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Simplified analysis of chromatographic-column dynamics
SIMPLIFIED
ANALYSIS
OF CHROMATOGRAPHIC-COLUMN DYNAMICS
Dipartimcntfl
MASSIMILIANO di Ingegneria Cbimica,
GIONA and ALESSANDRA Facolti di Ingegneria. UniversitP 00184 Ruma. Italy
Dipartimanto
di lngegneria
DIEGO BARBA FncoltB di lngegneria Univcrsiti 67040 L’Aquila. Italy
Chimica,
ADROVERt “La Sapietiea”.
dctl’Aquila.
Via Eudossiana
Monte
18.
Luco di Rob.
and
Consorzio
di Ricerche
Appticate
DANIELA SPERA alle Biotecnologie, Strada Provinciale
22, 67051 Avezzano.
Italy
Absbac-A sirnplihed analysis of chrumatsgraphic~dumn dynamics based oa a similatity equation (found ex&mentally) for outlet chromatogramv of size-exclusion columns is developed. A theoretical explanation OI the simiIarity equations is developed in the cam ala convectionxliffusion model. A critekn for validatina the similaritv results is aroacwd. As a corollary of the above analysis, an integral method for
molecular
INTRODUCTION
size,
Together with transport
in packed beds, chromatography is one of the most highly developed topics in chemical engineering, both From the chzmical point OF view as a laboratory tool (Yau et al., 1979). and in press development Tar the separation and purificntion of biomol~ules (Yau. 1974; Anspach cl ab. 1988; Davies, 1989; Pyle, 1990). Such a huge field of research can be approached in many different ways: from the mathematics of dispersion in packed beds (Aris and Amundsou. 1957; Whitakat, 1967; Bhattaeharya and Gupta, 1983); from the physics of solute transport in porous packings (Deen, 1987); from the analysis of chromatographic columns as separation units (Pyle, 1990x from detailed study of the port network and its influence in transport (Larson et al., 1981; Rasmuson, 1985; Reges and Jensen, 1935; Shah and Ottino, 1986; Reyes and Iglesia, 1991). In this article, we consider exclusively HPLC columns in which the diffusional partition between the mobile and the stationary phases depends on sizeexclusion
effects
(Determann,
1969).
Tn site-exclusion
chromatography (SEC), the diffusion coafit-ient depends on the steric interaction hectween solute molecules and porous matrix and deviates from the Stoke-Einstein diffusivity by a factor G(L) which dcpends exclusively on the ratio *I between the radius of the solute molecule and the pore radius, A = rs/rp. D = &G(I).
(11
The function G(A), known as the effective diffusion accessibility factor) figures in the- expression for the retention time r. u a function of
coefficient (OF
TAuthor to whom correspondence should be addressed.
since the equivalent void factor of a molecule written as #(4
= #o f (1 - $o)x@4
can be
(31
where & is the column void factor and x the internal porosity. A theoretic;il and experimental analysJs of the functional Form of G can be found in Determann (1969) and Deen (1987). The thrust of this article is twofold: to present a simplified characterization of solute transport in SEC columns, and to derive some analytical implications useful for an analysis of dispersion. The key point of the analysis lies in the expetimental evidence for an invariant principle vatid for the outlet conccntrarion profile in the case of impulse input: an invariant concentration profile characteristic of each biomolecule can k introduced indepndently of the eluent flow rate. Starting from the mathematical formulation of the invariant principle, it is possible to develop an integral method, alternative to moment analysis, For the evaluation of dispersion parameters. Some criteria for testing the validity of the invariant formulation arc developed. EXPERIMENTALSECTION
The columns used in performing the experiments were as fo116ws: Bio-SiI TSK 250 (packing G 3000 SW, avera=ge particle site 10 pm, average pore radius rp = 125 A) and Bio-SilTSK 125 (packing G 2ooO SW; avera particle size 10 m, average pore radius rp = 625 A” ) from Toyo-Soda (L = 60 cm, 541
542
MASSIMILIANCI GIONA at al.
S = 0.44 cm’)+ The elution solutions were 0.1 M Ns,SO,, 0.1 M NaW,PO* and 0.02% sodium aide (w/v) for the TSK 250 column; 0.05 M Na2S0&, 0.02 M NaH,PO, and 0.02% sodium tide (w/v) for the TSK 125 column. The eluent was adjusted at pH 6.8 by using an NaOH solution. The processing apparatus consisted of a twin-headed reciprocating pump, Waters model 510, a selectable-wavelength UV detector (481 Lambda Max), and a Rheodyne injector, model 7125, purchased from Millipore UK: The protein concentration varied from 0.1 to XI mg/ml and the injection volume from 20 lo 40 ~1. Al the experiments were perform& at 25°C. The biomoleeules considered were: thyroglobulin from bovine (669 kDa), apoferritin from horse spleen (443 kDa), ovalbumin from chicken egg (a kDa), myoglobin from horse (17 kDa) and cyanocobalamin (1.35 kDa), supplied by Sigma, PooIe U.K.
INVARIANT
PAlTERN
For large PeGlet numbers, the behaviour of the Green function can bz obtained by similarity analysis. More prtisely, st<, r) attains the simplified Form
where
Tn other words, it can be expressad in terms of a unique function 41v of the normalized dimensionless time 8. To prove eq. (4) and obtain the expression for the invariant function 4N, it is sufficient to substitute it inta eq. (5) and take into account the expression of d(i) given by eq. (8). After some atgebraic manipulation, eq. (5) can he rewritten as
FOR COhXECl’ION-DIFFUSION MODELS
In many problems related ta tubular reactor dynamics, chromatographic columns and dispersion in packed beds, the macrosr;opic fluid dynamics of the system (hereafter called for simplicity ccrlramn)involves the solution of a PDE of the convection-diffusion type:
By considering an infinite model OC column, x: E (0, co), the dynarni~ of transport inside the column is s@fied once the Green function g(& z) is known. The assumption of the infinite length of the column can be jusaed by considering that in our experimental apparatus the ratio between tbc length and the diameter of the column is L/a = 80 and the experimental conditions are such that Pe = I@-104. The Green function y(c, t) satisfies, in dimensianless fotm, the differential scheme (5)
s(C, 0) = 0
(6)
and with the Dirac delta function imposed at < = 0, SIO, T) = 6(r).
