Effect of Column to Particle Diameter Ratio on the Dispersion of Unsorbed Solutes in Chromatography John H. Knox and Jon F. Parcher’ Department of Chemistry, University of Edinburgh, U . K . The dispersion of unsorbed solutes eluted by water through 150-cm columns packed with spherical glass beads has been reexamined using large beads of various sizes and columns of different diameters. It i s shown that the dependence of the plate height on velocity, which i s similar to that previously found, follows the empirical equation h = (1/2 x l/C~)-l where the reduced plate height h = H/dp, and the The optimum reduced fluid velocity Y = d , / D , . value of n i s 0.3 f 0.1. Plots of log h VS. log were coincident for column t o particle diameter ratios, p < 6 and again for p > 8. The latter constancy i s in contrast t o previous results and it i s concluded that the high values of h obtained at high values of p in previous work were due to difficulties in packing the finer particle used. The sharp increase in h by a factor of about 1.8 for the range 6 < p < 8 i s consistent with some results obtained by Sternberg and Poulson for gas phase eluents, but has no obvious explanation. The constancy of h for high values of can be explained by the application of the Giddings nonequilibrium theory to a model column, possessing a discrete wall layer, in which a serious perturbation of velocity occurs. A hexagonal column showed poorer efficiency than the normal round column and an infinite diameter column showed higher efficiency than normal columns and less dependence of h upon v, the optimum value of n being about 0.2. It i s concluded that infinite diameter columns offer some advantages over conventional columns in high pressure, small particle, liquid chromatography.
+
JI
AXIALBAND
DISPERSION during
chromatography is measured in terms of the plate height, H , defined as the rate of increase of variance of the Gaussian peak profile per unit length of column. H = au,2/az (1) For uniform columns and incompressible eluents this becomes
H = uZ2/L (2) where L is the column length. In the construction and operation of columns it is generally desirable that H be as small as is consistent with the limitations of pressure and other operational parameters so that analysis can be achieved in minimum time. This is particularly important in liquid chromatography where the use of high pressures and small particles is essential. Much theoretical work has been carried out on the dependence of H upon fluid velocity and column parameters, see ( I ) and (2),and attempts have been made to define the requirements for minimum time analysis ( I , 3-5). There is still, however, a lack of high quality experimental data. Present address, Department of Chemistry, University of Mississippi, University, Miss. 38677 (1) J. C. Giddings, “Dynamics of Chromatography,” Part 1, Marcel Dekker, Inc., New York, N. Y . , 1965. 35,439 (1963). (2) J. C. Giddings, ANAL.CHEM., (3) Zbid., 34, 314 (1962). (4) J. H. Knox, J. Chem. SOC.,1961,433. (5) R. P. W. Scott in “Gas Chromatography 1958,” D. H. Desty, Ed., Butterworths, London, 1958, p 189.
In general, peak dispersion arises from the following factors: I. axial molecular diffusion; 11. a finite rate of equilibration of solute between the mobile and stationary phases in the column; and 111. nonequilibrium within the mobile phase which is continuously generated by the complex flow and velocity patterns within the column. For unsorbed solutes the second factor is absent, and Giddings (6) has argued convincingly that factors I and I11 give independent contributions to H which may be written as in Equation 3
where y, At, and w t are geometrical constants, d p = particle diameter, u = fluid velocity, and Dm = diffusion coefficient of solute in the mobile phase. This equation and other less sophisticated equations (7, 8) show that Hmay be scaled to the particle diameter and that the fluid velocity may be scaled to the rate of diffusion over a particle diameter. Thus, for comparative purposes, it becomes convenient to express H and u in dimensionless, or reduced, terms (9) as in Equation 4. Reduced plate height
h = H/d,
Reduced linear velocity v = udp/Dm
(4)
Plate height equations are then simplified. For example, Equation 3 takes the form : i
h = 2y/v
+ E (1/2Xt +
l/wtJI)-l
(5)
