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Gas—liquid partition chromatographic separations in columns packed with porous particles — a model for uniform thickness liquid film
The Chemical Engineering Journal, 23 (1982) 81 - 90 0 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
81
Gas-Liquid Partition Chromatographic Separations in Columns Packed with Porous Particles - a Model for Uniform Thickness Liquid Film
M. A. ALKHARASANI Department
and B. J. MCCOY
of Chemical Engineering,
(Received 24 October, 1980;in
University of California, Davis, CA 95616
(U.S.A.)
final form 15 April, 1981)
Abstract A linear mathematical model for gasliquid chromatography is presented for application to either analytical or larger-scale equipment. The model includes the effects of axial dispersion, interparticle mass transfer, intraparticle gas diffusion, and diffusion in a thin, uniform thickness liquid film. The governing differential equations are solved in the Laplace transform domain for a general input pulse. Temporal moments of the output pulses are expressed in terms of geometrical, equilibrium and transport pammeters. The moment theory is related to HETP and to resolution of separation. The optimization of the process and scale-up to industrial size are discussed.
INTRODUCTION
Separation by gas-liquid partition chromatography (GLPC) in columns packed with porous particles, where liquid is spread as a thin layer on pore surfaces, occurs because different gaseous solutes have different solubilities in the stationary liquid phase [ 11. To maximize the surface area of liquid and thus provide the most area for mass transfer, the liquid is spread on the surface of the pores of the porous particles. Because the diffusion of the solutes in the gas phase in the pores is orders of magnitude greater than that in liquid, this has obvious advantages over the case where the pores are filled with liquid. The thin liquid layer on pore surfaces is prepared by filling and saturating the pores with a solution of the liquid and a solvent. The solvent is then evaporated and removed
from the particles, leaving a layer of immobile liquid of average thickness about 3c( on the pore surface. Of course, the mathematical analysis of mass-transfer taking place in liquid-filled pores is considerably simpler than that for the situation considered in this paper where the immobile liquid phase is a thin layer on pure surfaces. The latter is the usual case in high resolution chromatographic processes. Chromatographic separations, including GLPC, have been moving toward large-scale use in the past few years. For example, Elf Aquitaine and Societe de Recherches Techniques et Industrielles (SRTI) have developed Production Scale Gas Chromatography (PSGC), a gas-liquid partition process that has now been commercialized in the U.S. [ 21. In the Elf-SRTI process columns up to 0.4 m internal diameter are being built with an annual output capacity ranging from 3 to 300 metric tons. Channeling is prevented by a proprietary packing technique [2, 31. This technology can separate close boiling point products, remove impurities from low and high-boiling point compounds, and separate azeotropes and thermally unstable substances. Purities of 99.9% and yields of 98% can be attained. The process is particularly suited to the essential oil industry [4]. In contrast with vacuum-distillation techniques, PSGC technology requires significantly less energy to accomplish difficult separations. The scale-up, optimization, and control of such large-scale chromatographic operations requires a deeper and more accurate mathematical understanding than has been necessary for the highly successful bench-scale chromatography. Especially the transport processes that contribute to band spreading should be under-
82
stood more thoroughly, we believe. This is the motivation for the present work. Two approaches to analyzing mass transfer processes in chromatographic columns have been used. One is the method known as the plate, or rate theory [5,6] and the other is a moment technique based on conservation and transport relationships between macroscopic variables described by differential balance equations and rate expressions [ 7 - lo] . The present treatment will follow this second approach; that is, we write the linear differential mass balance equations for solute in the interparticle voids in the column, in the pores of particles, and in the liquid film. These equations present the local average solute concentration in the column and in the intraparticle voids in terms of the average inter-particle void fraction (Y, the particle porosity p, the spherical radius R, and the liquid film thickness 6, and as a function of position z and time t. The transport processes in the column are described by means of effective coeffients, such as the axial dispersion coefficient De, gas film mass transfer coefficient for the spherical particle kf, the effective intraparticle gas diffusion coefficient Di, and the liquid film diffusion coefficient Di. It is important to note that Do, kf, Di and D1 may depend on geometrical parameters, e.g., (Y,(3, R, as well as on velocity u, but are assumed to be independent of time t, and column longitudinal position z. We restrict the theory to linear isothermal chromatography, i.e., where mass transfer is governed by linear transfer kinetics and the partition isotherm for equilibrium between gas and liquid phases is linear. The complete solution for the governing differential equations provides more information than is often actually needed for design and analysis. Frequently, knowing the first few temporal moments of the output response to an inlet pulse enables the engineer to design and optimize column operation [lo]. Because the partial differential mass balance equations are linear, they may be solved in the Laplace transform domain. The zeroth, first, and second moments of the solute concentrations are evaluated from derivatives of the Laplace-transformed concentration by means of well-known mathematical identities. Unless output concentration pulses are strongly skewed or
have large tails, the first three moments (i.e., up to the second moment) are sufficient to establish resolution criteria for the separation. We consider a packed column of length L and cross-sectional area A =, homogeneously packed with spherical porous particles of radius R. The interstitial carrier gas velocity u is given by u = L+,/(Ywhere u. is superficial velocity. The interparticle void fraction of the column is (IL,and the internal porosity of the particles is 0. Liquid as a thin layer of uniform thickness 6 is spread on the pore surfaces. In typical GLPC columns average pore diameter is in the range [ 111 of 10 p, and the average liquid thickness is in the range [l] of 3 p. The effective porosity of the liquid-coated porous particle, i.e. the ratio of gas volume to particle volume, is: fl=fl--6A,
(1)
where p. is the porosity of the dry particle, 6 is the film thickness, and A, is the dry surface area per unit particle volume. Although the following treatment refers to gas-liquid partition chromatography, the same equations should describe liquid-liquid partition chromatography, where the mobile and stationary phases are immiscible liquids.
MODEL
FOR GAS-LIQUID
CHROMATOGRAPHY
We begin with the point differential mass balance equation for concentration in the mobile phase, c(t, z):
ac
ac Do a2c
at
a2
-++-=---_
a
az2
3 (1 --a) ___ R a
kf
[C-(Ci)RI (2)
with the initial condition: c(O,z) = 0
(3)
and boundary conditions: c(t, 0) = co(t) c(t, -) = finite
(4)
(5) Continuity of flux at the spherical interface serves as a coupling boundary condition:
=
kf[C -ci(r
= R)]
(6)
Here kf is the interparticle mass transfer coefficient, Di is the effective intraparticle gas diffusion coefficient, and ci(r, t) is the intrapar-
83
The variable y is the (one-dimensional) position in the liquid film measured from the solid surface. Since there is no flux at the solid pore surfaces:
The treatment of the liquid film diffusion is an approximate description of real systems for two reasons. First, it is unlikely that the liquid forms a completely uniform thickness film throughout the porous particle. Because of the randomness of pore configuration and size, one expects unequal film thicknesses and perhaps pools of liquid to exist. However,’ such non-uniformities would cause increased band-spreading; best resolution is obtained when the film is of constant thickness as assumed in our model. Second, because the film thickness may not always be very small compared with the pore diameter, the rectangular form of the diffusion equation is an approximation. Nevertheless, even though the treatment of diffusion within the liquid phase is simplified we expect useful and reasonably accurate results since the treatment, although limited by the model, is systematic and rigorous. Because the complexity of the calculations for a more realistic model would be prohibitive, including the complications of unequal and cylindrical films does not seem justified at this stage. We have considered in other work the condition where liquid only partially covers the pore surfaces, leaving exposed solid surfaces for adsorption [ 131. For the case when pores are completely filled with liquid, the partitioning is described by eqns. (2) - (9) with k, = 0 and Di replaced with Di; the moments for this case are easily adapted from earlier publications [8, 9,121.
