Electrical Simulator of a Chromatographic Column for Linear Gas-Solid Chromatography Part II. Design of the model Elektrischer Simulator einer chromatographischen Siiule fiir lineare Gas-Test-Chromatographic ModUle ~l~ctrique simulant la marche d'une colonne pour chromatographie lin~aire gaz-solide Partie 2 - Presentation du module M. KoF.i~ik, P. Seidl, J. Dubsk~ Institute of Physical Chemistry of the Czechoslovak Academy of Sciences, Prague, C,SSR

Summary: The paper presents a design for a model of linear adsorption chromatography. Theoretical relations, taking into consideration the general case of adsorption kinetics in the particles of column packing, were described in paper 1. Here the arrangement of the model for a simple case is described, where the adsorption kinetics into a particle is simulated by a series combination of one resistor and capacitor. Controlling the function of the model it was found that the statistical moments calculated from model curves were in good agreement with theoretical values. Further an example is given of simulation of a chromatogram of a two. component mixture for the case where the components do not influence each other description of the instrument. The equipment for simulation of chromatographic curves includes a generator of input function, the analogue network proper, a recorder and a magnetic memory. The block circuit diagram is shown in Fig. 1.

Sommaire: On d6crit, dans ce travail, la construction d'un module simulant la chromatographie d'adsorption liniaire. Les relations thioriques, prenant en consid6ration le cas g6n&al de la cin6tique d'adsorption clans les particules du remplissage de la colonne, ont 6t~ mentionn6es dans la partie 1 du travail. Ici, on d6crit un module pour le cas simple, 06 la cin~tique d'adsorption dans la particule est simul6 par la combinaison d'une r6sistance et d'un condensateur en s6rie. En v6rifiant le fonctionnement du module, on a constat6 un bon accord entre les moments statistiques calcul6s ~ partir des courbes du module et les valeurs th~oriques. En outre on mentionne un exemple de simulation de chromatogramme pour un m61ange binaire dans le cas oh les constituants individuels ne s'influencent pas.

Zusammenfassung: In der vorliegenden Arbeit wird die Konstruktion des Modelles ftir die lineare Gas-AdsorptionsChromatographic beschrieben. Die theoretischen Beziehungen, die den allgemeinen Fall der Kinetik der Adsorption an den Teilchen der Siiulenf'tillung berticksichtigen, wurden in der Arbeit 1 angefithrt. Hier wird die Anordnung des Modells ftir den einfachen Fall, in dem die Kinetik der Adsorption zu den Teilchen durch die Serien-Schaltung eines Widerstandes und eines Kondensators nachgebildet wird, behandelt. Bei der Oberprtifung der Funktion des Modells konnte eine gute quantitative tYoereinstimmung der aus den ModeUkurven berechneten statistischen Momente mit den theoretischen Werten festgestellt werden. Ferner wird ein Beispiel eines Chromatogrammes angefiihrt, das aus einem 2.Komponentengemisch entstand, wobei sich beide Komponenten nicht gegenseitig beeinflufiten.

1. The generator of Input Function The generator of input function enables the simulation of different types of boundary conditions used in gas chromotography. The following boundary conditions were most frequently used: a) Boundary conditions for adsorption in frontal chromatography M (T,0) = 0 for T < T1 M (T,0) = 1 for T > / T t M (T,x) = 0 for T < Tl and for 0 < x < L b) Boundary conditions for desorption in frontal chromatography M (T,0) = 1 for T < T 1 M (T,0) = 0 for T >/Tl M'(To0 = 1 for T < Tl and for 0 < x < L Chrornatographia 3, 1970

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101

c) Boundary conditions for elution chromatography M(T,0) = 0 for T < T l and T > T 2 M (T,0) = 1 for TI --< T-,< T2 M'(T,x) = 0 for T < TI and for 0 < x <

L

where M is the dimensionless concentration of a given component in the gaseous phase. M' is the dimensionless concentration of a given component in the gaseous phase, which at equilibrium would correspond to a given concentration in the adsorbent according to the adsorbtion isotherm x is the coordinate measured in the direction of bed axis L is the length of bed T is model time

