Moment Analysis of Elution Curves from Gas Chromatography Columns Having a Known Size Distribution of Sorbent Particles Dwight W. Underhill Harvard University School of Public Health, 665 Huntington Avenue, Boston, Mass 021 15
The effect of a stationary phase consisting of spheres having a log-normal size distribution is examined. At high carrier gas velocities, the effect of the geometric standard deviation, d, is as follows: N , the number of theoretical plates, decreases as if the effective particle diameter y;l , the skewness, were increased by a factor of e 4 1 n ' 2 ) 0 and -12, the kurtosis, increase a s if the effective particle diameter were increased by factors of e81n'2)Liand e 1 0 1 n ( 2 ) drespectively. , The nondimensional factors, ( y 1 2 L / d m ) and ( Y 1 L / d m ) , where L is the length of the column and dm is the geometric mean particle diameter, as a function of the carrier gas velocity give hyperbolic plots which are similar to the van Deemter equation for the height of a theoretical plate. These calculations are extended to give corrections for the effect of pressure drop in the column.
Most theoretical and experimental analyses of the behavior of gas chromatography columns assume a uniform particle size for the stationary phase, or if a range of particle sizes is present assume with no further discussion that the results are controlled by a particle of some mean diameter. This paper is an effort to determine the effect of having a range of particle sizes. For these calculations, we assumed a stationary phase consisting of spheres having a log-normal size distribution, but the procedure given here is quite general and can be applied without significant modification to particles of other geometries and/or to other size distributions. At this time, for example, we have calculated the mass transfer effects for an unevenly distributed liquid phase where there is also a gas film present. Calculations have also been carried out for distributions which assume a sharp upper and lower bound on the size of particles present. These results will be published in a future paper. In this paper we made the assumption of spherical geometry in order to compare our results with the calculations of Rosen ( I ) for mass transfer in beds consisting of homogeneous spherical particles. This comparison is particularly easy because of the availability of tables, charts, and even Fortran programs giving Rosen's results. A second advantage of assuming spherical geometry for the sorbent particles is that we can, for this geometry, neglect the effect of a gas film. This has been stated before by Thiele ( Z ) , whose essential arguments are that: the distance through a gas film is less than the distance to the center of the sorbent particle. and intraparticle diffusion is likely to be retarded with respect to interparticle diffusion.
CALCULATIONS FOR SPHERICAL GEOMETRY These calculations begin with the partial differential equation for mass transfer in a gas chromatography column ( 1 ) J B Rosen J Chem Phys 20, 387 (1952) (2) E W Thiele, Ind Eng Chem 31, 916 (1939)
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ANALYTICAL CHEMISTRY, VOL. 45, NO. 7, JUNE 1973
bc(t) (1) at where c ( t ) = interparticle concentration of sorbate, mol/ cm3; D = interparticle diffusion coefficient, cm2/sec; V = superficial carrier gas velocity, cm/sec; = fractional interparticle void volume, dimensionless; and g ( t ) = interphase mass transfer function, sec-1. The terms on the left side of the above equation give, in order, the effects of interparticle diffusion, convection, and interphase mass transfer, on the interparticle sorbate concentration. The interphase mass transfer function, g ( t ) , is defined as the rate of mass transfer to a unit volume of stationary phase following its exposure to a constant unit concentration of sorbate starting a t time t = 0. Rosen ( I ) showed for spherical sorbent particles that the Laplace transform of g ( t ) is
or 2 s ) = h(1-
5 sd2 (&)+ E(&)
-
where k = equilibrium partition coefficient of sorbate between equal volumes of stationary and mobile phase, dimensionless; d, = sorbent particle diameter, cm; and D, = intraparticle diffusion coefficient, cm2/sec. If there is a distribution of sorbent particle diameters
where g(d,,t) = g ( t ) for particles of diameter, d, and f i = number of particles of diameter d, in a unit volume of sorbent, cm-3. Then
The size distribution function, straint that
fi,
is subject to the con-
(6)
Further the total number of sorbent particles per unit volume of sorbent, N,, is
(7)
T o carry these calculations further, some assumption must be made regarding the frequency distribution, f t . The log-normal distribution ( 3 )
The Laplace transform of Equation 1is
from which on the further assumption of a pulse input of sorbate a t time t = 0, integrates to yield where Np = number of sorbent particles in a unit volume of sorbent, cm-3; /3 = geometric standard deviation, dimensionless; and d , = geometric mean diameter, cm, is often observed in particle size distributions and is easy to handle mathematically. For this distribution it is well known that
-
3s) =($)e{(:
,/(r>' +
4sD(1 E- t)g(s)'
+ 4 s D ) x / 2 D } (13)
where A = the cross sectional interparticle area, cm2, and Q = moles of injected sorbate. As is well known, the ordinary temporal moments of c ( t ) , which when normalized with respect to the zeroth moment, are defined as
(9) and
Following the assumption of a log-normal distribution, the summation in Equation 5 yields
and are equal to
The ith central temporal moment (the ith moment about
Mi)is In applying this summation to a real distribution, we need only assume that the actual range of particle sizes is sufficiently broad that the absence of the tails at either end of the distribution makes no appreciable difference in the analysis. For the equation for the number of theoretical plates to be valid, the range of particle sizes present should extend from
These moments can then be used to calculate the number of theoretical plates, N
N = MI2/M2
(17)
t h e skewness, y 1
dme41n2P
d,
=to 3/3dme41n2B 3P and the kurtosis, y z .