wt, 0) ay
MOMENTS
title gas phase concentration. Equations (2) (6) are identical to equations presented by Schneider and Smith [ 121. The mass balance equation for the intraparticle gas concentration is: -A,k,(ci
P
-c,)
(7)
with initial condition: Ci(O, r) = 0
(8)
The boundary conditions are: aCi(tv
O) =
o
(9)
ar
and the coupling boundary condition given by eqn. (6). The coefficient in eqn. (7) describes mass transfer between the pore gas and the liquid coating on the pore surface, and may be estimated as the gas diffusion coefficient divided by the gas film thickness. If the liquid film is thin compared to the pore radius, the liquid phase transport equation for c,(t,y) is:
acl -= at
a2c, Dl
-
02
(10)
with initial condition c*(O, Y) = 0
--
(11)
o =
(12)
Continuity of flux at the interface serves as a boundary condition that couples eqns. (7) and (10): (13) where c, = q(t, 6)/K
(14)
is the concentration of the gas at the surface of the liquid of thickness 6, and R is the local equilibrium distribution coefficient. Equation (13) allows for variation of liquid concentration with radial position in the particle since ci is a function of r. We have ignored any small gradients that may occur transverse to the coordinate y.
Among the existing possibilities for the solution of partial differential equations are those procedurres based upon the Laplace transform. In this method the partial differential equations are transformed into ordinary differential equations, which can then usually be solved in terms of Laplace coordinates. Though the solutions do not always permit the inversion into normal coordinates, they permit calculation of the temporal moments. Having the first few moments is sufficient for most design and optimization analyses. In a chromatographic column where all processes are linear, c(t) after Laplace transformation is the transfer function, defined by :
84 C(s) =
J
dt
eesfc(t)
(15)
0
PC(~) dt,
k,A,6
l+
t”c(t)
dt (17)
9 -,
m0
c(t)
J
dt
0
J (t -PI)“c(~)
’
Pn
=
dt (18)
(25)
r=sm
(26)
Using eqn. (20) to obtain the zeroth moment, we find:
c(t) dt
6
l&z=I-(2 --PI2
anc
(1
(27)
= 0) +zu
+fio)/u
+3(1+y),
$--a)
(28)
(29)
a
+@I
+S2)l/u
+60)2/u2a+ (30)
where
=;(;
s=o
sml
S2 + 5 m2
+
S”
. . . (-l)n
-
and
+&)($+32R2(l+~)2
...
(21)
C(s) = Co(s) exp(h2)
(22)
where I
30 -a& CYR
112 [l
--OS)1
11
(23)
p--cw) -
(31)
Apti2K2(&+;) (32) a! Although the calculations needed to derive these moment expressions are lengthy, there is no conceptual difficulty in performing them. The second central moment expresses the variance of a probability density curve, and is a measure of band spreading. This moment is a function of all the parameters employed to characterize the model, i.e., those which describe the geometry of the bed (L, R, a, PO, 6 ), the equilibrium parameter K, and the mass ,transfer kinetic parameters (k,, Di, k,, 01). The first moment depends only on geometrical and equilibrium parameters. Simpler models can be evaluated by comparison with the present model. Ignoring 62
m,
For the non-equilibrium model, the set of differential equations (eqns. (2) - (14)) can be solved in the Laplace domain to give:
i
0),
p;(z) = /.&(z = 0) + 2z[Do(l
61
?l!
x s+
(2 =
as”
Then E(s) appears as the generating function of the moments, -
0
(19)
Knowledge of C(s) permits calculation of the moments by the relation:
= m.
= m.
where
The following relation for the second central moment is easily established:
C(s)
= Co(s)
@l(Z) = Pl(Z
0
mn = (-1)”
II
which shows that no solute is permanently retained in the column. We obtain the following expressions for the first reduced moment and second central moment:
0
J
coshI’ + r sinh l?]
and
ma(z)
and the nth central moment is
sinh r
$D;‘2&2[kg6/KDI)
the nth reduced moment is given as: J 0
sinh(bR)
$SSX i
(16)
0
m, I-in=_=
cosh(bR) + [ 1 -(Di/Rkf)]
(24) bR =
s
sinh(bR)
(Di b/k,)
The moments of order n of the chromatographic peak are defined as:
m, =
P(s) =
85
diffusion in the liquid film allows us to write instead of eqn. (7) and eqns. (10) - (14) the following:
where we have assumed that equilibrium is established instantaneously between the intraparticle gas and liquid according to K = ci/ci. The ratio of liquid volume to particle volume is given by V,, which is equivalent to 6A,. With this alteration, the expression for the first moment is identical to that of the more complete model, eqns. (28) and (29). The expressions for second moments are similar, except that 6 2 = 0. From eqn. (32) we note that 6 2 is small when 6 /D, and l/12, are small. Thus, the simpler model defined by eqn. (33) is adequate when liquid film diffusion is not a limiting process, not the most likely case for GLPC [ 51. A very simple model [l] for gas-liquid partition chromatography that ignores intraparticle diffusion as well as liquid film diffusion is defined by:
ac
ac2
(l+k);+ua2-=Dz
a22
(34)
with initial and boundary conditions given by eqns. (3) - (5). For a column packed with porous particles the partition ratio k is given by: Fz= KV,(l -a)/[c!
+p(l -cr)]
(35)
The first moment for this model is the same as the first moment for the more complete model, eqns. (28) and (29), when p = 0, i.e., for particles of small intraparticle gas phase capacity. The second moments are the same whenp=0,61 =Oandh2 =O.Fromeqn. (31) we note that s1 is small when R/Di and l/kf are small. Thus, although the very simple model is accurate for a limited class of processes, it does include partitioning and can be useful for analyzing systems in which intraparticle phenomena are of little consequence. Wong, McCoy and Carbonell [14] have studied capillary chromatography, where a liquid film coats the inner surface of a circular tube. First and second moments for their model agree with our results. The liquid film diffusion contribution to the second central moment is obtained from eqn. (32)
(36) D1 3 fg D1 where eL/eg is the ratio of the liquid to the gas volume in the column. The expression is identical to that obtained by Wong et al. [141, where the liquid film diffusion contribution to the second moment for a thin liquid film (6 Q column radius) is given by: 6,=-_
1 26 Kti2 -=--3 R D,
1 eL KS2 3 cg
D,
(37)
Andrieu and Smith [ 151 have developed a theory for absorption and chemical reaction in a liquid film coating nonporous (/3 = 0) spherical particles packed in a tubular column. Their first moment expression in the absence of chemical reaction agrees exactly with ours when p = 0. The second moment expressions, with the same conditions of no reaction and p = 0, are in excellent agreement. The only difference is that the AndrieuSmith work has 5/12 rather than l/3 as the coefficient of liquid film diffusion contribution to the second moment.
