Anal. Chem. 1989, 61, 1276-1287
1276
Theory of Optimization of the Experimental Conditions of Preparative Elution Using the Ideal Model of Liquid Chromatography Sadroddin Golshan-Shirazi and Georges Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, and Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 -6120
the extrapolation of knowledge acquired in analytical chromatography lack generality. Considerable interest has been devoted to issues such as volume or concentration overload. In most cases (3-5), the models used are fairly simple and the conclusions are based on the consideration of the classical resolution equation. They are rather straightforward in appearance, but they are erroneous. The mental fixation of many experimentalists on the apparent column efficiency and on the resolution between bands, which are two analytical concepts transported to the preparative area without much of a justification, results in very low production rates. Some authors have recognized this fact and recommended the use of large sample sizes, resulting in serious column overload, Le., in operating the column under nonlinear conditions (6-9). The degree of resolution that should be achieved between the bands a t the exit of the column remains controversial. Among the previous publications, the most perceptive has been the one by Knox and Pyper (7). An expression for the column throughput in the case of a binary mixture has been derived in the assumption that (i) the adsorption isotherm of each component of the mixture is Langmuirian, (ii) the elution profile of each band is a rectangular triangle standing on one side, (iii) the two bands are exactly resolved a t column outlet, and (iv) there is no competition between the two compounds for adsorption on the stationary phase. This system is very simple. Each compound is recovered totally pure, with a 100% yield, which is an ideal situation. The band broadening due to the nonlinear behavior of the equilibrium isotherm of each component between the two phases of the chromatographic system is treated as an independent contribution to the height equivalent to a theoretical plate. It is difficult to accept, however, than the chromatogram obtained for a mixture could be identical with the sum of the different chromatograms obtained successively for each component, injected pure but in the same amount as in the mixture. Column overloading arises because of non-linear behavior of the phase equilibrium isotherm and, by definition, nonlinear phenomena are not additive. Recently, Ghodbane and Guiochon (10) and Katti and Guiochon ( 1I ) used the semiideal model of chromatography and a simplex approach to study the optimization of the experimental conditions in preparative liquid chromatography. They have shown that the production rate can be increased considerably by accepting some losses in the recovery yield and the production of 99% pure fractions only. Their results showed that the optimum experimental conditions depend much on whether one is interested in purifying the first or the second eluted component of a binary mixture. In the former case the sample size injected during each cycle, hence the throughput, is much larger than in the latter. The optimum conditions (flow velocity, column length, particle size) are such that the injection of a very small size sample under these conditions gives two peaks with a rather low resolution, rarely exceeding unity.
Using the ideal model of chromatography, we have derived analytical expressions for the purity, the recovery yield, and the production rate in overloaded elution chromatography, in the case of a binary mixture, when the equilibrium isotherm of the components of the studied mixture can be described properly by the competitive Langmuir equations. The effects of the phase selectivity, the composition of the feed, the sample size, and the required degree of purity of the purified fractions are discussed. The optimum conditions are very different depending on whether one is more interested in the first or the second component of the mlxture. I n both cases, the production rate increases very rapidly with the selectivity of the phase system (almost as (a 1)2 for low a values). The production rate for the first component increases with increasing sample size up to very large sample loads; the optimum depends on the lowest acceptable value of the yield. For the second component there is an optimum sample size which can be calculated easily, using a very simple equation, and which remains practically unchanged for real columns with finite efficiencies. I f sample sizes larger than this optimum value are used, the production rate remains constant or decreases while the recovery yield drops. The production rate increases very rapidly with decreasing required degree of purity when the first component Is an impurity of the second one, to be eliminated. I f the second eluted compound is a minor component of the mixture which must be extracted, the production rate depends very llttle on the required degree of purity.
-
INTRODUCTION Preparative liquid chromatography is rapidly expanding as the method of choice to purify rapidly and economically moderate amounts of precious chemicals or biochemicals that are contaminated by closely related compounds or to extract them from complex matrices. Reviews ( I ) and books ( 2 ) dealing with preparative liquid chromatography have been published recently, in addition to an abundant literature. Most of the papers published, however, deal with particular applications of the method. Although each author strives for high-performance results, it is rarely possible to carry out a complete study under the constraints of method development in an industrial environment. When there is no theoretical guidelines to orient the selection of the conditions toward those that are most profitable for the solution of a given separation problem, the investigation of a wide range of experimental conditions is required. Accordingly, almost all the studies published so far have been empirical in nature. The conclusions derived in most cases from the analysis of these experimental results or by semiempirical elaborations based on *Author to whom correspondence should he addressed at the Crniversity of Tennessee. 0003-2700/89/0361-1276$01 50/0
C
1989 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989
One of the less useful controversies in the field has dealt with the definitions of the aim of an optimization process. Preparative liquid chromatography is a rather simple purification procedure, but the final cost of the products is a complex function of the investment and production costs and of the production rate and the recovery yield. Analytical accounting is sophisticated enough an art that the cost of the production of the same product, carried out with the same unit, operated under the same set of experimental conditions will depend on the company (e.g. on its overhead rate). The only thing we can do on a purely technical basis is to determine the experimental conditions under which the production rate of a certain component of a mixture a t a stated degree of purity is maximum and what are the trade-offs available to improve the recovery yield, if desired. The approach used in our previous papers consists of calculating the production rate a t a certain degree of purity, for a certain set of experimental conditions, and of changing these conditions by using the simplex procedure in order to maximize the production rate (IO, 11). A large number of numerical solutions of the system of partial differential equations corresponding to the multicomponent system under investigation must be calculated with the semiideal model for as many different sets of experimental conditions. The procedure is long and tedious, and its results are valid only for the particular problem under investigation. On the other hand, these results are accurate. In this paper, on the contrary, we present a theoretical discussion of the optimization of the experimental conditions for preparative liquid chromatography, based on the analytical solution we have recently derived for the ideal model of chromatography in the case of a binary mixture with a competitive Langmuir isotherm (12). This model assumes the column efficiency to be infinite. Comparison between the band profiles derived from the ideal model and those calculated by using the semiideal model which takes the finite column efficiency into account shows that modern HPLC columns used in preparative liquid chromatography, with efficiencies usually exceeding 5000 theoretical plates, with H E T P smaller than 50 pm, give profiles that are only slightly smoother than those predicted by the ideal model. Although the finite column efficiency will certainly lead to a lower production rate and a smaller recovery yield for the production of purified compounds at the same degree of purity, the trends demonstrated by considerations based on the analytical solution are well worth considering, because they are general, while the use of numerical methods of integration of the system of partial differential equations of chromatography cannot easily predict such trends. In a forthcoming paper, we discuss the corrections that have to be applied to the conclusions derived from the ideal model to take the finite column efficiency into account (13).