(7)
The function g(<. 7) is properly a Green function since the response to any inlet profile c,(r) is given by the mnvolution of Q(<, z) with c&l. The variance o’(t) for a value < of the dimensionless axial coordinate can k obtained from eqs (S)-(7) by moment analysis and is given by
In the liiit of PB -* CIS(i.e. in the fimit of Q tending to zero), 4. [I 1) can be simplified by considering the most significant terms, which prove to be those of order I/u and which cartespond to the terms in eq. (1 I) that are multiplied by the factor l/((a3Pc) once the detivatives of ~(5) are made explicit. In this way, one obtains an equation for the invariant fun&on 4,(B) of the form d24, _+fl% dB=
dir
“!-4N==O
which belongs to the class of integral-error-function equations (Zwillinger, 1989). The boundary conditions to be imposed on 4w are 4~~( & M) = 0 and d4,(O)/dB = 0. The solution of eq. (12) satisfying these boundary conditions, and whose integral is normalized to unity, is given by Abramowih and Steguu (1970):
The invariant principle expred by eq. (9) CB~ be stated alternatiwly by asserting that the function u(& [<, < + a(QB] is independent of c in the limit of large Peclet numbers. Figure 1 shows the behaviour 4+,(fi = u(l)g[i, 1 f 41)/?] vs fl for increasing values of Pe, together with the asymptotic limit ex-
Simplified analysis of chromatographic-column
1
0.5
am 0.4
dependent
0.1
2.0
0.0
d 4.0
B
invariant
a(l)fl YS p tar increasing values of Peclet nurnbe~~Fe = 10, 50, 100. 200. With the mceasing
Fig. t. GP, = a(t)g[l.
1+
of Pe, (pp, tends to the asymptotic
bchaviour (13) (-,
).
pression
(13). It can be noted that, even Tar P&et numbers of the order 200, the asymptotic expression (13) gives a fairly good approximation of the correct behavinur of the outlet concentration. To sum up, the macroscopic fluid dynamics of a tubular reactor with dispersion, as well as other Iransport situations described by means OF a convection-diffusion equation, are satisfactorily described by means of a unique invariant Green function for large Pe. This implies that a single ordinary differential equation in the limit of large P.c is wholly equivalent to the partial differential model expressed by eq. (4), It should he noted that the similarity principle expressed by eq. (9) is somewhat different from the constant pattern profile developed for adsorption columns by Garg and Ruthven (1975). In the latter case, the asymptotic constant pattern behaviour is expressed exclusively as a function of a shifted axiai coordinate by taking into account the macroscopic solute velocity. In the next section WC show how eq. (9) holds true for SEC data. INVARIAYT
PROE‘ILE
IN SEC
of the retention
time and depends
excIus-
the column and on the nature of the solute molecule. Figure Z(a) and (b) shows two normalid chromatograms obtained according to eq. (IS) from experimental data on TSK 125 and TSK 250 columns for an eluent flow rate in the range of 0.1, I.0 ml/min. Analogous invatiant outlet profdes were obtained For ail the biomolecules wnsidered. From these data, the invariant relation (14) seems to hold with sufficient accuracy. It is important to state in passing that the retention time t, introduced in eq. (14), corresponds to the peak retention time (Yau et al., 1979). i.e. the time instant corresponding to the peak of the outlet chromatogram. According to the points discussed in the case of the
0.2
-2.0
543
iveIy on
0.3
0.0 -4.0
dynamics
scheme,
formulation the
best
of
a
convection-diffusion
approximation
for
the
invariant
Green function is achieved for the higher eluenr flow rates. It is interesting to observe that the invariance of the outlet chromatograms seems to hold true even in 1.2
,
.:ii\__ -2 0.50
0
2
4
=CYa -
1
cy
5
I-
(b)
COLUMNS
Turning our attention to SEC columns, let u(t, t.) be the outlet concentration profile from a pulse injection associated with a given value of the retention time [associated via eq. (2) with the eluent flow rate]. Its integral with respect to the time r is normalized to unity. The similarity principle, equivalent to eq. (9), attains the form c(:, r,) = t
dh)
t- t, I
aJpJ [ 00,)
where 0 = ~{t,) depends on the retention tion (14) -an be put in equivalent form @H(Z) = art*)cCfr
+ a(t
r,7.
(14) time. Equa(15)
Equation (15) states that given an output chromatogram c,(t, r,), tbe Function b(t,) c[r, + a&) z, r,] is in-
CES 49:4-H
-5
-2.5
0.0
2.5
Fig. 2. Behaviour of the normalized dimensionless chromatogram dr, as a function of LI= it - t.#a(t.) for elutian Row rates varying from 0.1 to 1.0 ml/tnifi. (a] Thymglobulin on a TSK 250 column; (b) cyanocobalamin on
a TSK
125 colulnn.
MASSIMLL~ANCI
244
those cases in which the outlet chromafogram is intrinsically asymmetric and deviates strongly frOm being Gaussian. This implies that eq. (14) Seems to hold true not Only for Gaussian-shaped profiles of simple convection-diffusion models, but represents a more general feature of a broader class of trtinsport phenomena in packed beds. CBlTF,RlAOF YALIDITY OF THE INVARIANCE The validity of the similarity equation (15) requires analytical testing. Consistency criteria for the invariant representation, eqs (14) and (15)*can k developed from the analysis of the moments {JM,,_~~,] of the invariant function @Jrw)
Indeed,
and therefore the first- and secpnd-order .moments reduce to
= 1 + CM,,(,)Y where {M,) are the moments of c(t, L,). The last equation implies fl;> = 1.