FIRST MOMENT ANALYSIS
The first temporal moment indicates the location of the centroid of the area under the elution curve, which for chromatography is the (mean) retention time, tR, the position of the mean on the time axis. Providing retention data is the most widely used method of chromatographic data representation. The ability to predict the retention time is crucial for gas chromatography. In analytical gas chromatography, characterizing a substance by its retention behavior on one or more stationary phases is the most common identification method. In a full-scale industrial unit operation, separated components may be collected at different times, corresponding to retention time for each component, at the column outlet. The retention time is relatively very short in chromatography compared with countercurrent processes. This is mainly due to the very small thickness of the layer of stationary liquid in the column relative to countercurrent processes, where the liquid has sufficient thickness to flow. Because of the very low retention times associated with gas chromatography, products are rarely
86
subjected to thermal degradation; moreover, faster throughput results in increased productivity and reduces the need for large inventories of expensive materials [ 21.
L = 2u(R,/a)s(\k* + [49***
1’2l
=L+&,
RESOLUTION
(38)
To achieve satisfactory separation between two Gaussian peaks, it is often required that the inner tangents for the two peaks intersect at a point on or below the time axis [17]. This requirement leads to the condition that the resolution defined as:
(39) is greater than two. Often Rs = 1.5 or 1.0 is regarded as satisfactory resolution of chromatographic peaks. For PSGC even lower resolutions may be suitable, with the overlapping region of the chromatogram being recycled back to the column 1161. When moment expressions (28) and (30) for each component are substituted into eqn. (39) one can derive an equation for the column length:
(40)
(41)
+A2
U2
It is generally accepted that the height equivalent to a theoretical plate (HETP) is a good measure of column performance with respect to a particular set of operating parameters for a single chemical species. However, the degree of separation of two chemical species is best measured by the resolution. Thus, HETP is derived from parameters taken from a single chromatographic peak, whereas resolution involves two peaks. Many chromatographic output pulses closely approximate gaussian curves, and this is the basis for most discussions of the separation resolution, as well as the HETP. Many actual chromatograms of PSGC appear to be superimposed, nearly symmetrical peaks [ 161. The following discussion of GLPC is similar to that of Carbonell and McCoy [lo] in the study of adsorption and permeation chromatography. We represent the width W of a Gaussian pulse as the time-distance between the horizontal axis intersections of tangents to the inflection points of the curve, from which we obtain: w = 4(/,&l’s
+ &wdw21
where qJ
.SEPARATION
+ \ks +
(42)
a = 6,,B -60,
and X=
F(l
+*,)z
We have assumed that the input pulse is a homogeneous mixture so that at z = 0, filA =C(lB
(44)
=pl(O)
=/-&B =&to) (45) When the input pulse width is not negligible, eqn. (40) shows that the length of the column needed for a given separation increases with’ pa(O). Thus to minimize the length, the feed sample will enter as a narrow pulse. For rectangular pulses:
&?A
@h(O) = $
(46)
where the pulse width t, is given in terms of sample volume 5’ and column diameter d: to =
S
(47)
un (d/2)2
Thus & is smaller for’a larger column diameter d. The diameter d appears in the present analysis only in the expression for the injection time to because we exclude consideration of radial dispersion effects. When p;(O) is negligible in eqn. (40) : Rs = e(L/2,.#‘2/(\k,1’2
+ \k,1’2)
(48)
so that resolution increases as the square root of column length. The effect of mean particle radius R appears mainly through intraparticle diffusion. Correlations for the film mass transfer coefficient hi for gas systems and for liquid systems show that kfR is a weak function [lo] of R. Also, kf frequently has a much smaller effect than the pore diffusion coefficient D, , so that in eqn. (30) Fi increases as R2. Thus, resolution is enhanced for smaller particles, although the increased pressure drop through the column may be a disadvantage.
87
For large velocity and negligible p;(O), eqn. (39) reduces to: L = Z(R&z)%[
(6 1.4 + 6 &‘2
+
+ @ 1B + 6 2B)1’21 2
(49)
Thus pore diffusion, gas film mass transfer, and liquid film diffusion control the band spreading. This means that if velocity in eqn. (49) is increased, the residence time becomes too short to allow these rate processes to be as effective in aiding the separation. For small u, L varies inversely with U; i.e., eqn. (40) becomes: L = ~(R,/u)~(x *r’s + XB1’2)2/U
Uopt = [(6
1*
+
HEIGHT EQUIVALENT
(51)
TO A THEORETICAL
PLATE
The moment approach to chromatographic separation yields conclusions in general agreement with most chromatographic practice. The conventional analysis of chromatographic separation efficiency utjlizes the van Deemter equation [ 51 and the assumption that resolution is inversely proportional to the square root of the height equivalent to a theoretical plate. We will show [lo] the relationship between the moment theory of chromatography and the conventional approach by making use of the definition of the plate height: h =Lln
(52)
and the definition of the number of theoretical plates : n = (4Apl/u~)~ With eqn. (38) we see that:
h = LP;/@PI)~
h=A
(53)
+Blu+Cu
(55)
where now: A
=
PW)U2
(4S/nd 2)2
L(1 + &0)2 = lul(l+
2x B= (1 + ao)2 = 2Dola
(50)
For these low velocities axial dispersion and equilibrium coefficients dominate, and increasing u under conditions of eqn. (50) will reduce band spreading and therefore increase the resolving power. Since the effect of ~~(0) is usually small, the optimum velocity may be estimated approximately by setting equal eqns. (49) and (50) and solving for u to obtain:
(XA1’2 + XB1’2) 6 2*)1’2 + (S 1B + 6 2J.p2
is the relation between the plate height and moments. By substituting the moment expressions eqns. (28) and (30) into eqn. (54), one finds an equation corresponding to the van Deemter [ 51 equation:
c=
2(&l
+62)
(1 +60)2
&))i
(56)
(57)
(58)
The height equivalent to a theoretical plate (HETP) depends on both the equilibrium properties and geometrical properties of the column and packing, as well as the mass transfer kinetics. The coefficients A, B and C are interpreted as follows, according to eqns. (56) - (58). The quantity A accounts for the effect of the spread of the input pulse. Equation (56) shows that A increases with the square of the sample size S, and decreases as the fourth power of the column diameter D. For analytical chromatography the sample size may be so small that A is a negligible contribution to h. Since plate height is often interpreted as a measure of band spreading per unit column length, it is reasonable that eqn. (56) shows A a l/L. Thus h depends upon the sample volume, the column length, and its sorption capacity. The greater the column dimensions and its sorption capacity, the less is the effect of sample size on the efficiency. Experimental decrease of h with increasing column length has been observed by Perry [ 181. Verzele [ 191 found that in preparatory scale gas chromatography, sample size was the most important factor for determining relative band width, and that longer columns allowed larger samples. Sakodynskii and Volkov [20] show the effect in a plot of h versus column length obtained on a 15 mm diameter column when isopentane is analyzed. As a conclusion one can see that larger columns are more effective separators. However, longer columns lead to longer separation
88
time, as well as higher pressure drop. Thus in terms of capacity (i.e., the amount of substance separated per unit time) column length should be increased only up to a specific value
[201.