The equipment generating the input functions corresponding to the boundary conditions given above is shown in Fig. 2. It consists of the elements of the differential analyser MEDA TA.It contains two integrators 1 and 2 respectively and two differential relays A and B respectively. The output voltages of the integrators U1 = - kl T and U2 = - k2T are compared with a constant voltage U3. Whenever the sums of voltages U1 + U3 or U2 + U3 pass through zero, they activate the relays, which either supply the machine voltage (in our case + 5 V) to the input of the model or ground the model. 2. The Analogue Network

Generator I I I I

I I I I

IL JMczgnetictope~_l -|

memory /

Fig. I. 9

Block diagram of the simulator of chromatograms

9 Blockschema des Simulators tier chromatographischen Kurven 9 Sch6ma du circuit d'un simulateur de courbes chromatographiques

§

The analogue network consits of passive elements and of electronic separating networks. The circuit diagram for tile k-th section of the column is shown in Fig. 3. The elements: In most of the measurements the value of the capacity Cv could be neglected compared with the capacity Ck. The value of the Ck was chosen as 2.229 x 10-s F. See part I The values of the resistors Rx and Rk were changed in the range of 5 to 100 kOhms. See part I All elements were chosen with accuracy of +- 1%. The model consisted of 21 units connected in cascade connexion. The electronic separating networks: " Emitter followers were used. Their construction and properties have been described in detail elsewhere. [2,3] The basic requirement i.e. high input resistance was satisfied by the use of the Darlington's connection. The transfer factor of an emitter follower A = Uout / Uin is always less than unity. In our case it is of no harm, for the form of the studied output voltage curve remains unchanged, it is just reduced by the factor of 0.98 to 0.96. 3. The Recorder Recorders BAKII and BAK were used. The time base was generated by one integrating amplifier of the differential analyser MEDA TA.

Fig. 2

4. The Memory The memory serving for expansion of the capacity of the analogue network was tested in practice. The procedure is such that the output curve taken from the last unit is stored in the memory and the memory is used in the next step as a generator of input function for the analogue network. This process is cyclically repeated until a sufficient number of column sections is reached.

9 Generator of input function 1,2 integrating amplifiers A, B differential relays PI, P2, Ps potentiometers 3 inverter U1,U2,Ua voltages 9 Generator f~ die Eingangsfunktion 1,2 lntegration sverstlirker A, B Dffferentialrelais P1, P2, P3 Potentiorneter 3 Inverter U1, U~, U3 Ziihlsparmung

Simulation The proper function of the model was controlled by checking the agreement of the statistical moments calculated from the measured curves with theoretical values derived by solution of the system of equations (1) and

9 G6n6rateur de fonction d'entr6e 1,2 amplificateurs d'integration A, B relais differentiel Pl, P2, Pa potentiom6tres 3 inverseur U1, U2, Us voltages 102

Chromatographia 3, 1970

(2) 3M + ( 1 - a ) K

au-g-dx

aM'

-gi

0M

at

uAx02M

2

=0 (1)

Originals

~M' 0t - H (M - M')

Uk.]

R~

(2)

where a is the porosity o f bed o ~ Uk-"" - ~ ~

....

u is the linear velocity of the carrier medium Ax is the length o f the imaginary sections o f a column, in the gaseous phase of which thorough mixing is presumed t is real time

z

K is the distribution coefficient Fig. 3 9 Circuit diagram for the k-th section of the column E emitter follower 9 Schaltschema flit den k-ten Kolonnenabschnitt E Emitterfolger

H is the effective coefficient o f mass transfer characterizing the mass transfer from the mobile phase into the particles of the column packing uAx - - ~ = D is the effective coefficient o f longitudinal mixing

9 Sch6ma du circuit pour la ki~me section de la colonne E amplifieateur cathodique

The chromatographic curves are basically characterized by the first normal moment/at' and by the second central moment/a2, which are defined by the equations (3) and (4) respectively :

Y

o Mtdt /~t' = - O

(3)

Mdt

Oo

OO

(4)

/ a : --

~0

Mdt

0

The theoretical terms of these quantities were derived from the equations (1) and (2) by the procedure given in the paper o f E . Kucera [4] and are represented by the equations (5) and (6) respectively t

/at' = L ( K ( 1 - c+~c) t

I

1)

(5)

2DL ( 1 - ~ + 1 ) : +2L K(1-a) 1

(6)

From the transfer function of the four - po!e simulating the first section of the column the following terms for the changes of the moments of model curves A/~t' and A/a2, 3ccuring at the pass of the model curve through the given four - pole may be derived: "1 20 s e c .

Fig. 4

A/7; = Rx (Cv + Ck)

(7)

A/a2 = [Rx (C v -t- Ck)] 2 -]- 2 R x R k Ck 2

(8)

Simulated elution curves measured in the 3-rd, 6-th, 9-th, 12-th, 15-th, 18-th and 21-st section of the column for Rx = 10.0 kS2 and R k -- 10.7 kl2

It should be noted that the moments # t and/a2 are represented in real time units, whereas the moments fit' and ~2 are represented in units of model time.

Modellierte chromatographische Kurven l'fir den Fall der Elutionschromatographic, gemessen im 3, 6 , 9., 12., 15., 18. und 21. Abschnitt der Kolonne fiir R x = 10,0 ks2 und R k = 10,7 k~2

In part I relations were derived for the resistor Rx and the capacitor Cv respectively - see equation (9) and (113) and in the same way the equation (11) may be derived by specifying the impedance Z as a series combination of the resistor Rk and o f the capacitor Ck and by using equation (2) for description o f adsorption kinetics.

Courbes d'61ution sirnul6es mesur6es dans la 3-, 6-, 9-, 12-, 15-, 18-, et 21-i~me section de la colonne pour R x = 10,0 kS2 et Rk = 10,7 kI2

-

Ctttomatographia 3, 1970

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103

,u

Esec]

Further the relation between the effective coefficient of longitudinal mixing D and the length of the column section A x should be noted:

t

2

50

uAx D= ~

(12)

40

?

By substituting equations (9), (10), (11) and (12) into equations (7) and (8) and by taking into consideration that T = Kt t / Kt is the time scale and that the moments

30

20 5 10

[sec2]

6

~ I

I

I

I

I

3

6

9

12

15

I

I

18

21

7

150 -~ mN

Fig. s 9 Dependence o f the first normal moment #i o f the simulated curves on the number of the sections N present in the column. 100

The parameter R x is variable, R k = 18 kft. The solid lines represent the theoretical dependences, the dots represent the measured values. Curve 1 Curve 2 ChLrve 3 Curve 4 Curve 5 Curve 6 Curve 7

corresponds to corresponds to corresponds to corresponds to corresponds to corresponds to corresponds to

the the the the the the the

value value value value value value value

Rx Rx Rx Rx Rx Rx Rx

= = = = = = =

100 82 56 33 20 14 6,2

k~2 kS2 k~2 kS2 k~2 kS2 k~2

50

9 Abh~ingigkeit des ersten normalen Moments ~i der modellierten Kurven an der Zahl der Abschnitte in der Kolonne N. Es Lndert sich der Parameter Rx, R k = 18 kS2. Die yoU ausgezogenen Kurven stellen die theoretischen Abh~ingigkeiten dar, die eingezeichneten Punkte, die MeSwerte. Kurve Kurve Kurve Kurve Kurve Kurve Kurve

1 2 3 4 5 6 7

entspricht entspricht entspricht entspricht entspricht entsprieht entspricht

dem dem dem dem dem dem dem

Werte Werte Werte Werte Werte Werte Werte

Rx Rx Rx Rx Rx Rx Rx

= = = = = = =

100 82 56 33 20 14 6,2

kft kS2 kS2 k~ kS2 ks2 kS2

i 2 3 4 5 6 7

-

pour pour pour pour pour pour pour

Rx Rx Rx Rx Rx Rx Rx

= = = = = = =

100 82 56 33 20 14 6,2

Rk~

104

I

i

I

6

9

12

!

