1812
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ANALYTICAL CHEMISTRY, VOL. 50, NO. 13, NOVEMBER 1978
Speed of Analysis in Open Tubular Columns Gas Chromatography Georges Guiochon Laboratoire Chimie Analytique Physique, Route de Saclay, 9 1 128 Palaiseau Cedex, France
The optimization of capillary columns is discussed, and a number of practical rules are given. It is shown that small diameter open tubular columns offer large possibilities of improving current performances. The analysis time is proportional to column diameter, while the detection limits are practically constant, although the sample size has to decrease in proportion to d,'. The necessary inlet pressure, although higher than usual, is easy to achieve. For easy Separations, packed columns offer more reasonable experimental conditions. Although their performances can also be improved by the use of small particles, they cannot compete with open tubular columns in terms of analysis time. Small diameter capillary columns ( d , < 0.1 mm) demand specially designed equipment, contributing little to band broadening, but offer the possibility to achieve the separation of very closely related compounds.
The past year has seen a rebirth of capillary columns in gas chromatography. The interest in their use is rapidly increasing and a variety of new techniques of preparation have been published. T h e use of glass capillaries is favored, with a consequence, which has been perceived by only one group yet ( I ) , t h e possibility to select the inner diameter most suited t o any given analysis. Glass drawing machines are very flexible, and it is easy, using any of the recommended processes, t o prepare a column of any length and diameter. T h e use of columns of large inner diameter, with a porous layer of a n inert support of Chromosorb (2), Silanox ( 3 ) ,or graphitized Carbon black ( 4 ) or without such a layer ( 5 ) ,has already been recommended as a means to increase the sample size and thus to decrease the detection limits in GC or in GC-MS. Certainly caution is strongly advised in that matter, as the detection limits are functions of many parameters and the effect of column diameter on efficiency, gas flow-rate, etc. may reduce t h e actual effect much below what is expected. I t does not seem yet to have been realized how the analysis time is directly related to the column diameter and how the actual performances of equipment can be considerably improved by using small diameter columns. The present paper discusses the theoretical background of the problem. Some experimental results will be given in a forthcoming paper. We discuss first the parameters which determine the analysis time and t h e way to reduce it, then the effect on detection limits, and the requirements regarding equipment whose contribution t o peak broadening should not be too large.
THEORETICAL Preliminary. The analysis time increases rapidly with difficulty of separation. As we show later, it increases more rapidly that the necessary plate number N . This number is related t o the thermodynamics data by:
N
= 16R2
(
L)2
a-1
( 1 + k '7 ) (1)
where R is the resolution requested between two peaks, a the 0003-2700/78/0350-1812$01.OO/O
Table I. Relative Retention, Retention Free Energies, and Column Efficiency A(AG)~ 0
(calimol) 146 112 76 39 24 16 12 8.0 4.0
Nb
NC
1020 1420 1.2 1670 2 310 1.15 3 440 4 760 1.10 1.05 1 2 500 1 7 350 3 3 500 46 400 1.03 74 000 102 000 1.02 180 000 130 000 1.015 290 000 400 000 1.01 1150000 1590000 1.005 a A ( A G ) = A G , - A G , ; T = 400 K (127 "C). k ' = 3, R = 1. k ' = 1.76. R = l ( T h e choice of k ' = 1.76 is justified below in the discussion of Figure 1). relative retention of the corresponding compounds, and h'the column capacity factor of the second peak. This equation is used below, and we just want to point out here how much N increases rapidly with decreasing a - 1. Table I gives some examples. Now (Y is related (6) to the difference between the Gibbs free energies AG1, AGz, associated with the passage of one mole of solute from t h e stationary to the mobile phase: l ( A G ) = AG2 - l G , = RT In cy or in practice:
l ( A G ) = R T ( a - 1)
(3)