The skewness and kurtosis are more sensitive to the presence of larger particles and for the analysis of these facand tors to be valid the upper limit should be 3/3dme51n(2)B 3pdme'31n(2)B, respectively. The next step is to calculate the effect of the log-normal corrected g(s) on the shape of the breakthrough curve.
N =
YZ
=
(M4IMZ2)
-
3
(19)
Then carrying through the calculations for N , y1 and y2 gives
+
L { ( l - €)k t]~P,Z V ( 1 - t)Kdm2P,es'nzP/30Dp2Dt{(1 - E ) K
+ c)'B2/V
+
and
120t3((l -
+
+ Yz
=
t)k
v3
+ c ) ~ D +~ P12k(l ~ - t)((l -
t)K
+ t)2tZPdmzP3e8'n24
5VDp
+
+- -
2((1 - t)K t)eZohZP (1 ~ ) k tVD(1 ~ ~ -~t)Mm4F2 ~ ~ 2 105 150 D,2 (1 - ~ ) k ~ ~ d ~ ~ H ~ e ~ ~ ~ ~ ~ 0 4200Dd
{
1
- t)kdm2Ple81nZP + %D{(1 30Dp
(3) G. Herdan, "Small Particle Statistics," 2nd ed., ,t,cadernic New York, N.Y.. 1960.
press,
t)k V
~
+ E)~P,
( 22)
The PK factors are correction factors for the pressure drop; they Will be discussed later in the text, For small pressure drops, P K = 1. ANALYTICAL CHEMISTRY, VOL. 45, NO. 7 , JUNE 1973
1259
i
i 1
I
LJ
REDUCED VELOCITY
Figure 1. Effect of
the geometric standard deviation, id, on the reduced height equivalent to one theoretical plate, h
Assumed parameters: intraparticle diffusion coefficient, D, = 0.00002 cmi/sec; interparticle molecular diffusion coefficient, D, = 0.18 cm2/sec; fractional interparticle void volume, e = 0.45 (dimensionless): tortuosity factor for interparticle molecular diffusion, = 0.6: coefficient for eddy diffusion, ,\ = 1.2 (dimensionless): reduced carrier gas velocity, IJ = Vdrn/Dm; effective interparticle diffusion coefficient, D = yDm XVdp/c
-,
+
The reduced height equivalent to theoretical plate, h, is defined as
where L is the length of the column. The minimum in the factors, h, 71, and 7 2 , occurs a t carrier gas velocities a t which both interparticle and intraparticle mass transfer effects are important. The main difficulty in calculating precise values for the velocities a t which these minimums occur (as well as the minimum values of these functions) is our lack of precise knowledge of the behavior of the effective interparticle diffusion coefficient, D, as a function of velocity. This uncertainty arises both in the magnitude of the eddy diffusion as well as in the coupling between eddy and molecular diffusion. This problem, although intensively studied, has led to serious disagreement in studies of beds consisting of uniformly sized sorbent particles. Here we have the more complex problem of a bed of heterogeneous particles. I t is usual to assume in gas chromatography a coefficient for eddy diffusion which has a value of the order of 0.5-5.0. For our calculations we will assume a value of 1.2 for A, the coefficient for eddy diffusion, and make the additional assumption that the overall interparticle diffusion is the simple sum of eddy and molecular diffusion. Then
D
=
yD,
+
Ad,u/c
(24)
where y = tortuosity coefficient for interparticle molecular diffusion, dimensionless; D , = molecular diffusion coefficient for sorbate in the mobile phase, cm2/sec; X = coefficient for eddy diffusicn, dimensionless; V = carrier gas velocity, cm/sec; and I& = characteristic particle diameter, cm. For the problem a t hand, &, the characteristic particle diameter, seems best defined as the arithmetic mean particle diameter, as this diameter is proportional to the average mixing length. 1260
ANALYTICAL CHEMISTRY, VOL. 45, NO. 7, JUNE 1973
(25) If the dimensionless factors h, ( y 1 2 L / d m ) and (y2L/dm) are plotted as a function of the reduced velocity, Y = Vd,/D,, sets of hyperbolic curves are obtained. For the height equivalent to theoretical plate, h, the curve (neglecting the pressure factors) is a true hyperbola and for the other two sets of curves the deviations from a true hyperbolic curve are not great. At high carrier gas velocities, with the controlling factor interphase mass transfer, and neglecting the pressure drop effect
N = yI
~
30L{(l - t)h f tI2D, V{l - t)hdm2esln’J
(26)
d m e 8 ‘ n 2 d d T 14 (1 - c)kD,L
(27)
and
