JOURNAL OF CHROMATOGRAPHY 39

THEORY OF SORPTION CHROMATOGRAPHY

II. NUMERICAL CALCULATIONS

HANS VLNK

Imlilarle of Physical Ckevnistr~y, Univemity of Uppsala (Sweden)

(Received January Gth, ICJGG)

in mobile phase in stationary phase

SYMBOLS

f =

12 =

C =

161 =

122 =

=

b,, Dz = VI, v, = Jmt3 = hmg 3 = A$ =

P =

P2 =

M =

t =

Y =

V =

D = co =

a =

solute concentration solute concentration concentration of sorbent rate constant for sorption rate constant for desorption translational velocity of mobile phase diffusion coefficients in mobile and stationary phase respectively volumes per interphase area of mobile and stationary phase respectively matrix element representing f matrix element representing 12 ith moment of the concentration distribution mean of the concentration distribution variance of the concentration distribution mode of the concentration distribution duration of equilibration step partition coefficient peak velocity spreading coefficient velocity of concentration front in frontal analysis

2J31 -- tv3

,‘I =

112 =

YV2 _--- VC/'1 + yv2

(1 - e - W)

2D2 y

(

I -- -_

T/'2 Vl + -7; )

INTRODUCTION

The theoretical treatment of sorption chromatography in the preceding article1 ’ has been supplemented by numerical calculations performed on a digital computer.

As the basis of the calculations the following equations were used:

J. Cltromatog., 24 (1966) 39-55

40 1-I. VJNU

ah - = h, (c - II,) f - iz2h at

These equations were solved by a finite difference approximation method, leading to eqns. (40)-(42) in the Appendix. The numerical treatment of the problem followed the general outlines of the procedure in partition chromatography293. The results were obtained in the form of the .matrixes (J’“gg) and (1~~3) representing con- centration distributions in the mobile and stationary phases respectively of the chromatographic column. The data were abstracted from the computer in the form of a few selected columns of a matrix, representing the concentration distribution at different times. The zeroth, first and second moment with respect to the origin, with the cell width as unit length, were also calculated for every column. For the jth column they are defined as follows:

Al = 2 i fij (4) f

A2 = c i” fij (5! t

with corresponding definitions for the It-matrix. For a characterization of the concentration distributions the reduced moments,

the mean ,u and the variance ,u2 were used. They are defined as follows:

A2 p2 = -- _ k&2

A0

(6)

In addition the modeM, defined as the location of the maximum of the smoothed distribution curve, was also determined.

The primary results of the calculations are in the following given in terms of the parameters &, M, ,u and ,u2.

METHOD OF CALCULATION

In the present calculations the characteristic parameters of column operation were varied in order to determine their effect on the chromatographic process.

From the form of eqns. (I) and (2) it follows that not all parameters need be varied independently. The following transformations are seen to leave the equations unchanged :

J. Ckromalog., 24 (IgGG) 39-55

THEORY OF SORPTION CHROMATOGRAPHY. II. 41

(9)

(at, KI, AZ, Dl, v) + (t, a&, aha, aD1, av)

where a is an arbitrary constant.

(10)

Each of these transformations makes it possible to, change the value of one of the parameters via corresponding changes in some other parameters, In the calcula- tions therefore only the parameters c, D,, iz, and k, were varied, the others being kept constant and, when not otherwise stated, had the values:

v = 0.0x (cm set-1) (11)

'CI'l = 0.004 (cm) (12)

f = 100 (13)

t = 5 (set) (14)

The value of T/‘, was chosen to represent a column filling consisting of tightly packed spherical beads with a radius of approximately o.01 cm. The value of t may be fixed arbitrarily, but is related to the values of other variables ‘by formula (10). (Da, 12, and Jz2 enter the calculations in form of the combined parameters a = 2 DJrv2,

‘,

z/m Ii, and r/m 1~~). The. value in (14) may be used for, convenience, as it provides realistic operational conditions for the column. It gives a cell width ZZI = 0.05 cm.

All the matrixes were of the order 12 = 200 and in all cases the value nz = 5 was used.

