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Comments on the separation efficiency of asymmetrical flow field-flow fractionation in channels of constant channel and crossflow velocities leading to constant separation efficiency
Journal of Chromatography A, 1218 (2011) 6848–6850
Contents lists available at ScienceDirect
Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Discussion
Comments on the separation efficiency of asymmetrical flow field-flow fractionation in channels of constant channel and crossflow velocities leading to constant separation efficiency Karl-Gustav Wahlund ∗ Unit for Analysis and Synthesis, Department of Chemistry, Center for Chemistry and Chemical Engineering, Faculty of Engineeering LTH, Lund University, Box 124, SE-221 00 Lund, Sweden
a r t i c l e
i n f o
Article history: Received 24 February 2011 Received in revised form 15 July 2011 Accepted 18 July 2011 Available online 24 July 2011
a b s t r a c t It is shown theoretically that a claim in the literature about the overall separation efficiency of asymmetrical flow FFF channels being improved by geometries that permit a uniform channel flow velocity throughout the channel length is untrue. © 2011 Elsevier B.V. All rights reserved.
Keywords: Flow field-flow fractionation Asymmetrical flow FFF Separation efficiency Trapezoidal channel geometry Flow velocity gradient H-value
1. Introduction In two recent articles in this journal [1,2] it was proposed that two new designs of the asymmetrical flow field-flow fractionation (asymmetrical FlFFF) channel will permit both a constant carrier crossflow velocity through the accumulation wall ultrafiltration membrane (the crossflow velocity at the ultrafiltration membrane surface), and a constant carrier cross-sectional mean channel flow velocity along the length dimension. It was argued that this would result in uniform local separation efficiency along the channel and therefore improved overall separating efficiency. The first [1] of the new designs uses a novel possibility [3] to produce flat ultrafiltration membranes with controlled gradients of pore density and pore size (hydraulic conductivity gradient) over its length so that the permeability can be made to vary over the channel length. This was to be combined with the previously suggested use of an exponentially decreasing breadth of such asymmetrical flow FFF channels [4]. The second design [2] uses a variation of the height of the compartment below the ultrafiltration membrane/porous frit assembly and was, again, combined with an exponentially decreasing channel breadth.
Both designs lead to almost, but not perfectly [1], uniform channel flow velocity and crossflow velocity. Clearly, as stated, this will lead to “. . .the separation process to occur with the same efficiency over the whole length of the cell”. The main claim in the two articles is that with a constant separation efficiency the proposed channel designs “might contribute reducing the presently observed peak broadening”. The latter refers to the standard trapezoidal asymmetrical FlFFF channel [5,6]. The purpose of this discussion article is to show that this claim is not justified. There is no theoretical support or literature reference to it and, indeed, a literature search reveals nothing. Contrary, this article will show, by theoretical arguments, that a channel of constant separation efficiency has no benefits since the presence of a gradient in the local efficiencies does not infer any negative effect on the overall separation efficiency. The experimentally observed overall efficiency, as measured at the channel outlet, will simply be the result of a cumulation of all local efficiencies. Therefore, for any uniform efficiency it will be possible to find the same value for the overall efficiency by suitable choice of flowrates in a non-uniform channel. 2. Background
∗ Tel.: +46 46 222 8312; fax: +46 46 222 4525. E-mail address: [email protected] 0021-9673/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2011.07.060
The invention of the trapezoidal version of the asymmetrical flow FFF channel [5] was triggered by the fact that in the constant breadth rectangular asymmetrical channel [7–10] the channel
K.-G. Wahlund / J. Chromatogr. A 1218 (2011) 6848–6850