(28) The number of theoretical plates decreases as if the particle diameter increased by a factor of e4ln!*)13,but the increases in skewness and kurtosis are equivalent to particle diameter increases by factors of e81n(2)8 and P n ! 2 ) d , respectively. The tailing factors, skewness and kurtosis, are more sensitive to a distribution of sorbent particle diameters than is the number of theoretical plates. It may be somewhat surprising to realize that if we had any log-normal distribution of a set of particles-be they cubes or plates or cylinders-which differ from each other only in a three-dimensional scaling factor, we find the same dependence of the number of theoretical plates, the
10,000
m
3
$
(0 Y
1000
0 0 3
2
100
REDUCED VELOCITY
Figure 2.
Effect of the geometric standard deviation, p’, on the reduced skewness, -,, * L / d ,
skewness and the kurtosis on the geometric standard deviation, @.The point we stress here is that in all cases we would like to keep /3 as low as possible. In general, as long as /3 is 2, the loss of efficiency is dramatic.
PRESSURE DROP EFFECTS In any real gas chromatograph, the pressure drop across the column can affect, among the factors used here, the inter- and intraparticle diffusion coefficients, the carrier gas velocity, and the partition coefficient between the moving and stationary phases. Some interesting results can be obtained analytically following a procedure outlined irL an earlier paper ( 4 ) . All this procedure requires is knowledge of t(s) as a function of the pressure-dependent factors of mass transfer. Then the moments of the breakthrough curve can be calculated across the entire column by an integration procedure. For an adsorbent (such as charcoal), with macropore diffusion controlling intraparticle diffusion, D , 1/P. With the further assumptions of an ideal carrier gas ( D , 1 / P and V 1 / P ) and k independent of pressure, there results
-
-
-
where N , 71, and y2 are calculated based on the outlet pressure Po, and TI, ancb72 are calculated across the entire column and include the effect of the pressure drop. In the last three equations, the pressure drop effect is independent of the adsorbent particle size distribution and shape. The same result would have been obtained had the stationary phase consisted of cubes, or for that matter of a set of irregular particles. It is also important to recognize that if the mass transfer effects are from either inter-
n,
( 4 ) D W Underhill, Separ S c i , 5 , 219 (1970)
particle diffusion or from macropore intraparticle diffusion, we can correct the moments for the effect of pressure without having to know which mechanism controls mass transfer. For a microporous adsorbent, or in general for liquid absorbents, the intraparticle diffusion coefficient is expected to be independent of pressure. Then the correction factors for the pressure drop are given by the “ P K ” factors in Equations 20-22 where P, is the inlet pressure and Po is the outlet pressure.
The magnitude of the pressure drop correction now depends on the nature of mass transfer, as well as on the ratio ( P l / P o ) .At low carrier gas velocities, with interparticle diffusion controlling mass transfer, the correction factors are the same as those given in Equations 29-31, but a t higher carrier gas velocities, with mass transfer controlled by intraparticle diffusion.
DISCUSSION OF RESULTS This analysis has shown how sorbent shape parameters and the pressure drop effect can be integrated into simple analytical expressions. The figures below illustrate these results. Figure 1, 2 , and 3 give, respectively, the reduced height of theoretical plate, h , the reduced skewness, Ly12/dm, and the reduced kurtosis, Lyzld,, in a short gas chromatography column, as a function of the reduced velocity, Vdm/Dm. In this column, intraparticle mass transfer is assumed to be taking place in spheres having a log-normal size distribution. The strong effect of the geometric standard deviation is clearly shown in these figures. In a longer column, pressure effects may become important. Figure 4 gives the pressure drop correction factors for columns in which either interparticle diffusion or maANALYTICAL CHEMISTRY, VOL. 45, NO. 7, JUNE 1973 * 1261
10
0.8
1
1
I
I
c 1 1.m
I
i
I
1.01
1.1
1.2
1
I
l
l
i
I
2
3
4
6
10
20
40
I
1
60
100
pi / Po
RATIO OF ABSOLUTE INPUT TO ABSOLUTE OUTLET PRESSURE
Figure 4. Effect of pressure drop First example: mass transfer controlled either by interparticle diffusion or by intraparticle macropore diffusion, skewness, y 2 = kurtosis, and P , / P a = ratio of absolute input to absolute output pressure.
