Fundamentos jurisprudenciales

JO

(4.1)

(4.2)

w here L is the bed length, u the interstitial fluid velocity, e the colum n void fraction, £p the inclusion porosity' and t„ is the input pulse time. And the second central m om ent ^ m easure o f peak spreading (the variance) is expressed by

^2 = ---

I C(L,t)dt

(4.3)

15

(4.4)

w here is the axial dispersion coefficient, Rp the particle radius, the effective intraparticle diffusivity and Iq is the particle-to-fluid m ass transfer coefficient.

' This parameter represents the fraction o f intraparticle space that is available to a particular m olecule. It is also called intraparticle void fraction and it is equivalent to the equilibrium distribution coefficien t IQ.

F o r a sufficiently long colum n the exiting p eak is G aussian and the H E T P is given by

HETP

= —

(4.5)

t l

In this case the peak is sym m etric and the retention tim e t^ is identical w ith pi, w hile the variance is also accurately obtained from the peak w idth w ithout in terference from tailing o r sm all baseline shifts.

S ubstituting the first and second m om ents in the H ETP equation the follow ing expression is obtained

2 r>2

HETP= //= — ^ +

15(e+(l-€)€^)

(4.6)

E quations 4.2 and 4.6 constitute the basis o f the H ETP pulse response analysis. B y m eans o f these expressions and two peak properties, the retention tim e t^ and the v ariance , the equilibrium and rate constants can be estim ated.

The H ETP analysis is strictly valid only for colum ns with a large n u m b er o f plates (L/H ETP > 30) and w hen the spreading caused by end effects is sm all com pared w ith the spreading caused by the transport phenom ena, thus pulse techniques (H ETP m ethod) should be perform ed on a long colum n, even if the actual separation is to be perform ed on a short one (A rnold e t al, 1985). W hen the bed contains a sufficient num ber o f theoretical plates, G aussian peaks can be expected if the particle m ass transfer can be described by a linear concentration driv in g force (A rnold et al, 1985). The pulse analysis theory assum es linear equilibrium . T his is practically alw ays the case in SEC. E valuation o f the equilibrium distribution coefficient m akes use o f equation 4.2 w hich defines the first absolute m om ent p, o r retention tim e Ir. In this equation the equilibrium coefficient is represented by the inclusion porosity o r particle accessible void fraction T he evaluation requires know ledge o f the bed void fraction e.

From the shape o f eq. 4.2 it can be observed that a plot o f Uj-ty2 o r tf^-tJ2 vs L /u gives a straight line with a slope equal to l+ ep (l-e)/e w hich indicates the liquid volum e fraction that is available to a tracer m olecule. If the pores are not m uch larger than the tracer, entry o f the tracer into the particle m ay be hindered. If the pores are so sm all that no tra c e r can enter, the slope is the interparticle void fraction e. K now ledge o f £ allow s the evaluation o f £p from the slope o f the line. The estim ation o f the axial dispersion and the intraparticle diffusion coefficients involves the use o f the HETP expression (eq. 4.6) and the analysis o f the peak spreading cP as a function o f the interstitial fluid velocity, u.

In the experim ental conditions usually encountered in size exclusion chrom atography both the q uantity D j/u and the film m ass tran sfer coefficient are affected only slightly b y the interstitial velocity in the colum n (see figs. 3.1 and 3.2). T herefore it is clear from eq. 4.6 that a p lo t o f H E T P vs u should give a straight line w ith D^/u as the y intercept and the slope depending on the intraparticle diffusion D,, and the particle-to-fluid m ass transfer coefficient kf. Eq. 4.6 includes all the plate contributions inherent to the processes occurring inside the colum n, b u t the extracolum n contributions to peak spreading resulting from the initial sam ple bandw idth and the m ixing in tubing, valves, detector and especially in the distributors are not considered.

A rnold et al (1985) described a m ethod by w hich the extracolum n effects as w ell as the fluid film m ass transfer resistance contribution can be accounted fo r so that the in traparticle diffusivity could be evaluated. The m ethod is based on the assum ption that the variance o f the peak exiting the colum n is a sum o f the variance o f each o f the contributions:

w here o^extr is the contribution due to extracolum n effects, and o^coi the contribution due to the colum n non-equilibrium effects can in turn be separated into

2 2 2 2 2 2 M

^ c o l~ ^ /la w ^ ^masstransfer~ ^ fU r w ^ ^ f U m ^ ^ d if f

w here onflow represents the contribution o f axial dispersion, o \,„ , is the contribution o f the fluid film m ass transfer resistance and is the contribution resulting from intraparticle diffusion. T his equation can also be expressed as plate contributions,

» c o r

Initially the H ETP contributions from external sources are evaluated for a given bed and solute by m eans o f the follow ing equation:

(4.10)

t l

w here the extracolum n contribution is determ ined according to the experim ental procedure outlined in section 4.2.2.3.

T he extracolum n plate contribution is essentially constant o ver the range o f fluid velocities used in SEC as can be seen in fig. 4.1. S ubsequently this H E T P contribution is subtracted from the experim ental HETP, so that only the contributions due to the colum n non-equilibrium interactions are considered in the rem aining part o f the analysis.

(4.11)

u sing correlated values o f kf (see section 3.3.2). T his expression has been obtained from eq. 4.6 by assum ing the fluid film m ass transfer resistance to be the rate lim iting step (Dç=oo).

F inally by subtracting this mass transfer contribution from the overall H E T P colum n contributions (eq. 4.6) the follow ing equation is obtained:

If-ff . ^

+

2 6 > ;£ (l-e )

^

(4.12)

«

150,(e*(l-e)€^)^

As show n in this equation and in fig. 4.1, the difference o f the overall colum n H E T P (experim ental H m inus H^^J and the fluid film H ETP have a slope w hich is inversely proportional to the intraparticle diffusivity.

T he axial dispersion coefficient is usually divided into two term s: one for m olecular diffusion in the axial direction and the second an eddy m ixing term that is proportional to the fluid velocity (see section 3.3.1):

Z),= (4.13)

T he m olecular diffusion com ponent is negligible for liquid chrom atography o f large m olecules. T herefore the axial dispersion coefficient obtained from the y intercept o f eq. 4.12 considers only the effects o f the bulk m ixing in the interstitial spaces. T he w hole m ethod is graphically show n in fig. 4.1 for a typical data set.

T he present study involves the estim ation o f the changes in the param eter values that m ay be associated w ith the changes in the bed structure and with particle deform ation w hen the bed is subjected to different levels o f com pression. T hese param eters are the equilibrium constant, the axial dispersion and pore diffusion coefficients.

T he particle-to-fluid mass transfer coefficient has not been estim ated by m eans o f the pulse analysis. In a survey o f the application o f pulse chrom atography to the m easurem ent o f kinetic and transport properties in packed beds, G angw al et al (1978) dem onstrated that fo r low R eynolds num bers particle-to-fluid m ass-transfer coefficients could not be m easured w hen porous particles

H - Hgim - Hgxtr