I

15

18

i

21

Curve Curve Curve Curve

1 2 3 4

corresponds to corresponds to corresponds to corresponds to

the the the the

value value value value

Rx Rx Rx Rx

= 100 k a = 82 k f t = 56 kS-Z = 33 k a

9 Abh~_ngigkeit des zweiten Zentralmoments/~2 der modellierten Kurven yon der Zahl der Abschnitte N in der Kolonne. Parameter ist R x, R k = 18 kf~. Die voll ausgezogenen Kurven stellen die theoretischen Abh~ngigkeiten dar, die eingezeichneten Punkte die Megwerte. Kurve Kurve Kurve Kurve

(9)

1 Ck

(10)

K

Kt 1 H Ck

Chromatographia 3, 1970

I

3

9 Dependence o f the second central m o m e n t #2 o f the simulated curves o n the n u m b e r o f the sections N present in the column. The parameter R x is variable, R k = 18 kgZ. The solid lines represent the theoretical dependences, the dots represent the measured values.

ks2 kS2 kS2 kS2 kft ks2 kfZ

Rx - 1 -____~aaA___~XK u Kt ~k t~

0 [-

Fig. 6

9 Relation entre le premier m o m e n t n o r m a l / ~ des courbes simul6es et le nombre de sections (N) de la colonne. Le param6tre R x varie; R k = 18 kS2. Les courbes en traits pleins repr~sentent les relations th6oriques; les courbes en pointiUe repr6sentent les valeurs mesur$es. Courbe Courbe Courbe Courbe Courbe Courbe Courbe

-

(11)

Originals

1 2 3 4

entspricht entsprieht entspricht entspricht

dem dem dem dem

Werte Werte Werte Werte

Rx Rx Rx Rx

= = = =

100 82 56 33

ks2 ka ks2 ks2

9 Relation entre le deuxi~me m o m e n t central/~2 des courbes simul6es et le nombre de sections (N) de la colonne. Le paramgtre R x est variable; R k = 18 k ~ . Les courbes en traits pleins repr~sentent les relations th6oriques; les courbes en pointill6 repr6sentent le s valeurs mesur6es. Courbe Courbe Courbe Courbe

1 2 3 4

-

pour pour pour pour

Rx Rx Rx Rx

= = = =

100 82 56 33

kS2 kS2 ksq ks2

/21' and/a2 are proportional to the ordinal number o f the four - pole k, from which the m o d e l curve is taken [5], we obtain equations corresponding to equations (5) and (6). This agreement shows the i m p o r t a n t fact, t h a t the above described electrical model gives model curves, the first normal m o m e n t and the second central m o m e n t o f which are in agreement w i t h the corresponding moments of the solution of the system o f equations (1) and (2). A n y discrepancy in this line is therefore n o t o f principial character, b u t is a result o f technical imperfection o f the model.

These equations m a y be obtained from equations (7) and (8) b y substituting the value o f 2.229 x 10 -s F for capacity Ck and by multiplication by the ordinal number o f the four-pole k in which the model curve is measured.[2] The capacity Cv has been considered negligible in these calculations. F o r the height o f theoretical plate o f the column HETP the relation (15) derived from equations (13) and (14) t22

HETs =~-----~ L = A

After checking the technical perfection o f the realization it is sufficient to compare the moments o f the m o d e l curves with the m o m e n t s calculated from the equations (13) and (14) respectively #~' = k [ 2 , 2 2 9 . 1 0 -s ] R x

(13)

~2 = k . [ ( 2 , 2 2 9 ) 2 9 10 -1~ [Rx 2 + 2 R k R x ]

(14)

x~ [1

2Rk

(15)

)

An illustration of model curves for elution chromatography is shown in Fig. 4. The figure shows a rectangular pulse supplied to the input o f the column and the individual curves taken from the 3-rd, 6-th, 9-th, 12-th, 15-th, 18.th and 21-st section o f the column respectively. The curves are measured for Rx = 10 k Ohms and Rk = 10.72 kOhms.