cropore intraparticle diffusion controls mass transfer. At high pressure drops, the correction factors for N , 71, and 72 become 819, 4&/5, and 4 / 3 , respectively. Figure 5 shows the correction factors for pressure-independent intraparticle mass transfer. At high pressure drops, these factors become 2Pi/3Po, d 3 P o / 2 P j , and 3Po/2Pi, for N , 71, and 72,respectively.
EXPERIMENTAL TESTS Experimental tests of the effect of a particle size distribution require divising particle size distributions which 1262
ANALYTICAL CHEMISTRY, VOL. 45, NO. 7, J U N E 1973
N = number of theoretical plates,
-,
=
can be easily changed. This can be carried out by mixing of sorbates of two sizes. Suppose that in a bed we have a weight fraction, u ,of particles whose only difference from the smaller particles is that they are larger by a factor of r. We then find
"(1
-
U!
+
wT2) = N
+
71'
=
1- w wr4 ([1 + u:r%]3,p) 71
(36) (37)
0.01
I
I
1.001
1.01
1.1
pt 1 Po
I 2
I 3
I 4
1 5
I
I
I
10
20
40
1 60
100
RATIO OF ABSOLUTE INPUT TO ABSOLUTE OUTLET PRESSURE
Figure 5.
The effect of pressure drop
Second example mass transfer controlled by pressure independent intraparticle dlffuslon
1
-w
+ wr6
where N , 71, and 7 2 are the number of theoretical plates, skewness, and kurtosis in a bed containing only the smaller particles and the primed quantities are calculated from an otherwise similar bed under the same conditions consisting of a mixture of the particles. With wr2 > 1, the number of theoretical plates is controlled by the smaller particles but the skewness and kurtosis are increased by factors of wr4 and wr6, respectively. Also, it should be noted that the skewness and kurtosis from a bed containing a mixture of particle sizes can be much greater than from a bed consisting entirely of particles of the larger size. These results can be examined experimentally.
NOMENCLATURE A = cross sectional interparticle area, cm2 d t ) = interparticle concentration of sorbate, moles/cm3 Cfs) = Laplace transform of d t ) , mol sec/cm3 d, = sorbent particle diameter, cm d, = geometric mean value of d,, cm d, = characteristic particle diameter, cm I) = interparticle diffusion coefficient, cm2/sec U , = molecular diffusion coefficient for sorbate in the mobile phase, cm*/sec L), = intraparticle diffusion coefficient, cm2/sec f i = number of particles of diameter d, in a unit volume of sorbent, cm -3 g ( t ) = interphase mass transfer function, sec-1 gfs) = Laplace transform of g ( t ) , dimensionless h = reduced height equivalent to theoretical plate, dimensionless h = equilibrium partition coefficient of sorbate between equal volumes of stationary and mobile phase, dimensionless
L = length of column, cm Rl = ith ordinary temporal moment of c f t ) , normalized with respect to Go, (set)' M 1 = ith central temporal moment of c(t), normalized with respect to Ro,(set)' N = number of theoretical plates, dimensionless = N corrected for pressure drop effect N , = number of sorbent particles per unit volume of stationary phase, cm - 3 & = pressure correction factor, dimensionless P, = absolute input pressure Same units Po = absolute outlet pressure ! = moles of injected sorbate r = ratio of size of larger to smaller particles s = variable ofthe Laplace transform, sec-1 t = time following injection of sorbate, sec V = superficial carrier gas velocity, cm/sec u' = weight fraction of larger particles x = distance from inlet, cm /3 = geometric standard deviation, dimensionless y = tortuosity factor for interparticle molecular diffusion, dimensionless y1 = skewness, dimensionless = y1 corrected for pressure drop effect yz = kurtosis, dimensionless 72 = 7 2 corrected for pressure drop effect t = fractional interparticle void volume, dimensionless X = coefficient for eddy diffusion, dimensionless v = reduced velocity, dimensionless
m
Received for review September 25, 1972. Accepted January 29, 1973. This work was supported in part by Grant No. ES00002 between Harvard University and the National Institute of Environmental Health Sciences, U. s. Department of Health, Education, and Welfare. A N A L Y T I C A L C H E M I S T R Y , V O L . 45, NO. 7, J U N E 1973
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