DEFINITIONS For the sake of clarity, we want to state here the definitions of the main parameters employed to characterize the procedures used. Throughput. The throughput is the amount of the mixture to be treated (feed or feedstock) which is introduced in the column per unit time. The throughput is equal to the product of the sample size, or amount, by the frequency of the sampling. This frequency is the reverse of the cycle time. Cycle Time. The cycle time is the time that separates two successive injections during a production campaign. The cycle time is somewhat arbitrarily defined. It could be as long as the analysis time of the sample, Le., the time between injection and the entire elution of the last component band, t,, which includes the time necessary for the concentration of this last compound to return to an arbitrarily small value (typically,
1277
0 0
m
E
0 0
\
LD
- 0
i E
v
0
Q& 0
L
C
0 N 0
I 0 0
0
-A 10
E
r 0 22c
240
260
me
280
330
320
0
(sed
Flgure 1. Schematics of the chromatogram obtained for the purification of one component of a binary mixture. Definitions of the parameters used in the determination of the production rate, the yield, and the purity. See eq 1 to 5. The profile of component 1 is ABCDE; the profile of component 2 is FGHIJ. The first cutting time is in K, the second cutting time in N. The partial areas defined by eq 2a, 2b, 3a and 3b are A = KEO, A = FGMN, 5 = FGLK, B , = NEP.
two or three standard deviations of the last component band). In practice, a second sample can be injected after a shorter time. Nothing is eluted between the injection and the elution of a nonretained compound. Accordingly, the cycle time can be the corrected time, t A- t,. We have chosen this definition here, which is particularly easy to implement in a theoretical work where all the time values characterizing the different features of the elution profile of the band of a binary mixture contain to. Because we are using the ideal model which assumes infinite column efficiency, we have assumed t A to be equal to tR,*',however, and the cycle time becomes: tR,20to. In a previous work we have assumed that the cycle time is equal to the time between the moment when the concentration of the first component exceeds a threshold of 1 x M, until the concentration of the second component becomes smaller than this same threshold ( I O , 11). This second definition of the cycle time gives smaller values (i.e., the corresponding production rate is larger), which depends on the resolution between the two bands, Le., on most of the experimental parameters. Thus the results obtained with the two definitions can be slightly different, In practice, the cycle time will most often be intermediate between the values corresponding to the two definitions just given. Production Rate. The production rate is the amount of the purified fraction containing the corresponding component at the required degree of purity produced per unit time. The production rate is always referred to one of the components of the mixture. It is equal to the product of the throughput, the concentration of the corresponding component in the feed, and the yield. The production rate for component i is defined as follows (see Figure 1):
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989
p. =
nj - Ai -
9 0
tc
where (see also Glossary of terms at the end) n, is the amount of component i injected (Le., contained in the sample size). For i = 2, A , is the amount of component 2 that is eluted before the second cutting time and, accordingly, is lost for the production of the second component (see Figure I)
A2 = F ,
J::
a,
0 3
Cz d t 0
(D
For i = 1, A , is the amount of component 1 which is eluted after the first cutting time and is lost (if the intermediate fraction is wasted) or which has to be recycled
- 0
5 v
2
6s 0
where t B is the time when the first component concentration returns to 0 (Figure 1, point E). The production rate, P,, is the amount of the corresponding compound collected per unit time. In fact, the total amount collected for a fraction is n, - A, + R,, but the production rate of i is assessed a t its actual content in the corresponding fraction containing compound i and is given by eq 1. For i = 2, B z is the amount of the first component that is eluted after the second cutting time and, accordingly, is eluted with the second component, dilutes it, and contributes to increasing its production (see Figure 1)
B2 = F ,
stB C1 d t
tC.2
For i = 1, B1 is the amount of component 2 that is eluted before the first cutting point and is, therefore, collected with the first component and dilutes it
The concentrations c1and C, and the times t~ and tR,2 are given by the solution of the ideal model for two components, in the second region of the chromatogram (see next section, Figure 2 and Table 111). Recovery Yield. The recovery yield is the ratio between the amount of the component of interest that is collected in the product fraction and the amount injected in the column with the feed. It does not depend directly on the composition of the feed nor on the degree of purity a t which the products must be prepared. The recovery yield for component i is given by (see Figure I )
(4) Purity of a Component. The purity of a component, i, is the concentration of this component in the collected fraction. It is given by Pu, =
ni- A, ni- A i+ Bi
(5)
THEORY I. The Analytical Solution of the Ideal Model for a Binary Mixture. In a previous paper we have derived an analytical solution of the ideal model of chromatography for a binary mixture with a competitive Langmuir isotherm (12)
(see Glossary of terms a t the end). The analytical solution of the mathematical problem is composed of two concentration
3 h
0
0
0
3
Flgure 2. Schematics of the solution of the ideal model for a mixture of two compounds with competitive Langmuir isotherms. Case where the bands are partially resolved. The numbers refer to the equations in Tables 11-IV which give the relationships between the positions of the main features of the chromatograms and the characteristics of the chromatographic system, the column used, and the composition of the mixture separated.