In theory, columns of different particle diameters with different eluents may be directly compared by plotting h US. v. In particular, gas and liquid chromatography may be compared directly (IO). Such a comparison is particularly useful because one can readily obtain low values of v using a gaseous eluent and high values using a liquid eluent. A total range of about six orders of magnitude of v is thereby accessible with normal equipment. While Equation 5 has a sound theoretical basis, it is not obvious what values should be assigned to A t and w t , nor how many terms are required in the summation. Knox (II) has shown that the experimental data for unsorbed solutes using a liquid eluent can be fitted to this type of equation only if the summation is replaced by an integration in which the different values of A $ and w t are appropriately weighted, but the original simplicity and elegance of the model is thereby lost. Equations 3 and 5 contain no explicit reference to trans(6) J. C. Giddings, ANAL.CHEM., 35, 1338 (1963). (7) J. J . van Deemter, F. J. Zuiderweg, and A. Klinkenberg, Chem. Eng. Sci., 5,271 (1956). (8) . , J. F. K. Huber in “Advances in Chromatography 1969.” A. Zlatkis, Ed., Preston Technical Abstracts Co.,Evanston; Ill., 1969, p 283. (9) J. C. Giddings, J. Chromatogr., 13, 301 (1964). (10) J. C. Giddings, ANAL.CHEM., 37, 60 (1965). (11) J. H. Knox, ibid., 38, 253 (1966). VOL. 41, NO. 12, OCTOBER 1969
1599
column effects, although one or more terms in the summation can, of course, be used to accommodate them. The appropriate and w i should then contain the column to particle diameter ratio, p = d,/d,. In view of the difficulty in predicting a correct and useful theoretical expression for h, it is more reasonable at this state to use a general expression for h of the form
+
h = 2Y/V f ( V , P ) (6) This equation implies that for ideal columns, h should be a universal function of v for columns of similar geometric shape-Le., with the same ratios of length to diameter to particle diameter-but that the function may depend upon the column to particle diameter ratio, p . It also implies that factors I and I11 contribute independently to h. Previous work in gas and liquid chromatography with unsorbed solutes (3,11, 12) has indicated that the curves of h us. v for different values of p when plotted on a logarithmic scale, are parallel over at least a 100-fold range of v. At a constant value of v, h increases with p, at least up to a certain limiting value of p . Experiments with sorbed solutes likewise indicate a relatively strong dependence of h upon p (13), and it is a common practice in the preparation of efficient gas chromatographic columns to arrange them to be as narrow as is consistent with adequate capacity. Attempts to interpret the dependence of p upon h have concentrated attention upon trans-column velocity variations which are thought to arise from effects imposed upon the packing by the walls of the tube and by fractionation of particles during packing. Numerous equations have been derived (13-17) and fall into two classes. First, there are those which assign a contribution to H which is linearly dependent upon fluid velocity. Theories of this type (13, 14) assume that the dispersive effects of a nonuniform velocity profile are reduced by transverse molecular diffusion only. Second, there are theories which allow for the reduction of dispersive effects both by molecular diffusion and by anastomosis, or stream splitting (15-17). The second type of theory is a relatively simple extension of the first type and results in an equation of the coupled form which has the general form of Equation 7.
H = (1/2Xdp
+ Dm/wdP2u)-’
(7)
where X and w are now parameters which reflect the shape of the velocity profile over the column cross-section. The theories differ in the evaluation of these parameters. In principle, the best way of determining these parameters is by means of the Giddings nonequilibrium theory ( I , 16). In wide columns it appears that H is dominated by transcolumn effects. Thus, the simple Equation 7 ought to be able to accommodate the data if the theories are correct. In practice, this is not true ( I ] , 18). Even when H appears to be (12) J. C . Sternberg and R. E. Poulson, ANAL.CHEM., 36, 1492 (1964). (13) F. H. Huyten, W. van Beersum, and G. W. A. Rijnders in “Gas Chromatography 1960,” R. P. W. Scott, Ed., Butterworths, London, 1960, p 224. (14) P. C. van Berge, P. C. Haarhoff, and V. Pretorius, Trans. Faraday SOC.,58,2272 (1962). (15) S. M. Gordon, G. J. Krige, P. C. Haarhoff, and V. Pretorius, ANAL.CHEM., 35, 1537 (1963). (16) A. B. Littlewood in “Gas Chromatography 1964,”A. Goldup, Ed., Institute of Petroleum, London, 1965, p 77. (17) S. T. Sie and G. W. A. Rijnders, Anal. Chem. Acta., 38, 3 (1967). (18) D. S. Horne, J. H. Knox, and L. McLaren, Sep. Sci., 1, 531 (1966). 1600
