EL CÓDIGO CIVIL Y EL ESTADO DE FILIACIÓN EXTRAMATRIMONIAL

T he classical equation describing the phenom enon o f size exclusion chrom atography is based on the fact that a partition or equilibrium distribution coefficient Kj com pletely governs this process. T hus, the elution o r retention volum e Vr o f a given chrom atographic species is given by the follow ing equation:

(3.50)

w here is the colum n interparticle void volum e and Vj is the intraparticle void volum e. F or a m acrom olecule which does not bind to the gel m atrix, it is clear from the previous equation that the partition coefficient betw een the gel and the surrounding solution is equal to the volum e fraction effectively accessible to the m acrom olecule in the gel (L aurent and K illander, 1964)

V - V V - V

K,

K ^ r - K -

A ccording to this definition, this param eter is usually denom inated as inclusion porosity and denoted by £p in the SEC rate m odels, so that it is directly associated to the gel m atrix characteristics.

Partition equilibrium is linear up to relatively high concentrations (Jonsson, 1987), this param eter is therefore a constant w ithin the w hole range o f concentrations norm ally used in SEC. Ky is readily m easured by chrom atographic m ethods and finite bath experim ents, and it can be estim ated by pulse analysis techniques. The latter techniques are dealt with in the previous section and w ith m ore detail in sections 4.2.1. and 5.2.4.

T he analysis o f the param eters involved in the description o f size exclusion chrom atography is now com plete and in the follow ing section the num erical m ethod used in the solution o f the SEC m odel is presented.

3.5. N u m e ric a l m e th o d : F a st F o u r ie r T ra n s f o r m (F F T ) te c h n iq u e .

T h ere is at present no analytical solution in the case o f a pulse injection, to the general rate m odel o f chrom atography (G olshan-Shirazi and G uiochon, 1992), o f w hich the size exclusion chrom atography m odel used in the present study is a special case (eqs. 3.11 to 3.18). A lthough C arta (1988) has derived an analytical solution for this m odel, he did no t con sid er axial dispersion (Dl= 0) and his approach requires the assum ption o f a periodical injection. It is therefore necessary to con d u ct the solution o f the general rate m odel o f chrom atography num erically.

L en h o ff (1987) im plem ented a solution o f the general rate m odel fo r a pulse injection in the L aplace dom ain, and accom plished the inversion by contour integration in the com plex plane. The integral to carry out the inversion into the tim e dom ain w as conducted num erically.

In the particular case o f size exclusion chrom atography num erical inversion o f the L aplace dom ain solution o f different versions o f the general rate m odel has been accom plished by several w orkers. Y am am oto et al (1979) neglected the fluid phase m ass transfer resistance and carried ou t the inversion into the tim e dom ain either by integration o f the com plex function or by F ourier series approxim ation (Crum p, 1976). Davies (1989a) used a num erical technique fo r the inversion o f the L aplace transform solution o f a gel filtration m odel used in his m ethod fo r the determ ination o f protein diffusion coefficients.

N um erical approaches to carry out the inversion o f Laplace transform s hav e been developed by several authors, am ong them D ubner and A bate (1968) presented a m ethod w hich relates the inverse Laplace transform to the finite F ourier cosine transform . T heir resulting inversion form ula is easy to program for digital com putation (involving only cosines and exponential functions) but the convergence o f the series is generally slow and the valid region in the tim e dom ain is restricted to the interval 0 < t ^ / 2 (where the period is 2T). In their analysis D ubner and A bate (1968) first suggested the use and the im plem entation o f the F FT algorithm to significantly reduce the com putation tim e for their num erical m ethod.

B ased on the approach o f D ubner and A bate, C rum p (1976) developed a num erical m ethod w hich inverts the Laplace transform by m eans o f a F ourier series approxim ation. By involving the inform ation contained in the sine function o f the F ourier series. C rum p (1976) reduced the error in the approxim ation, com pared to that o f D ubner and A bate (1968), and doubled the interval u nder w hich the inverse function is approxim ated, i.e. 0 < t ^ .

H su and D ranoff (1987) have theoretically justified why the sine function term s should be retained in the final inversion formula. Based on the com plete F ourier series approach developed by C rum p (1976) these researchers have directly adapted the FFT algorithm to invert L aplace transform s num erically. T his technique was found to be very sim ple, accurate, efficient and generally superior

w hen com pared to other conventional m ethods (H su and D ranoff, 1987).