Table I shows that for difficult separations A(AG) is in the range of a few tens calories per mole or less, which is not much considering that both AG1 and AG, are in the order of 10 to 30 kcal/mol or about lo3 times larger. I t is reasonable t o think t h a t changing the nature of the stationary phase will affect AG1 and AG, in a slightly different way and modify largely their difference. This is indeed the basic reason why analysts are using so many different stationary phases and why, in spite of various suggestions to the opposite, the situation is not going to change much with the development of capillary columns: increasing the range of column efficiency practically available by one order of magnitude decreases by a mere factor of 4 t h e difference in free energies of retention between two compounds which can be resolved (cf. Table I). Practically all chromatographic separations have to be made in the efficiency range of 103-106. For easier separation, no optimization is really interesting. The analysis is so fast any way. More difficult separations are almost impossible. This corresponds to a small range of 40 in A(AG), or a - 1,which shows how limited is the field of optimization: in many cases it is better t o change the stationary phases as analysts have found experimentally. There are cases, however, when it is just impossible, either because all available phases have been tried unsuccessfully or because the mixture is complex enough and interferences between some compounds on one phase are replaced by interferences between other compounds on another phase. C 1978 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 50, NO. 13 NOVEMBER 1978
Any time it is important to reduce the analysis time, the kinetic theory of chromatography can give useful clues. Flow Velocity t h r o u g h the Column. The integrated Poiseuille law relates the outlet column flow velocity, u,, to the column length, L , and to the outlet and inlet pressures, Pi and Po, respectively:
(4) where q is the gas velocity and Bo the column permeability, which for an empty tube is:
1813
In practice it is possible to prepare columns with CL coefficients less than 2-3 times the value calculated from Equation 15, using for df,the average film thickness, the ratio V L / r d J of the volume of liquid phase coated on the column wall to the wall surface. In such a case, CIAis negligible compared to CG. It has been observed that with capillary columns, even with relatively large phase ratio, the resistance to mass transfer in the liquid phase is negligible (9). A similar relationship holds for packed columns prepared with low coating ratios. In these cases we may write:
A 2
B = -U C 32 or using the reduced plate height and velocity:
while for a packed column it is about ( 7 ) :
dP2 1000
(sa) (18)
Because of the compressibility of gases, the gas velocity varies largely along the column. The average velocity, u, is given by:
L =
t,
In the following we use sponding to u,.
. = Juo
U,
for the reduced velocity corre-
with:
.
J
3P2-1 Pi =P i- 13 p 9= -P O
(7)
t , is the transit time of an inert tracer, or the retention time of a nonretained compound. As we are interested in difficult and rapid analysis, we can assume in most cases that the inlet pressure is large compared to the outlet pressure so Equations 4, 6, and 7 become respectively: u, =
B3,* W'P,
-
(8)
with y = 1 for an open tubular column and C is easily derived from Equation 14. The packing term AVO^^) conventional in the plate height equation of a packed column is neglected here. T h e M i n i m u m P r e s s u r e . This concept has been suggested by Giddings ( I O ) who has shown that for each analysis, characterized by a chromatographic system, hence a number of plates to achieve, and a column type (including column diameter for an open tubular column), there is a minimum pressure. If the equipment does not permit work at this pressure, the analysis is impossible. The analysis is possible a t higher pressures; the higher the pressure, the faster the analysis. It results from the definition of the plate number and plate height that:
L=NH (10)
Combination of Equations 8 and 20 gives:
BZi*
Equation 10 can be compared to the similar equation written for liquid flow:
u=-
BJP
SL For the same inlet and outlet pressures, the average flow rate a t large pressure drop is only 3 / 4 as large with a gas than with a liquid of the same viscosity. Plate Height a n d Gas Velocity. For a capillary column, the plate height is given by the Golay (8, 9) equation:
H =
B
-
C G U , -I- C
~ii
(12)
uo
uo =
For a given chromatographic system and sample, N is given by Equation 1, Bo and Po depend on the column characteristics. Only P, and H can be adjusted. But as shown by Equation 16, H is a function of u,. Eliminating H between Equations 16 and 21 and solving for u, gives:
which is meaningful only i f
with:
B = 2DG
(13) (14)
(20)
with:
Pi* =
Pi L Pi*
(23)
6;
(24)
2qP,NB
Before discussing the value of this characteristic pressure, we want to stress its importance. No analysis is possible if
1814
ANALYTICAL CHEMISTRY, VOL. 50, NO. 13, NOVEMBER 1978
the inlet pressure is not a t least equal to Pi*. The optimum flow-rate and the minimum plate height are easily derived from Equation 16.