ANALYTICAL CHEMISTRY
dominanted by trans-column effects the dependence of h upon v is very similar to that when H is not dominated by transcolumn effects. An equation of the form of Equation 7 predicts a change from a linear dependence of h upon v to h independent of v over a velocity range of about 100-fold. The actual transition occurs over a much wider range; the gradient of the log h against log v curve changing only from about 0.33 to 0.25 over such a range. It is apparent that the current theories are unable to predict, except qualitatively, the dependence of h upon Y or p , and that progress will be made only when more extensive and precise experimental data become available. Previous work by Knox (11) on unsorbed solutes employed columns of fixed diameter packed with glass beads of different diameters, and showed that h for a given Y increased by a factor of about five when p increased from about 6 to about 20; for higher values of p , h was more or less constant. Horne, Knox, and McLaren (18) have shown that the increase probably was not associated with an increase in the particle size range as the particles became smaller. Experiments by Gordon et a(. (15) with wider columns and larger beads suggest that the effect might be due to difficulties in packing small beads rather than to genuine wall or trans-column effects. The present work was undertaken to examine this point. Experiments were carried out with several diameters of column from 0.3 to 1.1 cm and with relatively large beads from 0.05 to 0.2 cm in diameter. We believe that there is no problem in packing such beads uniformly. We have also investigated the properties of a hexagonal column in which more regular packing might be expected, and what we have termed an infinite diameter column. A column which is effectively infinite in diameter, that is, one which is free from wall effects, may be constructed on the basis of the following reasoning. Lateral dispersion, as shown by Horne, Knox, and McLaren (18) may be characterized by a radial or lateral plate height H , which is defined by Equation 8.
H,
= aur2/az
= u,~/L
(8)
where a,*is the variance of the transverse Gaussian concentration profile which results from a point injection a distance z upstream. Numerous chemical engineering studies have established that the profile is indeed Gaussian and that the dependence of H, upon fluid velocity may be given within 50% by the equation:
H,
or
h,
=
2yD,/u
=
2y/v
+ 0.15 d p
+ 0.15
(9)
The first term is identical to that appearing in the equation for the longitudinal plate height, and results from transverse molecular diffusion. The second term arises from anastomosis or stream splitting. At the velocities of interest in the present study the first term is negligible and so H, = 0.15 d p , or h, = 0.15. If a sample injected centrally at the head of a column fails to reach the disturbed packing region close to the walls by the time the sample reaches the foot of the column, then the c01umn is effectively free from wall effects, particularly-if only a central cut of the sample is taken for analysis. In practical terms, if we specify that not more than 5 of a centrally injected samples reach the walls, the condition for a column to be infinite in diameter becomes: d,
=
4c7
(10)
Capillary Restrictors ----
-/-I 7
L----J\
r'
25.0 O C
I
I ycolumn I
.
lN21
1
,
Mercury I Manometer1, II
Flame Detector Figure 1. Schematic diagram of the apparatus
Elimination of H and ur between Equation 8, 9, and 10, assuming that the first term in Equation 9 is unimportant, gives: dcz/L = 2.4 d p or Lid, = 0.4 p
Figure 2. Graph of Hu us. u2 for the elution of acetone in water through a 1.00-mm precision bore glass tube at 25.0 "C
(1 1)
Thus, with d p = 0.05 cm and dc = 3.0 cm-Le., p = 60the column would be effectively infinite if its length were less than 75 cm. If the particle size is greatly reduced, as is desirable in high pressure liquid chromatography, quite narrow infinite diameter columns may be constructed. Thus, if d p = 0.003 cm (30 microns) and d, = 0.5 cm, then L 5 33 cm. A column of such dimensions has 1.1 X l o 4 particles to its length and is therefore expected to achieve considerable resolution when working under optimum conditions for which h should be well below 10. In order to ensure that only those molecules which traverse the central part of the column are sampled, a central take-off tube must be inserted which samples approximately 10% of the total flow of eluent, the remaining 90 being rejected, or possibly saved for preparative purposes. As will be seen from the following discussion infinite diameter columns have several advantages over conventional walled columns. EXPERIMENTAL