The coefficient B includes band spreading due to axial dispersion. Here we have assumed II,, is independent of velocity, although the more general case, where Do(u) leads to useful results [lo] . The coefficient C includes spreading due to particle film mass transfer, pore diffusion and liquid film diffusion. The minimum plate height occurs when dh/du = 0, or at Uopt = (B/C) ‘I2 . Since the gas diffusion coefficient of a component varies inversely with pressure, B decreases with increasing average column pressure, while C increases. Thus at low velocities, the plate height decreases with increasing column pressure. The opposite is true at high flow velocities. Therefore, for fastest throughput subatmospheric outlet pressures are beneficial [21]. The nature of the carrier gas affects the diffusion coefficient; the carrier gases of low molecular weight (H2 and He) will give appreciably higher values of diffusion coefficient than, e.g., Na or Ar. Thus at low velocities, carrier gases of high molecular weight will yield the lowest h, while at high velocities the reverse is the case [21]. An important parameter determining the carrier gas velocity at a given pressure drop is the viscosity of the selected gas. The carrier gas velocity is inversely proportional to the viscosity as a result of Darcy’s law [21] . Reduction of particle size leads to significant decrease of plate height, as well as the decrease of the slope of h uersus u curve [22] . As R decreases, C decreases and leads to the decrease in h. Column efficiency decreases by increasing the temperature; as T increases, K decreases, and therefore A and C increase. When the sample size decreases, this dependence is less significant [ 231. Production scale gas chromatography is based on achieving the separation at increased carrier gas velocity with minimal loss of separation efficiency. This can be achieved with columns packed with porous particles of small size, because porous packing has large surface area which permits the application of large sample amounts, and increases the sorption capacity (a increases in eqn. (48)). Small
particle diameter leads to increased efficiency as explained earlier. Experimental evidence indicates that scaling-up leads to purity enhancement [2] . Two possible explanations for the purity enhancement follow. The A term in the h expression decreases as the fourth power of the column diameter d, so as d increases, h decreases, thus purity increases. At high velocities, this reduction in h is less significant; thus purity difference due to increasing the column diameter is larger at high velocities although purity itself is lower. The increased purity of the larger diameter columns could also be explained by the reduction in the relative contribution of the wall. The wall causes an increased void fraction, which is an inhomogeneity that increases the dispersion in the immediate vicinity of the wall. Because the C term in h dominates the high velocity region the purity decreases with increasing gas velocity [2]. However, the larger diameter column allows the same purity at a larger flow rate. CONCLUDING
REMARKS
Apart from the approximation of a uniform thin liquid film, which we have already discussed, several other limitations of the present treatment deserve attention. We have assumed that the equilibrium partition coefficient is constant, which is accurate for ideal solutions or low concentrations. A related assumption is that sorption effects for two or more solutes can be considered independently. Numerical computations to account for these nonlinearities have been based on first-order partial differential equations that ignore all transport effects such as dispersion and particle mass transfer processes [24]. We have also ignored the pressure drop along the column, as well as any pressure effects due to injection of the sample. Admittedly, the approximations in the present work impose limitations on the use of the results for detailed design of large-scale processes. However, our approach has the decided avantage of accurately and consistently describing the transport phenomena which occur in fixed beds of porous
89
particles. Thus, the present results and their extension would seem to complement other treatments of PSGC, which indeed have their own approximations and limitations.
Vl
W X
Y z
NOMENCLATURE
coefficient in eqn. (56), cm column cross-sectional area, cm2 4 particle surface area per unit volume, A* cm2/cm3 coefficient in eqn. (42) coefficient in eqn. (57), cm2/s coefficient in eqn. (58), s concentration of solute, g/ml intraparticle gas concentration, g/ml concentration in liquid, g/ml concentration at column inlet, g/ml gas concentration at liquid surface, g/ml effective intraparticle. diffusion coefficient, cm2/s Dl liquid binary diffusion coefficient, cm2/s Do axial dispersion coefficient, cm2/s column diameter, cm d height equivalent to a theoretical plate h (HETP), cm equilibrium partition coefficient K partition ratio k gas film mass transfer coefficient in the kf interparticle voids, cm/s ki3 intraparticle gas film mass transfer coefficient, cm/s column length, cm L m, nth temporal moment number of theoretical plates II volumetric flow rate of carrier gas, Q cm3/s particle radius, cm R resolution R, radial coordinate for a spherical r particle, cm input sample volume, cm3 s Laplace transform parameter, s-l s time, s t time of injection for rectangular input to pulse, s interstitial carrier gas velocity, cm/s U = alp, superficial carrier gas velocity, cm UO empty column volume, cm3 VC A
volume of liquid per unit volume of particle time width of a Gaussian pulse, eqn. (38), s coefficient defined in eqn. (43), cm”/s liquid film coordinate normal to surface, cm axial coordinate, cm
Greek symbols interparticle void fraction in the column gas inter-particle void fraction of the FO particles internal porosity of the particles P 6 liquid film thickness, cm reduced moment I&, reduced central moment Pn coefficient defined in eqn. (41) \k Sub&rip ts A first solute luted from the column second solute eluted from the column B
REFERENCES
1 S. Dal Nogare, R. S. Juvet, Jr., Gas-Liquid Chromafogmphy: Theory and Practice, Interscience, New York, 1962. 2 R. G. Bonmati, G. Chapelet-Letourneux, J. R. Margulis, Chem. Eng., 24 (1980) 70. 3 Anon., Anal. Chem., 52 (1980) 448A. 4 R. Bonmati, G. Guiochon, Perfum. Flavor. 3, (1978) 17. 5 J. J. Van Deemter, F. J. Zuiderweg, A. Kiinkenberg, Chem. Eng. Sci., 5 (1956) 271. 6 J. C. Giddings, Dynamics of Chromatography, Part I, Dekker, 1965. 7 M. Kubin, Coil. Czech. Chem. Commun., 30 (1965) 1104,290O. 8 K. Kucera, J. Chromatog., 19 (1965) 237. 9 M. Suzuki, J. M. Smith, Chem. Eng. Sci., 26 (1971) 221. 10 R. G. Carboneii, B. J. McCoy, Chem. Eng. J., 9 (1975) 115. 11 E. J. Boneili, H. M. McNair, Basic Gas Chromatography, 5th edn., Varian, 1969. 12 P. Schneider, J. M. Smith, AZChE J., 14 (1968) 762. 13 M. A. Aikharasani, B. J. McCoy, J. Chromatog., 213 (1981) 203. 14 A. K. Wong, B. J. McCoy, R. G. Carbonell, J. Chromatog., 129 (1976) 1. 15 J. Andrieu, J. M. Smith, Chem. Eng. J., 20 (1980) 211.
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16 B. Roz, R. Bonmati, G. Hagenback, P. Valentin, G. Guiochon, J. Chromatog. Sci., I4 (1976) 367. 17 C. J. King, Separation Proceseee, McGraw-Hill, New York, 1971. 18 J. A. Perry, J. Gus Chromatog., 4 (1966) 194. 19 M. Verzele, J. Gas Chromatog., 4 (1966) 180. 20 K. Sakodynskii, S. Volkor, J. Chromatog., 49 (1970) 76.