Hmi,= 2 Uopt
=
@
6
Table 11. Minimum Pressure for Various Separations hydro- nitroargon helium gen gen Capillary columnsa N = 105
d,=
50
3.5
3.0
11.1
9.6
5.2
5.5
4.8
-
4.8
2.6
3.1
-
~
100 N=
1 0 6
d, = 50
Introduction of these values in Equation 21 gives:
5.4
100
250 Packed columnsb
2.2
N = 104
d,
tR
= $1
2.7
50 125 N = 106 d, = 8 0
8.5 3.4
17.65 7
7
17.2 7
35 14.5
12
N = 10’
The analysis time is given by:
L
d,
+ h’)
or
=
200
uo
Bs12 -
B0Pl2CG
2?lP$vUO2 BBI2 2qpfl[ w - B ]
or, using Equation 24:
H -_ U,
cg12 P12- PI*2
Combination of Equations 9, 29, and 31 gives:
I t is easy to show by differentiation of Equation 32 that t~ is minimum for:
Pi = Pi“ &i
15 31
8.5 3.3 17 7
a DG = 0.2 with hydrogen. Po = l o 6 (1 atm). Pi = (8/dC)(2riD G P ~ N ) “ * .~ D =G0.2 with hydrogen. Po = l o 6 ,y = 0.8. Pi = ( 2 0 / d q ) ( 1 0 ~ ~ ~ y P o N )Diameters ”2. are in wm. N o data are given for low pressures.
Combination of Equations 21 and 22 gives:
H -_
5.5 3.6
50 80
=
The first parameter is a function of the analysis to be made and cannot be changed much. The minimum pressure is proportional to the reverse of the column diameter. We show in the following section, however, that the smaller the column diameter, the faster the analysis. This is already apparent in Equation 34. This leads to the use of rather large pressures. The nature of the carrier gas determines the product ~ D G . Thus hydrogen in preferred to helium (cf. Table I11 and below). Optimization of analytical conditions cannot be really achieved after the minimum pressure, although this is a very and P,”are all functions of more important parameter. N, CG, basic parameters and the optimization had better be carried out on the explicit equation. The Minimum Analysis Time. Combination of Equations l, 5 , 13, 14, 24, 34, gives:
(33)
The variation of t R is slow around the minimum. The pressure dependent term, P?/(P; - PI**) is 2.60 P,* for PI = PI*\ and 2.83 P,* for PI = PI*\’?. In conclusion the column should be operated around 1.5 times the minimum pressure for optimum performance. Then:
5
tR
= 1.8N (1
C81*
+ h’) P O
(34)
As results obviously from Equations 27, 33, and 34 the minimum pressure PI*is an important parameter. Table I1 gives some results. Only values of PI*larger than 3 are given. Otherwise the assumption that PI>> Po would not be valid. It is important to observe t h a t the values are much larger with packed columns than with capillary columns, because of the larger permeability of the latter. Finally, the values obtained even for very large plate numbers are not very difficult to achieve. Indeed gas chromatography has been carried out in many instances with inlet pressure larger than 20 atm. T h e minimum pressure is independent of the outlet pressure, since B is proportional to DG (Equation 13) and the product P$p, constant in a large pressure range. Thus the three factors which determine the minimum pressure are the plate number, the column permeability, and the nature of the carrier gas.
This equation is the same as the one obtained previously in the case of packed columns, replacing B and C by the values these coefficients have for capillary columns (10). I t shows that a good resolution factor R is very costly in term of analysis time: it takes 3.4 times longer to achieve a resolution of 1.5 than a resolution of unity. Similarly the analysis time is proportional to the reverse of (cu - 1)3which is a very heavy dependence. In fact, as already shown (IO), the analysis time is proportional to I@’*: an analysis requiring lo6 plates will be 32000 times longer than one needing lo3 plates only. Time is the limiting factor to achieve very difficult separations. Differentiation of tR with respect to h ’ permits the calculation of the optimum value of h’: 1.76, a value in close agreement with the one derived earlier through a different approach (Ref. 11,p 1027). The analysis time is smaller than with k’= 3, although the plate number is about a third larger as well as the column length (cf. Table I). If we calculate the analysis time by mere combination of Equations 1and 28 and neglect the influence of h’on H and U , we find an optimum value of h’equal to 2. As H decreases rapidly with increasing
ANALYTICAL CHEMISTRY, VOL. 50, NO. 13, NOVEMBER 1978
1815
Table IV. Retention Times“ and Gzngth of Capillary Columns Which Can Separate a Pair of Compounds of Given a d, = 0.25 mm
d, = 65 pm t R (s)
L(m)
26 X 1 0 - 3 5 3 x 10-3 160 X
0.11
&
1.2 1.15
1.10 1.05 1.03 1.02 1.015
-
3
I
3
4
5
0.23 0.85 2.26 5.0
4.8 15.7 37 8.8 1.01 122 19.6 1.005 966 78 “ t~ = 0.0184(0/(a - l ) ) 3 d , (d,, N d , = 29.5(a/(a - 1 ) ) 2 d , .