Thy apparatus is shown schematically in Figure 1. Deaerated water from a pressurized reservoir was forced through the column at a stabilized rate by nitrogen maintained at constant pressure by a pressure regulator (Negretti and Zambra, London) in series with a Cartesian manostat (Edwards High Vacuum, Crawley, Sussex, England). By using different pressures of nitrogen (up to 2 atm) and a series of thermostated capillaries (0.25 mm diameter nylon) before the column, flow rates from 10 pl/min to 20 ml/min could be realized, corresponding to reduced velocities from 5 to 5 X lo4. The columns were straight glass tubes 150 cm in length and 0.33 to 1.11 cm in diameter. Stainless steel columns showed serious absorption of the acetone which was used as the solute. The columns were thermostated at 25.0 "C. They were packed with dry, clean glass beads by tapping endwise until no further settling occurred. Packing columns already filled with water using the slurry method gave significantly poorer results. The column exits were closed with 400 mesh wire gauze attached by Araldite epoxy resin. After packing, the air was dissolved out by passage of deaerated water. Samples of acetone (0.1 to 0.5 pl) were injected directly into the center of the packing by means of a microsyringe. Eluted bands were detected by a membrane detector (18). Although
this detector has a time constant of about 5 sec, it is stable over long periods of time, has negligible dead volume, and is very sensitive to volatile organic materials. For liquid chromatography with the particle sizes used here, it was satisfactory at all but the highest flow velocities. Eluted peaks were highly symmetrical except at high linear velocities when the time constant of the detector became important. The reduced plate height was obtained from the chromatogram by Equation 12. h
=
(L/dv) (Ut/tr>'
(12)
where u t is the standard deviation of the peak and tr the elution time in the same units. The standard deviation was obtained in three ways: from the base width of the peak, from the peak width at half maximum height, and the peak width at maximum height/2.718. The three values of h differed by less than 2 %, confirming that Gaussian peaks were obtained. The contribution of the apparatus to peak variance was determined by plotting u z z against column length for four different column lengths. The apparatus contribution was below 2% of the total at the highest velocities for which peaks were truly symmetrical with 150 cm long columns. Repacking the columns with identical beads gave values of h within 2 of previous values. Repacking with beads from the same batch gave values of h within 8 % of previous values at the worst. In order to evaluate the reduced velocity, the diffusion coefficient of acetone in water at 25.0 "C was required. This was obtained by Taylor's method (19) using a 1.00-mm diameter precision bore glass tube. According to Taylor's equation the plate height of an unsorbed solute eluted from an empty tube is:
H = 2D,/u f r2u/24D, or Hu
=
20,
+ (r2/24Dm)u 2
(13)
The plot of H u against u2, shown in Figure 2, was a good straight line from which D, = 1.06 + 0.05 X 10-5 cmz sec-1. (19) G . Taylor, Proc. Roy. SOC.,Ser. A , 219, 186 (1953).
VOL. 41, NO. 12, OCTOBER 1969
1601
1.o
0.5 c m
s
0.0
x
-0.5 1.o
1.5
2.0
2.5
3.0
3.5
Log v
I/
Figure 5. Plot of log h us. log p for 150-cm columns P =
cm
-
U I
0.48cm-+
Figure 3. The infinite diameter column
The value calculated by the Wilke-Chang equation (1, 20) is 1.15 x cm* sec-1. We have accordingly taken a mean value for the present work.
D, (acetone in water at 25.0 "C)
= 1.10 X
cm2 sec-l
Spherical glass beads (English Glass Co., Ltd., Liecester, England) were sieved to give cuts of small diameter ranges between 0.48 and 2.04 mm. The majority of experiments were carried out with 0.48-mm beads. The average bead diameter of each batch was determined by measuring the lengths of a row of 50 beads aligned in a groove using a travelling microscope. The standard deviation of the diameter of a single bead was about 0.03 mm. Two particularly uniform batches of beads were prepared by collecting those beads which stuck in 30 and 35 mesh sieves. For these two batches, the standard deviation for the diameter of a single bead was 0.015 mm. No significant difference in (20) C. R. Wilke and P. Chang, Amer. Inst. Chem. Engr., J., 1, 264 (1955).
