced and a diffusion-induced molecular transport within the interparticle or external volume across which significant velocity biases can take place. This lead Knox to propose an empirical plate height equation still accepted by most. Recently, a noninvasive experimental protocol35 confirmed clearly that the coefficient a depends on the mobile phase velocity in both packed36−38 and monolithic columns.12,39,40 This work progressively lead to a complete re-evaluation of our understanding of the mechanisms of band broadening and of the relative importance of the different contributions involved. Actually, most physical effects that contribute to the term a are absorbed by the classical c term. Although being correct experimentally, the van Deemter equation is based on incorrect approximations. Influence of the Molecular Weight of the Solute. All the coefficients of the plate height equation depend on the diffusion coefficients of the molecules of the sample components. Depending on the sample origin and on the purpose of the analyst, the molecular weights of the analytes may vary in a wide range of molecular weights, from inorganic ions, to organic molecules with molecular weights below 500 Da, to proteins or larger molecules of biochemicals that may be several hundreds times heavier. This has important consequences on the experimental conditions under which samples may be analyzed. Systematic measurements made by countless investigators have shown that the reduced plate height at which the efficiency of a column is maximum is between 5 and slightly more than 10. Thus the optimum column efficiency, or efficiency corresponding to the minimum value of the HETP, is not constant for any given column but depends on the nature
and it can easily yield peak capacities of the order of 40 in less than a minute and requires an inlet pressure around 1000 bar.9 A comparable level of performance can be reached with various column types packed with sub-2 or sub-3 μm fully porous or core-shell particles,10,11 with silica monolithic columns of the second generation.12−15 Details on this evolution were previously reported.16 The efficiency of a column of given length and bed characteristic scale is inversely proportional to its height equivalent to a theoretical plate (HETP). This parameter is mainly a function of the linear velocity of the mobile phase during isocratic runs. The importance of the HETP was demonstrated by van Deemter et al.17 in a landmark paper on band broadening in linear chromatography. These authors simplified the solution of the kinetic model derived earlier by Lapidus and Amundson18 by assuming that the shape of the injection profile is a Dirac pulse injection. They established that the column HETP can then be expressed as the sum of the contributions of three contributions, those of (1) longitudinal diffusion along the mobile phase percolating through the bed, which is inversely proportional to the mobile phase velocity since the lesser time the band spends in the column, the lesser time it has to diffuse; (2) eddy dispersion, due to the anastomosis of the channels conveying the sample band across the bed of particles; and (3) the mass transfer resistance, due to the time molecules need to diffuse in and out of the particles and through their pores. These three contributions account for all the band broadening due to the mass transfer processes encountered in any type of chromatographic column (open tubular, packed, or monolithic columns), independent of the physical state of the mobile phase (gas, liquid, or supercritical fluid), the flow regime (laminar to turbulent), or the nature of the stationary phase (liquid, solid, or narrow pores where molecular movements are hindered). Over the last fifty years, the van Deemter HETP model was proven to successfully fit to all plots of the experimental plate height versus the mobile phase velocity. This explains why this equation is considered as the best one in chromatographic sciences. Strikingly, however, a scrupulous investigation of the most reliable values found for the coefficients of the van Deemter HETP equation reveals considerable difficulties in providing detailed, accurate explanations regarding their meaning and their relationships with physical chemistry.19 Recent investigations were able to clarify a number of difficult issues, particularly by explaining that the eddy dispersion term is not independent of the mobile phase velocity, as was believed earlier, and why it is not. They also showed a profound difference between the relative importance of the contributions to band broadening depending on the size of the molecules analyzed. van Deemter Equation. The initial expression of this equation, as it has been used for long, is the first of the following two equations. The second one, dimensionless, is more elaborate: H=A+
h=a+
B + Cu u
2γ + cν ν
(1)
(2)
where h = H/dp is the reduced plate height and ν = udp/Dm is the reduced mobile phase velocity, with dp the particle size, Dm is the diffusion coefficient of the solute in the mobile phase, u is the interstitial linear velocity, and γ is a geometrical coefficient 3018