k’ 2
1.1
0.07
L(m)
tR (s)
0.27 0.21 0.14 0.6 0.90 4.3 3.26 18.6 8.7 61 19.2 142 34 473 76 3750 299 CITI; t ~s)., L = 0.75 x 0.1
6
Figure 1. Variation of retention time with the column capacity factor for the second component of the pair of compounds to separate (cf. Equation 35). The minimum is at k ’ = 1.76 for which the k’dependent factor in Equation 35 is worth 63.76
if long columns can be prepared. T h e column length is calculated from Equation 20. Combination of Equations 16, 22, 24, and 33 shows that:
Table 111. Speed Characteristic of Carrier Gasa gas argon carbon dioxide helium hydrogen nitrogen
r?xm (PP x gl’,)
Nd,
/1
+ 6k’+
llh’2
-.
.
1138
995 440 90 741
a Product of viscosity by square root of molecular weight.
h’ in this range, it is normal t h a t t h e correct optimum is somewhat smaller. As shown on Figure 1,however, the analysis time does not vary rapidly around t h e optimum. While h ’ i s between 1.1 and 3, the analysis time is smaller than 1.1times the minimum and this variation may be neglected. Note, however, that the column length necessary t o separate the two compounds decreases with increasing k’. If h’is adjusted by changing the temperature of the column for example, assuming that N does not change with temperature, the analysis time is proportinal to 1 + li’, the resolution is larger than 1.0 a t k’values above 1.76 and smaller for lower h’values, if a is independent of the temperature. As a first approximation, t h e diffusion coefficient is proportional to the square root of the reverse of the molecular weight of the carrier gas. Table Ill gives the value of of the most conventional carrier gases and shohs t h a t hydrogen is considerably better than all other gases (12). It also gives the lowest critical pressure. Finally Equation 35 shows t h a t t h e analysis time is proportional to the column diameter and, thus, that t h e use of 65-pm i.d. columns, which is quite feasible, permits a 4-fold reduction of t h e analysis time compared to the conventional 0.25-mm i.d. columns. T h e analysis times corresponding t o t h e optimum values of t h e parameters (hydrogen carrier gas, h’= 1.76) are given in Table I\’ for a pair of compounds with various relative retention (same as for Table I) and for columns with 65-pm and 0.25-mm i.d. The values obtained are remarkably small: an analysis requesting about 1.5 million plates is achieved in about 1 h in t h e optimum conditions with a 0.25-mm i.d. column. Note t h a t it would take 5 h with helium carrier gas. Even taking into account the fact that practical columns are not as efficient as theory predicts, it remains that performances much better than currently obtained could be achieved
or:
(38) The values of L are given in Table IV. Except for the most difficult separation calculated, there i.s no serious problem to foresee in the preparation of the column. There are two last problems to discuss: first, we are interested in the analysis of complex nnixtures, riot usually in t h e separation of a pair of compounds. Second, we need a more phenomenological approach to optimization to be able to deal with the real columns used in practical applications. Multicomponent Analysis. The most difficult pair t o separate is usually somewhere in the chromatogram, preceded by compounds not too difficult to separate, although as h’ decreases the separation power ot’ the column drops sharply, and followed by much more retained compounds. T h e column must generate enough plates to separate t h e critical pair of compounds while giving the mininium retention time for the last component of the mixture (13). If kS’ and h,’ are the column capacity factors for the second component of the pair and for the last component of the mixture respectively, let n he:
T h e analysis time is then:
L,
T h e column length and the gas velocity are dcltermined by the condition that the pair 1,2 be s e p a a t e d , so !V is given by Equation 1, H l u , by Equation 3 1 and tR by Equation 35 modified as shown in Equation 40:
tn = 9.6R3
(
3
a-1
ANALYTICAL CHEMISTRY, VOL. 50, NO. 13, NOVEMBER 1978
1816