4.9; 0 p = 7.3; 0 p = 18.9; 0 p = 20.8;
0p =
23.0
performance was observed with columns containing these specially selected beads, confirming previous conclusions (18). A hexagonal glass column was constructed by shrinking a round glass tube into a hexagonal brass former by heating it under vacuum. After cooling, the grass rod was readily removed because of its greater coefficient of thermal expansion. An infinite diameter column containing 0.48-mm diameter beads was constructed as shown in Figure 3. Injection was made centrally into the wide part of the column using a syringe with a long needle. The flow rate from the central take-off tube was adjusted so that there was no disturbance of the flow in the main part of the column. However, it was found that the precise fraction removed by the sampling tube was not critical and could be varied by a factor of at least two on either side of the calculated value without introducing additional peak spreadings. RESULTS AND DISCUSSION
Plots of log h against log v for twelve circular columns are shown in Figures 4,5, and 6. A notable feature of the data is that the curves for p < 6 are nearly coincident and those for p > 8 likewise. This is illustrated convincingly by the composite Figure 7, which includes all of the data for columns with p outside thelimits 6 to 8. These data compare well with those of Edwards (21), who examined columns containing beads from 0.1 to 6 mm in diameter, in a column 8.3 cm in diameter and 115 cm in length. (21) M. F. Edwards, ANAL.CHEM., 41, 383 (1969).
"O 0.5
1
0.5
c
-c
m _J
0.0
0.0
-0.5
1.o
1.5
2.0 Log
2.5
3.0
-0.5 1 1.o
I 1.5
v
Figure 4. Plot of log h us. log p for 150-cm columns O p = 6.3; 0 p = 6.9; 0 p = 7.8; 0 p = 12.9
1602
3.5
ANALYTICAL CHEMISTRY
I 2.0
I 2.5
I
3.0
Log I,
Figure 6. Plot of log h us. log 0 p =
P
for 150-cm columns
7.0; 0 p = 7.9; 0 p = 10.1
I
3.5
I
"O
0.75
0.5
r
s" 0.0
q. 0.50 2 z
Is)
0
_I
1.0
1.5
2.5
2.0
3.0
3.5
Log v
8
0.2 5
Figure 7. Comprehensive plot of all data for p < 6 (represented by 0 ) and p > 8 (represented by 0). The solid line is one form of Equation 14 with n = 0.33
0.00 Agreement is well within the experimental error of Edwards' data. Edwards also observed no dependence of h upon p for p 2 12. Whereas, Edwards worked with reduced velocities from about 1 to 100, the data presented here represents a range of velocities from about 10 to 5000. Edwards' data can be fitted by the simple coupled equation, that is, Equation 3 with a single term in the summation. However, the data over a more extensive and higher range of reduced velocity show that this equation is entirely inadequate. As noted previously, an integral form of Equation 5 may be fitted to the data. However, a much simpler empirical equation, a form of which has been proposed by Huber (8), fits the data equally well. In Figure 7, Equation 14 has been fitted to the data using n = 0.33. h = (1/2X 1jCv")-' (1 4)
+
While Huber has taken n = 0.5, there seems to be no theoretical justification for such a value and it seems best to regard n as a purely empirical parameter. While Equation 14 does not seem very satisfactory from the theoretical point of view, it would have great practical value, particularly if it could be established to have validity for sorbed solutes. The advantages of having an analytical expression for h which can be used in the derivation of optimum conditions for analysis would be inestimable. The present results contrast with those obtained previously (ZI) by showing much lower values of h for high values of p although the agreement is good at low values. Because large beads were used in both studies for low values of p but smaller beads used in the original study to obtain high values of p, we conclude that the effect previously noted was mainly the result of the difficulty in packing smaller particles. From Figures 4 to 7, it is clear that the curves of log h against log v are essentially parallel with the possible exception of some of the curves for the range 6 < p < 8. It is therefore possible to indicate the dependence of h upon p by plotting values of h at fixed v against p. Figure 8 shows that there is a sharp increase in h (v = 300) in the critical region of p between 6 and 8. The sharp change results in an increase in h by a factor of about 1.8. Experiments carried out in the gas phase by Sternberg and Poulson ( 1 2 ) show similar features, though less marked, as illustrated by the comparative Figure 9. It therefore seems likely that the effect is genuine and must result from a marked change in packing mode in the region of p = 7. The effect may be rationalized by assuming that the walls impose their structure on about four layers of beads in contact
1.o
0.5
Log Figure 8. Plot of log h3W at This work: 0 Knox in (11); Gordon et nl. in (15)).
