Third International Symposium on Advances in Gas Chromatography
University of Houston, Houston, Texas, October 18-21, 1965.
Evidence for Turbulence and Coupling in Chromatographic Columns JOHN
H. KNOX
Department o f Chemistry, University of Edinburgh, Edinburgh, Scotland The spreading of bands of potassium permanganate in aqueous 10% potassium nitrate solution during elution through columns packed with glass beads has been studied as a function of fluid velocity and the column particle diameter ratio, p. Plots of the reduced plate height, h = H/dp, against the reduced fluid velocity, v = udp/D,, provide conclusive evidence for the coupling of eddy diffusion and transverse diffusion processes over a 10,000-fold range of reduced fluid velocity. The effect of changing p shows that the main contributors to peak broadening in the mobile phase at reduced velocities in excess of 20 are trans-column processes. The column efficiency declines with increase of p until p = 20, and thereafter increases slowly. Particle size distribution i s important in governing column efficiency and more work on its effect i s required. At high fluid velocities and at Reynolds numbers in excess of about 3 turbulence becomes important and reduces H. Turbulence may be expected in gas chromatographic columns at gas velocities typically used in analyses. It will therefore be difficult to separate the effects of coupling and turbulence b y experiments with gas chromatographic columns alone.
-
A
to the “classical” theories of peak dispersion in chromatographic columns-e.g., those of van Deemter (4, Golay ( l a ) , and Jones (l~)-contributions to the height of equivalent theoretical plate (HETP) from various processes in the column are regarded as additive and a linear equation such as 1 may accordingly be written for the dispersion of a band of an unsorbed substance in any packed column. CCORDING
turbulence is the Reynolds number defined in Equation 5. Giddings (9) has, however, pointed out that eddy diffusion and lateral mass transfer in the mobile phase actually cooperate and reduce the dispersion which is continually being generated by uneven flow patterns across and along the column. iiccordingly their contributions to H should not be added as in Equation 1 but combined in a more complex “coupled” form to give a combination which is less than either term alone. The coupled form, as first proposed, was :
The equation has since been extended (7) to take account of five distinguishable irregularities in packing and flow structure by replacing the first term in Equation 2 by a summation-Equation 3.
Dm/Wtdp2u)-1
+ 2 ~ D m / u (3)
The five terms in the summation represent contributions from trans-particle, trans-channel, short-range interchannel, long-range interchannel, and trans-column processes in the mobile phase. Equations 1 to 3 may conveniently be cast into reduced forms (10) by defining the dimensionless parameters Reduced plate height, h = H / d , Reduced linear velocity,
Y
=
udp/Dm
(4)
A further reduced parameter which is valuable in assessing the importance of
Reynolds number, Re
=
upmdp/g (5)
I n terms of reduced parameters Equations 1 and 3 take the simplified forms:
h 5
h
=
+ 2y/v + (1/2A; + l/w;v) + =
2x
WY
(6)
-1
1
The main advantage in formulating HETP equations in terms of reduced parameters is that data obtained in experiments with different particle sizes and different eluents can be compared directly. Thus if Equation 6 correctly represented all data and if A, y, and w were constants dependent only upon the type of packing but not upon its size, all plots of h against Y would coincide. The use of reduced parameters is particularly valuable in comparing data obtained with liquid and gaseous eluents. The different forms of Equations 6 and 7 are most readily demonstrated by plots of log h against log Y , since the linear form (Equation 6) gives a symmetrical curve about a minimum h, while the coupled form (Equation 7) gives a curve becoming asymptotic a t high reduced velocities to h = 22hi (15). This flattening is the most characteristic feature of data obeying the coupled equation or any of its modifications, and it is a feature which many workers have tried to detect. To obtain a satisfactory discrimination it is necessary to work over a wide range of reduced velocity, preferably covering a factor of 100 above the value for minimum h. Although there is little doubt as to the theoretical correctness of the general concept of coupling, experimental evidence from gaseous systems has been VOL. 38, NO. 2, FEBRUARY 1966
253
Table
I.
Viscosity, c.g.s. units C2H4in Ht C2H4in N? Aq, KhhO4
0.9 X 1.7 X 0.7 X
Diffusion coefficient, sq. cm./sec. 0 . 6 (estd.) 0.16 0 . 8 X IO-'
disappointingly inconclusive. Perrett and Purnell (18) and Knox and McLaren (16) reported no clear evidence for any deviation from the linear type of equation. Giddings and Robison (11) considered that on balance the evidence favored a coupled form of equation. Van Berge, Haarhoff, and Pretorius ( 1 ) concluded that wall effects can explain the available data. Sternberg and Poulson (19) in their work on unsorbed bands obtained further evidence which could have been interpreted in terms of a coupled HETP equation, but they concluded instead that the apparent reduction in H at high velocities was due t o the gradual onset of turbulent flow. Some recent data obtained by RIcLaren (17) confirm that H at high velocities is lower than expected from a linear HETP equation, but it is difficult to assess the relative importance of coupling and turbulence. By far the best evidence for coupling has been obtained with liquid chromatographic systems using unsorbed solutes. Terry, Blackwell, and Rayne (2.3) assembled the available data, and these have been quoted by Giddings. The data, although rather scattered, show clearly that H a t high reduced velocities is much less than that expected from simple linear equations such as 1. Much better data were obtained by Gordon et al. (IS), who studied the spreading of bands of 1-butanol in n-heptane in columns 50 cm. long and 0,925 em. in diameter containing spherical beads 0.045 and 0.0195 cm. in diameter, Their data show clearly that H rises less than linearly with fluid velocity in a region where longitudinal diffusion is negligible. The main reason for the inconclusiveness of the work on gaseous systems is the difficulty of working at sufficiently high reduced velocities to obtain a significant difference between the coupled and linear forms of equation. However, as Giddings (6,10)has pointed out, very high reduced velocities are readily accessible in liquid systems. This arises because D , is typically lo-' for gases and 10-5 to 10-6 sq. em. per second for liquids. The same absolute velocities thus correspond to lo4 to lo5 times higher reduced velocities. I t is indeed difficult to work with liquids a t sufficiently low reduced velocities to overlap data obtained from gaseous systems. 254
that s l f ( h , u ) d h d w = unity.
Relation between Reduced Velocity and Reynolds Number
ANALYTICAL CHEMISTRY
Density, g./cc. 0 . 8 X lo-' 1.1 1.0
x
10-3
tion 3 would be equivalent to writing
v/Re Exact 1.9 1.0 0.8
Approx. 2
x
103
Equa-
1 1000
Working with liquids has two other advantages. First, liquids are incompressible and in calculating H no pressure corrections need be applied to the experimentally determined values. Second, turbulence will become important at very much higher reduced velocities in liquids than in gases. The onset of turbulence in packed beds occurs at Reynolds numbers between 1 and 100 (2, 3). To assess the importance of turbulence relative to coupling it is useful to examine the ratio of reduced velocity to Reynolds number. v/Re = q / D m P n (8) Typical values of viscosity, density, and diffusion coefficient are given in Table I for gaseous and liquid systems. The last column gives the calculated values of v/Re. Since the reduced velocity for minimum H is normally between 2 and 5, signs of turbulence, which appear a t Re between 1 and 10, may be expected even a t the relatively low flow rates currently used in gas chromatography. Much of the confusion as to the importance of coupling may have arisen from the uncertain role of turbulence in the experiments of different workers. If liquids are used as the mobile phase in place of gases, turbulence need not be considered at reduced velocities below l o 3 to 104. The region between unity and 103 should therefore provide clear evidence for or against coupled forms of the H E T P equation. The present work was initiated to obtain clear evidence regarding the coupling phenomenon, to determine more accurately the exact form of the H E T P equation at high velocities, to examin the importance of turbulence, and to determine the role of transcolumn processes at high velocities.