outlet flowrate, and therefore also the channel outlet flow velocity, became extremely low in the case that the crossflow rate was chosen to be a very large fraction of the channel inlet flowrate. This left only a small fraction of the channel flow to leave by the outlet end. The result is a very steep linearly decreasing gradient of the channel flow velocity. It was suspected that such very low channel flow velocity close to the outlet end could cause increased zone broadening due to longitudinal diffusion. Moreover, prolonged retention times, or even immobilisation on the ultrafiltration membrane [11] might occur. Later [6], however, it was shown that the longitudinal diffusion is in fact negligible under normal conditions. The trapezoidal channel geometry with its linearly decreasing breadth was intended to assist in keeping up the channel flow velocity at the channel outlet end and thereby level out the otherwise steep gradient occurring in the constant breadth rectangular channel. The reduction in breadth has the effect of opposing the reduction of flow velocity occurring in the rectangular channel. This was confirmed in a theoretical analysis of the rectangular and trapezoidal asymmetrical channels [5]. It was also shown that the velocity gradient could take different shapes, such as continuously decreasing or increasing, or with a minimum. Most importantly it was also concluded that “Yet a constant flow velocity along the channel can never be obtained. This would require some nonlinear decrease of the breadth”. This prediction was confirmed later by the theoretical development of a channel design with exponentially decreasing breadth [4] and the statement “It has been shown that a constant mean channel flow velocity may be obtained in asymmetrical FlFFF with a channel of exponential geometry and with suitable volumetric flow rates at inlet and outlet” [4]. Here, “suitable” should be interpreted as “specific” with the meaning that only a specific setting of the ratio of channel inlet to channel outlet flowrates leads to constant channel velocity. Its experimental application would therefore be limited since this condition would be immediately ruined if other ratios of flowrates were chosen. A theoretical description of the nonequilibrium zone broadening in the trapezoidal channel, taking into account the channel flow velocity gradient, was given [5]. Based on this the overall nonequilibrium zone broadening can be predicted.
w2 v D
Hl,i =
(1)
where is a complex function of the retention factor , w is the channel thickness, and D is the diffusion coefficient of a sample component. If the local separation efficiency is uniform throughout it means that the local H-value is constant throughout channel ¯ as measured at the outlet end length. Then, the observed H-value, H, of the separation channel, equals the local H-values. This is characteristic of uniform separation systems where the carrier channel velocity and the crossflow velocity are uniform. A good example of this is the symmetrical FlFFF channel. Contrary, in a non-uniform system the carrier velocity varies along the length dimension. A typical example is gas chromatography where there is a pressure drop across the column leading to increased carrier velocity [16]. Another example is the rectangular and trapezoidal asymmetrical FlFFF channels where there is a gradient in the mean channel flow velocity [5–7].
w2 vl,i D
(2)
This function can also be expressed in the usual way by distancebased parameters in the form Hl,i =
2 l,i
Ll,i
(3)
2 is the distance-based local peak variance. If expressed by where l,i the corresponding time-based parameters it will become
Hl,i = L
2 t,l,i 2 tr,l,i
(4)
2 is the time-based local peak variance and t where l,i r,l,i is the local retention time in a segment. The observed standard deviation peak width is a result of the cumulative effect of zone broadening on passing from one segment to the next [16]. It was shown that the average H-value is best derived from time-based parameters by
¯ =L H
t2 tr2
(5)
where L is the channel length, t2 is the observed peak variance in time units, and tr is the observed peak retention time [16]. The way that the local time-based zone widths cumulate is expressed by the rule of additivity of variances [16] and after a complex mathematical treatment [16] results in the following surprisingly simple relationship for the observed average H-value [5] ¯ = H