1.5
p Y
= 300 against log p
Horne et ul. in (18);
with the walls. Thus, the column may be regarded as being made up of a central, circular core, whose structure is that of a random bed, surrounded by a layer of about three particle diameters in which the packing structure is markedly different and dominated by the requirements of a smooth, circular wall. Columns with p < 6 are composed almost entirely of the wall layer, while those with p > 8 contain both types of structure. The problem of the plate height contribution from such a structure may be treated by the nonequilibrium theory ( I , 22). This has been applied to trans-column effects resulting from various types of velocity profiles by Littlewood (16). He assumed various forms of velocity profiles for the general expression :
u(r)
= uo{1
+ bf(r) ]
(15)
where r is the distance from the center of the column; uo is a base velocity; and b is a dimensionless constant. Very similar results were obtained for circular and parallel plate models. For qualitative purposes, the parallel plate model gives an adequate approximation, and the mathematical treatment is simpler. Littlewood assumed profiles which expanded as the column diameter expanded. That is,f(r) could be written asf(r/R), where R is the column diameter. For such profiles, the plate height contribution was of the form:
H = Constant
{EJ a
Where D is the total transverse diffusion coefficient given by D = y D , 0.05 udp;andg(b) is a weak function of b. Thus, the plate height contribution increases with the square of the column radius for the profiles proposed by Littlewood. When the nonequilibrium theory is applied to a column with a distinct wall layer, the result is somewhat different. Equation 17 .represents a hypothetical velocity profile for such a situation. It may be applied either to a circular column of diameter 2R or to a parallel plate column of width 2R.
+
(22) J. C . Giddings, J . Gus Chromatogr., 1(4), 38 (1963). VOL. 41, NO. 12, OCTOBER 1969
1603
2.0
1.5
h
R ladp Figure 10. Velocity profiles corresponding to Equation 17 for p values from 2 to 40
1.0
velocity variation, with only the numerical constant being different for different profiles. The evaluation of the plate height contribution of the transcolumn effect is based on Giddings’ nonequilibrium theory ( I , 22). The velocity at any point on the cross section of a column is given by Equation 15 and the mean velocity is determined from the equation:
0.5 0
I
I
I
I
5
10
15
20
P
+ (buo/R)s, f ( 4dr R
Figure 9. Effect of p on h at fixed values of v
.ii = uo
(19)
0 This work (liquid eluents at Y = 300
Sternberg and Poulson (12) (gaseous eluents at corresponding to the minimum plate height)
f(,.) = 1 + e-R/adn(e-Wadp - ,-r/a&
- eT/ad,)
Y
(17)
The dimensionless scale factor, a , measures the effective thickness of the wall layer in particle diameters. Figure 10 shows the form of the profile for different values of R/adp = p/2a. The wall effect decays exponentially toward the center of the column, and for large values of p, there is a distinct wall layer within a few particle diameters of the wall where the velocity is significantly greater than in the center of the column. This profile bears some resemblance to those determined experimentally by Schwartz and Smith (23). They found that the velocity passed through a maximum at a distance of a few particle diameters from the wall before falling to zero at the wall. Depending upon the value of p , the velocity excess profile is difficult to represent mathematically, but could be developed from Equation 17 by the addition of a further term, for instance, if g(R, r, a) represents the second term in Equation 17, a profile with a hump of roughly the correct shape would be generated by Equation 18. f’(r)
=
1
+ a:g(R, r ,
UI)
- (a:
- 1 ) g(R, r , UZ)
(18)
Where a is a constant greater than unity, and a2> al. Because of the algebraic complexity of the nonequilibrium treatment, we have chosen the simpler profile represented by Equation 17. This is probably justifiable in view of Littlewood’s (16) results. He showed that the transcolumn contribution to the plate height was always given by an equation of the form of Equation 16, irrespective of the details of the transcolumn (23) C. E. Schwartz and J. M. Smith, Znd. Eng. Chem. 45, 1209 (1953). 1604
ANALYTICAL CHEMISTRY
Giddings has derived a general equation for the nonequilibrium factor, E , which he has applied to various sources of nonequilibrium ( I ) , including, for example, that arising from transcolumn effects due to velocity profiles caused by thermal lag between the center and the walls of columns in programmed temperature gas chromatography (24). -7%= (u
- ii)
a hF
-
Daz
Where is the mean concentration over the column cross section at a distance z from the head of the column and Vzt is the Laplacian operator. For columns with circular symmetry, the Laplacian operator is - The integration of Equation 20 is performed subject to the boundary conditions that ac/dr = 0 at r = 0 due to symmetry and
lR
e dr = 0, which arises from the definition of
t
as the
fractional excess of the local concentration over the mean concentration. Substitution of Equations 15 and 19 into 20 with subsequent integration leads to the value of E :
r2 g (1 - x ) +
(9
( R 2 - 6u2dp2)) (21)
The plate height contribution is obtained from the equation:
(24) J. C. Giddings, ANAL,CHEM., 34, 722 (1962).