dx.doi.org/10.1021/ac3033307 | Anal. Chem. 2013, 85, 3017−3035
Analytical Chemistry
Perspective
because intraparticle diffusivity increases with increasing temperature while the small thickness of the porous shells in these particles reduces the average diffusion path through the particles and accelerates mass transfer.4,53,54 This has been thoroughly studied and already reported in the literature.11,16,55−57
of the compound considered, through its diffusion coefficient. Since ν=
ud p Dm
(3)
■
The flow velocity must be chosen so as to obtain a value of ν between 5 and 10 (the exact value is not critical since the HETP varies slowly around its minimum). The diffusion coefficients of small molecules are of the order of 1 × 10−5 cm2/s. The practical values of the reduced linear velocities used today in chromatography are typically smaller than 20. As discussed later, for reduced velocities around and beyond the optimal velocity, more than 80% of the reduced plate height is controlled by the eddy diffusion term because the contribution of the cν term is negligible. Any further improvement of column performance should come from reducing the value of the A coefficient in the HETP equation. This shows why axial dispersion in porous beds has become of paramount importance in various other fields of chemical engineering, including the transport of contaminants in water tables, oil and gas extraction from underground rocks, and reactant and product transport in packed bed reactors. The physics of flow and mass transport through porous media was extensively reviewed in several monographs and review papers.41−45 Four main dispersive regimes can be distinguished for liquid flows through porous media, as a function of the Peclet number:44,45 the diffusion regime (ν < 0.3), the powerlaw regime (0.3 < ν < 300) in which both diffusion and convection contribute to dispersion, the mechanical or pure convection regime (300 < ν < 105), and the turbulent regime (ν > 105), in which case, the Reynolds number should be added to the Peclet number as a relevant parameter to predict the dispersion coefficient. In chromatography, the Peclet number is the reduced velocity; it ranges between unity and a few hundreds, depending on the analyte diffusion coefficient. The dispersion regime depends on the combined effects of diffusion and convection on the analyte dispersion. The overall band broadening and the column efficiency are essentially governed by the axial dispersion of the sample zone along the packed bed46−49 because the Cu term in eq 1 is practically negligible if the molecular weight of the analyte does not exceed a few thousand Dalton.49 In fast liquid chromatography, the axial dispersion coefficient reduces to the eddy dispersion coefficient since the contribution of longitudinal diffusion to the overall band broadening becomes negligible. Therefore, it is of fundamental importance in liquid chromatography to understand, predict, and quantify eddy dispersion in packed beds. For molecules larger than insulin, diffusion coefficients are equal to or smaller than 1 × 10−6 cm2/s, and the use of the mobile phase velocities classically applied in the separation of small molecules lead to reduced velocities ν in excess of 20 for 2.7 μm particles50−52 and may even reach up to 750 for large proteins such as immunoglobulin (IgG).9 For such high ν values, the Cν HETP term would become significantly larger than the B/ν and A(ν) reduced HETP terms. Analysts have therefore to operate their columns at low actual flow rates, hence low velocities, to achieve high resolution. They must trade resolution for analysis time. Significant improvement in column efficiency for high molecular weight biochemicals can be achieved only by increasing the effective intraparticle diffusivity of these molecules. High-temperature chromatography using core-shell particles is clearly the simplest option
BEGINNING OF STUDIES ON MASS TRANSFER THEORIES The first fundamental attempts at developing a theory of axial dispersion in packed beds were made by Gunn who investigated dispersion in the homogeneous bulk region of packed beds58 and in poorly packed beds.59 The assumptions made by Gunn for this model are as follows: (1) the elementary flow cell has the size of a particle diameter; (2) the eluent is segmented into two regions, a fast stream and a slow stream, the velocity of which is supposed to be equal to zero; and (3) the exchange of analyte molecules between the slow and the fast streams is controlled by a diffusive exchange process. However, Gunn was a chemical engineer and he ignored the earlier empirical work of Giddings, an analyst, on dynamics in chromatography,1 in which Giddings suggested an analogy between eddy dispersion in packed beds and the simple mathematical model of random walk. This earlier model was more sophisticated than Gunn’s in terms of its physical assumptions but easier to solve from a mathematical viewpoint due to this