Table VI. Separation Power of a Capillary Column - 1)
k‘ m
1.76 0.70 0.50 0.33 0.22 0.13 0.06
M
O
qs
n
P
100 (arbitrary unit) 80 60 50 40 30 20 10
As the efficiency of capillary columns decreases steadily with increasing k’, it is interesting to calculate the variations of the relative retention of a pair of compounds separated by a resolution 1 as a function of h ’. For packed columns it is given with a fairly good approximation by solving Equation 1 for cy. For a capillary column, we have to take into account the variation of the plate height with k ’ a t constant uo. As the column length is constant:
L = N(k,’)H(h,’) = N ( k ’ ) H ( k ? Figure 2. Variation of the optimum value of k’(minimum analysis time) for the second component of the most difficult pair of compounds to resolve in a mixture with the relative retention of the last component of the mixture to the second compound of the pair
Table V. Analysis Time of a Complex Mixtureu n
hiopt,*
t R , z (s)
tR,n (s)
The conclusions are the same except as regards the optimum value of k’. Differentiation of tR in respect to h’gives the equation: -
rn
uo=vC(h,l)
b
47 3 47 3 1.5 476 620 (624) 2 480 763 (775) 3 1.30 489 1042 (1076 4 1.22 497 1317 (1378) 6 1.13 509 1860 (1980) 8 1.08 517 2400 (2600) 10 1.05 523 2935 (3200) 13 1.02 530 3740 (4100) 20 0.99 537 5610 (6200) a a = 1.01; column d , = 0.25 mm. In parentheses, analysis time if k’ is kept to 1.76 for the second compound.
-3
where k2’ is the column capacity factor for the second compound of the pair. N(h,’) is given by Equation 1 with k ’ = 1.76, assuming the analysis has been optimized. H(h,’) is given by Equation 36 (with h i = 1.76). Combinations of Equations 22 and 33 show that:
1.76 1.55 1.43
1
+
( 2 r ~ 14)k’- (7n
+ 17)k‘ + 1 l n k ‘
(43)
(44)
Hence:
We assume in the following t h a t DG is the same for all compounds, hence B = B2. Hence:
Equation 43 can thus be rewritten:
= 0 (42)
T h e optimum value of k’is a function of n, decreasing with increasing value of n. T h e variation of kbp, with n is given on Figure 2. Obviously for n = 1, k ’ = 1.76. For the sake of comparison, Table V gives the analysis time of a mixture of 3 compounds, with a pair of compounds with cy1,* = 1.01 and a third compound with increasing values of from 1 to 20. Obviously the analysis time increases markedly, but so does the number of compounds which can be eluted and resolved, the complexity of the mixture which can be analyzed. T h e data in the last column of Table V compares the minimum analysis time achieved by optimizing the column capacity factor, and the time obtained by merely keeping the same h’for the most difficult pair of compounds to resolve and waiting. For complex mixtures with a large range of retention times, the reduction can be 10% which is not too rewarding. The separation of the compounds eluted before the critical pair of compounds is achieved only if their relative retention is large enough. This can be a problem with pair of compounds having a large cy value but eluted soon after the unretained tracer, and this is why the analysis of complex mixtures has often to be made in temperature programming.
Equation 47 can easily be solved for a. I t gives as a function of the column capacity factor for the second compound, the relative retention of a pair of compounds which can be resolved with a resolution unity by a column which has N theoretical plates for the compound with k’ = 1.76. Figure 3 shows a plot of cy - 1 vs. k’for the columns (cf. Table I) which give a resolution 1 for the pair of compounds with cy between 1.005 and 1.05 and k2‘ = 1.76. This figure shows that the separation power of a capillary column drops much more slowly than usually believed. From the numerical data used to draw Figure 3 we can calculate (cf. Table VI) the loss of separation power, using l/(a - 1) for the pair of compounds separated with a resolution 1 as a measure of the separation power. The corresponding value of A(AG) would perhaps be a more meaningful measure, but it is somewhat too foreign to the analyst problems. Excellent results are obtained for k’ larger than 0.4.