HETty8 [crn]

7-4

/

I

0 I~. ~

I

5

/2

!

10

I

15

I

"~, 100

9 Dependence of the height of theoretical plate, HETP, on the quantity I/R x 9R k is parameter. The solid lines represent the theoretical dependences, the dots represent the measured values. Curve 1 corresponds to the value R k = 33 k~2 Curve 2 corresponds to the value R k = 18 k[2 Curve 3 corresponds to the value R k = 10,7 k ~ 9 Abh~ingigkeit der H6he einer theoretisehen Stufe HETP yon der Gr6t~e 1/Rx. La variable est Rk. Les courbes en traits pleins repr6sentent les theoretisehen Abhgngigkeiten dar, die eingezeielmeten Punkte die Megwerte. Kurve 1 entsprieht dem Werte R k = 33 kll Kurve 2 entsprieht dem Werte R k = 18 k~2 Kurve 3 entsprieht dem Werte R k = 10,7 kll 9 Hauteur th6orique (HETP) de l'etage en fonetion de la grandeur 1/R x 9la variable est R k. Les eourbes en traits pleins repr6sentent les relations th6oriques, les courbes en pointill6 tept6sentent les valeurs mesur6es. Coutbe 1 - pour R k = 33 k~2 Courbe 2 - pour R k = 18 k12 Coutbe 3 - pour R k = 10,7 k ~ Chtomatographia 3, 1970

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105

[generator~-- ~ modelRxt [ [ model l

Fig. 8 9 Block diagram of the circuit for simulation of a two-component mixture 9 Blockschaltung zur Modellierung eines Zweikomponentengemisehes

9 Sch6ma du circuit destin~ ] simuler le chromatogramme d'un m~lange binaire

The relation between o f / ~ and ~2 and the n u m b e r of sections (i.e. the length of the column) for different resistors Rx and Rk as well as the corresponding theoretical relations calculated from the equations (13) and (14) are plotted in Fig. 5 and 6 iespeetively.

The dependence of the height of theoretical plate, calculated from model curves, on the q u a n t i t y 1 / Rx is plotted in Fig. 7. This dependence corresponds to the well k n o w n van Deemter's equation 6) for the case when the term inversely proportional to linear velocity is negligible. The parameter of this dependence is the resistance Rk. An example is presented of a case where the two components have different distribution coefficients, b u t the transfer of b o t h materials into grains is simulated b y the same impedance. For the simulation two chains of four poles each comprising ten units differing b y the magnitude of the resistance Rx were connected in parallel. The inputs of b o t h chains were connected in parallel with the generator of a rectangular pulse. The o u t p u t voltages were summed up in an operational amplifier. The circuit diagram is shown in Fig. 8.

M

o,s

Fig. 9 9 Simulated elution curves of a two- component mixture for R k = 10 k~2. Curve 1 Chromatogram of one of the components (Rxl = 56 k~2) Curve 2 Chromatogram of the second component (Rx2 = 6,2 ks2) Curve 3 The resultant ehromatogram of the composition 9 Modelliertes Chromatog~amm eines Zweikomponentengemisches Rk = 10 k~2 Kurve t - Chromatogmmm der einen Komponente (Rxl = 56 kS2) Ktuwe 2 - Chromatogramm der zweiten Komponente (Rx2 = 6,2 kS2) Kurve 3 - resultierendes Chromatogramm des Gemisehes 9 Courbes d'61ution simul~es pour un m61angebinaire (R k = 10 k~2). Courbe 1 - chromatogramme du premier constituant (Rxl = 56 ks2) Courbe 2 - chromatogramme du deuxi~me constituant (Rx2 = 6,2 kS2) Courbe 3 - chromatogramme correspondant au m61ange

I

20sec.