T he F FT algorithm w as first used in the field o f liquid chrom atography by Jônsson (1984) to invert the L aplace transform solution o f the linear non-ideal m odel developed by G rushka (1972). In gas chrom atography V illerm aux (1974) m ade used o f this algorithm to study the influence o f m ass tran sfer processes and kinetic retention m echanism s on p eak asym m etry. T hese researchers, how ever, did not take advantage o f the sim plicity and easiness o f program m ing o f the F ou rier series approach.

S oon after the introduction o f the FFT algorithm by H su and D ran o ff (1987) the num erical m ethod proposed by these w orkers was applied to the field o f chrom atography. H su and C hen (1987) solved the general rate m odel by adopting the F FT technique in ord er to study the influence o f intraparticle diffusion and sorption kinetics on the peak shape in linear chrom atography. T hey have also show n how this m ethod can be applied to the prediction o f breakthrough curves o f a fixed- bed adsorber (C hen and Hsu, 1987), and com pared its accuracy and speed w ith an exact analytical solution and the orthogonal collocation m ethod. W u et al (1991) developed a m ethod that com bines the FFT technique and the orthogonal collocation m ethod to calculate breakthrough curves o f nonlinear adsorption system s. B ecause the FFT technique has advantages o f accuracy and com puting speed, besides being used to analyze and to sim ulate chrom atographic system s, it has also been used to perform param eter estim ation by fitting the tim e dom ain solution to experim ental chrom atographic peaks (Ernst and Hsu, 1991).

Very recently B oyer and Hsu (1992) em ployed the F ast F ou rier T ransform (FFT) technique to perform the num erical inversion o f the Laplace dom ain solution o f the general rate m odel for the special case o f SEC.

As already m entioned in section 3.2.1 the general rate SEC m odel as presented by B oyer and H su (1992) has been selected to be used in this investigation (The m odel is described in section 3.2.1). The F FT num erical m ethod w hich was also follow ed by these researchers has been chosen to carry out the solution o f the SEC model due to its accuracy, easiness o f program m ing and its high efficiency. T he F ourier series inversion formula (C rum p, 1976) is given by Hsu and D ranoff (1987) in an adapted form to be im plem ented with the F FT num erical m ethod, while the F F T algorithm has been clearly presented by Hsu (1979).

In the follow ing chapter the effects o f m atrix com pression on chrom atographic perform ance w ill be looked at, while the effects o f m atrix fouling on colum n perform ance w ill be dealt w ith in chapter 5. In both o f these investigations the SEC m odel and the FFT num erical m ethod presented here will be used to aid the analysis.

3.6 C o n clu sio n s.

T hroughout the present study, size exclusion chrom atography (SEC ) has b een used because o f its sim plicity. It does not involve adsorption and its perform ance is characterized only by axial dispersion, particle-to-fluid m ass transfer and intraparticle diffusion. B esides the partition equilibrium relationship is linear.

D ue to its com prehensiveness and sophistication the general rate m odel fo r lin ear chrom atography, as defined by B oyer and Hsu (1992) for size exclusion chrom atography, h as been em ployed in the present study to sim ulate outlet chrom atographic peaks. T he num erical F ast F o u rier T ransform (FFT) technique developed by Hsu and D ranoff (1987) has been selected to solve the S E C model. T his technique is easy to program , accurate and very efficient.

E xisting m odels o f differential chrom atography are capable o f predicting the behaviour o f polydisperse packings if they have a narrow size distribution (A thalye et al, 1992) and the surface average particle diam eter is used in the calculations (R asm uson, 1985b).

M ore research is needed to im prove the understanding and the prediction o f the convective axial dispersion coefficient under the conditions prevailing in liquid chrom atography, in particular for reduced velocities R eSc betw een 1 and 100. To date the correlation derived by H ejtm ânek and S chneider (1993) seem s to be the m ost appropriate for the estim ation o f this param eter in the chrom atographic range o f operation.

O n the o ther hand for the prediction o f the particle-to-fluid m ass-transfer coefficient it appears that the correlation developed by Ohashi et al (1981) follow s the asym ptotic b ehaviour expected at the low R eSc range and it seem s to be the m ost suitable to be used in the creeping flow conditions encountered in the chrom atography field. N evertheless it is unlikely that the fluid phase mass transfer resistance is the rate controlling step within these conditions.

A lthough sophisticated m ethods have been developed for the m easurem ent o f protein intraparticle diffusivities, these require advanced skills and equipm ent. As a result traditional m ethods, which are easy to use, such as the pulse techniques are still w idely used to estim ate n o t only this param eter but also the axial dispersion coefficient and the equilibrium distribution coefficient.