ANALYTICAL CHEMISTRY, VOL. 50, NO. 13, NOVEMBER 1978 d
1817
I C
-1
1
J
2
1
9
Figure 4. Variation of plate height of a capillary column as a function
I 0
of velocity. The resistance to mass-transfer in the liquid phase is neglected. Reduced plate height h = H / d , . vs. reduced velocity, v = ud,/D,. The numbers on the vertical scale are the values of k‘. (cf. Equation 14). The dotted line is the locus of the minima of the h vs. u curves I
I
I
1
2
3
* k’
Figure 3. Variation of the relative retention of a pair of compounds separated with a resolution unity, as a function of the column capacity factor of the second component of the pair. The segment at right gives the asymptote of each curve. The different columns correspond to those described in Tables I and V. The values of CY for k’= 1.76 are respectively (1) 1.005, (2) 1.010, (3) 1.015, (4) 1.020, (5) 1.030, (6) 1.040, (7) 1.050
Optimization for Real Columns. Practical columns often have an efficiency which is somewhat lower than the one predicted from the Golay equation (cf. Equations 12 to 15), although the efficiency is often between 0.5 and 0.8 times the predicted maximum. Lower values tend to be obtained for polar compounds, but various treatments are available to prepare good columns even for these compounds (14). In this case, we cannot rely any more on a theoretical equation for predictions and we have to use a phenomenological approach. We can safely predict however, that the optimum value of k’is between 1.7 and 3.0 (11,151. As regards the effects of column diameter we can proceed as follows. Knox (16) has shown that the plate height of a chromatographic column can be written:
h = Au0,33+
B
-
U
+ Cu
(48)
the reduced plate height and velocity being given by Equations 17 and 18, respectively. A is a characteristic of the quality of the column packing and is zero for capillary columns. To be exact and reconcile Equations 1 2 and 48, we should add a second term to account for the mass-transfer in the stationary phase, a term function of j and v. As our approach is empirical, we shall not do it. However we cannot completely ignore this phenomenon. Depending whether the resistance to mass-transfer in the stationary phase is important or not, we have to keep constant the average flow velocity or the outlet flow velocity (15). Although I am of the opinion that the very efficient columns used in difficult problems where optimization is really useful are lightly loaded and consequently have low resistance to mass-transfer in the liquid phase, the two series of equation are given. We observe that, in practice, packed columns have a maximum efficiency at a value of the reduced velocity slightly below 2, the reduced plate height being somewhat below 3. As columns are used a t a velocity larger than the optimum,
we shall consider that for a packed column h,, = 3, For a capillary column, Equation 48 is written:
2 1+ 6k’+ I l k Q h=-+ v
96(1
+ k’)*
vo =
2.
(49)
Plots of h vs. v are given on Figure 4 for various values of h’. The optimum value of v is between 5 and 14 while the minimum value of h is between 0.3 and 1. Indeed values of h lower than 1 for retained peaks have been published (17). We have selected conservative values of 1.5 and 5 for h and u, respectively. It w ill thus be possible to achieve performances better than predicted below. The approach used here is similar to the one we have already applied to the optimization of liquid chromatography (18). The column length is: L = NH = Nd,h (50) where d, is for the column diameter or the particle diameter. If we keep constant the average flow rate to maintain constant the plate height, we derive from Equation 10:
4vLu
p , = -3B, and from Equation 28:
128 3
A;
hu -
dCL
The equations corresponding to the numerical application discussed later are given in Table VII. If we keep constant the outlet flow velocity to achieve a constant reduced plate height while changing column length and diameter, we can derive the inlet pressure from Equation 8:
The retention time is obtained by a combination of Equations 10 and 28:
Similar equations (cf. Table VII) are obtained for packed
1818
ANALYTICAL CHEMISTRY, VOL. 50, NO. 13, NOVEMBER 1978
--
.-
Table VII. Equations for the Optimization of Practical Columnsa A. Capillary columns,b L = 1 . 5 N d , -
" = cte
v o = cte pi = 93" i d c - ' =
7 6 Lll2d
t R = 3.94
10-4
p i = 5.76 x 10-3~ d ; ' = 3.84 x Ld;3 t, = 4.5Nd,'
-312 C
x
d,
B. Packed columns,c L = 3 N d , -
" = cte
v o = cte
Pi= 465N"'dP-' =
Pi = 0.144 NdP-'
=
268 L' I2dp-3 1 2 0.048 Ld,-3 t R = 1.31 % t R = 30 N d p 2 10-"2 i 2 dP a Hydrogen q = 9 x 10-5P,D G = 0.2 cmz/s,Po = l o 6 (1 atmL h h = 1 . 5 . ~ = 5 , k ' - 2 . C h = 3 . ~ = 2 . k ' = 3 .
Figure 6. Optimization of real columns operated at constant average flow velocity (to keep hconstant). Pressure gradient ( P , / L )vs. diameter. (1) Capillary columns, (2) packed columns. The dotted lines correspond
to the examples discussed in the text
d
3.2 81
L