w2 v¯ D
(6)
where v¯ is the carrier time-average channel velocity. This can be calculated from the expression
v¯ =
Under normal experimental conditions the band broadening in FFF channels is dominated by nonequilibrium phenomena [12–15]. Then, the plate height H (distance-based zone variance per length unit), hereafter termed the H-value, is proportional to the crosssectional mean axial channel velocity, , according to H=
For non-uniform separation systems the observed average H¯ has to be some function of the local H-values, Hl . The latter value, H, depend on the local carrier velocity so that in each infinitely thin length segment, i, of the channel the local H-value will be given by [5,16]
3. Discussion
6849
L t0
(7)
where t0 is the channel void time. The latter can be calculated as shown previously for rectangular and trapezoidal channels [5,7]. Hence, the observed average H-value can be calculated by an equation (Eq. (6)) having the same mathematical form as for a uniform separation system but using the time-average carrier mean velocity instead of the constant uniform velocity (Eq. (1)). It can be concluded that for a given sample component (a given D) in asymmetrical channels having identical lengths L, identical thicknesses w, identical void times t0 , and identical -values (i.e., identical retention factors ) the same overall separation efficiency will be obtained whether the channel is rectangular, trapezoidal, or having an exponentially decreasing breadth. Identical -values can be obtained if the channel membrane areas and the crossflow rates are chosen to be identical. Moreover, the observed average Hvalue can be modulated by adjusting the channel inlet and outlet flowrates so that suitable values of the dead-time will be obtained. Therefore, it will almost always be possible to adjust the timeaverage cross-average mean channel flow velocity of a trapezoidal channel to equal the uniform cross-average mean channel flow velocity in a symmetrical FlFFF channel and an “exponential” asymmetrical FlFFF channel. Hence, the overall separation efficiency will be identical. This shows that the claim made in the mentioned articles [1,2] are not justified.
6850
K.-G. Wahlund / J. Chromatogr. A 1218 (2011) 6848–6850
Regarding the suspected non-uniformity of the crossflow velocity at the membrane surface it can be said that this can be treated in similar fashion as the non-uniform channel velocity discussed above. However, experiments have shown that this kind of nonuniformity is negligible [6]. 4. Conclusions In evaluating, discussing, and predicting the overall separation efficiency of asymmmetrical FlFFF channels of different breadth geometries it is necessary to base the arguments on a sound theoretical model for the zone broadening and/or to perform relevant experiments. The two articles deliver none. They appear as theoretical exercises. It appears that some authors in the literature on asymmetrical FlFFF perceive that the presence of a channel velocity gradient is in itself detrimental to separation efficiency. This is untrue. Acknowledgement This work was financed by the Swedish Research Council.
References [1] P. Déjardin, J. Chromatogr. A 1187 (2008) 209. [2] P. Déjardin, J. Chromatogr. A 1203 (2008) 94. [3] S. Sirijarukul, E. Balanzat, E.N. Vasina, P. Dejardin, J. Membr. Sci. 296 (2007) 185. [4] P.S. Williams, J. Microcolumn Sep. 9 (1997) 459. [5] A. Litzén, K.G. Wahlund, Anal. Chem. 63 (1991) 1001. [6] A. Litzen, Anal. Chem. 65 (1993) 461. [7] K.G. Wahlund, J.C. Giddings, Anal. Chem. 59 (1987) 1332. [8] K.G. Wahlund, A. Litzen, J. Chromatogr. 461 (1989) 73. [9] A. Litzen, K.G. Wahlund, J. Chromatogr. 476 (1989) 413. [10] J.J. Kirkland, C.H. Dilks, S.W. Rementer, W.W. Yau, J. Chromatogr. 593 (1992) 339. [11] K.G. Wahlund, H.S. Winegarner, K.D. Caldwell, J.C. Giddings, Anal. Chem. 58 (1986) 573. [12] J.C. Giddings, J. Chem. Phys. (1968) 81. [13] M.E. Hovingh, G.H. Thompson, J.C. Giddings, Anal. Chem. 42 (1970) 195. [14] J.C. Giddings, Y.H. Yoon, K.D. Caldwell, M.N. Myers, M.E. Hovingh, Sep. Sci. 10 (1975) 447. [15] J.M. Davis, in: M. Schimpf, K. Caldwell, J.C. Giddings (Eds.), Field-Flow Fractionation Handbook, Wiley-Interscience, New York, 2000, p. 49. [16] J.C. Giddings, Anal. Chem. 35 (1963) 353.
