Prevención de accidentes

3.1 Introduction.

T his chapter review s the m odelling o f chrom atographic operations as w ell as som e im portant aspects o f transport in fixed porous beds. T he m odelling analysis is centred on the description o f the m odel selected for study in this thesis, i.e. the general rate m odel o f size exclusion chrom atography, and on the description o f the num erical m ethods used in its solution. These m odelling and num erical tools will be used in the study o f the effects o f m atrix com pression and fouling on colum n perform ance in the next two chapters.

Initially the m odelling o f colum n chrom atography focusing on gel filtration o r size exclusion chrom atography (SEC ) is outlined (section 3.2). This particular type o f chrom atography has been selected to be used throughout the present study because o f its sim plicity as com pared w ith the other types o f chrom atography. The only rate param eters needed fo r sim ulation o f SEC chrom atographic peaks are intraparticle diffusivities, convective dispersion coefficients, and to a lesser degree fluid-phase m ass-transfer coefficients. Furtherm ore, partition equilibrium is linear up to relatively high concentrations (Jonsson, 1987). Solute-m atrix interactions in this type o f chrom atography are negligible and therefore adsorption constants are not to be considered. The linearity of the partition equilibrium relationship is o f im portance since lin ear chrom atography m odels are m ore easily solved than the non-linear types. B esides in SE C th e solute concentration in the applied sam ple is generally low , being limited by the viscosity o f the sam ple w hich m ust not be so large as to cause hydrodynam ic instability (H agel, 1989). T his m eans that n o t only the equilibrium distribution coefficient is constant, but all the transport param eters and fluid properties are considered to be independent o f concentration and constant.

S ubsequently, since axial dispersion, fluid-phase m ass-transfer resistance and the solute intraparticle diffusivity are the transport param eters needed to describe the chrom atographic process and to characterize its perform ance, they are looked at w ith som e detail in respect to fixed porous beds. T he correlations and m ethods used in their determ ination and estim ation are briefly

review ed. Particular em phasis is given to those m ethods and correlations w hich are applicable w ithin the range o f reduced velocity o f interest to m od em liquid chrom atographic practice. To conclude the num erical m ethod em ployed in the solution o f the size ex clusion chrom atography m odel used in this thesis is presented.

It is intended that the theory presented here w ill provide a basis fo r the interpretation o f the com pression and fouling experim ents described in the next tw o chapters.

3.2 Size exclusion c h r o m a to g ra p h y m odels.

T he developm ent o f theoretical m odels for the prediction o f the perform ance o f size exclusion chrom atography has follow ed different approaches.

T he m ajority o f chrom atographers now agree that size separation o f m olecules can b e fully explained on the purely steric basis that large m olecules can only partially perm eate the pore volum e o f the support. A ccording to this view , a series o f theoretical m odels have been derived in ord er to predict SEC calibration curves, also term ed exclusion or selectivity curves. In these m odels it is assum ed that the equilibrium distribution coefficient K^, depends only upon the size and shape o f the solute m olecules and upon the size and shape o f the pores in the colum n packing m aterial. Using statistical theory for the equilibrium distribution o f rigid m olecules in inert porous netw orks, G iddings et al (1968) obtained theoretical relationships to estim ate K^, fo r solutes o f spherical to thin-rod shape and for pores w hose shape ranged from circular cross section to an infinite slab. However, these m odels are o f regular geom etry and consists o f a single uniform pore size, therefore they cannot accurately represent real SEC packings.

M ore com plex m odels o f non-regular geom etry and random -sized pores have been derived, am ong them the random -sized touching sphere m odel and the uniform -sized random -sphere m odel give excellent quantitative fits to experim ental data (Knox and Scott, 1984). T hese researchers have also show n that SEC calibration curves can be predicted w ith rem arkable accuracy from m ercury porosim etry data, by using a m odel consisting o f an assem bly o f cylindrical pores having the pore size distribution given experim entally.

A lthough all these m odels are capable o f predicting K^ and therefore the retention volum e Vr o f a solute given its m olecular w eight M, they do not provide any inform ation regarding peak shape and band spreading.

Since the equilibrium relationship in SEC is linear, all the m odels derived to describe and to predict the elution profile in linear chrom atography are applicable. T hese m odels can be classed into three broad types: plate or tank-in-series m odels, statistical m odels and transport m odels

(G olshan-Shirazi and G uiochon, 1992b).