f(~,w) with five roughly equal peaks at specific values of x and W . As u
tends to infinity, the integral form, like the summed and simple forms, reduces to H = constant. I n this respect all the coupled forms are similar. I n terms of reduced parameters Equation 9 becomes:
Plots of log h against log v should then reveal the form of the integral, since the second term on the right may be allowed for or made negligible by working at sufficiently high reduced velocities. Since d p does not enter explicitly into Equation 10, one might expect plots of log h against log v to be coincident for all types of packing. However, transcolumn effectsproduce a contribution to H which depends not upon d, but upon d,, the column diameter; thus h will contain a part which is dependent upon the ratio p = d,/d,. This will turn up as a dependence of ~ ( x , w )on p in a certain region of the +J field. I n a similar way ~ ( x , w ) will depend upon the type of particle, the particle size range, and the density of packing. Variation in these parameters will tend to emphasize different combinations of x and W . Thus we expect in general that plots of log h against log v will show dependence upon p , the particle type, the particle size distribution, and the packing density. Ideally in well prepared columns the effects of unequal packing density and variable particle size distribution should be small. With a given type of particle noncoincidence of log h against log v plots should be due mainly to transcolumn effects as p is varied. Following Giddings (7) it may therefore be possible to split the integral in Equation 10 into two parts, one for trans-column processes, and one for all other nonequilibrium processes. Considering only h,, the nonequilibrium contribution h, we could then write:
h,
=
(h - 2Y/Y) =
THEORY
The experimental data described below show that further generalization of the coupled equation is desirable. Complete generalization may be obtained by replacing the summation in Equation 7 by a weighted integral
2 ~ D m l u (9)
where f ( ~ , w ) is a weight'ing factor such
x
=
h,
+ hc
where A, and up are functions of p ; h , is the trans-column contribution to h, and h, is the contribution from all other nonequilibrium processes. According to the random walk theory (8),which is admittedly based upon a highly simplified model, the transcolumn parameters may be expressed as
'nject Ion
Column
1 Inch Figure 2.
Detector Figure 1.
Diagram of apparatus
A, = 5wa*p2 w p =
where
'/*WP2P2
is the fractional velocity bias, Au/u, between the fluid in the wall region and that in the central core of the column. Equation 12 illustrates the idea, which may or may not be generally correct, that the trans-column parameters contain a common factorizable dependence upon p . Generalizing, we may be able to write u p
= wo.gl(P)
(13)
Thus the trans-column contributions to h could be expressed as
where x0 and w 0 are trans-column parameters with their dependence upon p factored out. The integral in Equation 14 is now a function of v only and we can write hi, =
or log hre and hm
=
=
(15)
.Bl(P) .82(Y)
1%
gl(P)
+
+ log
hc(~) .gx(P).gdV)
worked a t reduced velocities a t which 2 7 / ~is negligible and therefore the experimental plate heights can be identified with h,. A glance at the later figures shows that the plots of log h against log v are nearly parallel over a wide range of conditions. This broadly confirms the assumption of factorizability and the view that trans-column processes are the major source of band dispersion a t high reduced velocities.
(12)
wp
A, = Aogl(P);
Section of photometric detector
6r&)
(16)
Plots of log h , against log Y should then be parallel lines : plots of log h, against log v may not be parallel if h, is a significant fraction of h,. The degree to which the experimental plots are parallel for different values of p will therefore show how far the assumption of factorizability is correct and how far transcolumn processes are dominant. In the present work we have consistently
EXPERIMENTAL
Samples (1 to 3 pl.) of a potassium permanganate solution (5YC w./w. KMn04, 7YC w./w. KNOI in water) were injected into columns packed with glass beads and eluted with 10% aqueous potassium nitrate. The columns were mostly 0.30 cm. in diameter and 115 cm. in length. They were filled with water and packed by adding a slurry of beads in water until no further settling occurred when the columns were tapped lengthwise. The columns were closed a t the bottom with disks of glass fiber paper. Samples were injected directly onto the top of the column by a Hamilton 10-111. microsyringe. Immediately after injection the bands occupied less than 5 mm. of the column length; the calculated contribution of this initial dispersion to the observed H E T P was less than 1%. The flow of solvent through the column was controlled by a head of mercury adjustable up to 130 cm., which drove the solvent through a suitable constriction (see Figure 1). For high flow rates and with the smaller beads the column itself served as the constriction; otherwise a 150-cm. length of 0.004-inch i.d. steel capillary tubing was used. Aqueous potassium nitrate (10%) was chosen as the solvent, since the diffusion coefficient of potassium permanganate in pure water is strongly concentrationdependent at low concentrations (5, Zl). The use of a solvent of high ionic strength should eliminate this dependence. Since diffusion coefficients depend on temperature, the column was thermostated a t 36" i 2" C., corresponding to a change of about &57, in D,.