shocks (14), which define three zones in the elution profile of an incompletely separated chromatogram and of five concentration curves (see Figure 2 ) . Equations are given in Tables I1 (third zone), I11 (second zone), and IV (first zone) for the times and the heights of the shocks and for the concentration profiles of the two components in the three zones. The first shock involves only the concentration of the first component, the concentration of the second component being naught before and after this first shock. The second shock involves the concentrations of both components. The concentration of the first one drops from a finite value to another finite one, while the concentration of the second component jumps from zero to a finite value (see Figure 2). Only pure mobile phase is eluted before the first shock, where the first component concentration jumps from zero to the maximum concentration of the first component band. Only a solution of pure component 1 is eluted between the two shocks. This constitutes the first zone, during which the concentration of the first component decreases monotonically. The second zone of the chromatogram begins at the second shock and lasts until the elution of the first component is completed. At the second shock, the concentration of the first eluted component drops to a value that is different from zero, if the separation is incomplete, A t the same time, the concentration of the second component jumps from zero to the maximum concentration of the second band. During the elution of the second zone of the chromatogram, the eluate is a mixture of the two components of variable composition. I t has been shown that the concentration of the solution of the second component in the mobile phase which leaks out of the mixed (second) zone of the chromatogram is constant as long as there is a mixed zone (12,15,16). So, the third zone, during which a solution of the pure second component in the mobile phase is eluted, begins with a Concentration plateau.
ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989
Table 111. Analytical Solution of the Ideal Model of Chromatography (Zone 2)
Table I. General Equations
Isotherms alcl Q1 = 1 + blCl + b2C2
Retention Time of the Second Shock (Beginning of Zone 2) (111-1) tR,2 = t p + t o + Y(tR,Z0 - t o ) (1 - Lfl")' End of the Profile of the First Component (End of Zone 2) (111-2) tB = t p + t o + (Y/ff)(tRJ0- t o )
a2C2 Q2 = 1 + blCl + b2C2 Loading Factor for the Mixed Band Lf = (1
+9
Band Profile of the Second Component 1
c2 = b2 + ablrl
L f , 2
Loading Factor for the Second Component
[(
tR,2°
- t0
ablrl
)1'2
-
(111-3)
' t - t, - to
Band Profile of the First Component
[
Y=
1279
)"'
1 tR,1° c1 = bl + b2/arl (1 a t - tp - to
+ b2
blrl + bz
-
1]
(111-4)
Table IV. Analytical Solution of the Ideal Model of Chromatography (Zone 1)
(rl is the positive root of eq 1-6) Loading Factor for the First Component
Retention Time of the Second Shock (End of Zone 1) (Iv-1) t R , 2 = t, + t o + Y(tR,2' - tO)(l Elution Profile of the First Component
Equation Describing the Interactions between the Two Band Profiles ablC20r2- ( a - 1 + ablClo - b2CZ0)r- b2C10= 0 (1-6)
t = t,+ t o +
I ' c
(tR,10
-
+
Table 11. Analytical Solution of the Ideal Model of Chromatography (Zone 3)
End of the Profile of the First Component (Beginning of Zone 3) (11-1) tB = tp + t o + (Y/ff)(tR,1°- t o ) Retention Time of the Second Component (End of Zone 3) (11-2) t E = tR.2' + t , Length of the Rear Plateau of Component 2 (11-3) Lit = tg' - t B = [y(Y - 1)/a2](tR,2' - to) Elution Profile of the Second Component after the End of the Rear Plateau
[(
c 2 = 1b ,
ttR,ZO - t , - -t ot0 )1'2
- 1]
(11-4)
Concentration of the Second Component on the Rear Plateau a-1
CH = b2 + ablrl
Eventually this plateau ends and the concentration of the third component starts to decrease monotonically and becomes zero at the limit retention time of this compound at infinite dilution. Each feature of the chromatogram can be calculated algebraically, except the retention time of the first front (see Tables 11-IV). The root rl of eq 1-6 is the positive one; in most cases it is approximately equal to Clo/CZo,Le., to the ratio of the amounts of the two compounds injected into the column. Accordingly, it is possible to derive algebraic expressions of the production rate and the yield as a function of (i) the characteristics of the chromatographic system studied, (ii) those of the column used, and (iii) those of the mixture investigated. Differentiation of these expressions permits the simple determination of the optimum conditions. We present here the results of this approach, which, as a first approximation, neglects the effect of column efficiency. 11. Purification S c h e m t . Obviously, since during the elution of the first and third zones of the chromatogram the
blC1)2
a-1 1 - L,2 a [ ( a - 1 ) / a + b1CJ2
.
(IV-2) Retention Time of the First Shock (Beginning of Zone 1) Is obtained by writing that the integral of eq 111-4 (profile of component 1 between the second shock at tR,? and the end of that profile at t B ) plus the integral of eq IV-2 from the first shock to the second one is equal to the amount of the first component injected in the sample. Concentration of the First Component on the Front Side of the Second Shock (IV-3) ((1 - & ) / a )+ L p 2 C1,A'
=
b l ( l - L:/')
mobile phase is a solution of the pure first and second component, respectively, the collection procedure will be based on the recovery of these two zones which contain 100% pure products. A part of the second, central zone will also be collected and added to the neighbor pure band. This addition increases the recovery yield of each component and its production rate. At the same time it decreases its purity. There is, therefore, a compromise to determine, depending on the aim of the purification scheme in the particular case considered. This compromise is defined by the cutting times, tc,l and tc,2. We have investigated four possible strategies for the preparation of purified fractions containing one of the components of the mixture: Strategy I. In the most general case, sample size is adjustable (i.e., it is one of the parameters to be optimized). Band overlap is allowed, and optimization is performed to maximize the production rate. Two cuts will be performed, between which a mixed band w ill be isolated and either wasted or recycled. Before the first cut a fraction containing component 1 at the required degree of purity is collected. Similarly, after the second cut, a fraction containing the second component a t a sufficient degree of purity is collected. The degree of purity and/or the yield can be considered either as variable or as parameter. Usually, the degree of purity is stated and the yield is calculated.
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989
Strategy 11. One hundred percent pure components 1 and 2 are recovered, by collecting only the first and the third zone of the chromatogram, respectively. In this case the yield is lower than with the former strategy, in exchange for the highest possible purity. In practice, because real columns have a finite efficiency, this is not a very realistic approach. Some compromise, Le., the acceptance of a purity lesser than 100% will be necessary. Strategy 111. Both 100% purity and 100% yield are required. The sample size is limited to the maximum size compatible with total resolution between the two bands a t column exit. This is the case considered by Knox and Pyper ( 7 ) .with the difference that we take into account here the fact that the two compounds interfere during their migration along the column, since if their bands are just resolved a t column outlet, they must overlap all the time during their migration. Strategy IV. Column overload is limited to the extent to which the two cuts corresponding to the selected degree of purity (see strategy I) coincide, Le., all the material injected is collected. This means in practice that the position of the unique cut depends on whether we are more interested in component 1 (in which case the cut time is equals to t B ) or in component 2 (cut time equal to tR,2). One component can be collected with a 100% yield, a t the degree of purity required. The other component is collected totally pure, but with a reduced yield. 111. Production Rate, Purity, and Recovery Yield of the Second Component. Strategy I. The collection of the fraction containing the purified component 2 will begin at the second collecting time, tc,,, before the end of the elution of the first component, a t tg. We can calculate the areas A, and Ri(eq 2 and 3, above) as a function of t,,> The concentrations of the two components during the second zone are given by eq 111-3 and 111-4 (see Table 111), where rl is the positive root of the second degree equation 1-6. tR,2 and t B are given by eq 111-1 and 111-2 (Table 111), respectively. Inserting these equations for the concentrations in eq 2 and 3 results in the following equation for the Durity of the second component of the binary mixture studied:
where tc,2is the second cut time, t p ,the width of the rectangular injection pulse, to,the dead time, and tR,,', the retention time of the second component a t infinite dilution. y is given by eq 1-4. The recovery yield and production rate are given bs
II\
ersely, by solving eq 7 for a certain, stated value of the
p u r i t \ (specitication of the final product), it is possible to (
alculate the collection time:
with (11)
in the new binary mixture prepared; usually Pu, is close to unity). The yield and production rate become
The recovery yield and the production rate depend on the required degree of purity through the value of x , given by eq 11. This result is discussed below. If we choose, as explained above, the cycle time as the corrected analytical retention time of the second component, tR,'' - t o ,we obtain a particularly simple result for the production rate
When the specified purity of the collected fraction containing the second component is known, we can calculate x and derive the collection time from eq 10 and the recovery yield and the production rate from eq 12 and 14, respectively. IV. Production Rate, Purity, and Recovery Yield of the Second Component. Strategies I1 to IV. In these particular cases (see above), the purity or the yield take some simple value. The production rate is easy to derive from the previous equations. In the case of the second strategy, the purity required for the second component is 1 (Le., 100% purity). Accordingly, x is equal to 0 (eq 11). The second collection time, the recovery yield, and the production rate become respectively tc,Z
=
(16)
In the case of the third strategy, we require no overlapping between the elution profiles of the two components, which provides for a recovery yield equal to unity. Again, we also have a purity unity, which means that x is 0 and the collection time is given by eq 15. But, whereas in the previous cases (strategies I and 11),the sample size is not defined, it is now given by the condition that R2 (eq 16) is unity, hence
The sample size can be derived from eq 1-3,by solving it for n2 and calculating the corresponding feed amount. In this case, the production rate is still given by eq 17. The difference between the strategies I1 and I11 is that the latter provides the optimum conditions under which the production of the second component should be run, since, when the sample size is increased beyond the value predicted by eq 18 (as strategy I1 would require) the production rate remains constant and the increased throughput results merely in a decreased recovery yield. Finally, in the fourth strategy we have a recovery yield equal to 1, a purity equal to a certain stated value, Pu,, and the collection time is equal to tR,2. Now. x is given by x = x m = ( 1-
where P u 2 is the required degree of purity of the second component (it.., the concentration of the second component
(15)
tB
P u2,,o r
j
ltL
(19)
ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989
The sample size, or rather the loading factor for the second component is given by (see eq 12)
Lfp2= 1 +
1 blrl/b2
[
a - l)/a ( 1 - x,
]
(20)
FvLf32
FV
b,
b2 + blr,
-
demand Pl = 1 and z = 0. Equation 22 gives tc,l
=
tR,2
(26)
which is obvious from Figures 1 or 2. The recovery yield and production rate in this case become, respectively
R1 =
and the production rate is
p2=--
1281
[
a - 1)/a ( 1 - x,
]
(21)
which is actually the same as eq 14. These equations are summarized in Table V. Comparison between eq 14 and 21 shows that strategy IV is better than strategy I. Zf a totally pure product is not demanded but the degree of purity of t h e final product must exceed a certain value, the highest production rate will be achieved when using the sample size derived from eq 20 and I-3. Increasing further the sample size (as done in strategy I) will not change the production rate of the second component (with our definition; actually it may decrease it with other definitions) but will result in a decrease of the recovery yield, which has no economic justification. V. Production Rate, Purity, and Recovery Yield of the First Component. Strategy I. The collection of the purified fraction of the first component involves recovery of the first zone of the chromatogram and a first part of the second one, until a first collection time determined by the specifications of purity or yield. The elution profile of the first component during the first zone is given in Table IV. The elution profiles of the two components during the second zone are given in Table 111. From these profiles, it is easy to calculate the integrals Aiand Biand to derive the equations for the purity, recovery yield, and production rate of the first component. Following the same procedure as in the previous sections, we obtain these equations, for the purity
and