rl

0

'M Fig. 10 9 Simulated elution curves of a two component mixture for Rk = 18 kfL Curve 1 Chromatogram of one of the components (Rxl = 56 k~2) Curve 2 C~omatogram of the second component (Rx2 = 6,2 k~2) Curve 3 The resultant chromatogram of the mixture 9 Modelliertes Chromatogramm eines Zweikomponentengemisehes Rk = 18 ks2. Kurve 1 - Chromatogramm der einen Komponente (Rxl = 56 kS2) Kurve 2 - Chromatogramm der zweiten Komponente (Rx2 = 6,2 ksq) Kurve 3 - tesultierendes Chromatogramm des Gemisches 9 Courbes d'61ution simul~e pour un m61angebinaire (R k = 18 k~2). Courbe 1 - ehromatogramme du premier constituant (Rxl = 56 ks2) Courbe 2 - chromatogramme du deuxi~me eonstituant (Rx2 = 6,2 kS2) Coutbe 3 - chromatogramme eorrespondant au m61ange

,~Osec. I

T

106

Cl~omatogmphia 3, 1970

Originals

-05

M

-O,5

Fig. 11 9 Simulated elution curves of a two-component mixture for R k = 33 ks2. Curve 1 Chromatograrn of one of the components (Rxl = 56 k ~ ) Curve 2 Chromatogram of the second component (Rx2 = 6,2 kl2) Curve 3 The resultant ehromatogram of the mixt~e 9 Modelliertes Chromatograrnm eines Zweikomponentengemisehes Rk = 33 ks2. Kurve 1 - Chromatogramm der einen Komponente (Rxl = 56 k~2) Kurve 2 - Chromatogramm tier zweiten Komponente (Rx2 = 6,2 kS2) Kttrve 3 - resultierendes Chromatogramm des Gemisches 9

Courbes d'~lution simul6e pour un m61ange binaire (R k = 33 kl2). Courbe 1 - ehromatograrnme du premier eonstituant (Rxl = 56 kS2) Courbe 2 - ehromatogzamme du deuxi~me eonstituant (Rx2 = 6,2 k~2) Courbe 3 - chromatogramme correspondant au m61ange

T ~O sec.

Simulation o f the chromatogram o f a two-component mixture for the case when the c o m p o n e n t s do n o t influence each other. The simulated chromatograms o f the t w o - c o m p o n e n t mixture are shown in Fig. 9, 10 and 11. They differ b y the magnitude o f the resistance Rk. Thus they show the effect o f the mass transfer coefficient o n separation o f the components. The curves 1 and 2 show the chromatograms of the individual components while the curve 3 represents the t o t a l chromatogram.

r,

0

m o m e n t o f the central model curve corresponded exactly to the m o m e n t s derived b y the solution o f chromatographic equations including the term o f longitudinal diffusion. To check the technical perfectness o f the model the moments calculated from model curves were c o m p a r e d w i t h their corresponding theoretical dependences and were found to be in good agreement.

Acknowledgements The authors are indebted to Dr. OTTO GRUBNER for his encouragement and interest in this work.

Conclusion The present paper describes the design o f a linear adsorption chromatography model for the special ease when the impedance Z simulating the adsorption kinetics into a particle o f the adsorbent is formed b y series combination of a resistor and a capacitor. Theoretical relations between the statistical m o m e n t s o f m o d e l curves and the number of four-poles connected in cascade were derived. It was found at the same time that the relations derived for the first m o m e n t o f the normal curve and for the second

References [1] M. Ko~i~ik andP. Seidl: Chromatography - sent to print [2] M. Ko~itik: A contribution to simulation of dynamic sorption processes, Postgradual thesis, 1968 [3] M. Ko~iYik andP. Seidl: Automatisation, prepared for print [4] E. Ku~era: J. Chromatog. 19, 237 / 1965 / [5] M. Ko~iiik: J. Chromatog. 30, 459 [ 1967 / [6] J. Z Van Deemter, F. J. Zuiderweg and A. Klinkenberg: Chem. Eng. Sei. 5, 271 / 1965 /

Received: August 1, 1969 Accepted: November 26, 1969

Chromatographia 3, 1970

Originals

107