The calculations were carried out with the following initial conditions :

JOi1 = l 1.00 Ior i = 1

ofori = 2,....,200 (IS)

JO13 = 1

Iooforj= ~,....,n

0 for j =n+I,,...,200 (16)

(17)

h”ilio = 0 lori = I,....,zpo (18)

In the case of isolated peaks in general the value 32 = 5 was used, though for matrixes 22, 24, 25 and 26 the value of 32 was 2, 10, 15 and 20, respectively. In the case,of frontal analysis, for matrixes 21 and 22, the value of n was 200..

‘The values of the characteristic, parameters for the different matrixes are listed in Table I, and the primary results of the calculations are given in Tables II and IV.

In Table II, the matrixes may be grouped together. according to ,the following .’ scheme; In. I, 2 and 3 the longitudinal diffusion coefficient is varied ; in 4; 5,2 arid.6 the equilibrium constant is varied; in 7,8, 9; IO, 2 and L I the concentration of the sorbent

; :’

.is v‘aried’; in i2, 13,,14, 2, 15 and 16 the reaction rate is varied and in,I7, 18, ? and Ig :

‘: the feed concentration. isvaried. Finally, the matrixes. 20, and 21 ‘represent frontal ‘, ,.. ,

: .‘:. (,’ ,. ;.

‘L,,‘.’ ., J, Clrro~zilo~., 24 (+GG) 39-&j I ,,i, ,; ” ; ..’ ,.

.,” ‘.

,;, :, / ., .,,’ ;.. ‘, : ; ,,’ : ‘j’irj

42 IX. VINK

TABLE I

VALUES OF CNARACTERISTIC PARAMETERS

I 100 094 0 0.005 0.05

2 100 o-4 0.1 0,005 0.05 3 100 0.4 0.2 0.005 0.05 d 100 100 100 0.4 0.4 0.X 0,X 0.1 0.005 0.005 0.005 0.001

0.4 0,005 0.25 z 100 100 0,02 0 0.1 0.1 0.005 0.05

0.005 0.05 9 100 0. I 081 0,005 0,05

IO 100 0.2 0.1 0,005 0,05 II 100 0.7 0.1 0.005 0.05 12 100 004 0.1 o.oooJj 0.005

I3 100 004 0.1 0,001 0,OI

I4 100 0.4 0.1 0,002 0.02

15 100 004 0.1 0.007 0.07 IG 100 0.4 0.1 o.oog 0.09

I7 ;:

0.4 0.1 0.005 0.05 IS o-4 0.1 0.005 0.05 19 400 004 0.1 0.005 0.05 20 100 004 0 0.0005 0.005 21 100 004 0.8 0.0005 0.005 22 100 014 0.08 0.005 0.05 23 100 0.4 0.2 0.002 0.02

24 100 04 0.4 0.001 0.01 25 100 0.4 0.6 0.00047 0.0067 26 IO0 004 0.8 0.0005 0.005

analysis with constant feed concentration, and there the longitudinal diffusion coefficient is varied.

RESULTS AND DISCUSSION

We will first consider isolated peaks. From the results in Table II it appears that in sorption chromatography steady state conditions are approached much more slowly than in partition chromatography, Therefore, under ordinary conditions, plots of ,U and ,u2 against time yield curved lines and hence ,the peak velocity Y and spreading coefficient D are variable quantities. However, if the sorption isotherm has a finite slope at the origin, as is the case with Langmuir isotherm, the conditions of partition chromatography are approached as a limit. We will therefore first study the asymptotic behaviour of isolated peaks.

In a column of infinite length the spreading of a peak will cause the concen- tration in the peak to decrease indefinitely. Thus, asf tends to zero eqn. (2) takes the asymptotic form :

Eqns. (I) and (19) may be compared with those of partition chromatography, eyns. (I) and (2) in ref. 6. To make a direct comparison possible we delete the term for

J. Chvomato~., 24 (1966) 39-55

THEORY OF SORPTION CWROMATOCRAPWY. II, 43

longitudinal diffusion in the stationary phase in the latter equations and put V, = I, Then, by identity:

Is2 = 2n2 (20)

and IZlC x; = Y (21)

It then becomes possible to use the exact expressions for peak velocity and peak spreading, which were derived for the partition case, eqns. (36) and (39) in ref. 6. With proper values of the parameters (V, = I and D, = o in the last term in the expression for D) we get:

I

v = --G-c- IfIzV 2 1

and

12 = --3- + -__~%L-j VW (I -v)

RlC 1+--

h2Vl 1222Vl

( Y. + &)'

= Dp + --lz__

2 (23)

These relations are amenable to simple physical interpretations. Thus, 1’ is equal to the fraction of solute in the mobile phase, and is independent of the rate of the sorption reaction (12,/12, is the equilibrium constant), D, on the other hand, is strongly dependent on the reaction rate. For an infinitely fast reaction the chromatographic dispersion vanishes, and the spreading is solely due to longitudinal diffusion in the mobile phase. The spreading coefficient then equals the diffusion coefficient times the fraction of solute in the mobile phase.

In order to show the deviation from asymptotic conditions’for different column characteristics, v and D values were calculated for the matrixes in Table II according to eqns. (22) and (23)) and from finite differences of the data in Table II, according to:

(24)

The results for the mobile phase are listed in Table III. They are expressed in local units (z and vr as units of time and length respectively) and refer to the mid- points of the respective intervals.

The data in Table III show that v generally is rather close to its asymptotic value, whereas for D pronounced deviations occur. The deviations are small if the initial concentration is low, as in matrixes 17 and 18. Also, in the case of large D values the asymptotic conditions are rapidly approached. Then the peak spreads out rapidly and its concentration falls to a level where asymptotic conditions prevail. This is the case in matrixes 12, 13 and 14 where the reaction rate is low and hence D is large. In cases when the concentration in a peak remains high, usually pronounced deviations from asymptotic conditions occur. This happens when the column is overloaded, matrix 19, and also when the reaction rate is high, matrixes ,15 and 16.

J. Chromatog,, 24 (1966) 39-55

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THEORY OF SORPTION CHROMATOGRAPEW. II. 47

However, it should be noted that the use of finite differences in the calculations in- volves an approximation which becomes less satisfactory at high reaction rates (see Appendix). The deviations in the latter case are therefore exaggerated.

VALUl3S OF RELATIVE PEAK VELOCITIES AND SPREADING COEFFICIENTS

For each matrix Y is given in the first row and D in the seconcl.

Matrix Time (in zmits of z) No.

30 75 125 175 CL,

I

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0.02546 0.01884

0.4176 0.1701

1.0000 0.05000

0.9052 0.3102

o-5393 0.5342

o-3147 0.2787

o.094o5 0.044S1

0.1526 0.505s

0.1270 0.2332

O-1437 0.1100

o.IGG~ 0.0805

0.1673 0.1013

0.107s 0.0503

0.1303 0.0632

o..3764 0.5228

0.1287 0.0355

0.1277 o.osgg

0.1267 o.osg7

0.003695

0.00339

o.023GG 0.01752

0.3788 0.1722

1.0000 0.05000

o.sgoo 0.6705

0.4010 0.5468

0.2360 0.23G7

0.07384 0.04033

0.1064 0.3312

0.1136 0.1800

0.1210 0.12gg

0.1284 0.0566

0.1255 0,0867

o.og94 0.0499

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0.1rgr 0.1144 0.0817 0.0799

0.11Q3 0,1x37 0.0840 0.0822

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0.2182 0.22GG

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0.039I5

0.1032 0.3240

0.1081 0.1751

0.1135 0.1ogr

0.1187 0.0531

0.115s 0.053r

0.0973 0.0497

0.x0& 0.0589

0.1619 0.3051

0.1130 0.0845

0.003373 0.00245

0.01854 0.01342

0.3619 0.1717

0.7841 o.g8G5

0.3557 0.5201

o.zogG 0.22zs

o.oG613 0.03882

0.1013 0.3203

0.1053 0.1726

o.1og8 0.1073

0.1140 0.0813

0.1140 0,ogogr 0.0812 0.02124

o.ogG2 o.o4gG

0.1036 0.0583

0.1495 0.2857

o.ogog1 0.03005

O.OgOQI 0.03460

o.ogogs 0.04II4

o.ooIggG o,ooosg5

o.ooggo1 0.004378

0.3333 0.07592

1.0000 0.05000

0.6667 o.G260

0.2857 0.2476

0.1667 0.1oog

o*o5405 0.01376

o.ogogr 0.3051

o.ogogr 0.1548

o.ogogr 0.07968

o.ogogr 0.02600

o.ogogr 0.03460

o.ogog1 0,034Go

0,ogogr 0.03460

J, Clwomatog., 24 (1966) 39-55

4s W. VINIC

Peak asynznzetry The form of the concentration peaks was found to be rather similar in all cases