(191
Equation (19) should be satisfied independently of the shape Of the invariant Green function. Equation (19) constitutes a simple criterion for testing the validity of the similarity equation (14). In the application of this criterion to experimental data, it is important to note that the experimental outlet chromalogram should converge to the invariant Green function in the limit of high eluent flow rates. Figure 3 shows the behaviour of u& a5 a function Of the eluent flow rate FO. As
1.4 I
41r 1.21
r-- .---
d) thyreglobulin a) ap&srrllln bl myoalabln c) cyanocobrlamln
al walbumln
= c-L-\.__ b t’-__ E
1.0
l
0-8
*-
-s---._
_,___ d
T/-c-
---__ _----c-____ -&_
cc al.
expected; u& converges to unity as the elution flow rate increases, even if the convergence depends On the solute. In the case of the ideal model represented by a c0~vectiO~-d~lMiOn equation, u& is prOpOrtiona to l/P*. The behaviour of 03, as a function of FO (or equivalently of Fe) can be analgsed in order to use the invariant principle as constraints to be imposed on the functional form of multiphase transport models of chromatographic columns. This topic rcquircs further investigatiOn+ For the purpose Ofthe present work, the data in Figure 3 can be regarded as a strong indication’of the validity of the invariance hypothesis.
FUNCTIONAL
PREDICHON
OF
DlSPERSlON
The similarity equations (14) and (15) not only simplify the treatment of chromatographic column dynamics, since a single function of a normalized time 3~= (r - r,)/u(t,) is representative for all the flow conditions. but also enable us to formulate alternative methods of analysis of chromatographic data. A consequence of this is related to the analysis of solute dispersion. It is known that the evaluation of dispwsion from moment analysis is highly sensitive on the cut (i.e. on the truncation point) in the outlet chromatogram (Yau es tit., 1979). Let us assume that aq. (15) holds true, and let cco,Ct, b,oJ cc,,Ct. 5r1, 1 be two ChrOmatograms with the respective retention times t+,,? t,(1) and variances u(“o,,o(:, . Let us further assume that the variance O& is known and the variance O:~, is to be determined. According to the invariant equation (15), the integral Its> =
s
i+lrC&cll
+ W t&l
+ ~(0, m, trd2
- flW?o>Ct,(o)
da
(20)
is idcnticaIIy equal to zero for’the value of the parameter Y equal to D,,~_Allowing for experimental error, it is convenient to define fiGI, as the value of the parameter s which minimizes l(s). By developing I(s) and putting s, = sfeaco,, one obtains I($,) = J,,(s,)s,Z
- 211 ,I&>& + Ire
’
_-zc.-_ -_-----k-G -.
_-_-~;~-T=--= ,_-_/a--
CWNA
./‘-
(a)
0.6 0.1
0.3
0.5
0.7
o~gFJml/min~~l
Fig. 3. t$, YS Fv rc~otthe biomnlecules
exwidered:
(a) TSK !25 cotumn;
(b)
TSK
250
column.
(21)
Simplifiedanalysisof chromzttographic-column dynamics
545
where the integrals appearing in eq. (21) read as
sitnutions, the integration limits are not 0 aad ca but PI’ = min (L m, L&j and r(l) = max I’M, ~MD}, t,0 and rMO Mng the termination points of the reference chromatogram ctO,Et, I,(~) 1. The evaluation of u by means of the integral equation (26) is more robust than the calculation of u directly from its definition. This GUI be intuitively explained by observing that the right-hand side of eq. (26) corresponds to a normalited correlation integral of +,) and cILj which proto b less sensitive to the trunation points than the integral of t”c,,,(t, t*,).This aspect can be also checked from the behaviour of the sensitivity function
From definition (22), by taking into account Schwa= inequality (Stakgold. 1979). it follows that
of 0~1, with respect to fM, Fig. 5(b). A theoretical analysis of the sensitivity of the integral method represented by eq. (26) with respect to the termination points could also be carried out, but is left for further investigation because it is of secondary physical interest in the analysis. Of course, eq. (26) is valid as long as eqs (14) and (151 hoId true. ConsequerMy, future attention should
P I 1WI2
G 12012
I (%I
(241
and therefore the only real solution ofeq. (24) is given
by the value of 8. which nullifies the expression under square root. Cr,,(s.)la = Jd2iIS,). Consequantly, the ratio 4 = us,,/+, is given by
which can be further simplified as
1x
be addresd to both experimental and theoretical analysis of the physica vatidity of eqs (14) and (15) in other transport conditions on chromatograpbhic WIumns.
Q s
; h,C~,h,l~“d~ (26)
Therefore, given one chromatogram of known variance and another chromatogrsm whose variance is to k estimati. the corresponding value of a, can be obtained through eq. (262 In practice, u, can be evaluated, e.g. a least-squares fit. In order to apply the functional method discussed above to a practical example. let us consider the inffuence of terminatio,a points &, tM, on the estimate ofdispersion, Fig, 4. This topic has been developed by Skopp (1984) in terms of an optimal estimator, by means of weighted moments. and has be~?ndiscussed by de Lass et ai. (1986) in the case of adsorption in packed beds. Figure 4 shows the expetimental chromatogram of thyroglobulin on the TSK 250 column (r&tad upon normalization and elimination of the instrumental bias to cgl) [r, r.,,,] > as measured by the UV detector for an eluenf flow rafe F0 = 0.8 ml/min. Let us assume as reference chromatogram, ctO,[t, rrlolJ, the correspending chromatogram for F,, 9 1.0 mt,bnin. Figure 5(a) shows the estimate of btl, as a function of tar, by taking fixed I, = 11.5 min. as obtained from eq. (26) and from the direct evaluation of the variance from its definition+ Of purse. in applying q. (24) to practica1
INFLUENCE
OF SOLUTE QUANTrTiES
In order fo analyse the influence of the solute concentration on the invariant structure of the outlet chromatogram, experiments were performed for diff-t biomolecules with molecular weights ranging C,lVl 0.90
I,i
0.20
0.10
Fig. 4. Experimental outlet pr&te (measured in Y by the WV detector) as a function of time for thyrdglobulin on the TSK 250 colunln. Fo = 0.8 ml/tin. The meaning or the rcrIninalivn points c,, LHm pointed out in the text.
-I-_--
a) b)
--
-
from ea.
IPB) from the deflnltbm ___A---
b
/----
0.08
.
---b
-.I
Q.41 1 l.6
_
16.6
16.6
17.6t, tminl
Fig. 6. d
w injection solute concentration y. for sweral
biomolecules
1
(TSK
125 column); F, = 1.0 d/mh.
a) from w. (26) b3 from the deflnltlon
ti4.S
16.5
161
17-5 t,
[mid
Fig. 5. (a) Behariuut ofqll(fdd) VBthe termination point zM. (b) Sensitivity function Ss[tMa)vs 6M for the cuwes of Fig. W.
frm 1 to 700 kDa, keeping the elution volume (33 4) constant and varying the inlet mncentration in the range 0.1, 20.0 mg/ml. It was observed that, for each biomolecule, the outlet chromatogram at different levels of concentration practically mincide and the variance square root is mnstant for the considered concentration range, Fig. 6. The invariance behaviout observed in the outlet chromatogram is therefore independent of the concentration. Tt should also be noted that the concentration range considered is larger than those typical of analytical chromatography and goes beyond the range in which preparative cdumns usually work. Consequently, the independence of the outlet chromatogram from concentration GUI be a valid assumption in the design of industrial separation units for the purification of biomolecules in SEC columns. Experiments were also performed to analyse the influence of solute quantities on dispersion, k-in& the concentration (20 mg/ml) constant and varying the elution volume in the range 20,4O ~1, Even in this case, the variant and the invariant behaviour of the outlet chromatograms are independent of the injection volume.
This article shows a simplified way of treating chromatographic data base&on the invariance analytically expressed by eqs (14) and (15). The method
discussed leads to severat intertAng masequences in the analysis of dispersion and can be further applied to the analysis and control of the separation performance of SEC cAumns. These topics will be discussed elsewhere. Several opm questions remains for further theoretial analysis_ In particular, It is important that the solute transport should be studied in more detail to obtain better understanding ol the physical nature involved in the experimentally found invariant principle. and consequently to derive the range of validity and the physical limitations of eqs (14) and (15).
outlet concentration profile starting from an impulsive injection solute concentration distribution column diameter, cm diffusion cweffident. cmZ/min Stokes-Einstein diffusion coefficient, cm2/min eluent flow rate, ml/min Green function of convactionafhision scheme {or acx&.seffective diffusion cdlitient ibility factor), dimensionless integrals defined by eq. (17) length, cm tith-order moment of c(t, r,) nth-order moment of the invariant function mrv Pec!et number average pore radius solute pore radius column section, cm2 sensitivity hmction defined by eq. (22) time, min termination point in the evaluation 01 outlet chromatograms, min peak retention time, min solute velocity, em/min axial coordinate, cm
column
Simplilkd
analysis
of chromatogmphic-column
Greek letters
; Y s :: fl2 fl:,
dimensionless normalized time (r - 5)IcIO injection solute concentration, m&n1 firac delta function r&P x/L variance of the outlet chromatogram invariant variance M,,+, - (n*i,+,,#
dynamics
547
the moment method technique for the definition of adsorption parameters in a packed bed. Chem. Eww Sci. 41, 123>1242. Determann, H., 1969, Ge/ Chromatography. Springer, Berlin. Gar& D. R. and Ruthven. I). M., 1975. Performance of m&c&r
sieve adsorption
columns:
combined
elkzta
of
mass fransfer and longitudinal difhtaion. Chut. Engng Sci. 30, 1192-f 194. Larson, R. G, &riven. L. E. and Davis, H. T., 1981, Pcrcolaticu theory of two phase flow in poreus media. Chem. Engng Sci. x 5673.
“L/V
equivalent
%0
void factor void factor invariant Green function graphic response internal porosity internal
of chromato-
Abramowie M. and Stegun, I. A, lP70, Handbook olMathmrical Functions. Dover, New York. Anspach, B., Gkrlich, H. U. and Ungcr, K. K., 1988, Cornparative study of Zorbax Bio Series GF 250 and GF 450 and TSK-GEL Moo and SWXL caluin sire-exclusion chromatography of proteins. 1. Chrnmot. 44% 4554 Aris, R and Aruuudson, N. R., 1957, Some remarks on longitudinal mixing or diffusion in lixed beds. A.1.Ch.E. J. 3.280-282. Bhattacharya, R. N. and Gupta, V. K.. 1983, A theoretical explanation of solute diswrsion in saturated porous media at the Druey scale. Water Remrces Res. 19. 93%944. Davies, P. A., 1989, Determination ofdiffusion ce&&uts of proteins in beaded agarose by gel filtration. J. CRrau~ut. 433,22t-237. Deen, W. M., lQ87, Hindered transport of large molecules in liquid-filled pores. A.lLXE. J. 33, 140%14X. de Lass, H., Hazlctt, J. and Fuller, 0. M., 1986, Evaluation of
Rasmuson, A., 1985, The effect of particles of variable size, shape and properties on the dynamics of fixed beds. Chsm Engng Sci. Qo, 621429. Reyes, S. C. and Iglesia, E., 1991, Monte Carlo simulation$ of structural properties of packed beds. Chem Engng Sci. 46, 108~1099. Reyes, S. and Jensen, K. F., 1985, Estimation of effective transport cocfl%knts in porous solids based on percolation concepts. Chem. Enptg Sci. 90, 1723-1734. Shah, N. and Ottino, J. M., 1986, Effective transport properties of disordered multi-phase composites: application of real-space rertormalization group theory. Ckeut. Eagug Sci. 41, 283296. Skopp, J., 1984, Estimation of true moments from truncated data. A.1.Ch.E. J. 30, 151-155. Stakgold, I., 1979, G~.+II’s Functions ud Boundary Value Problems. Wiley, New York, Yau, W. W., Kirkland. J. J., Bly, D. D. and Stoklosa, H. J., 1976, Effect of cdumu performance on the aeeuraey of molecular weisrhts obtained from size exclusion chromatography). J. chrernatography (gel pwneation Chromur. 125,219-230. Yau, W. W., Kirkland, J. J. and Bly. D. D.. 1979. Modera Size-Exchsion Liquid Chrom~mgm~&u. Wiley, T&u York. Whitaker, S., 1967, Diffusion and dispersion in porous media. A.1.CL.E. J. 13, 420-427. Zwillinger, D., 1989, Ham&ok of Diflermtial Epuatims. Academic Press, New York.
ANALYSIS
OF CHROMATOGRAPHIC-COLUMN DYNAMICS
Dipartimcntfl
MASSIMILIANO di Ingegneria Cbimica,
GIONA and ALESSANDRA Facolti di Ingegneria. UniversitP 00184 Ruma. Italy
Dipartimanto
di lngegneria
DIEGO BARBA FncoltB di lngegneria Univcrsiti 67040 L’Aquila. Italy
Chimica,
ADROVERt “La Sapietiea”.
dctl’Aquila.
Via Eudossiana
Monte
18.
Luco di Rob.
and
Consorzio
di Ricerche
Appticate
DANIELA SPERA alle Biotecnologie, Strada Provinciale
22, 67051 Avezzano.
Italy
Absbac-A sirnplihed analysis of chrumatsgraphic~dumn dynamics based oa a similatity equation (found ex&mentally) for outlet chromatogramv of size-exclusion columns is developed. A theoretical explanation OI the simiIarity equations is developed in the cam ala convectionxliffusion model. A critekn for validatina the similaritv results is aroacwd. As a corollary of the above analysis, an integral method for
molecular
INTRODUCTION
size,
Together with transport
in packed beds, chromatography is one of the most highly developed topics in chemical engineering, both From the chzmical point OF view as a laboratory tool (Yau et al., 1979). and in press development Tar the separation and purificntion of biomol~ules (Yau. 1974; Anspach cl ab. 1988; Davies, 1989; Pyle, 1990). Such a huge field of research can be approached in many different ways: from the mathematics of dispersion in packed beds (Aris and Amundsou. 1957; Whitakat, 1967; Bhattaeharya and Gupta, 1983); from the physics of solute transport in porous packings (Deen, 1987); from the analysis of chromatographic columns as separation units (Pyle, 1990x from detailed study of the port network and its influence in transport (Larson et al., 1981; Rasmuson, 1985; Reges and Jensen, 1935; Shah and Ottino, 1986; Reyes and Iglesia, 1991). In this article, we consider exclusively HPLC columns in which the diffusional partition between the mobile and the stationary phases depends on sizeexclusion
effects
(Determann,
1969).
Tn site-exclusion
chromatography (SEC), the diffusion coafit-ient depends on the steric interaction hectween solute molecules and porous matrix and deviates from the Stoke-Einstein diffusivity by a factor G(L) which dcpends exclusively on the ratio *I between the radius of the solute molecule and the pore radius, A = rs/rp. D = &G(I).
(11
The function G(A), known as the effective diffusion accessibility factor) figures in the- expression for the retention time r. u a function of
coefficient (OF
TAuthor to whom correspondence should be addressed.
since the equivalent void factor of a molecule written as #(4
= #o f (1 - $o)x@4
can be
(31
where & is the column void factor and x the internal porosity. A theoretic;il and experimental analysJs of the functional Form of G can be found in Determann (1969) and Deen (1987). The thrust of this article is twofold: to present a simplified characterization of solute transport in SEC columns, and to derive some analytical implications useful for an analysis of dispersion. The key point of the analysis lies in the expetimental evidence for an invariant principle vatid for the outlet conccntrarion profile in the case of impulse input: an invariant concentration profile characteristic of each biomolecule can k introduced indepndently of the eluent flow rate. Starting from the mathematical formulation of the invariant principle, it is possible to develop an integral method, alternative to moment analysis, For the evaluation of dispersion parameters. Some criteria for testing the validity of the invariant formulation arc developed. EXPERIMENTALSECTION
The columns used in performing the experiments were as fo116ws: Bio-SiI TSK 250 (packing G 3000 SW, avera=ge particle site 10 pm, average pore radius rp = 125 A) and Bio-SilTSK 125 (packing G 2ooO SW; avera particle size 10 m, average pore radius rp = 625 A” ) from Toyo-Soda (L = 60 cm, 541
542
MASSIMILIANCI GIONA at al.
S = 0.44 cm’)+ The elution solutions were 0.1 M Ns,SO,, 0.1 M NaW,PO* and 0.02% sodium aide (w/v) for the TSK 250 column; 0.05 M Na2S0&, 0.02 M NaH,PO, and 0.02% sodium tide (w/v) for the TSK 125 column. The eluent was adjusted at pH 6.8 by using an NaOH solution. The processing apparatus consisted of a twin-headed reciprocating pump, Waters model 510, a selectable-wavelength UV detector (481 Lambda Max), and a Rheodyne injector, model 7125, purchased from Millipore UK: The protein concentration varied from 0.1 to XI mg/ml and the injection volume from 20 lo 40 ~1. Al the experiments were perform& at 25°C. The biomoleeules considered were: thyroglobulin from bovine (669 kDa), apoferritin from horse spleen (443 kDa), ovalbumin from chicken egg (a kDa), myoglobin from horse (17 kDa) and cyanocobalamin (1.35 kDa), supplied by Sigma, PooIe U.K.
INVARIANT
PAlTERN
For large PeGlet numbers, the behaviour of the Green function can bz obtained by similarity analysis. More prtisely, st<, r) attains the simplified Form
where
Tn other words, it can be expressad in terms of a unique function 41v of the normalized dimensionless time 8. To prove eq. (4) and obtain the expression for the invariant function 4N, it is sufficient to substitute it inta eq. (5) and take into account the expression of d(i) given by eq. (8). After some atgebraic manipulation, eq. (5) can he rewritten as
FOR COhXECl’ION-DIFFUSION MODELS
In many problems related ta tubular reactor dynamics, chromatographic columns and dispersion in packed beds, the macrosr;opic fluid dynamics of the system (hereafter called for simplicity ccrlramn)involves the solution of a PDE of the convection-diffusion type:
By considering an infinite model OC column, x: E (0, co), the dynarni~ of transport inside the column is s@fied once the Green function g(& z) is known. The assumption of the infinite length of the column can be jusaed by considering that in our experimental apparatus the ratio between tbc length and the diameter of the column is L/a = 80 and the experimental conditions are such that Pe = I@-104. The Green function y(c, t) satisfies, in dimensianless fotm, the differential scheme (5)
s(C, 0) = 0
(6)
and with the Dirac delta function imposed at < = 0, SIO, T) = 6(r).
(7)
The function g(<. 7) is properly a Green function since the response to any inlet profile c,(r) is given by the mnvolution of Q(<, z) with c&l. The variance o’(t) for a value < of the dimensionless axial coordinate can k obtained from eqs (S)-(7) by moment analysis and is given by
In the liiit of PB -* CIS(i.e. in the fimit of Q tending to zero), 4. [I 1) can be simplified by considering the most significant terms, which prove to be those of order I/u and which cartespond to the terms in eq. (1 I) that are multiplied by the factor l/((a3Pc) once the detivatives of ~(5) are made explicit. In this way, one obtains an equation for the invariant fun&on 4,(B) of the form d24, _+fl% dB=
dir
“!-4N==O
which belongs to the class of integral-error-function equations (Zwillinger, 1989). The boundary conditions to be imposed on 4w are 4~~( & M) = 0 and d4,(O)/dB = 0. The solution of eq. (12) satisfying these boundary conditions, and whose integral is normalized to unity, is given by Abramowih and Steguu (1970):
The invariant principle expred by eq. (9) CB~ be stated alternatiwly by asserting that the function u(& [<, < + a(QB] is independent of c in the limit of large Peclet numbers. Figure 1 shows the behaviour 4+,(fi = u(l)g[i, 1 f 41)/?] vs fl for increasing values of Pe, together with the asymptotic limit ex-
Simplified analysis of chromatographic-column
1
0.5
am 0.4
dependent
0.1
2.0
0.0
d 4.0
B
invariant
a(l)fl YS p tar increasing values of Peclet nurnbe~~Fe = 10, 50, 100. 200. With the mceasing
Fig. t. GP, = a(t)g[l.
1+
of Pe, (pp, tends to the asymptotic
bchaviour (13) (-,
).
pression
(13). It can be noted that, even Tar P&et numbers of the order 200, the asymptotic expression (13) gives a fairly good approximation of the correct behavinur of the outlet concentration. To sum up, the macroscopic fluid dynamics of a tubular reactor with dispersion, as well as other Iransport situations described by means OF a convection-diffusion equation, are satisfactorily described by means of a unique invariant Green function for large Pe. This implies that a single ordinary differential equation in the limit of large P.c is wholly equivalent to the partial differential model expressed by eq. (4), It should he noted that the similarity principle expressed by eq. (9) is somewhat different from the constant pattern profile developed for adsorption columns by Garg and Ruthven (1975). In the latter case, the asymptotic constant pattern behaviour is expressed exclusively as a function of a shifted axiai coordinate by taking into account the macroscopic solute velocity. In the next section WC show how eq. (9) holds true for SEC data. INVARIAYT
PROE‘ILE
IN SEC
of the retention
time and depends
excIus-
the column and on the nature of the solute molecule. Figure Z(a) and (b) shows two normalid chromatograms obtained according to eq. (IS) from experimental data on TSK 125 and TSK 250 columns for an eluent flow rate in the range of 0.1, I.0 ml/min. Analogous invatiant outlet profdes were obtained For ail the biomolecules wnsidered. From these data, the invariant relation (14) seems to hold with sufficient accuracy. It is important to state in passing that the retention time t, introduced in eq. (14), corresponds to the peak retention time (Yau et al., 1979). i.e. the time instant corresponding to the peak of the outlet chromatogram. According to the points discussed in the case of the
0.2
-2.0
543
iveIy on
0.3
0.0 -4.0
dynamics
scheme,
formulation the
best
of
a
convection-diffusion
approximation
for
the
invariant
Green function is achieved for the higher eluenr flow rates. It is interesting to observe that the invariance of the outlet chromatograms seems to hold true even in 1.2
,
.:ii\__ -2 0.50
0
2
4
=CYa -
1
cy
5
I-
(b)
COLUMNS
Turning our attention to SEC columns, let u(t, t.) be the outlet concentration profile from a pulse injection associated with a given value of the retention time [associated via eq. (2) with the eluent flow rate]. Its integral with respect to the time r is normalized to unity. The similarity principle, equivalent to eq. (9), attains the form c(:, r,) = t
dh)
t- t, I
aJpJ [ 00,)
where 0 = ~{t,) depends on the retention tion (14) -an be put in equivalent form @H(Z) = art*)cCfr
+ a(t
r,7.
(14) time. Equa(15)
Equation (15) states that given an output chromatogram c,(t, r,), tbe Function b(t,) c[r, + a&) z, r,] is in-
CES 49:4-H
-5
-2.5
0.0
2.5
Fig. 2. Behaviour of the normalized dimensionless chromatogram dr, as a function of LI= it - t.#a(t.) for elutian Row rates varying from 0.1 to 1.0 ml/tnifi. (a] Thymglobulin on a TSK 250 column; (b) cyanocobalamin on
a TSK
125 colulnn.
MASSIMLL~ANCI
244
those cases in which the outlet chromafogram is intrinsically asymmetric and deviates strongly frOm being Gaussian. This implies that eq. (14) Seems to hold true not Only for Gaussian-shaped profiles of simple convection-diffusion models, but represents a more general feature of a broader class of trtinsport phenomena in packed beds. CBlTF,RlAOF YALIDITY OF THE INVARIANCE The validity of the similarity equation (15) requires analytical testing. Consistency criteria for the invariant representation, eqs (14) and (15)*can k developed from the analysis of the moments {JM,,_~~,] of the invariant function @Jrw)
Indeed,
and therefore the first- and secpnd-order .moments reduce to
= 1 + CM,,(,)Y where {M,) are the moments of c(t, L,). The last equation implies fl;> = 1.
(191
Equation (19) should be satisfied independently of the shape Of the invariant Green function. Equation (19) constitutes a simple criterion for testing the validity of the similarity equation (14). In the application of this criterion to experimental data, it is important to note that the experimental outlet chromalogram should converge to the invariant Green function in the limit of high eluent flow rates. Figure 3 shows the behaviour of u& a5 a function Of the eluent flow rate FO. As
1.4 I
41r 1.21
r-- .---
d) thyreglobulin a) ap&srrllln bl myoalabln c) cyanocobrlamln
al walbumln
= c-L-\.__ b t’-__ E
1.0
l
0-8
*-
-s---._
_,___ d
T/-c-
---__ _----c-____ -&_
cc al.
expected; u& converges to unity as the elution flow rate increases, even if the convergence depends On the solute. In the case of the ideal model represented by a c0~vectiO~-d~lMiOn equation, u& is prOpOrtiona to l/P*. The behaviour of 03, as a function of FO (or equivalently of Fe) can be analgsed in order to use the invariant principle as constraints to be imposed on the functional form of multiphase transport models of chromatographic columns. This topic rcquircs further investigatiOn+ For the purpose Ofthe present work, the data in Figure 3 can be regarded as a strong indication’of the validity of the invariance hypothesis.
FUNCTIONAL
PREDICHON
OF
DlSPERSlON
The similarity equations (14) and (15) not only simplify the treatment of chromatographic column dynamics, since a single function of a normalized time 3~= (r - r,)/u(t,) is representative for all the flow conditions. but also enable us to formulate alternative methods of analysis of chromatographic data. A consequence of this is related to the analysis of solute dispersion. It is known that the evaluation of dispwsion from moment analysis is highly sensitive on the cut (i.e. on the truncation point) in the outlet chromatogram (Yau es tit., 1979). Let us assume that aq. (15) holds true, and let cco,Ct, b,oJ cc,,Ct. 5r1, 1 be two ChrOmatograms with the respective retention times t+,,? t,(1) and variances u(“o,,o(:, . Let us further assume that the variance O& is known and the variance O:~, is to be determined. According to the invariant equation (15), the integral Its> =
s
i+lrC&cll
+ W t&l
+ ~(0, m, trd2
- flW?o>Ct,(o)
da
(20)
is idcnticaIIy equal to zero for’the value of the parameter Y equal to D,,~_Allowing for experimental error, it is convenient to define fiGI, as the value of the parameter s which minimizes l(s). By developing I(s) and putting s, = sfeaco,, one obtains I($,) = J,,(s,)s,Z
- 211 ,I&>& + Ire
’
_-zc.-_ -_-----k-G -.
_-_-~;~-T=--= ,_-_/a--
CWNA
./‘-
(a)
0.6 0.1
0.3
0.5
0.7
o~gFJml/min~~l
Fig. 3. t$, YS Fv rc~otthe biomnlecules
exwidered:
(a) TSK !25 cotumn;
(b)
TSK
250
column.
(21)
Simplifiedanalysisof chromzttographic-column dynamics
545
where the integrals appearing in eq. (21) read as
sitnutions, the integration limits are not 0 aad ca but PI’ = min (L m, L&j and r(l) = max I’M, ~MD}, t,0 and rMO Mng the termination points of the reference chromatogram ctO,Et, I,(~) 1. The evaluation of u by means of the integral equation (26) is more robust than the calculation of u directly from its definition. This GUI be intuitively explained by observing that the right-hand side of eq. (26) corresponds to a normalited correlation integral of +,) and cILj which proto b less sensitive to the trunation points than the integral of t”c,,,(t, t*,).This aspect can be also checked from the behaviour of the sensitivity function
From definition (22), by taking into account Schwa= inequality (Stakgold. 1979). it follows that
of 0~1, with respect to fM, Fig. 5(b). A theoretical analysis of the sensitivity of the integral method represented by eq. (26) with respect to the termination points could also be carried out, but is left for further investigation because it is of secondary physical interest in the analysis. Of course, eq. (26) is valid as long as eqs (14) and (151 hoId true. ConsequerMy, future attention should
P I 1WI2
G 12012
I (%I
(241
and therefore the only real solution ofeq. (24) is given
by the value of 8. which nullifies the expression under square root. Cr,,(s.)la = Jd2iIS,). Consequantly, the ratio 4 = us,,/+, is given by
which can be further simplified as
1x
be addresd to both experimental and theoretical analysis of the physica vatidity of eqs (14) and (15) in other transport conditions on chromatograpbhic WIumns.
Q s
; h,C~,h,l~“d~ (26)
Therefore, given one chromatogram of known variance and another chromatogrsm whose variance is to k estimati. the corresponding value of a, can be obtained through eq. (262 In practice, u, can be evaluated, e.g. a least-squares fit. In order to apply the functional method discussed above to a practical example. let us consider the inffuence of terminatio,a points &, tM, on the estimate ofdispersion, Fig, 4. This topic has been developed by Skopp (1984) in terms of an optimal estimator, by means of weighted moments. and has be~?ndiscussed by de Lass et ai. (1986) in the case of adsorption in packed beds. Figure 4 shows the expetimental chromatogram of thyroglobulin on the TSK 250 column (r&tad upon normalization and elimination of the instrumental bias to cgl) [r, r.,,,] > as measured by the UV detector for an eluenf flow rafe F0 = 0.8 ml/min. Let us assume as reference chromatogram, ctO,[t, rrlolJ, the correspending chromatogram for F,, 9 1.0 mt,bnin. Figure 5(a) shows the estimate of btl, as a function of tar, by taking fixed I, = 11.5 min. as obtained from eq. (26) and from the direct evaluation of the variance from its definition+ Of purse. in applying q. (24) to practica1
INFLUENCE
OF SOLUTE QUANTrTiES
In order fo analyse the influence of the solute concentration on the invariant structure of the outlet chromatogram, experiments were performed for diff-t biomolecules with molecular weights ranging C,lVl 0.90
I,i
0.20
0.10
Fig. 4. Experimental outlet pr&te (measured in Y by the WV detector) as a function of time for thyrdglobulin on the TSK 250 colunln. Fo = 0.8 ml/tin. The meaning or the rcrIninalivn points c,, LHm pointed out in the text.
-I-_--
a) b)
--
-
from ea.
IPB) from the deflnltbm ___A---
b
/----
0.08
.
---b
-.I
Q.41 1 l.6
_
16.6
16.6
17.6t, tminl
Fig. 6. d
w injection solute concentration y. for sweral
biomolecules
1
(TSK
125 column); F, = 1.0 d/mh.
a) from w. (26) b3 from the deflnltlon
ti4.S
16.5
161
17-5 t,
[mid
Fig. 5. (a) Behariuut ofqll(fdd) VBthe termination point zM. (b) Sensitivity function Ss[tMa)vs 6M for the cuwes of Fig. W.
frm 1 to 700 kDa, keeping the elution volume (33 4) constant and varying the inlet mncentration in the range 0.1, 20.0 mg/ml. It was observed that, for each biomolecule, the outlet chromatogram at different levels of concentration practically mincide and the variance square root is mnstant for the considered concentration range, Fig. 6. The invariance behaviout observed in the outlet chromatogram is therefore independent of the concentration. Tt should also be noted that the concentration range considered is larger than those typical of analytical chromatography and goes beyond the range in which preparative cdumns usually work. Consequently, the independence of the outlet chromatogram from concentration GUI be a valid assumption in the design of industrial separation units for the purification of biomolecules in SEC columns. Experiments were also performed to analyse the influence of solute quantities on dispersion, k-in& the concentration (20 mg/ml) constant and varying the elution volume in the range 20,4O ~1, Even in this case, the variant and the invariant behaviour of the outlet chromatograms are independent of the injection volume.
This article shows a simplified way of treating chromatographic data base&on the invariance analytically expressed by eqs (14) and (15). The method
discussed leads to severat intertAng masequences in the analysis of dispersion and can be further applied to the analysis and control of the separation performance of SEC cAumns. These topics will be discussed elsewhere. Several opm questions remains for further theoretial analysis_ In particular, It is important that the solute transport should be studied in more detail to obtain better understanding ol the physical nature involved in the experimentally found invariant principle. and consequently to derive the range of validity and the physical limitations of eqs (14) and (15).
outlet concentration profile starting from an impulsive injection solute concentration distribution column diameter, cm diffusion cweffident. cmZ/min Stokes-Einstein diffusion coefficient, cm2/min eluent flow rate, ml/min Green function of convactionafhision scheme {or acx&.seffective diffusion cdlitient ibility factor), dimensionless integrals defined by eq. (17) length, cm tith-order moment of c(t, r,) nth-order moment of the invariant function mrv Pec!et number average pore radius solute pore radius column section, cm2 sensitivity hmction defined by eq. (22) time, min termination point in the evaluation 01 outlet chromatograms, min peak retention time, min solute velocity, em/min axial coordinate, cm
column
Simplilkd
analysis
of chromatogmphic-column
Greek letters
; Y s :: fl2 fl:,
dimensionless normalized time (r - 5)IcIO injection solute concentration, m&n1 firac delta function r&P x/L variance of the outlet chromatogram invariant variance M,,+, - (n*i,+,,#
dynamics
547
the moment method technique for the definition of adsorption parameters in a packed bed. Chem. Eww Sci. 41, 123>1242. Determann, H., 1969, Ge/ Chromatography. Springer, Berlin. Gar& D. R. and Ruthven. I). M., 1975. Performance of m&c&r
sieve adsorption
columns:
combined
elkzta
of
mass fransfer and longitudinal difhtaion. Chut. Engng Sci. 30, 1192-f 194. Larson, R. G, &riven. L. E. and Davis, H. T., 1981, Pcrcolaticu theory of two phase flow in poreus media. Chem. Engng Sci. x 5673.
“L/V
equivalent
%0
void factor void factor invariant Green function graphic response internal porosity internal
of chromato-
Abramowie M. and Stegun, I. A, lP70, Handbook olMathmrical Functions. Dover, New York. Anspach, B., Gkrlich, H. U. and Ungcr, K. K., 1988, Cornparative study of Zorbax Bio Series GF 250 and GF 450 and TSK-GEL Moo and SWXL caluin sire-exclusion chromatography of proteins. 1. Chrnmot. 44% 4554 Aris, R and Aruuudson, N. R., 1957, Some remarks on longitudinal mixing or diffusion in lixed beds. A.1.Ch.E. J. 3.280-282. Bhattacharya, R. N. and Gupta, V. K.. 1983, A theoretical explanation of solute diswrsion in saturated porous media at the Druey scale. Water Remrces Res. 19. 93%944. Davies, P. A., 1989, Determination ofdiffusion ce&&uts of proteins in beaded agarose by gel filtration. J. CRrau~ut. 433,22t-237. Deen, W. M., lQ87, Hindered transport of large molecules in liquid-filled pores. A.lLXE. J. 33, 140%14X. de Lass, H., Hazlctt, J. and Fuller, 0. M., 1986, Evaluation of
Rasmuson, A., 1985, The effect of particles of variable size, shape and properties on the dynamics of fixed beds. Chsm Engng Sci. Qo, 621429. Reyes, S. C. and Iglesia, E., 1991, Monte Carlo simulation$ of structural properties of packed beds. Chem Engng Sci. 46, 108~1099. Reyes, S. and Jensen, K. F., 1985, Estimation of effective transport cocfl%knts in porous solids based on percolation concepts. Chem. Enptg Sci. 90, 1723-1734. Shah, N. and Ottino, J. M., 1986, Effective transport properties of disordered multi-phase composites: application of real-space rertormalization group theory. Ckeut. Eagug Sci. 41, 283296. Skopp, J., 1984, Estimation of true moments from truncated data. A.1.Ch.E. J. 30, 151-155. Stakgold, I., 1979, G~.+II’s Functions ud Boundary Value Problems. Wiley, New York, Yau, W. W., Kirkland. J. J., Bly, D. D. and Stoklosa, H. J., 1976, Effect of cdumu performance on the aeeuraey of molecular weisrhts obtained from size exclusion chromatography). J. chrernatography (gel pwneation Chromur. 125,219-230. Yau, W. W., Kirkland, J. J. and Bly. D. D.. 1979. Modera Size-Exchsion Liquid Chrom~mgm~&u. Wiley, T&u York. Whitaker, S., 1967, Diffusion and dispersion in porous media. A.1.CL.E. J. 13, 420-427. Zwillinger, D., 1989, Ham&ok of Diflermtial Epuatims. Academic Press, New York.