simple analogy. Giddings did not limit velocity biases to those existing in Gunn’s flow cell volume, the one between the packed particles (trans-channel velocity bias), but included those that potentially take place over interparticle distances between a few particle diameters and the column diameter. Other theoretical models of eddy dispersion were suggested by Huber,60,61 Horvath and Lin,62 and Bouchaud and Georges.63 They failed to properly account for dispersion data recorded for large (210 μm) solid, dry-packed particles.64 Analysts preferred empirical models, like the one proposed by Knox65−67 to account for the complex variation of eddy dispersion with the reduced velocity. For nearly 40 years, the problem of eddy dispersion in packed beds remained an empirical exercise, obfuscated by a lack of accurate data and a dearth of reliable, fundamental models. The first suggestion that eddy dispersion is far more complex than presumed by the assumptions made by van Deemter et al.17 and by Giddings1 was made by Knox who reported in 1969 that eddy dispersion of unretained analytes depends strongly on the column bed aspect ratio or ratio of the column to the particle diameter, Dc/dp.68,69 This should have drawn attention to the importance of the confinement of the particles in chromatographic columns, but this isolated observation remained ignored for 40 years. It is only recently that calculations of mass transport rates by computer-generated70 and reconstructed71 packed beds of nonporous particles confirmed the impact of the packing microstructure,72−74 the particle size distribution,75,76 the trans-channel velocity bias,77 the short-range interchannel velocity bias,77 the bed aspect ratio,78,79 and the bed confinement80 on the overall eddy dispersion term. However, the lack of relevant chromatographic data prevented the possibility of a direct comparison of these results with HPLC data while the assumptions made in these calculations that the particles are nonporous and that the sample molecules are not retained limited their applications. Most of the properties identified by Tallarek’s group are controlled by the interparticle diffusivity, hence the rate of 3019
dx.doi.org/10.1021/ac3033307 | Anal. Chem. 2013, 85, 3017−3035
Analytical Chemistry
Perspective
exchange between streams having extreme velocities and the contribution of eddy dispersion to the plate height. The faster molecular exchange, the smaller the expected eddy dispersion, as predicted by the general dispersion theories of Giddings1 and Aris.81 Recently, a noninvasive protocol was designed to measure accurate values of eddy dispersion in modern, high-resolution HPLC columns for small molecules having different retention factors.33,35 This protocol is based on (1) the use of the peak parking method to measure the axial diffusion HETP term;82 (2) the use of an accurate model of effective diffusion in chromatographic columns24,25,83 to estimate solid−liquid mass transfer resistance;46 and (3) the measurement of the true first and second central moments of elution bands by numerical integration and correction for the band broadening contributions of extra-column volumes.84−86 Experimental results have demonstrated that eddy dispersion does depend on the bed properties. This was in contrast with the long assumed misconception that eddy dispersion depends only on the distribution of axial velocities through the stream of mobile phase only. Experimental data have proven that other factors do influence axial dispersion in packed beds.32,33,87,88 The bed porosity and the retention factor of analytes strongly affect the eddy dispersion term. For instance, the reduced eddy dispersion term of a 4.6 mm × 100 mm column packed with core-shell particles decreased from about 4 to 2 and 1 when (1) the access of analyte molecules to the mesoporous volume was blocked by n-nonane; (2) the pores were not blocked but the analyte was nonretained; and (3) the pores were unblocked and the analyte was retained (k ≃ 2).37 This significant decrease of eddy dispersion with increasing internal porosity of the particles and analyte retention factor confirms the importance of intraparticle diffusivity on column performance in RPLC.28,32 The goal of this work is to use these accurate values of eddy dispersion and classical models of axial dispersion including the Gunn and Giddings models, to discuss their relevance and to predict the most important source of band broadening in liquid chromatography, the eddy dispersion contribution to mass transfer. Comparison of the experimental data and the values predicted by these models will lead to reintroduction of the concept of trans-column eddy dispersion in packed beds. Two limiting cases will be considered, whether the column is radially equilibrated or not. Experimental data will be provided in both cases, and ways to cope with and minimize the trans-column eddy dispersion term will be proposed. Possible solutions to further improve column efficiency for the analysis of small molecules will also be proposed.