studied. From the data in Table II it appears that generally p < M. Thus, the peaks exhibit negative skewness (according to Pearson’s measure S = (p - M)/ d,u2), This behaviour can be explained as an effect of the nonlinear sorption isotherm, which has the tendency to compress the leading boundary of a peak. This effect is reduced when asymptotic conditions are approached. It is realized from eqns. (zz) and (23) that in the limit of partition chromatography the operational conditions are symmetric, as the equations are invariant under the reversal of the velocity of the mobile phase. Under these conditions an originally symmetric peak will remain symmetric. Some typical peaks are reproduced in Figs. I, 2 and 3, In Fig. I the peaks both in the mobile and stationary phases are shown, whereas in Figs. 2 and 3 the variation of shape with time is shown.

Co~avergence of i?he natnzerical solattio7as

In order to investigate the dependence of the numerical solutions on the size of the finite differences, some calculations were carried out in which the number of cells for a given length of column was varied. Thus, in the matrixes 22, 23, 24, 25 and 26, the initial peak is confined to 2, 5, IO, 15 and 20 cells, respectively, and the operational conditions of the corresponding chromatographic columns are identical if z is assigned the values 10, 4, 2, 4//3 and I set, respectively. The results for the mobile phase are listed in Table IV in the form of z,u and 72~~ values for two columns of each matrix,

TAl3LE IV

CONVERGENCE OFTWE NUMERICAL SOLUTIONS

ibfairh t (see) Column t/t s/.1 2 V D NO. NO.

22

23 4

24 2

25

2c

412

I

IO 28.336 126,952 20 41.665 244.283

0.1333

26834 115,947 40.305 214.411 0.1347

50 26.633 100 40.092

75 26.595 15“ 40.041

100 26.5S3 200 40.020

116.94G 214.814. 0.1346

117.672 215.939 0.1345

IIS,ZIG ZIG.GsQ 0.1344

0.5867

0.4923

"-4893

O-4913

o-4929

representing the situations at the same time instances. It also contains 1’ and D values, calculated from the differences between the two sets of values according to eqn. (24). Finally, in Fig. 4 the concentration distributions for a peak, resulting from some of these matrixes, are compared. It may be concluded that the convergence of the numerical solutions is quite satisfactory.

J. Clwomalog., 24 (IQGG) 39-55

THEORY OF SORPTION CNROMRTOGRAPWY. II.

Fig. I. Concentration distribution in the mobile and stationary phases. Column j = zoo of matrix No. 2.

Fig, 2. Concentration clistribution in the mobile phase. Columns j = IOO and j = 200 of matrix No. 14.

O 0 50 i

f

3 -

2.

I-

0 0 50 1

Fig. 3. Concentration distribution in the mobile phase of nn overloaded chromatographic column Columns i = IOO ancl j = zoo of matrix No. 19.

Fig. 4. Concentration distributions in the mobile phnse resulting from calculations with finite cliffcrenccs of varying size, The curve represents columnj = 200 of rnatrix 26, filled circles column j= I00 of matrix 24 and unfilled circles column j = 20 of matrix 22.

J. Ckromafog., 24 (1966) 39-55

50 H. VINK

Frontal analysis We will next consider a column fed with a solution of constant concentration.

This case is amenable to a straightforward analytical treatment and has been studied +

by earlier investigators 49 6. We will indicate here a more direct approach where longi- ,:

tudinal diffusion is also taken into account. We start with eqns. (I) and (2) and inves- tigate their solution for a stationary boundary. The existence of such a boundary is guaranteed by the nonlinearity of the sorption isotherm, which makes the movement of different points of the boundary a function of concentration.