:”r 2.0
Ik4
c 0.5
2 .o
1.5
2.5
3.0
3.5
v
Log
Figure 12. Comparison of round and hexagonal columns with roughly equivalent values of p
P 1.o
i
Evaluation of H pfor the profile given by Equation 17 leads to:
Ho =
o Circular column
0 Hexagonal column
Figure 11. Dependence of hamupon p , calculated from Equation 24 at various values of b
(adp)2b2 -X 37
D
R(l
- b) + abd,
- x(Rb
+ abdp)
ti (23)
1 -0.5
1.o
A more convenient form of the equation is in terms of reduced parameters h and v (Equation 4) and p = 2R/dp: 2(ab) h=X 3Y
[p(l
I
i
1.5
2.0
Log
’
v
(24)
2.5
I
30
35
u
Figure 13. Comparison of infinite diameter column with a conventional walled column 0 Walled column
- 6 ) + 2ab - b(p + 2a) e-”laI2
I
0
Infinite diameter column
R/adp, and hence p , increases, the averaged value over any annulus will approximate better to a smooth profile, such as that suggested. The calculated values of h are therefore likely to become more realistic when R/adpis large. It is tempting to identify the rise in h noted experimentally with the rise predicted by theory for values of R/adp above those giving the minimum in h. But because the experimental rise is much sharper than the predicted rise, the identification is doubtful, and it is possible that a discontinuity in the flow mechanism occurs at a particular value of p . Theory nevertheless predicts that for high values of p the trans-column contribution to h , which arises from a finite wall layer, becomes independent of column diameter. This is in complete agreement with the current experiments. The agreement for high R/adp values suggests that it might be worthwhile to test other models of trans-column velocity profiles, for example a damped wave, which might reproduce the experimental data better at low values of p . Data obtained with the hexagonal column are shown in Figure 12. The column is less efficient than the equivalent round column. While the hexagonal column may be expected to give a more regular packing arrangement over restricted regions, any disturbance of the arrangement will extend over a large part of the column cross section (22) so that major voids and imperfections will arise if there is not a perfect fit of the particles to the column cross section. The experimental result suggests that the dispersive effects of such major packing faults more than counteracts any beneficial effect which arises VOL. 41, NO. 12, OCTOBER 1969
1605
from the areas of regular packing. It may well be true that a perfectly regular packing offers the highest efficiency,but when it is remembered that a normal chromatographic column contains upward of lo6 particles it seems unlikely that such columns will ever become a practical possibility. Data obtained with the infinite diameter column are shown in Figure 13. They differ in one important respect from those obtained with walled columns. The dependence of log h upon log v is much less than for the walled column. This must arise because of the absence of genuine trans-column effects. Over the range of reduced velocities 20 to 2000, h increases by a factor of only two. If such a shallow depend-
ence can be extended to sorbed solutes, there is clearly great potential for the use of infinite diameter columns in high speed analytical chromatography, particularly in high speed liquid chromatography, where the use of high pressures and very small particles makes it possible to construct relatively long narrow columns which still have the properties associated with an infinite bed. RECEIVED for review June 6,1969. Accepted August 11,1969. This investigation was supported by a Public Health Service Special Fellowship (No. IF3 AP 32,964-01) from the Division of Air Polution.