u=
Fν εeπrc 2
(4)
The reduced interstitial linear velocity, ν, or Peclet number reported to the bulk diffusion coefficient, Pem, is given by ν = Pem =
ud p Dm
(5)
The Peclet number reported to the axial dispersion coefficient DL, PeL, is written as PeL =
ud p DL
(6)
The Reynolds number, Re, is defined as Re =
εeρud p η
(7)
where ρ and η are the density (kg/m ) and the viscosity (Poise or Pa s) of the moving eluent, respectively. The Reynolds number represents the ratio of the inertial to the viscous forces. Laminar flow is expected for Reynolds number smaller than about 2000. Beyond this value, the flow is nonlinear-laminar (viscous-inertial) then turbulent and chaotic. Finally, the Schmidt number, Sc, represents the ratio of the viscous diffusion rate to the molecular diffusion rate. This number is directly related to the Peclet and Reynolds numbers, Pem and Re, by 3
Sc =
εePem η = ρDm Re
(8)
The Schmidt number accounts for the temperature and the physical state (gas or liquid) of the moving fluid. Sc is usually around 1000 for liquids and small analyte molecules (η ≃ 10−3 Pa s, ρ ≃ 103 kg/m3, and Dm ≃ 10−9 m2/s). Dimensionless Analysis. The coherent interpretation of axial dispersion data in packed beds involves a maximum number of four independent dimensionless group numbers that combines the impact of the column length (L), the column inner diameter (D = 2rc), the average interstitial linear velocity (u), the average particle diameter (dp), the density of the moving fluid (ρ), the viscosity of the fluid (η), and the bulk molecular diffusion coefficient of the analyte (Dm). It can be shown that44 ⎛ L D ud p η ⎞ ⎟⎟ , , PeL = f ⎜⎜ , ⎝ D d p Dm ρDm ⎠
■
THEORY We list below a series of theoretical models of axial dispersion along packed beds. Their assumptions are clearly formulated, which will allow us to discuss later the relevance of these models for the prediction of the eddy dispersion coefficient in modern packed LC columns. Definitions. In this work, the bulk diffusion coefficient of analytes in the moving eluent is Dm. Their axial dispersion coefficient is DL, The packed bed is characterized by its porosity or bed porosity, εe, equal to the ratio of the interparticle volume to the column tube volume. The average diameter of the particles in the packed bed is dp. The column length is L. The flow rate is Fv, and the column inner radius is rc. The interstitial linear velocity, u, is given by
(9)
In other words, the Peclet number, PeL, is solely a function of the column geometry through its length-to-diameter ratio, L/D, the particle size, through the bed aspect ratio, D/dp, the reduced interstitial linear velocity, Pem, and either the Schmidt, Sc, or the Reynold, Re, numbers. If the column dimension, the particle size, the physical state, and the temperature of the eluent are fixed, the Peclet number with respect to the axial dispersion coefficient D L is a function of only one dimensionless parameter, the reduced velocity or the Peclet number reported to the bulk molecular diffusion coefficient. Figure 1 shows a typical representation of PeL vs Pem for a Schmidt number around 1000. It represents the variation of the axial dispersion coefficient of small analytes with speed using water at ambient temperature as the carrier fluid. Two 3020
dx.doi.org/10.1021/ac3033307 | Anal. Chem. 2013, 85, 3017−3035
Analytical Chemistry
Perspective
average migration length, z,̅ and the variance of the migration length, σ2, after a total number, n, of steps are given by89 z ̅ = nl − + np(l + − l −) = npl + + n(1 − p)l − −
(11)
+
where l and l are the backward (slow stream) and forward (fast stream) migration lengths, respectively. The variance, σ2, of the random walk model is given by89 σ 2 = np(1 − p)(l + − l −)2
(12) 1
In a turbulent regime, p = /2, the average migration length and the variance of the migration length reduce to n z ̅ = (l + + l −) (13) 2 and
σ2 =
Figure 1. Representation of the Peclet number relative to the axial dispersion coefficient, DL, as a function of the Peclet number relative to the bulk molecular diffusion coefficient, Dm, for a liquid fluid. The Schmidt number is typically around 1000. Note the existence of two transition regimes separating the diffusion (orange color), the convection (blue color), and the turbulent (black color) regimes. The domain that corresponds to liquid chromatography (for which the Reynolds number varies typically between 10−3 and 10−1) is shown in red.
n + (l − l −)2 4
(14)
If we assume that a very large number of steps were taken, the variance of the binomial distribution tends toward that of a Gaussian distribution. So, combination of eqs 13 and 14 provides z σ 2 =2DL (Pem → ∞) ̅ u = 2DL (Pem → ∞)
transition regimes separate three dispersion regimes controlled by diffusion (Pe < 0.3), convection (500 < Pe < 4 × 104), and turbulence (or chaotic regime, Pe > 107). The transition between the convection and turbulent regimes is a laminar flow regime, in which the law of permeability is no longer linear. The other transition regime, between the diffusion and the pure convection regimes lies within the domain of application of liquid chromatography and is often called the power law regime. In this transition, the axial dispersion coefficient increases from a horizontal asymptotic value at infinitesimally small velocities (DL → γeDm) to a linear asymptotic straight line at relatively high velocities (DL → λudp, where λ depends on the quality of the bed packing). The next section lists a series of models of axial dispersion in packed beds that can be used in liquid chromatography in the power law regime. Limiting Additive Model. A first model of axial dispersion in packed beds can be constructed by adding the contribution of molecular diffusion at infinitely small flow rate (Pem → 0) and the contribution of turbulent flow at infinitely large flow rate (Pe → ∞). Assuming that the particles are nonporous, the axial dispersion coefficient is written as44 DL = γeDm +
=
n + (l − l −)2 4
(15)
Therefore, DL (Pem → ∞) =
1 (l + − l −)2 u 4 l+ + l−
(16)
Returning to the microscopic level, the smallest physical average microscopic displacement of a molecule is given by the average particle diameter. So, l+ + l− = dp 2
(17)
Finally, assuming that a molecule is either at rest or in movement, the backward and forward migration lengths are l− = 0
and
l + = 2d p
(18)
and DL (Pem → ∞) =
ud p PeL(Pem → ∞)
n + (l + l −) 2u
2 1 4d p 1 u = ud p 4 2d p 2
(19)
The axial dispersion coefficient, DL, is then given by
(10)
DL = γeDm +
where γe is the obstruction factor of the packed bed and PeL(Pem → ∞) is the limit of the Peclet number reported to DL when the flow velocity tends toward infinity. Under turbulent flow conditions, the velocity of the sample molecules is chaotic in time and space. Therefore, a simple analogy can be made between axial dispersion and the random walk model,1 in which the probabilities for a molecule to move forward and backward with respect to the migration distance run at the mean velocity, u, are equal to p = 1/2. According to the properties of the general random walk model with a probability p for a molecule to move forward, the
1 ud p 2
(20)
or after dividing both sides of eq 20 by the product udp, γ 1 1 = e + PeL Pem 2 (21) A similar expression can be derived for the transverse dispersion coefficient, Dt. Let assume that a molecule has the same probability of 1/2 to contour a particle by its left or by its right along the distance r = (dp/2) (two-dimensional problem). Again, a random walk model can be applied. In this case, l+ = 3021
dx.doi.org/10.1021/ac3033307 | Anal. Chem. 2013, 85, 3017−3035
Analytical Chemistry
Perspective
−l− = r because there is no net transverse displacement of the molecules across the packing. Accordingly, nr 2 = 4Dt t = 4Dt
Assuming that the initial (t = 0) distributions of the sample molecules in the slow and the fast streams are uniform, the Laplace transform of this system of coupled mass balance equations permits the calculation of the zeroth, first, and second central moments of the eluted band, providing the following expression of the Peclet number PeL:58
nd p u
(22)
Therefore, γ 1 1 = e + Pet Pem 16