Denoting the velocity of the stationary boundary by CO, we may determine it directly from the mass balance equation:

vtV1.fo = cdV1fo + c&h, (25)

where fO and 12, refer to feed concentration and equilibrium concentration off and k, respectively. From (25) and (2) (with dIz/dt = o) we get:

co fo I -=---= ------- (26)

We will next switch to a new coordinate system, which follows the movement of the boundary. Thus we make the transformation:

g=x---cd (27)

Eqns. (I) and (2) take the form:

af a2f af 1 aJJ all

at= _gl---((v-~w)-_--_ --“J-

a62 &f ( T/l at ae >

For a stationary boundary we have:

Hence :

d”f Dl - -

de2

(25)

(30)

(3r)

(32)

A first integration of (31) gives:

(331, df -- D1 d$ (v - w) f + E 7t = I<

J, CJwotnalog., ,24 (IgbG) 39-55

THEORY OF SORPTION CHROMATOGRAPHY. II. s=

For an originally empty column we have:

f’h+o c

hence :

(34)

Ii = 0

Thus, in this case the stationary boundary is determined by the following equations :

df II1 - - (v - w) f + --

ClF w 7L = t-I Vl

(35)

0) y; + hlf (c - IL) - i221L = 0 (36) c,

These equations may be solved directly for D, = o. Then according to (35) :

With (37) and (26), the integration of (36) yields:

f I - = --_-

fo hfrl

1-t-e G-6

(35)

The case D, # o is more troublesome. However, owing to the small value of D,, the solution for D, = o is a good first order approximation. It is therefore pos- sible to solve the full equations by iteration, inserting the approximate solution into the non-linear term in eqn. (36). The resulting linear equations may then be solved by standard methods.

In Table II numerical solutions are given for the case a = o and a = OS. Choosing t = I sec. this gives D, = o and D, = 4’10-5 cm2 set-1 respectively. In Table II only the values of A, are given, They determine the first moment 1~ of the boundary. We have:

clfl=i&i(ft-fft.l-l) =,i~xifa-i~I(i + I)f~+l+,ig~fr+l =iztfr = A0 (39)

Heref, is the constant concentration in the plateau region. In the mobile phase we havef, = IOO, hence ,U = A,/Ioo. From the data in Table II we see that the velocity of a stationary boundary is constant. It has exactly the value predicted by eqn. (26).

* In Fig. 5, the boundaries for the two cases are shown in detail. We see that in the case i. of non-vanishing lbngitudinal diffusion the boundary is not symmetrical. The effect

of diffusion is seen to be rather small, however, and the translational velocity of the boundary remains unaffected. .

J. Chromaiog., 24 (1966) 39-55

52 H.VINK

Fig. 5, Concentration profiles in frontal analysis. Columns j = zoo of msltrixcs Nos. 20 (line) nncl 2 I (filled circles).

~ APPENDIX

Some aspects concerning the errors involved in the np.plication of the finite difference method to chromatography are now considered. First, for the sake of generality, the recursion formulae in partition and sorption chromatography are reformulated on a common basis, and then take the form:

f%j = fr3 4 l/2 a (Js - 1, 3 - * fcj + fz + 1. $) (40)

fz e I.. 3 + 1 = f%j -pi, (41)

h, 3 -I- 1 = hj f 813 (42)

Here the term &j represents the exchange of solute between the mobile and stationary phases in a cell, and is thus determined by the kinetics of the chromato- graphic process. The parameter p has the values V,/V, and I/V, for partition and sorp- tion chromatography respectively. In the case of partition chromatography we get according to eqns. (23)) (24) in ref. 2 :

(here kg3 is the solute concentration in the stationary phase, but is ref. 2).

In sorption chromatography with Langmuir kinetics we eqns. (12)-(16) in ref. I

(43)

designated ~~g’rj in

get according to

(44)

(here, by comparison to ref. I, the indices have been changed for convenience). The object is now to establish the variation with time in the first and second

moments of a concentration distribution and compare the results with exact formulae. Such formulae are available in partition chromatography and the treatment will therefore be restricted to this case only, the results also being valid asymptotically

J. Cirromalog., 24 (1966) 39-55

THEORY OF SORPTION CHROMATOGRAPHY. II. 53

for sorption chromatography. We will consider isolated peaks and, in the formulae below, let all summation limits refer to points on both sides of the peak, in regions of zero concentration. For the second moment at time j + I we then get:

-42, j + 1 = c ww -1- 1 = F (i + I)2 Jz + J., j -I- 1 = (I -q) T (i + I)2 fOi3 + i