The emergent bands were detected photometrically by a detector of low dead volume, shown in Figuie 2 . Light from a 6-volt 24-watt bulb powered by a 6-volt lead accumulator, passed through 2-mm.-diameter windows in the detection and balancing cells, and fell on to two balanced LS223 photovoltaic light sensors (Texas Instruments, Inc.) whose maximum rated output was 270 pa, into 1000 ohms. The outputs of the two photocells dropped across two potentiometers set a t between 500 and 1000 ohms; under these conditions the output with pure water in both cells was about 70 pa. The potential difference across the ends of the two potentiometers was fed to it 1-mv. */,-second Fisher recorder; an out-of-balance signal could thereby be recorded after adjustment of the potentiometers to give null deflection when both cells contained pure solvent. The photocells were sensitive to temperature; this led to base line drift nhich limited the duration of runs. Oiily under favorable conditions rould elution times greater than about 100 minutes be used. Because of the limited sensitivity of the photocells, the low detection volume, and the thermal instability of the equipment, it was necessary to use somen hat large1 >amples of permanganate than necessary to give an esperimental HETP independent of sample size. All peak heights were therefore corrected to zero sample size by means of a calibration graph relating the fractional excess in H to the maximum concentration of the permanganate in the emergent band. The correction was geneially about 15%. The largest coirection was 30%. The experimental plate height was obtained from the column length and measurements of the retention distance and peak width on the recorder chart. Three measurements of peak width were made, the width a t half maximum height, the width at l / e of maximum height, and the base line width obtained by drawing tangents a t the inflection points. The three values of H were nearly always within 5% of each other and the mean value was taken. The peaks were always symmetrical, except for a slight tailing which did not affect the meawrements of width. The experimental VOL. 38, NO. 2, FEBRUARY 1966
* 255
I
I
3.0
of them in a right-angled groove, counting the beads under a microscope, and measuring the length of the row. Although there was normally a spread in bead sizes of about 50%, this procedure gave mean diameters reproducible to about 3%. One particular batch of beads had a much narrower particle size range. It was obtained by sieving beads larger than 0.130-mm. diameter out of a batch whose average diameter was 0.110 cm.; they formed less than 5% of the whole batch. The diameters of beads smaller than 0.1 mm. could not be determined by the above method because of static electricity; they were graded by sieving and their diameter was taken as the mean sieve opening.
I
I
4.0
3.5 Log v
Figure 3.
Plots of log h against log v RESULTS
Effect of detector design and reproducibility of column packing 0, C) Runs with standard detector on duplicate columns 0 , Runs with straight through detector Tap band d, = 3.0 mm., p = 12.8 Lower band d, = 5.7 mm., p = 11.6
plate heights were corrected to zero sample size as described above. Most of the values of H recorded below are the mean of three values obtained a t the same fluid velocity. Duplicate determinations of H rarely differed by more than 5%. The elution velocity was obtained from the retention distance, column length, and chart speed. This method of determining the plate height depends for its accuracy upon the assumption that the elution velocity is the same as the average velocity of movement of the peak along the column between injection and detection. The two are not necessarily identical. In the present system, which depends upon the flow of liquid under pressure through a constriction, variations in flow rate can result from changes in the temperature of the constriction and in the pressure head as the liquid flows out of the column. I t was therefore essential to check by duplicate runs that the elution velocity had not changed or, if it had, to make a suitable allowance. In order to be able to calculate the reduced velocity it was necessary to know the diffusion coefficient of potassium permanganate in the solution used. An attempt was made to measure this by Knox and Illclaren's method (16). X band of permanganate was eluted half way down the column and allowed to diffuse statically for up to 15 days before final elution. Assuming y = 0.6, a value of D, = 1.7 X sq. cm. per second was obtained a t 36" C. This compares with the literature value of D, = 0.5 X a t 18" C. in water a t a concentration of about 1% (22). Assuming an activation energy for viscosity of about 5 kcal. per mole yields D, = 0.8 X a t 36" C. Since the diffusion coefficient of KMn04 in lOyo KKOo can hardly be twice that in pure water, the value of 1.7 X must he high. Slight movements of 256
ANALYTICAL CHEMISTRY
A number of check experiments are first described. These were designed to assess the validity of some of the implicit assumptions made in the study: (1) The detector gave a reliable measure of the peak spreading and did not itself introduce any significant peak spreading; ( 2 ) the method of column packing was reproducible; and (3) plots of log h against log Y were the same for columns of different diameters but with the same ratio of column to particle diameter, p . Assumption 3 is equivalent to assuming that the packing density and particle size distribution are the same for different sizes of beads or that those parameters have no effect on h, which is unlikely. The first assumption was checked by constructing a detector in which the column passed through the device, so that the band was viewed while still in the column. This is in theory the ideal geometry for the detector, but the model used was less sensitive and less stable than that shown in Figure 2 . Figure 3 shows the results obtained with the same column using the two detectors
the band up and down the column could have occurred as a result of expansion and contraction when the thermostat switched off and on. This would cause an additional dispersion and result in a More accurate high value of D,. temperature control is required in order to obtain reliable diffusion coefficients by the arrested elution method with liquids. I n deriving the value of the reduced velocity we have therefore taken the lower value