Further Studies on the DC Carbon Arc, Gas-Chromatographic Technique for the Determination of Gases in Metal Royce K. Winge and Velmer A. Fassel Institute for Atomic Research and Depart. of Chemistry, Iowa State University, Ames, Iowa 50010 A summary of new observations on several important phenomena related to the dc carbon arc-extraction and gas-chromatographic measurement of the oxygen and nitrogen content of metals is presented. Homogenization and temperature control of the chamber gases are discussed with respect to obtaining representative aliquot samples for gas-chromatographic analysis. Data indicate that oxygen can be determined in steels containing up to 10 per cent manganese by the addition of 30-40 mg of silver or tin per gram of sample. A platinum flux allows the same determination with up to about 5 per cent manganese. The nitrogen determination appears not to be affected by manganese content. The actual operating blank of the dc arc method is the algebraic sum of a number of positive and negative components. The positive components can be controlled more readily than the negative components which result from reactions that convert a portion of the extracted gases into unmeasurable products. The dc arc-extraction technique shows promise of being able to determine the oxygen and nitrogen content of certain refractory metal sulfides, materials which are rather recalcitrant toward other analytical methods. Also included is a description of a versatile extraction chamber designed for use with both gas-chromatographic and optical emission spectrometric determinations of the extracted gases.
PREVIOUS STUDIES ( I , 2) have established the value of arcextraction techniques in the rapid determination of the gaseous element content of many metals and alloys. These studies have been prompted by the profound effects of gases, in the form of inclusions and interstitial impurities, on the physical and chemical properties of metals. This paper is concerned with two aspects of the arc-extraction technique: (a) a description of a versatile extraction chamber designed for use with both gas-chromatographic and spectrometric methods of measuring extracted gases and (b) a summary of new observations on several important phenomena related to the extraction and measurement of the gaseous element content of metals.
(1) F. M.Evens and V. A. Fassel, ANAL.CHEM.,35, 1444 (1963). (2) R. K.Winge and V. A. Fassel, ibid., 37,67 (1965). 1606
ANALYTICAL CHEMISTRY
EXPERIMENTAL. FACILITIES
Extraction Chamber. Several chambers for the arc-extraction of gases from metals have been described. The typical chamber consists of a vacuum tight enclosure containing a horizontal rotating electrode stage with which a number of sample electrodes can be rotated into arcing position (3). Regardless of whether a gas-chromatographic or spectrometric analytical method is used, it is advantageous in theory to keep the chamber volume small to reduce dilution of the extracted gases. Chamber volumes as low as 25 cc were evaluated by Applied Research Laboratories ( 4 ) but this small size proved impractical because of a tendency of the arc to strike to the chamber wall. This problem was eliminated by increasing the chamber volume to 50 cc, but another problem became apparent with this chamber. More than 50 per cent of the carbon monoxide and nitrogen added to the chamber was lost to what was attributed to adsorption on the walls, on graphite deposits, or to gettering by vaporized metal (4). Internal volumes of other chambers have ranged from 150 cc (5) to approximately two liters (6). The newly developed chamber, shown in Figure 1, is in the form of a short cylindrical section [l] with a horizontal axis. With the exception of a few external fittings the chamber is constructed entirely of stainless steel. “Dead volume appendages” such as vacuum gauges, extension tubes for viewports, valve body volumes, and the associated manifold have either been eliminated or separated from the chamber by a valve. The resulting consolidated volume of the chamber, with its maximum dimension in the vertical direction, allows the extracted gases to homogenize very rapidly with the extraction atmosphere through convection currents set up by the arc discharge. The internal volume of the chamber is approximately 650 cc. The internal diameter is 11.4 cm, and the depth is 6.35 cm. The electrode stage is rotated in a vertical plane through a gear set [2] (recessed into cover plate) and knob [3]. The chamber was used initially with the 5-electrode stage [4]. (3) F.M.Evens and V. A. Fassel, ANAL.CHEM.,33, 1056 (1961). (4) J. L. Jones, Applied Research Laboratories, Hasler Research Center, Goleta, California, personal communication, 1964. (5) H. Goto, S. Ikeda, K. Hirokawa, and M. Suzuki, 2. Anal. Chem. 228(3), 180 (1967). M (6) J. Artaud and C. Berthelot, Mem. Sci. Rev. Met. LVII(S), 338 (1960).