21 C. A. Cramers, J. A. Rijks, Adv. Chromatog., 17 (1979) 101. 22 J. F. K. Huber, H. H. Lauer, H. Pope, J. Chromatog., 112 (1975) 377. 23 K. Sakodynskii, S. Volkov, Chromatognaphia, 5 (1972) 330. 24 P. Valentin, G. Guiochon, Sep. Sci., 10 (1975) 245, 260, 275.
81
Gas-Liquid Partition Chromatographic Separations in Columns Packed with Porous Particles - a Model for Uniform Thickness Liquid Film
M. A. ALKHARASANI Department
and B. J. MCCOY
of Chemical Engineering,
(Received 24 October, 1980;in
University of California, Davis, CA 95616
(U.S.A.)
final form 15 April, 1981)
Abstract A linear mathematical model for gasliquid chromatography is presented for application to either analytical or larger-scale equipment. The model includes the effects of axial dispersion, interparticle mass transfer, intraparticle gas diffusion, and diffusion in a thin, uniform thickness liquid film. The governing differential equations are solved in the Laplace transform domain for a general input pulse. Temporal moments of the output pulses are expressed in terms of geometrical, equilibrium and transport pammeters. The moment theory is related to HETP and to resolution of separation. The optimization of the process and scale-up to industrial size are discussed.
INTRODUCTION
Separation by gas-liquid partition chromatography (GLPC) in columns packed with porous particles, where liquid is spread as a thin layer on pore surfaces, occurs because different gaseous solutes have different solubilities in the stationary liquid phase [ 11. To maximize the surface area of liquid and thus provide the most area for mass transfer, the liquid is spread on the surface of the pores of the porous particles. Because the diffusion of the solutes in the gas phase in the pores is orders of magnitude greater than that in liquid, this has obvious advantages over the case where the pores are filled with liquid. The thin liquid layer on pore surfaces is prepared by filling and saturating the pores with a solution of the liquid and a solvent. The solvent is then evaporated and removed
from the particles, leaving a layer of immobile liquid of average thickness about 3c( on the pore surface. Of course, the mathematical analysis of mass-transfer taking place in liquid-filled pores is considerably simpler than that for the situation considered in this paper where the immobile liquid phase is a thin layer on pure surfaces. The latter is the usual case in high resolution chromatographic processes. Chromatographic separations, including GLPC, have been moving toward large-scale use in the past few years. For example, Elf Aquitaine and Societe de Recherches Techniques et Industrielles (SRTI) have developed Production Scale Gas Chromatography (PSGC), a gas-liquid partition process that has now been commercialized in the U.S. [ 21. In the Elf-SRTI process columns up to 0.4 m internal diameter are being built with an annual output capacity ranging from 3 to 300 metric tons. Channeling is prevented by a proprietary packing technique [2, 31. This technology can separate close boiling point products, remove impurities from low and high-boiling point compounds, and separate azeotropes and thermally unstable substances. Purities of 99.9% and yields of 98% can be attained. The process is particularly suited to the essential oil industry [4]. In contrast with vacuum-distillation techniques, PSGC technology requires significantly less energy to accomplish difficult separations. The scale-up, optimization, and control of such large-scale chromatographic operations requires a deeper and more accurate mathematical understanding than has been necessary for the highly successful bench-scale chromatography. Especially the transport processes that contribute to band spreading should be under-
82
stood more thoroughly, we believe. This is the motivation for the present work. Two approaches to analyzing mass transfer processes in chromatographic columns have been used. One is the method known as the plate, or rate theory [5,6] and the other is a moment technique based on conservation and transport relationships between macroscopic variables described by differential balance equations and rate expressions [ 7 - lo] . The present treatment will follow this second approach; that is, we write the linear differential mass balance equations for solute in the interparticle voids in the column, in the pores of particles, and in the liquid film. These equations present the local average solute concentration in the column and in the intraparticle voids in terms of the average inter-particle void fraction (Y, the particle porosity p, the spherical radius R, and the liquid film thickness 6, and as a function of position z and time t. The transport processes in the column are described by means of effective coeffients, such as the axial dispersion coefficient De, gas film mass transfer coefficient for the spherical particle kf, the effective intraparticle gas diffusion coefficient Di, and the liquid film diffusion coefficient Di. It is important to note that Do, kf, Di and D1 may depend on geometrical parameters, e.g., (Y,(3, R, as well as on velocity u, but are assumed to be independent of time t, and column longitudinal position z. We restrict the theory to linear isothermal chromatography, i.e., where mass transfer is governed by linear transfer kinetics and the partition isotherm for equilibrium between gas and liquid phases is linear. The complete solution for the governing differential equations provides more information than is often actually needed for design and analysis. Frequently, knowing the first few temporal moments of the output response to an inlet pulse enables the engineer to design and optimize column operation [lo]. Because the partial differential mass balance equations are linear, they may be solved in the Laplace transform domain. The zeroth, first, and second moments of the solute concentrations are evaluated from derivatives of the Laplace-transformed concentration by means of well-known mathematical identities. Unless output concentration pulses are strongly skewed or
have large tails, the first three moments (i.e., up to the second moment) are sufficient to establish resolution criteria for the separation. We consider a packed column of length L and cross-sectional area A =, homogeneously packed with spherical porous particles of radius R. The interstitial carrier gas velocity u is given by u = L+,/(Ywhere u. is superficial velocity. The interparticle void fraction of the column is (IL,and the internal porosity of the particles is 0. Liquid as a thin layer of uniform thickness 6 is spread on the pore surfaces. In typical GLPC columns average pore diameter is in the range [ 111 of 10 p, and the average liquid thickness is in the range [l] of 3 p. The effective porosity of the liquid-coated porous particle, i.e. the ratio of gas volume to particle volume, is: fl=fl--6A,
(1)
where p. is the porosity of the dry particle, 6 is the film thickness, and A, is the dry surface area per unit particle volume. Although the following treatment refers to gas-liquid partition chromatography, the same equations should describe liquid-liquid partition chromatography, where the mobile and stationary phases are immiscible liquids.
MODEL
FOR GAS-LIQUID
CHROMATOGRAPHY
We begin with the point differential mass balance equation for concentration in the mobile phase, c(t, z):
ac
ac Do a2c
at
a2
-++-=---_
a
az2
3 (1 --a) ___ R a
kf
[C-(Ci)RI (2)
with the initial condition: c(O,z) = 0
(3)
and boundary conditions: c(t, 0) = co(t) c(t, -) = finite
(4)
(5) Continuity of flux at the spherical interface serves as a coupling boundary condition:
=
kf[C -ci(r
= R)]
(6)
Here kf is the interparticle mass transfer coefficient, Di is the effective intraparticle gas diffusion coefficient, and ci(r, t) is the intrapar-
83
The variable y is the (one-dimensional) position in the liquid film measured from the solid surface. Since there is no flux at the solid pore surfaces:
The treatment of the liquid film diffusion is an approximate description of real systems for two reasons. First, it is unlikely that the liquid forms a completely uniform thickness film throughout the porous particle. Because of the randomness of pore configuration and size, one expects unequal film thicknesses and perhaps pools of liquid to exist. However,’ such non-uniformities would cause increased band-spreading; best resolution is obtained when the film is of constant thickness as assumed in our model. Second, because the film thickness may not always be very small compared with the pore diameter, the rectangular form of the diffusion equation is an approximation. Nevertheless, even though the treatment of diffusion within the liquid phase is simplified we expect useful and reasonably accurate results since the treatment, although limited by the model, is systematic and rigorous. Because the complexity of the calculations for a more realistic model would be prohibitive, including the complications of unequal and cylindrical films does not seem justified at this stage. We have considered in other work the condition where liquid only partially covers the pore surfaces, leaving exposed solid surfaces for adsorption [ 131. For the case when pores are completely filled with liquid, the partitioning is described by eqns. (2) - (9) with k, = 0 and Di replaced with Di; the moments for this case are easily adapted from earlier publications [8, 9,121.
wt, 0) ay
MOMENTS
title gas phase concentration. Equations (2) (6) are identical to equations presented by Schneider and Smith [ 121. The mass balance equation for the intraparticle gas concentration is: -A,k,(ci
P
-c,)
(7)
with initial condition: Ci(O, r) = 0
(8)
The boundary conditions are: aCi(tv
O) =
o
(9)
ar
and the coupling boundary condition given by eqn. (6). The coefficient in eqn. (7) describes mass transfer between the pore gas and the liquid coating on the pore surface, and may be estimated as the gas diffusion coefficient divided by the gas film thickness. If the liquid film is thin compared to the pore radius, the liquid phase transport equation for c,(t,y) is:
acl -= at
a2c, Dl
-
02
(10)
with initial condition c*(O, Y) = 0
--
(11)
o =
(12)
Continuity of flux at the interface serves as a boundary condition that couples eqns. (7) and (10): (13) where c, = q(t, 6)/K
(14)
is the concentration of the gas at the surface of the liquid of thickness 6, and R is the local equilibrium distribution coefficient. Equation (13) allows for variation of liquid concentration with radial position in the particle since ci is a function of r. We have ignored any small gradients that may occur transverse to the coordinate y.
Among the existing possibilities for the solution of partial differential equations are those procedurres based upon the Laplace transform. In this method the partial differential equations are transformed into ordinary differential equations, which can then usually be solved in terms of Laplace coordinates. Though the solutions do not always permit the inversion into normal coordinates, they permit calculation of the temporal moments. Having the first few moments is sufficient for most design and optimization analyses. In a chromatographic column where all processes are linear, c(t) after Laplace transformation is the transfer function, defined by :
84 C(s) =
J
dt
eesfc(t)
(15)
0
PC(~) dt,
k,A,6
l+
t”c(t)
dt (17)
9 -,
m0
c(t)
J
dt
0
J (t -PI)“c(~)
’
Pn
=
dt (18)
(25)
r=sm
(26)
Using eqn. (20) to obtain the zeroth moment, we find:
c(t) dt
6
l&z=I-(2 --PI2
anc
(1
(27)
= 0) +zu
+fio)/u
+3(1+y),
$--a)
(28)
(29)
a
+@I
+S2)l/u
+60)2/u2a+ (30)
where
=;(;
s=o
sml
S2 + 5 m2
+
S”
. . . (-l)n
-
and
+&)($+32R2(l+~)2
...
(21)
C(s) = Co(s) exp(h2)
(22)
where I
30 -a& CYR
112 [l
--OS)1
11
(23)
p--cw) -
(31)
Apti2K2(&+;) (32) a! Although the calculations needed to derive these moment expressions are lengthy, there is no conceptual difficulty in performing them. The second central moment expresses the variance of a probability density curve, and is a measure of band spreading. This moment is a function of all the parameters employed to characterize the model, i.e., those which describe the geometry of the bed (L, R, a, PO, 6 ), the equilibrium parameter K, and the mass ,transfer kinetic parameters (k,, Di, k,, 01). The first moment depends only on geometrical and equilibrium parameters. Simpler models can be evaluated by comparison with the present model. Ignoring 62
m,
For the non-equilibrium model, the set of differential equations (eqns. (2) - (14)) can be solved in the Laplace domain to give:
i
0),
p;(z) = /.&(z = 0) + 2z[Do(l
61
?l!
x s+
(2 =
as”
Then E(s) appears as the generating function of the moments, -
0
(19)
Knowledge of C(s) permits calculation of the moments by the relation:
= m.
= m.
where
The following relation for the second central moment is easily established:
C(s)
= Co(s)
@l(Z) = Pl(Z
0
mn = (-1)”
II
which shows that no solute is permanently retained in the column. We obtain the following expressions for the first reduced moment and second central moment:
0
J
coshI’ + r sinh l?]
and
ma(z)
and the nth central moment is
sinh r
$D;‘2&2[kg6/KDI)
the nth reduced moment is given as: J 0
sinh(bR)
$SSX i
(16)
0
m, I-in=_=
cosh(bR) + [ 1 -(Di/Rkf)]
(24) bR =
s
sinh(bR)
(Di b/k,)
The moments of order n of the chromatographic peak are defined as:
m, =
P(s) =
85
diffusion in the liquid film allows us to write instead of eqn. (7) and eqns. (10) - (14) the following:
where we have assumed that equilibrium is established instantaneously between the intraparticle gas and liquid according to K = ci/ci. The ratio of liquid volume to particle volume is given by V,, which is equivalent to 6A,. With this alteration, the expression for the first moment is identical to that of the more complete model, eqns. (28) and (29). The expressions for second moments are similar, except that 6 2 = 0. From eqn. (32) we note that 6 2 is small when 6 /D, and l/12, are small. Thus, the simpler model defined by eqn. (33) is adequate when liquid film diffusion is not a limiting process, not the most likely case for GLPC [ 51. A very simple model [l] for gas-liquid partition chromatography that ignores intraparticle diffusion as well as liquid film diffusion is defined by:
ac
ac2
(l+k);+ua2-=Dz
a22
(34)
with initial and boundary conditions given by eqns. (3) - (5). For a column packed with porous particles the partition ratio k is given by: Fz= KV,(l -a)/[c!
+p(l -cr)]
(35)
The first moment for this model is the same as the first moment for the more complete model, eqns. (28) and (29), when p = 0, i.e., for particles of small intraparticle gas phase capacity. The second moments are the same whenp=0,61 =Oandh2 =O.Fromeqn. (31) we note that s1 is small when R/Di and l/kf are small. Thus, although the very simple model is accurate for a limited class of processes, it does include partitioning and can be useful for analyzing systems in which intraparticle phenomena are of little consequence. Wong, McCoy and Carbonell [14] have studied capillary chromatography, where a liquid film coats the inner surface of a circular tube. First and second moments for their model agree with our results. The liquid film diffusion contribution to the second central moment is obtained from eqn. (32)
(36) D1 3 fg D1 where eL/eg is the ratio of the liquid to the gas volume in the column. The expression is identical to that obtained by Wong et al. [141, where the liquid film diffusion contribution to the second moment for a thin liquid film (6 Q column radius) is given by: 6,=-_
1 26 Kti2 -=--3 R D,
1 eL KS2 3 cg
D,
(37)
Andrieu and Smith [ 151 have developed a theory for absorption and chemical reaction in a liquid film coating nonporous (/3 = 0) spherical particles packed in a tubular column. Their first moment expression in the absence of chemical reaction agrees exactly with ours when p = 0. The second moment expressions, with the same conditions of no reaction and p = 0, are in excellent agreement. The only difference is that the AndrieuSmith work has 5/12 rather than l/3 as the coefficient of liquid film diffusion contribution to the second moment.