Contents lists available at ScienceDirect
Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Discussion
Comments on the separation efficiency of asymmetrical flow field-flow fractionation in channels of constant channel and crossflow velocities leading to constant separation efficiency Karl-Gustav Wahlund ∗ Unit for Analysis and Synthesis, Department of Chemistry, Center for Chemistry and Chemical Engineering, Faculty of Engineeering LTH, Lund University, Box 124, SE-221 00 Lund, Sweden
a r t i c l e
i n f o
Article history: Received 24 February 2011 Received in revised form 15 July 2011 Accepted 18 July 2011 Available online 24 July 2011
a b s t r a c t It is shown theoretically that a claim in the literature about the overall separation efficiency of asymmetrical flow FFF channels being improved by geometries that permit a uniform channel flow velocity throughout the channel length is untrue. © 2011 Elsevier B.V. All rights reserved.
Keywords: Flow field-flow fractionation Asymmetrical flow FFF Separation efficiency Trapezoidal channel geometry Flow velocity gradient H-value
1. Introduction In two recent articles in this journal [1,2] it was proposed that two new designs of the asymmetrical flow field-flow fractionation (asymmetrical FlFFF) channel will permit both a constant carrier crossflow velocity through the accumulation wall ultrafiltration membrane (the crossflow velocity at the ultrafiltration membrane surface), and a constant carrier cross-sectional mean channel flow velocity along the length dimension. It was argued that this would result in uniform local separation efficiency along the channel and therefore improved overall separating efficiency. The first [1] of the new designs uses a novel possibility [3] to produce flat ultrafiltration membranes with controlled gradients of pore density and pore size (hydraulic conductivity gradient) over its length so that the permeability can be made to vary over the channel length. This was to be combined with the previously suggested use of an exponentially decreasing breadth of such asymmetrical flow FFF channels [4]. The second design [2] uses a variation of the height of the compartment below the ultrafiltration membrane/porous frit assembly and was, again, combined with an exponentially decreasing channel breadth.
Both designs lead to almost, but not perfectly [1], uniform channel flow velocity and crossflow velocity. Clearly, as stated, this will lead to “. . .the separation process to occur with the same efficiency over the whole length of the cell”. The main claim in the two articles is that with a constant separation efficiency the proposed channel designs “might contribute reducing the presently observed peak broadening”. The latter refers to the standard trapezoidal asymmetrical FlFFF channel [5,6]. The purpose of this discussion article is to show that this claim is not justified. There is no theoretical support or literature reference to it and, indeed, a literature search reveals nothing. Contrary, this article will show, by theoretical arguments, that a channel of constant separation efficiency has no benefits since the presence of a gradient in the local efficiencies does not infer any negative effect on the overall separation efficiency. The experimentally observed overall efficiency, as measured at the channel outlet, will simply be the result of a cumulation of all local efficiencies. Therefore, for any uniform efficiency it will be possible to find the same value for the overall efficiency by suitable choice of flowrates in a non-uniform channel. 2. Background
∗ Tel.: +46 46 222 8312; fax: +46 46 222 4525. E-mail address: [email protected] 0021-9673/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2011.07.060
The invention of the trapezoidal version of the asymmetrical flow FFF channel [5] was triggered by the fact that in the constant breadth rectangular asymmetrical channel [7–10] the channel
K.-G. Wahlund / J. Chromatogr. A 1218 (2011) 6848–6850