In the third strategy, we demand no overlapping between the two bands, so both the purity of the collected fraction and their recovery yield are unity, for both compounds. Equation 26 is still valid, and the loading factor for the second component is given by eq 18 (see eq 1-5, Table I for the loading factor of the first component). The production rate for the first component is given by
Finally, in the case of the fourth strategy, the first collection time is equal to t g ,in order to permit collection of the first component with a 100% yield. The recovery yields are still unity, but not the purity. If the required purity of the first component in the corresponding fraction is F',,,, z becomes
1- P U l m 2,
=
Pu1,m
(30)
The maximum loading factor is equal to
and the production rate becomes
P, = F,
the recovery yield
1 [ ( a - 1)/a12 b2 + blrl C 2 0 / C 1 0- z,
(32)
T o the contrary of what was shown above in the case of the
and the production rate
with (25) Knowing the required degree of purity of the fraction containing the first component, we derive z from eq 25, then the first cutting point by solving eq 22, and, finally, the recovery yield and the production rate from eq 23 and 24, respectively. VI. Production Rate, Purity and Recovery Yield of the First Component. Strategies I1 to IV. We now want to impose certain values to the purity or yield. In the second strategy, we want to collect the first component pure. So we
production of the second component of the binary mixture, when we are interested in the production of the first component the strategies 111 and IV are not better than the strategies I1 and I, respectively. This result from the strong displacement effect of the second component band on the first one. Stretegy I now provides a higher production rate. As a matter of fact there does not seem to be an optimum sample size in this case. The maximum sample size will rather be determined by the requirement, which must be arbitrarily made on the lowest acceptable recovery yield. This in turn depends on the ease and cost of the reprocessing of the intermediate fractions collected. This conclusion is derived from calculations based on the ideal model. Because the column efficiency is finite, the degree of separation between bands which is observed is lesser than that predicted by the ideal model and, as a consequence, the production rate of material a t a certain degree of purity is somewhat lower than that predicted (see Figures 3 and 10). When the column is highly overloaded in order to achieve high production rate of component 1 with a low yield (hence a high degree of interference
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989
Table V. Equations Giving the Purity, Recovery Yield, and Production Rate of the Second Component
strategy
purity
recovery yield
production rate
specified
I
I1 111
100%
100%
Iv
specified
100%
P2 = F,
[ ( a- 1)/.12
bz + b f ,
--[
F, ~y - 1)/a P2 = b2 + burl ( 1 - x
l2
Table VI. Equations Giving the Purity, Recovery Yield, and Production Rate of the First Component
strategy
purity
recovery yield
I
specified
I1
100%
I11
100%
100%
IV
specified
100%
between bands), the predictive value of the ideal model falters and extreme caution is needed in extrapolating its conclusions. The various sets of equations permitting the optimization of the experimental conditions of a separation and the calculation of the maximum production rate are summarized in Tables V (second component) and VI (first component). We discuss the consequences of these results in the next section.
DISCUSSION I. Collection of the Completely Pure Second Component. In this section, we assume that we want to collect the second component at a 100% purity. In practice, a small amount of the first component will always be accepted, but the discussion of this first case presents some interest. First, it is striking that eq 17 gives the production rate of 100% pure second component, whether the column is merely overloaded just enough to achieve total resolution, which permits both a yield and a purity of 100% (strategy 111), or if the column is more strongly overloaded (strategy 11). Thus, if we want to prepare 100% pure second component, the third strategy is better, since the yield is 100% and the production cost, which depends on the throughput, is minimum. When the sample size is increased, the production rate increases proportionally to the throughput as long as the resolution between the two bands a t column outlet is total. When the two bands begin to interfere, the production rate remains constant. The optimum values of the loading factor for the second compound and of the sample size of the mixture are given by eq 18 and 1-3, respectively. We refer to the maximum loading factor compatible with the production of 100% pure second component with a recovery yield unity as the first critical loading factor. 11. Optimum Sample Size for the Collection of Highly Pure Second Component. If a certain amount of the first component is allowed as an impurity in the processed product (strategies I and IV), a degree of interference between the bands can be tolerated and the maximum production rate
production rate
increases. This increase takes place a t first without a loss in the recovery yield of the second component. When the sample size is increased beyond the first critical loading factor, the collected fraction contains an increasing amount of the first component. As long as the concentration of the second component in this fraction exceeds its stated degree of purity (specification), however, the amount of second component injected in the column can be entirely recovered. The second critical loading factor is reached when the collected fraction has just the required purity. Any further increase in the sample size results in the necessity to move the second collection time beyond the retention time of the second shock, in order to eliminate the excess of the first component. Thus, the yield begins to decrease. Equation 21 shows that the production rate depends on the purity accepted for the second component fraction and increases with decreasing degree of purity. However, the production rate does not depend on the sample size and will remain constant when the loading factor increases beyond the second critical loading factor corresponding to the required degree of purity for the product (cf. eq 14 or 21). In the same time, however, the recovery yield decreases. The second critical loading factor (eq 20) is thus the optimum loading factor. Figure 3 shows that the production rate of the purified second component increases linearly with the loading factor, Le., the sample size injected, until the second critical loading factor is reached, then remains constant. The maximum production rate increases rapidly with decreasing degree of purity, as predicted by eq 14. Figure 4 shows a plot of the recovery yield versus the reverse of the loading factor. As predicted by eq 12, under constant experimental conditions, when the loading factor is decreased, the recovery yield increases linearly with the reverse of the loading factor, until the reverse of the second critical loading factor is reached, a t which point the recovery yield becomes unity and stays so.
ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989
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Lf;: Flgure 3. Plot of the production rate for the second component (P2/Fv),versus L,,*, the loading factor for this component (eq 1-3): saki lines, eq 14; points calculated by using the semiideal model (77), with a 5000-plate column; relative retention, 1.2; b , = 2.5, b z = 3.0, k ,'O = 6; composition of the feed, 1:9; required purity of the fraction containing the second component, (1) 99.5 % , (2) 99 % , (3) 98 % .
It is remarkable that the results derived from the semiideal model (13,using a 5000 theoretical plate column, agree so closely with the results predicted by the ideal model (see the points on Figures 3 and 4). Of major significance is the fact that, although the maximum production rate decreases somewhat with decreasing column efficiency, the second critical loading factor seems to be essentially unaffected, a t least for efficiencies above 5000 theoretical plates. Even the decrease in production rate is moderate. With a column efficiency of 5000 plates, this decrease is only 7% for a required 99.5% purity of the second component, and 2% for a purity of 98%. The results described here are also in excellent agreement with those predicted by a previous study on the optimization of the experimental conditions for gas chromatography ( I 8). Finally, there are other definitions of the production rate, corresponding to different choices made for the cycle time (see above and ref 10). Figure 5 shows a plot of the production rate versus the loading factor, where the cycle time is defined as the time between the elution time of the first shock, tR,l, and the end of the chromatogram, tR,20. The production rate increases almost linearly with increasing loading factor at fiist, goes through a maximum, reaches the second critical loading factor, and then decreases rather rapidly. This decrease is due to the rise of the cycle time caused by the broadening of the elution band of the unresolved mixture. 111. Influence of the Selectivity on the Production Rate of the Second Component. Equations 14 and 17 show that the maximum production rate varies rapidly with the selectivity of the phase system. The selectivity of a chromatographic system for two compounds which exhibit a Langmuir-type adsorption behavior is independent of the concentration, since the selectivity is the ratio of the apparent adsorption coefficients, Qi/Ci. This ratio is of critical im-
20
30
40
50
60
70
80
90
100
l k 2 Flgure 4. Plot of the recovery yield of the second component versus the reverse of its loading factor, same experimental conditions as for Figure 3: solid lines, eq 12; points calculated by using the semiideal model ( 77), with a 5000-plate column. 0
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Lf2 Figure 5. Plot of the production rate of the second component of a binary mixture versus the loading factor, same conditions as for Figure 3, except the definition of the production rate, which uses fR.2' - fR,, as value of the cycle time: purity of the product (1) 99.9%, (2) 99.5%, (3)99%, (4) 98%.
portance for the achievement of a high production rate by preparative chromatography. It is as important in preparative
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ANALYTICAL CHEMISTRY, VOL. 61,NO. 1 1 , JUNE 1, 1989 0
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Figure 6. Variation of the maximum production rate of the second component with the selectivity of the chromatographic system, plot of P,IF, versus [(a- 1)/0(]~: same conditions as for Figure 3; 119 mixture, required purity (1) 100YO,(2)99YO,(3)98YO; 9/1 mixture, required purity (4) loo%, (5) 98% (the curve for a purity of 99% is between curves 4 and 5 and cannot be distinguished).
chromatography as it is in analytical applications. Figure 6 illustrates the influence of the selectivity on the production rate. The production rate increases very rapidly, in proportion to ( a - 1)2, for values of a lower than about 1.50. However, the variation becomes slower a t higher values of a; because the dependence of P2 (eq 14) on a is more complex than the explicit relationship (e.g., x is a function of a and bz is changed in the same time as u2, i.e. with cy, in order to keep constant the column saturation capacity for both components), the plots on Figure 6 are not linear. Their curvature depends on the exact numerical values chosen for the parameters. For example, if the ratio C1"/Cz0 is large, the term b2 + blr, = b, ( a r l ) varies very slowly with a , but it varies almost as cy if this ratio is small. The production rate tends toward an asymptotic limit a t large values of a. The contrast between the results obtained for the 1/9 and the 9 / 1 mixtures is striking (see Figure 6). In the former case, the production rate increases rapidly with decreasing required degree of purity. In the latter case, the production rate is almost unchanged by decreasing the required purity of the product from 100% to 98%. This can be understood by considering eq 14 and the definition of x (eq 11). rl is practically equal to the ratio of the concentrations of components 1 and 2 in the feed. If the first component is in large excess, the value of x will be very small. On the contrary, if the second component is the major one, x will not be much smaller than unity, even a t high degrees of purity. For example, with Pu, = 0.99, x will be 0.3 for a ratio Clo/Czo= 0.10 and only 0.03 for a value of this ratio of 10 ( a ca. 1.1). In the former case the production will be twice as large if we accept a 1% concentration of the first component than if we want a 10070pure second component. In the latter case, the increase will be only 6%.
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Physically, the explanation is related to the very strong change that takes place in the effect of the interaction between the two components when the composition of the feed changes. At high concentrations of the second component, the displacement effect of the second component on the first one is very strong. A decrease in the required purity of the second component permits an earlier opening of the collection valve for the second component, hence a larger production. On the contrary, a t low relative concentrations of the second component, the displacement effect vanishes; the second band tags along the first one, while the band of the first component tails just a little more than it would for a band of the same amount of first component pure. Accordingly, relaxing the purity requirement does not permit a significant change in the collection time nor an increase in the production rate. Such large differences between the conclusions derived from slightly different experimental situations make rather difficult the task of the separation chemist who cannot rely too heavily on past experience in the development of a new purification procedure. IV. Influence of the Composition of the Feed on the Production Rate of the Second Component of a Binary Mixture. Equations 14 and 17 show that the production rate of the second component increases with increasing concentration of this compound in the feed. The positive root rl of eq 1-6 is practically equal to the ratio C1'/CZo and decreases with increasing concentration of the second component. The production rate depends on rl for two reasons: explicitly first, through the denominator (b, + blrl), and implicitly, through the value of x (eq 11). The influence of the latter is very strong when rl is small and decreases a t large values of the relative concentration of the first component (see above). It is much easier to purify a feed from its impurities by liquid chromatography than to extract a minor component and prepare it a t a high degree of purity. This is a known fact in the field.
ANALYTICAL CHEMISTRY, VOL. 61,NO. 1 1 , JUNE 1, 1989
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(5)90%.
Equations 14 and 17 tell exactly how much easier it is. Figure 7 shows the variation of the production rate as a function of the concentration of the second component. As predicted by eq 21, the production rate increases linearly with the reverse of 1 + blC10/b2C20when the required purity is 100%. It is remarkable, however, that for lower required degrees of purity, the production rate increases very fast with increasing concentration of the second component (see eq 14, with x derived from equation 11). This arises from the very rapid increase of x with decreasing value of the relative concentration of the first component in the mixture (i.e,, of rl). V. Influence of the Required Degree of Purity on the Production Rate of the Second Component. Equations 14 and 17 show that the production rate of the second component increases rapidly with decreasing required degree of purity when the feed is rich in the second component. Figures 8 and 9 show plots of the production rate versus the reverse of (1 - x ) ~where , x is given by eq 11. The main plot on Figure 8 (1/9 mixture) is a very steep straight line. When total separation is required, x becomes equal to 0 and the production rate of 100% pure second component can be varied from 0 to its maximum (strategy 111),depending on the sample size injected. The production rate increases very rapidly with decreasing required degree of purity for this feed which has a large concentration of the second component. The plots on Figures 8 and 9 corresponding to the 9 / 1 mixture show much lower values of the production rate and a much lesser influence of the required degree of purity. This is due to the low concentration of the second component in this feed. The values predicted for the production rate depend very strongly on the composition of the feed, because the production rate depends on the reverse of b2 + blrl and on x which depends itself on rl, Le., practically, on the ratio
CIo/C2". It turns out that both dependences play in the same direction, so the production rate increases rapidly with decreasing value of rl. VI. Other Factors Influencing the Production Rate of the Second Component. Obviously, the production rate is proportional to the volume flow rate of the mobile phase through the column. There are two ways to increase F,, either by increasing the column diameter at constant mobile phase velocity or increasing the mobile phase velocity. The first method is classical. The second leads in practice to a decrease in the column efficiency. We have shown above (see Figures 3 and 4) that the predictions of the ideal model are in surprisingly good agreement with the numerical results derived from the semiideal model. This agreement, however, becomes less than satisfactory when the column efficiency is reduced. The ideal model is certainly not suited for an optimization of the mobile phase velocity or the column efficiency. The column efficiency is considered as infinite in the ideal model, and the separation between the bands of two compounds at infinite dilution is always achieved, provided only that the width of the injected pulse, t,, is smaller than the difference between the two limit retention times, i.e., ( a - l)kl'oto. The influence of a finite column efficiency on the results derived from a study of the ideal model will be discussed in a forthcoming paper (13). The influence of the column length can be derived from careful consideration of eq 9. If we keep constant the mobile phase velocity, in order to keep the value of the plate height constant (but, within the framework of the ideal model, this concept is not very clear), the cycle time increases in proportion to the column length. The production rate remains constant, a result that was already reported previously (18). If the mobile phase velocity is increased, in order to keep the
Figure 9. Variation of the production rate of the second component with the degree of purity, plot of P,IF, versus the reverse of (1 - x ) ~ : same conditions as for Figure 3; case of a 911 mixture (10% component 2 in the feed); the numbers on the line indicate the corresponding degree of purity of the product, under the experimental conditions described Figure 3, (1)loo%, (2)99%, (3)98%, (4)95%,
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The points on Figure 10 show results obtained under the same experimental conditions by calculations made by using the semiideal model, with a 5000 theoretical plate column. There is excellent agreement between the predictions of both the ideal and the semiideal models at low values of the loading factor for the first component. At large values, however, an increasingly large discrepancy appears, the production rate achieved with the actual column increasing more and more slowly than the production rate predicted by the ideal model, which is expected.
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actual column efficiency constant, the cycle time increases less rapidly than the column length and the production rate increases. Obviously, the ideal model does not permit any definite conclusion on this point. A detailed discussion of the optimization of the column characteristics, including its length, will be presented in a forthcoming paper (13). VII. Collection of the Purified First Component. The comparison between eq 28 and 29, corresponding to the second and third strategies, respectively, i.e., to a 100% purity with a yield of 100% (strategy 111) or lower (strategy 11))shows that the production rate of the first component increases constantly with increasing loading factor; hence strategy I1 is better. If we compare now eq 24 and 32, we see the production rate given by eq 24 is the larger. Strategy I appears to be better than strategy IV. The gain in production rate is moderate, however, and is paid for by a serious decrease in the recovery yield. The final choice will depend on the value of the recovery yield that can be afforded. This is illustrated by Figure 10. This phenomenon is due to the displacement effect of the first component by the second one. Increasing the size of the feed sample injected in the column increases the intensity of the displacement effect and permits an increase in the production rate. Equation 32 shows that the influence of the required purity on the production rate decreases when the relative concentration of the second component increases. This is also due to the displacement effect of the band of the second component on that of the first one. Equation 32 shows also the production rate to be inversely proportional to CZof Cl0 - z . Thus, the required degree of purity may have an influence on the production rate only if C20/C10is not very large compared to the maximum concentration accepted for the second component in the purified first one (i.e., 1 - Pu,). As long as the concentration ratio exceeds 0.1, the influence of the required degree of purity remains practically negligible.
CONCLUSION The optimum experimental conditions for maximum production rate of the components of a binary mixture, with a certain required purity, with or without a minimum yield requirement depend very much on whether one is more interested in the f i t or in the second component of the mixture. For the first component, there is a constant increase in the production rate with increasing sample size. Beyond a certain value of the loading factor, the recovery yield becomes less than total and decreases with increasing loading factors. Since the production rate keeps increasing, however, there might be reasons to accept a moderate yield and an economic optimum must be looked for. On the contrary, the production rate of the second component reaches a maximum when the recovery yield begins to fall below 100% and remains constant (or even decreases, depending on the exact definition of the production rate chosen) for higher values of the loading factor. In this case, there is no incentive to use large sample sizes. Equation 20 gives the exact value of the optimum sample size for maximum production of the second component of a binary mixture at a certain stated degree of purity. T h e sample size predicted by this equation is independent of the column efficiency within a very wide range (Figure 3 and ref 13). Larger sample sizes result in a constant or decreasing production rate and a decrease in the recovery yield. When the relative concentration of the second component is very high, relaxing the purity requirement for the second component permits a large increase in the production rate (e.g. the production can be multiplied by 2 and 3.3 when the required purity is lowered from 100% to 99% or 9870,respectively. In other cases, e.g., when the concentration of the second component in the feed is low, or when the first component is concerned, the purity requirement has little or almost no effect on the production rate (see Figures 6 to 9). There lies the basic justification in column overloading for the purification of a major component from impurities eluted before it. When the impurities are eluted after it, only a change in chromatographic system, leading to an inversion in the elution order or an important increase in the relative retention, may permit an increase in the production rate. The results of this work illustrate the difficulties encountered by an entirely empirical approach to the optimization of the column design and operation parameters. A theoretical framework is necessary to guide the experimentalist. Unfortunately, the ideal model cannot provide all the necessary answers and a more sophisticated approach is necessary, especially to deal with those experimental parameters that control the column efficiency. This will be the topic of a forthcoming paper (13).
GLOSSARY amount of component i found in the part of zone 2 that is rejected (eq 2) coefficient in the Langmuir isotherm (eq 6) ai amount of component i that is collected with the Bi other component and reduces its purity (eq 3) coefficient in the Langmuir isotherm (eq 6) b, concentration of component i in the mobile phase Ci CIo, CZo concentrations of the two components in the sample pulse
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989
volume flow rate of mobile phase column capacity factor at infinite dilution for the second component column length loading factor, defined by eq 1-2 (Table I) loading factor for the component i amounts of the two components in the sample pulse production rate of component i purity of component i amount of component i adsorbed at equilibrium with the concentrations C, and C2of the first and second components in the mobile phase recovery yield for component i positive root of eq 1-6 (Table I) column cross section area end of the first component elution profile retention time of the maximum concentration of the elution band of the second component retention time of the second component at infinite dilution (Le., limit at zero sample size) cycle time first and second cutting times, when the collection of the first fraction is ended and the collection of the second fraction begins, respectively retention time of an unretained compound (dead time) width of the iniected band (in time unit) reduced va1ues"of the requir'ed purity (eq 11 and 25, respectively)
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selectivity, Le. ratio k i 0 / k l ' O = a2/al ratio (cublrl + bz)/(blrl+ b2),used to simplify the writing of equations volume void fraction of the column
LITERATURE CITED Guiochon, G.; Kattl. A. Chromatographia 1987, 2 4 , 165. Bidlingmeyer. B. A. Preparative Liquid Chromatography; Journal of Chromatography Library 38; Elsevier: Amsterdam, 1987. Scott, R. P. W.;Kucera, P. J . Chromatogr. 1976, 719, 461. Coq, B.; Cretier. G.; Gonnet, C.; Rocca, J. L. Anal. Chem. 1982, 5 4 , 2277. Personnaz, L.; Gareil, P. Sep. Scl. Techno/. 1981, 16, 135. Gareil, P., Ph.D. Thesis, Universite Pierre et Marie Curie, Paris, 1983. Knox, J. H.: Pyper, H. M. J . Chromatogra. 1988, 363, 1. Guiochon, G.: Colin, H. Chromafogr. Forum 1986, 7(3). Eble, J. E.; Grob, R. L.: Antle, P. E.; Snyder, L. R. J . Chromafogr. 1987, 384, 2 5 . Ghodbane, S . ; Guiochon, G. Chromatographia 1988, 2 6 , 53. Katti, A,; Guiochon, G. Anal. Chem. in press. Golshan-Shirazi, S.;Guiochon, G. J . Phys, Chem ., in press. Golshan-Shirazi, S . : Guiochon, G. Anal. Chem. Chem., in press. Lin, B.; Golshan-Shirazi. S.;Ma, Z.; Guiochon, G. Anal. Chem. 1988, 6 0 , 2647. Glueckauf. E. Proc. R. SOC.London, A 1946, A186, 35. Helfferich, F.; Klein, G. Mufiicornponent Chromatography-Theory of Interface; Marcel Dekker: New York. 1970. Guiochon, G.; Ghodbane, S . J . Phys. Chem. 1988, 92, 3682. Guiochon, G.; Jacob, L.: Valentin, P. Chromatographia 1971, 4 , 6.
RECEIVED for review November 9,1988. Accepted March 10, 1989.