FIRST MOMENT ANALYSIS
The first temporal moment indicates the location of the centroid of the area under the elution curve, which for chromatography is the (mean) retention time, tR, the position of the mean on the time axis. Providing retention data is the most widely used method of chromatographic data representation. The ability to predict the retention time is crucial for gas chromatography. In analytical gas chromatography, characterizing a substance by its retention behavior on one or more stationary phases is the most common identification method. In a full-scale industrial unit operation, separated components may be collected at different times, corresponding to retention time for each component, at the column outlet. The retention time is relatively very short in chromatography compared with countercurrent processes. This is mainly due to the very small thickness of the layer of stationary liquid in the column relative to countercurrent processes, where the liquid has sufficient thickness to flow. Because of the very low retention times associated with gas chromatography, products are rarely
86
subjected to thermal degradation; moreover, faster throughput results in increased productivity and reduces the need for large inventories of expensive materials [ 21.
L = 2u(R,/a)s(\k* + [49***
1’2l
=L+&,
RESOLUTION
(38)
To achieve satisfactory separation between two Gaussian peaks, it is often required that the inner tangents for the two peaks intersect at a point on or below the time axis [17]. This requirement leads to the condition that the resolution defined as:
(39) is greater than two. Often Rs = 1.5 or 1.0 is regarded as satisfactory resolution of chromatographic peaks. For PSGC even lower resolutions may be suitable, with the overlapping region of the chromatogram being recycled back to the column 1161. When moment expressions (28) and (30) for each component are substituted into eqn. (39) one can derive an equation for the column length:
(40)
(41)
+A2
U2
It is generally accepted that the height equivalent to a theoretical plate (HETP) is a good measure of column performance with respect to a particular set of operating parameters for a single chemical species. However, the degree of separation of two chemical species is best measured by the resolution. Thus, HETP is derived from parameters taken from a single chromatographic peak, whereas resolution involves two peaks. Many chromatographic output pulses closely approximate gaussian curves, and this is the basis for most discussions of the separation resolution, as well as the HETP. Many actual chromatograms of PSGC appear to be superimposed, nearly symmetrical peaks [ 161. The following discussion of GLPC is similar to that of Carbonell and McCoy [lo] in the study of adsorption and permeation chromatography. We represent the width W of a Gaussian pulse as the time-distance between the horizontal axis intersections of tangents to the inflection points of the curve, from which we obtain: w = 4(/,&l’s
+ &wdw21
where qJ
.SEPARATION
+ \ks +
(42)
a = 6,,B -60,
and X=
F(l
+*,)z
We have assumed that the input pulse is a homogeneous mixture so that at z = 0, filA =C(lB
(44)
=pl(O)
=/-&B =&to) (45) When the input pulse width is not negligible, eqn. (40) shows that the length of the column needed for a given separation increases with’ pa(O). Thus to minimize the length, the feed sample will enter as a narrow pulse. For rectangular pulses:
&?A
@h(O) = $
(46)
where the pulse width t, is given in terms of sample volume 5’ and column diameter d: to =
S
(47)
un (d/2)2
Thus & is smaller for’a larger column diameter d. The diameter d appears in the present analysis only in the expression for the injection time to because we exclude consideration of radial dispersion effects. When p;(O) is negligible in eqn. (40) : Rs = e(L/2,.#‘2/(\k,1’2
+ \k,1’2)
(48)
so that resolution increases as the square root of column length. The effect of mean particle radius R appears mainly through intraparticle diffusion. Correlations for the film mass transfer coefficient hi for gas systems and for liquid systems show that kfR is a weak function [lo] of R. Also, kf frequently has a much smaller effect than the pore diffusion coefficient D, , so that in eqn. (30) Fi increases as R2. Thus, resolution is enhanced for smaller particles, although the increased pressure drop through the column may be a disadvantage.
87
For large velocity and negligible p;(O), eqn. (39) reduces to: L = Z(R&z)%[
(6 1.4 + 6 &‘2
+
+ @ 1B + 6 2B)1’21 2
(49)
Thus pore diffusion, gas film mass transfer, and liquid film diffusion control the band spreading. This means that if velocity in eqn. (49) is increased, the residence time becomes too short to allow these rate processes to be as effective in aiding the separation. For small u, L varies inversely with U; i.e., eqn. (40) becomes: L = ~(R,/u)~(x *r’s + XB1’2)2/U
Uopt = [(6
1*
+
HEIGHT EQUIVALENT
(51)
TO A THEORETICAL
PLATE
The moment approach to chromatographic separation yields conclusions in general agreement with most chromatographic practice. The conventional analysis of chromatographic separation efficiency utjlizes the van Deemter equation [ 51 and the assumption that resolution is inversely proportional to the square root of the height equivalent to a theoretical plate. We will show [lo] the relationship between the moment theory of chromatography and the conventional approach by making use of the definition of the plate height: h =Lln
(52)
and the definition of the number of theoretical plates : n = (4Apl/u~)~ With eqn. (38) we see that:
h = LP;/@PI)~
h=A
(53)
+Blu+Cu
(55)
where now: A
=
PW)U2
(4S/nd 2)2
L(1 + &0)2 = lul(l+
2x B= (1 + ao)2 = 2Dola
(50)
For these low velocities axial dispersion and equilibrium coefficients dominate, and increasing u under conditions of eqn. (50) will reduce band spreading and therefore increase the resolving power. Since the effect of ~~(0) is usually small, the optimum velocity may be estimated approximately by setting equal eqns. (49) and (50) and solving for u to obtain:
(XA1’2 + XB1’2) 6 2*)1’2 + (S 1B + 6 2J.p2
is the relation between the plate height and moments. By substituting the moment expressions eqns. (28) and (30) into eqn. (54), one finds an equation corresponding to the van Deemter [ 51 equation:
c=
2(&l
+62)
(1 +60)2
&))i
(56)
(57)
(58)
The height equivalent to a theoretical plate (HETP) depends on both the equilibrium properties and geometrical properties of the column and packing, as well as the mass transfer kinetics. The coefficients A, B and C are interpreted as follows, according to eqns. (56) - (58). The quantity A accounts for the effect of the spread of the input pulse. Equation (56) shows that A increases with the square of the sample size S, and decreases as the fourth power of the column diameter D. For analytical chromatography the sample size may be so small that A is a negligible contribution to h. Since plate height is often interpreted as a measure of band spreading per unit column length, it is reasonable that eqn. (56) shows A a l/L. Thus h depends upon the sample volume, the column length, and its sorption capacity. The greater the column dimensions and its sorption capacity, the less is the effect of sample size on the efficiency. Experimental decrease of h with increasing column length has been observed by Perry [ 181. Verzele [ 191 found that in preparatory scale gas chromatography, sample size was the most important factor for determining relative band width, and that longer columns allowed larger samples. Sakodynskii and Volkov [20] show the effect in a plot of h versus column length obtained on a 15 mm diameter column when isopentane is analyzed. As a conclusion one can see that larger columns are more effective separators. However, longer columns lead to longer separation
88
time, as well as higher pressure drop. Thus in terms of capacity (i.e., the amount of substance separated per unit time) column length should be increased only up to a specific value
[201.