outlet flowrate, and therefore also the channel outlet flow velocity, became extremely low in the case that the crossflow rate was chosen to be a very large fraction of the channel inlet flowrate. This left only a small fraction of the channel flow to leave by the outlet end. The result is a very steep linearly decreasing gradient of the channel flow velocity. It was suspected that such very low channel flow velocity close to the outlet end could cause increased zone broadening due to longitudinal diffusion. Moreover, prolonged retention times, or even immobilisation on the ultrafiltration membrane [11] might occur. Later [6], however, it was shown that the longitudinal diffusion is in fact negligible under normal conditions. The trapezoidal channel geometry with its linearly decreasing breadth was intended to assist in keeping up the channel flow velocity at the channel outlet end and thereby level out the otherwise steep gradient occurring in the constant breadth rectangular channel. The reduction in breadth has the effect of opposing the reduction of flow velocity occurring in the rectangular channel. This was confirmed in a theoretical analysis of the rectangular and trapezoidal asymmetrical channels [5]. It was also shown that the velocity gradient could take different shapes, such as continuously decreasing or increasing, or with a minimum. Most importantly it was also concluded that “Yet a constant flow velocity along the channel can never be obtained. This would require some nonlinear decrease of the breadth”. This prediction was confirmed later by the theoretical development of a channel design with exponentially decreasing breadth [4] and the statement “It has been shown that a constant mean channel flow velocity may be obtained in asymmetrical FlFFF with a channel of exponential geometry and with suitable volumetric flow rates at inlet and outlet” [4]. Here, “suitable” should be interpreted as “specific” with the meaning that only a specific setting of the ratio of channel inlet to channel outlet flowrates leads to constant channel velocity. Its experimental application would therefore be limited since this condition would be immediately ruined if other ratios of flowrates were chosen. A theoretical description of the nonequilibrium zone broadening in the trapezoidal channel, taking into account the channel flow velocity gradient, was given [5]. Based on this the overall nonequilibrium zone broadening can be predicted.
w2 v D
Hl,i =
(1)
where is a complex function of the retention factor , w is the channel thickness, and D is the diffusion coefficient of a sample component. If the local separation efficiency is uniform throughout it means that the local H-value is constant throughout channel ¯ as measured at the outlet end length. Then, the observed H-value, H, of the separation channel, equals the local H-values. This is characteristic of uniform separation systems where the carrier channel velocity and the crossflow velocity are uniform. A good example of this is the symmetrical FlFFF channel. Contrary, in a non-uniform system the carrier velocity varies along the length dimension. A typical example is gas chromatography where there is a pressure drop across the column leading to increased carrier velocity [16]. Another example is the rectangular and trapezoidal asymmetrical FlFFF channels where there is a gradient in the mean channel flow velocity [5–7].
w2 vl,i D
(2)
This function can also be expressed in the usual way by distancebased parameters in the form Hl,i =
2 l,i
Ll,i
(3)
2 is the distance-based local peak variance. If expressed by where l,i the corresponding time-based parameters it will become
Hl,i = L
2 t,l,i 2 tr,l,i
(4)
2 is the time-based local peak variance and t where l,i r,l,i is the local retention time in a segment. The observed standard deviation peak width is a result of the cumulative effect of zone broadening on passing from one segment to the next [16]. It was shown that the average H-value is best derived from time-based parameters by
¯ =L H
t2 tr2
(5)
where L is the channel length, t2 is the observed peak variance in time units, and tr is the observed peak retention time [16]. The way that the local time-based zone widths cumulate is expressed by the rule of additivity of variances [16] and after a complex mathematical treatment [16] results in the following surprisingly simple relationship for the observed average H-value [5] ¯ = H
w2 v¯ D
(6)