D,
=
0.8 X lod6sq. cm. sec.
Glass beads were obtained from the English Glass Co., Ltd. The majority of batches contained a high proportion of spherical beads. Only one batch showed more than about 10% nonspherical beads and it was necessary to select spherical beads by rolling down a slowly rotating tube. The mean diameters of beads larger than 0.1 mm. were determined by aligning 50 to 200
1
0.0I 2.0
I
2-5
I
I
3.5
30 Log v
Figure 4.
Effect of column diameter
l o p line, 0 d, = 3.0, p = 12.8 Lower line, 0 d, = 3.0, p = 10.2 0 do = 5.7, p = 11.6 Broken line interpolated for d, = 3.0 p = 1 1.6
I
4.0
I
I
I
I
I
2.0 Figure 5. ratios
1
I
I
25
Plots of log h vs log
Y
4.0
3.5
3.0
stants. The present data therefore provide the clearest possible evidence that Equation 1 is incorrect, and that some form of coupled equation is required to describe the dependence of plate height upon velocity at high velocities. The dependence of reduced plate height upon p is most simply represented by a plot of log h against log p a t a specific reduced velocity, say Y = 300. Figure 7 shows that hw is more or less constant when p changes from 3 to 6 but rises rapidly in the region 6 < p < 18. The gradient of the logarithmic plot in this region is between 1.5 and 2. The value predicted by the Giddings (8) random walk theory is 2. At higher values hm declines gradually with further increase in p. In this region, 18 < p < 60, there is some scatter from one column to another; and with the particularly uniformly sized beads of 0.130-mm. mean diameter (the point 0)a column of unusually high efficiency was obtained. This last observation casts some doubt upon the reality of the decrease in hw with p a t the higher values. The trend might be due to an improvement in particle size uniformity with particles smaller than about 0.17mm. diameter. The strong dependence upon particle size distribution was unexpected, and more work is required to investigate this parameter. All forms of the coupled H E T P equation predict that H should tend to a constant value as the fluid velocity increases. At reduced velocities around 3000 the plate height for many of the columns in fact became more or less constant, but a t still higher velocities, where these were accessible, the plate height showed a decline, as seen in Figure 8. This decline cannot be explained by any form of coupling, nor by instrumental errors, since these
1
I
I
Log Y
1
I
for different column-particle diameter
Volues of p given on lines
and with duplicate columns prepared from the same batch of beads. The reproducibility from duplicate columns is good, but the straight-through detector gives a slightly lower plate height than the detector used in the rest of the work. The third assumption was checked by experiments with a column 0.57 cm. in diameter and p = 11.6. The results are compared with those obtained on 0.3O-cm.-diameter columns in Figure 4. Because of the fourfold larger volume of the wider column, some difficulty was experienced in obtaining the correct elution velocity, because of changes in the driving pressure as the band was eluted. Thus the values of H showed a considerably larger spread. Nevertheless Figure 4 shows that there is general agreement between the results for the wider column and the two narrower columns with similar ratios of column to particle diameter, although the curve for the wider column ( p = 11.6) should have fallen between those for the narrower columns ( p = 10.2 and 12.8). That there is a significant discrepancy probably indicates differences in packing density or in particle size distribution. The range of particle size was smaller the larger the beads and an increase in the uniformity of particle size may lead to a marked increase in column performance. The assumptions are clearly not completely valid, but the error introduced by accepting them is not likely to exceed 0.1 in log h. Figures 5 and 6, which contain the main body of the data, show reduced plots for 0.30-cm.-diameter columns containing beads ranging in mean diameter from 0.975 to 0.048 mm., corresponding to values of p from 3.1 to 62. The plots are essentially parallel, indicating that the effect of changing p