The coefficient B includes band spreading due to axial dispersion. Here we have assumed II,, is independent of velocity, although the more general case, where Do(u) leads to useful results [lo] . The coefficient C includes spreading due to particle film mass transfer, pore diffusion and liquid film diffusion. The minimum plate height occurs when dh/du = 0, or at Uopt = (B/C) ‘I2 . Since the gas diffusion coefficient of a component varies inversely with pressure, B decreases with increasing average column pressure, while C increases. Thus at low velocities, the plate height decreases with increasing column pressure. The opposite is true at high flow velocities. Therefore, for fastest throughput subatmospheric outlet pressures are beneficial [21]. The nature of the carrier gas affects the diffusion coefficient; the carrier gases of low molecular weight (H2 and He) will give appreciably higher values of diffusion coefficient than, e.g., Na or Ar. Thus at low velocities, carrier gases of high molecular weight will yield the lowest h, while at high velocities the reverse is the case [21]. An important parameter determining the carrier gas velocity at a given pressure drop is the viscosity of the selected gas. The carrier gas velocity is inversely proportional to the viscosity as a result of Darcy’s law [21] . Reduction of particle size leads to significant decrease of plate height, as well as the decrease of the slope of h uersus u curve [22] . As R decreases, C decreases and leads to the decrease in h. Column efficiency decreases by increasing the temperature; as T increases, K decreases, and therefore A and C increase. When the sample size decreases, this dependence is less significant [ 231. Production scale gas chromatography is based on achieving the separation at increased carrier gas velocity with minimal loss of separation efficiency. This can be achieved with columns packed with porous particles of small size, because porous packing has large surface area which permits the application of large sample amounts, and increases the sorption capacity (a increases in eqn. (48)). Small
particle diameter leads to increased efficiency as explained earlier. Experimental evidence indicates that scaling-up leads to purity enhancement [2] . Two possible explanations for the purity enhancement follow. The A term in the h expression decreases as the fourth power of the column diameter d, so as d increases, h decreases, thus purity increases. At high velocities, this reduction in h is less significant; thus purity difference due to increasing the column diameter is larger at high velocities although purity itself is lower. The increased purity of the larger diameter columns could also be explained by the reduction in the relative contribution of the wall. The wall causes an increased void fraction, which is an inhomogeneity that increases the dispersion in the immediate vicinity of the wall. Because the C term in h dominates the high velocity region the purity decreases with increasing gas velocity [2]. However, the larger diameter column allows the same purity at a larger flow rate. CONCLUDING
REMARKS
Apart from the approximation of a uniform thin liquid film, which we have already discussed, several other limitations of the present treatment deserve attention. We have assumed that the equilibrium partition coefficient is constant, which is accurate for ideal solutions or low concentrations. A related assumption is that sorption effects for two or more solutes can be considered independently. Numerical computations to account for these nonlinearities have been based on first-order partial differential equations that ignore all transport effects such as dispersion and particle mass transfer processes [24]. We have also ignored the pressure drop along the column, as well as any pressure effects due to injection of the sample. Admittedly, the approximations in the present work impose limitations on the use of the results for detailed design of large-scale processes. However, our approach has the decided avantage of accurately and consistently describing the transport phenomena which occur in fixed beds of porous
89
particles. Thus, the present results and their extension would seem to complement other treatments of PSGC, which indeed have their own approximations and limitations.
Vl
W X
Y z
NOMENCLATURE
coefficient in eqn. (56), cm column cross-sectional area, cm2 4 particle surface area per unit volume, A* cm2/cm3 coefficient in eqn. (42) coefficient in eqn. (57), cm2/s coefficient in eqn. (58), s concentration of solute, g/ml intraparticle gas concentration, g/ml concentration in liquid, g/ml concentration at column inlet, g/ml gas concentration at liquid surface, g/ml effective intraparticle. diffusion coefficient, cm2/s Dl liquid binary diffusion coefficient, cm2/s Do axial dispersion coefficient, cm2/s column diameter, cm d height equivalent to a theoretical plate h (HETP), cm equilibrium partition coefficient K partition ratio k gas film mass transfer coefficient in the kf interparticle voids, cm/s ki3 intraparticle gas film mass transfer coefficient, cm/s column length, cm L m, nth temporal moment number of theoretical plates II volumetric flow rate of carrier gas, Q cm3/s particle radius, cm R resolution R, radial coordinate for a spherical r particle, cm input sample volume, cm3 s Laplace transform parameter, s-l s time, s t time of injection for rectangular input to pulse, s interstitial carrier gas velocity, cm/s U = alp, superficial carrier gas velocity, cm UO empty column volume, cm3 VC A
volume of liquid per unit volume of particle time width of a Gaussian pulse, eqn. (38), s coefficient defined in eqn. (43), cm”/s liquid film coordinate normal to surface, cm axial coordinate, cm
Greek symbols interparticle void fraction in the column gas inter-particle void fraction of the FO particles internal porosity of the particles P 6 liquid film thickness, cm reduced moment I&, reduced central moment Pn coefficient defined in eqn. (41) \k Sub&rip ts A first solute luted from the column second solute eluted from the column B
REFERENCES
1 S. Dal Nogare, R. S. Juvet, Jr., Gas-Liquid Chromafogmphy: Theory and Practice, Interscience, New York, 1962. 2 R. G. Bonmati, G. Chapelet-Letourneux, J. R. Margulis, Chem. Eng., 24 (1980) 70. 3 Anon., Anal. Chem., 52 (1980) 448A. 4 R. Bonmati, G. Guiochon, Perfum. Flavor. 3, (1978) 17. 5 J. J. Van Deemter, F. J. Zuiderweg, A. Kiinkenberg, Chem. Eng. Sci., 5 (1956) 271. 6 J. C. Giddings, Dynamics of Chromatography, Part I, Dekker, 1965. 7 M. Kubin, Coil. Czech. Chem. Commun., 30 (1965) 1104,290O. 8 K. Kucera, J. Chromatog., 19 (1965) 237. 9 M. Suzuki, J. M. Smith, Chem. Eng. Sci., 26 (1971) 221. 10 R. G. Carboneii, B. J. McCoy, Chem. Eng. J., 9 (1975) 115. 11 E. J. Boneili, H. M. McNair, Basic Gas Chromatography, 5th edn., Varian, 1969. 12 P. Schneider, J. M. Smith, AZChE J., 14 (1968) 762. 13 M. A. Aikharasani, B. J. McCoy, J. Chromatog., 213 (1981) 203. 14 A. K. Wong, B. J. McCoy, R. G. Carbonell, J. Chromatog., 129 (1976) 1. 15 J. Andrieu, J. M. Smith, Chem. Eng. J., 20 (1980) 211.
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