To evaluate the first term on the right hand side we substitute for fO+j from eqn. (40) and use the identities:

(i + I>2 = i2 f 2 i -I- 1

(i + I)2 = (i - x)2 + 4 (i - I) + 4

Then :

- Az,jel.= (I - 17) (A2 j + 2 illj 4 14oj + doj) + q/y c (i + I)2 hij (47) i

Using the same procedure we get for the first moment:

Al,j 41= (I - 7) (Al j + Aoj) + v/r zl (i + I) hj (48) i

In general these expressions are dependent on the original concentration distributions (ft,,, hro) and hence become exceedingly complicated for high values of j. However, when the reaction rate is so high that equilibrium between the two phases in a cell is established in an equilibration step, this dependence disappears and the equations take simple forms. We may use eqns. (21)-(24) in ref, 2 and, as then :

m = oo in the expressions for r and 6, we get

r = E, which implies:

Also, as the concentration tributions and put :

6

AOj = I

(49)

in a phase is now constant, we use normalized dis-

(50)

Inserting the last two equations into (47) and (48) we get:

Az,j+l=Azj + (I ---7pJ)(2.41j + I+ 4

Al.j+l =Alj + 1-q

(51)

(52)

For the variance we get:

(~2,j+1=~2,j~L-~21,j+l=CC2;1+~(~-~77) +77(I -7) (53)

J. Clwomatog., 24 (1966) 39-55

54 El. VINX

We are now in the position to write clown expressions for the peak velocity v and spreading coefficient D. In local units they take the form:

Substituting the values of a and q~ and comparing the formulae with eqns. (36) and (39) in ref. 6 we find that no error is involved in the expression for V, and that in the expression for D the term representing longitudinal diffusion is exact, while the chromatographic dispersion is subject to the error 1/z q (I - 7).

This is strictly valid for an infinitely fast equilibration reaction and it is there- fore of interest to consider the error at lower reaction rates. Although a general theoretical analysis of this problem is impracticable, some information may be ob- tained from the numerical data in this paper and in ref. 3* Thus, it appears that for steady state conditions the error in v always is very small and that the error in D generally decreases with decreasing reaction rate. This indicates that 1/z r (I: - 7) represents the upper limit of the error of the chromatographic dispersion. Further the fact that no error is involved in the longitudinal diffusion is of great interest, If pure diffusion is considered, this implied that the finite difference method (Schmidt’s formula, cf. ref. 7) leads to macroscopically correct results (with respect to ,u2, cf. ref. S), This result may be generalised and, with the help of the formula in ref. 8, it may be shown that the result is correct even when the diffusion coefficient is a linear function of concentration.

ACXNOWLEDGEMENTS

The author expresses his gratitude to Mr. T. H~~GLUNG, who carried out the programming and supervised the computations. Financial support from the Swedish Office of Organization and Management and from the Swedish Natural Science Research Council is gratefully acknowledged.

SUMMARY

The operation of a chromatographic column with a sorption reaction following Langmuir kinetics has been simulated by numerical calculations on a digital computer. The operational conditions of the column are varied within wide limits and the results are related to theoretical considerations. It is shown that in sorption chromatography the conditions of partition chromatography are asymptotically approached and the process may then be described by the exact analytical formulae of linear partition chromatography, The errors in the finite difference method are discussed and eval- uated for some special cases.

REFERENC’ES

1 I-1. VINx, J. Clwomalo~., 20 (1965) 496.

2 El. VINIC, J. Clavomalog., 15 (1964) 4SS. 3 H. VINIC, J. Clzvomalog., IS (Ig65) 2.5.

J. Chromalog., 24 1x964) 39-55

THEORY OF SORPTION CNROMATOGRAPW\-, II. 55

4 E. GL~E~I~A~~ AND J. 1. COATES, J. Chm. Sm., (1947) 1315. 5 L. G. SILL$N, Arltiv I&wzi, 2 (1950) 477 zmd 499. Q 1-E. VINIC, J. Clwomalog., 20 (1965) 305. 7 J. CRANK, The Mallaensatics of Di~usio~a. Cl~~rcnclon, Oxford, x956. 8 I-E. VINK, ~~U~ZCYC, (1965) 205 73.

: J. Clwomatog., 24 (19%) 39-55