is to multiply h by a factor which is independent of velocity-that is, Equation 11 holds to a good approximation where h, is replaced by the experimentally determined reduced plate height. This implies that trans-column processes are the only ones which contribute significantly to H in the current range of reduced velocities, and that to a good approximation and w contain common factors which are functions of the ratio of column to particle diameter. The gradients of the plots of log h against log Y are about 0.3 a t v = 100 and 0.25 a t v = 2000: They are much less than unity, the value required by any linear H E T P equation. Indeed, the experimental plate heights a t Y = 3000 are approximately one hundredth of those predicted by Equation 1, assuming reasonable values of the con-
“Ot I 0.5
I
1.0
I
I
1.5
Figure 6. Plots of log h vs. log ameter ratios
Y
Log v
2.0
I
‘2.5
1I
for different column-particle di-
Volues of p given on lines
VOL. 38, NO. 2, FEBRUARY 1966
257
1.0
-
We have attempted to fit various forms of coupled equation to the experimental data. As shown in Figure 9, the simple coupled form bears little relation to the observed dependence of h upon Y. It predicts a too rapid change from a linear dependence of h upon Y to constant h. Various unweighted summation forms have been considered, but they give only slight improvement over the simple coupled form 2. A better fit is obtained only by weighting terms in the summation so as to give more emphasis to lower values of W . A good fit is obtained with an integral form in which is constant while contributions from different w's are weighted inversely with w-that is,
A
8
c en 0 2
0.5-
1
Log Figure
1
1
1.0
e
2.0
1.5
7. Effect of p on h at constant reduced velocity v = 300
h=
@ refers to 0.1 30-mm.-diometer beads
invariably produce high values of H. The decline is probably due to the onset of turbulent flow which effectively increases D, and thereby decreases transcolumn mixing times as well as reducing velocity differences within the column. The Reynolds number a t which the decline sets in is between 3 and 30. It is lower the higher the value of p. By using particles 0.05 mm. in diameter it was possible to obtain data a t a reduced velocity not far above that expected for minimum H. Figure 6 includes the data for this column. The point at the lowest reduced velocity shows a higher plate height than that for the next lowest, and it might appear that longitudinal diffusion is becoming important. Unfortunately, if D, = 0.8
~~
~~~
~
Table II. Values of trans-Column Parameters for Various p
Column diam. partlcle diam,, P 3.1 6.2 7.7 10.2" 12.8 17.0 18.1 28 33 37 62 22.6c 0
Integration limits for w 2x
WI
W)
0.00007 0.00012 0.00027 0.00027
0.07 0.12 0.27 0.27
p
DISCUSSION
The following points are clearly established by the experiments. With a liquid as eluting fluid the mobile phase contributions to H must be compounded in a coupled form for columns containing nonporous supports. At Y > 20 plots of log h against log v are roughly parallel for different values of p and hence h can be expressed approximately in the form of Equation 11. The major contributions to h must be from trans-column processes in this velocity range. At high reduced velocities and a t Reynolds numbers in the region of 10 turbulence becomes important and results in a decrease in h. Turbulence sets in a t Reynolds numbers which are lower as p is higher.
4.4 0.027
5 . 5 0.050 0.030 0.038 0.066 0.070 0.10 0.088 0.13 0.10 0.10
0;o
0;5
Log
Re
< 18.1 2X is close
to p . beads used in this series were very close1 graded compared to those used
in otXer series.
258
this expression and the data for one column is shown in Figure 9. The best values of the constants A, w1, and wz for the various columns are recorded in Table 11, which shows that within the range 6 < p < 18 parameter 2 X is close to p-that is, the eddy diffusionparameter, A , is close to the column diameter, do. However, in view of the uncertain role of particle size distribution, too much importance should not be attached to this coincidence. There may be little genuine relation between A and do. For columns with p larger than 18, data were obtainable a t low reduced velocities; below log Y = 1.8 the fit between Equation 17 and the experimental data was less good. In this region the plots of log h against log Y were nearly straight. If the upper points were fitted to the curve, the lower points fell above the theoretical curve. This could indicate that Equation 17 is an inadequate expression of the trans-column effect or that other
w2/2x
Data shown in Figure 9.