⎤2 ⎡ γ εePem εePem 1 2 ⎥ ⎢ (1 p ) = e + − + PeL Pem 4(1 − εe)α12 ⎣ 4(1 − εe)α12 ⎦
(23)
⎞ ⎛ ⎡ 4(1 − ε )α 2 ⎤ e 1 ⎥ − 1⎟⎟ p(1 − p)3 ⎜⎜exp⎢ − ⎠ ⎝ ⎣ p(1 − p)εePem ⎦
This expression is acceptable for two-dimensional ordered packed beds, such as pillar arrays. In real three-dimensional random packings, however, this limiting value of the reciprocal of Pet was measured at around 1/12 in gas flow experiments.44 Expressions better relevant to chromatography were recently determined in the boundary-layer dispersion regime, which required the numerical solution of the Navier−Stokes equation and the simulation of the advection-diffusion transport process.90 Gunn’s Model. The previous model of axial dispersion is not adapted to predict experimental axial dispersion data over a wide range of reduced velocity (or Peclet number, Pem) for either gas or liquid flow through packed beds.44 In fact, the transition between the diffusion and turbulent (chaotic) regime is far more complex, with the existence of local minima and maxima for the Peclet number reported to the axial dispersion coefficient. In the 1960s, chemical engineers recorded a wealth of experimental data and attempted to mathematically solve this complex power law regime by building more sophisticated models of axial dispersion along packed beds. Gunn suggested to consider a unit cell of size close to that of the particle diameter. The eluent volume is segmented into two flow streams: the fast stream (the volume fraction is equal to p, velocity Uf = U/p, where U is the average interstitial linear velocity) and the slow stream (the volume fraction is equal to 1 − p, velocity Us = 0). Gunn also assumed that the transfer of the sample molecules from the slow stream to the fast stream is governed by a pure stochastic diffusive exchange process, leading to the random Poisson process in which the probability that a molecule remains in a flow stream decreases exponentially with time, with the rate constants τs (slow stream) and τf (fast stream). τs =
τf =
The value of the parameter p is directly related to the Reynolds number for spheres: ⎛ 24 ⎞ p = 0.17 + 0.33 exp⎜ − ⎟ ⎝ Re ⎠
γ Pe Pe 2 1 = e + m (1 − p)2 + m p(1 − p)3 5 25 PeL Pem ⎛ ⎡ ⎞ ⎤ 5 ⎜⎜exp⎢ − ⎥ − 1⎟⎟ ⎝ ⎣ p(1 − p)Pem ⎦ ⎠
p=
(25)
where α1 = 2.4048 is the first root of the zero order Bessel function. Finally, he assumed that the axial dispersion coefficient in both streams is governed by the obstruction factor (tortuosity), in order for the model to be consistent with a diffusioncontrolled regime at extremely low flow rates. The mass balances in both streams are written: (26)
c ∂c f ∂c c U ∂c f = γeDm f2 − f + s + ∂t τf τs p ∂z ∂z
(27)
⎛ 75Sc ⎞ ⎛1 0.48 0.48 ⎞ + ⎜ − 0.15 ⎟ exp⎜ − ⎟ 0.15 ⎝2 Sc Sc ⎠ ⎝ Pem ⎠
(31)
The agreement between eq 30 and the experimental data covers a large range of Peclet (Pem < 105) and Schmidt (0.88 < Sc < 1930) numbers because eq 31 now depends on the Schmidt number. Axial dispersion data measured for both gas and liquid flows along packed beds agree qualitatively well with eq 30 in a double logarithmic-logarithmic scale for all values of the Schmidt number from about 1 (gas flow) to 2000 (liquid flow). Giddings’s Model. A few years before Gunn formulated his model, Giddings elaborated a stochastic theory to quantitatively predict the axial dispersion coefficient in chromatographic columns.1 This approach consisted in finding an analogy between the sample dispersion under the influence of both diffusion and convection exchange processes and the mathematical random walk model that was previously used in the Limiting Additive Model section. Giddings refined the analysis of eddy dispersion in packed beds by considering various characteristic lengths along and across the packed bed. They cover the interparticle distance (