where v¯ is the carrier time-average channel velocity. This can be calculated from the expression
v¯ =
Under normal experimental conditions the band broadening in FFF channels is dominated by nonequilibrium phenomena [12–15]. Then, the plate height H (distance-based zone variance per length unit), hereafter termed the H-value, is proportional to the crosssectional mean axial channel velocity, , according to H=
For non-uniform separation systems the observed average H¯ has to be some function of the local H-values, Hl . The latter value, H, depend on the local carrier velocity so that in each infinitely thin length segment, i, of the channel the local H-value will be given by [5,16]
3. Discussion
6849
L t0
(7)
where t0 is the channel void time. The latter can be calculated as shown previously for rectangular and trapezoidal channels [5,7]. Hence, the observed average H-value can be calculated by an equation (Eq. (6)) having the same mathematical form as for a uniform separation system but using the time-average carrier mean velocity instead of the constant uniform velocity (Eq. (1)). It can be concluded that for a given sample component (a given D) in asymmetrical channels having identical lengths L, identical thicknesses w, identical void times t0 , and identical -values (i.e., identical retention factors ) the same overall separation efficiency will be obtained whether the channel is rectangular, trapezoidal, or having an exponentially decreasing breadth. Identical -values can be obtained if the channel membrane areas and the crossflow rates are chosen to be identical. Moreover, the observed average Hvalue can be modulated by adjusting the channel inlet and outlet flowrates so that suitable values of the dead-time will be obtained. Therefore, it will almost always be possible to adjust the timeaverage cross-average mean channel flow velocity of a trapezoidal channel to equal the uniform cross-average mean channel flow velocity in a symmetrical FlFFF channel and an “exponential” asymmetrical FlFFF channel. Hence, the overall separation efficiency will be identical. This shows that the claim made in the mentioned articles [1,2] are not justified.
6850
K.-G. Wahlund / J. Chromatogr. A 1218 (2011) 6848–6850
Regarding the suspected non-uniformity of the crossflow velocity at the membrane surface it can be said that this can be treated in similar fashion as the non-uniform channel velocity discussed above. However, experiments have shown that this kind of nonuniformity is negligible [6]. 4. Conclusions In evaluating, discussing, and predicting the overall separation efficiency of asymmmetrical FlFFF channels of different breadth geometries it is necessary to base the arguments on a sound theoretical model for the zone broadening and/or to perform relevant experiments. The two articles deliver none. They appear as theoretical exercises. It appears that some authors in the literature on asymmetrical FlFFF perceive that the presence of a channel velocity gradient is in itself detrimental to separation efficiency. This is untrue. Acknowledgement This work was financed by the Swedish Research Council.
References [1] P. Déjardin, J. Chromatogr. A 1187 (2008) 209. [2] P. Déjardin, J. Chromatogr. A 1203 (2008) 94. [3] S. Sirijarukul, E. Balanzat, E.N. Vasina, P. Dejardin, J. Membr. Sci. 296 (2007) 185. [4] P.S. Williams, J. Microcolumn Sep. 9 (1997) 459. [5] A. Litzén, K.G. Wahlund, Anal. Chem. 63 (1991) 1001. [6] A. Litzen, Anal. Chem. 65 (1993) 461. [7] K.G. Wahlund, J.C. Giddings, Anal. Chem. 59 (1987) 1332. [8] K.G. Wahlund, A. Litzen, J. Chromatogr. 461 (1989) 73. [9] A. Litzen, K.G. Wahlund, J. Chromatogr. 476 (1989) 413. [10] J.J. Kirkland, C.H. Dilks, S.W. Rementer, W.W. Yau, J. Chromatogr. 593 (1992) 339. [11] K.G. Wahlund, H.S. Winegarner, K.D. Caldwell, J.C. Giddings, Anal. Chem. 58 (1986) 573. [12] J.C. Giddings, J. Chem. Phys. (1968) 81. [13] M.E. Hovingh, G.H. Thompson, J.C. Giddings, Anal. Chem. 42 (1970) 195. [14] J.C. Giddings, Y.H. Yoon, K.D. Caldwell, M.N. Myers, M.E. Hovingh, Sep. Sci. 10 (1975) 447. [15] J.M. Davis, in: M. Schimpf, K. Caldwell, J.C. Giddings (Eds.), Field-Flow Fractionation Handbook, Wiley-Interscience, New York, 2000, p. 49. [16] J.C. Giddings, Anal. Chem. 35 (1963) 353.