P late m o d els. In the plate m odels, it is assum ed that the colum n consists o f a n u m b er o f identical equilibrium stages or theoretical plates, placed in series, and that the stationary and m obile phases are in equilibrium at each stage.

T here are tw o categories o f plate m odels. In the discrete stage distribution o r C raig m o d el (C raig, 1944; E ble et al 1987), a finite volum e o f eluent is equilibrated step by step, w ithin one theoretical plate in the colum n after another. In the second category o f plate m odels, there is continuous flow o f the m obile phase through a series o f stages (M artin and Synge, 1941), this is w hy it is called

the continuous pla te model. All plate m odels are by nature approxim ate since the equilibrium assum ption requires a m ixing m echanism w hich is clearly absent from the physical system . Statistical m odels. In the second class o f linear chrom atography m odels, a m icroscopic statistical m ethod is used to derive the probability density function o f a single solute m olecule at a p articular colum n position and time. G iddings (1965) used the random w alk approach to calculate the profile o f the chrom atographic band in a sim ple way. A nother probabilistic approach, the stochastic m odel, w as introduced by G iddings and Eyring (1955), for the description o f the m igration o f a single solute m olecule in chrom atography. T h eir m olecular dynam ic approach based on statistical ideas treats chrom atography as a Poisson distribution process. T hey derived an expression for the elution profile, o r distribution o f the residence tim e o f a m olecule in the colum n, assum ing random adsorption/desorption processes, with a single type o f site on the stationary phase and fo r a pulse injection. T his latter approach did not consider the colum n axial dispersion and the m ass transfer kinetics, and therefore its predictive capabilities are lim ited.

T ransport m o d els. The third approach followed in the m odelling o f chrom atography is the transport approach. T his has been widely used to calculate the chrom atographic response to a given input function. In this m ethod the transport processes that take place betw een the stationary phase and the m obile phase as w ell as the additional dispersion o f the solute in the interstices o f the packing m atrix due to eddy diffusion and m olecular diffusion are built into the m odel, and the chrom atographic process is described by a set o f partial differential equations w hich result from the differential mass balance o f the solute in a slice o f colum n and its kinetics o f m ass transfer. R ate m odels o f different levels o f com plexity have been developed. The m ost im portant o f them are the equilibrium -dispersive model, the lum ped kinetic m odel and the general rate m odel o f chrom atography (G olshan-Shirazi and G uiochon, 1992b). M ost o f these m odels have been developed for adsorptive chrom atography but size-exclusion chrom atography could be treated as a special case, with the adsorption/desorption rate constants and the equilibrium isotherm constant all equal to zero.

constant equilibrium . T he contributions o f non-equilibrium effects, e.g. axial dispersion and m ass transfer resistances are lum ped together in an apparent dispersion coefficient D , (adsorption/desorption kinetics could also be considered but they are no t relevant to the present study dealing w ith SEC). W ith these assum ptions the m odel is reduced to a single equation, the differential m ass balance o f the solute in a slice o f colum n given by:

O ^

(3.1)

‘*32^

dz

m dt

dt

where:

C = solute concentration in m obile phase, m g /c m \

q= solute concentration in stationary phase in equilibrium w ith m obile phase conc. C, mg/cm^. t= tim e, s.

z= axial distance from colum n inlet, cm. u= interstitial fluid velocity, cm/s.

D .= apparent axial dispersion coefficient, cmVs. m = phase ratio, e /(l-e )

e= bed void fraction.

T his m odel has been discussed by Yam amoto et al (1988), Jonsson (1984) and G olshan-Shirazi and G uiochon (1992a) am ong others. Its analytical solution has been derived by Lapidus and A m undson (1952) and has a sim ple G aussian form w hen the sam ple volum e is sm all and the plate num ber is large.

Y am am oto and Sano (1992) discussed the criteria under w hich this sim plified m odel is applicable to SEC o f proteins and found that for a small sam ple volum e Vp, w hen

N > 2 2 l i m * K J ) I K j ]

(3.2)

w here N is the plate num ber and the distribution coefficient, the calculated elution curves by the three rate m odels discussed in this section and the plate theory w ere hardly distinguishable. T hey stated that this condition is usually fulfilled in current protein SEC.

In the lum ped kinetic m odel o f chrom atography, the contributions to band broadening due to all the mass transfer resistances are lumped in a single coefficient and a lin ear driving force m odel is assum ed. The contribution o f axial dispersion to peak spreading is treated independently and accounted for by the axial dispersion coefficient D^. T herefore the behav io u r o f the chrom atographic system is described by the follow ing m ass balance equation: