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Dynamic simulation of gas chromatographic column
Chemical Engineering Science, 1967,
Vol. 22, pp. 1299-1304.PergamonPress Ltd., Oxford. Printed in Great Britain.
Dynamic simulation of gas chromatographic column CHENG-LIANGLNt and J. A. ROTH VanderbiltUniversity,Nashville,Tennessee (Received 25 August 1966; accepted 7 March 1967) Abstract-The dynamic behavior of a gas-liquid chromatographic column was investigated to develop methods for estimation of the pulse response and dead time. A mathematical model describing the column behavior was proposed and the resulting response curves compared with the experimental column behavior. This study indicated that the models based on assumptions of perfect mixingand plug flow could be used to predict the dead time which could represent the residence time of the solute gas through the column if enough subdivisions were used for the simulation. Van Deemter’s model was verified experimentally and adequately describes the gas-liquid chromatographic column in process control computer simulation of the system. The values of the system parameters characterizing the dynamic behavior were obtained from the simulation. The pulse response curve simulation is unique for the estimation of dead time just as is the transient response curve for the steady-state time.
INTRODUCTION To PREDICTthe influence of process variables upon a gas chromatographic column, mathematical models must be developed which simulate the dynamic behavior. Uses of such models are the determination of the length of dead time, the development of transfer functions for process analysis and control and the simulation of interrelated processes in a dynamic computer control system. Gas chromatography involves a homogeneous volatile mixture of n-components to be separated. The mixture is injected into a carrier gas stream which passes through a packed column on which the n-components are separated. Figure 1 illustrates a typical section of a gas-liquid chromatographic column. The mathematical and experimental investigation of the dynamic behavior .of the column is reported. Experimental arbitrary pulse response data were obtained on a packed chromatographic column with helium as carrier gas and normal butane or ethylene as solute gas. The resulting response curves were compared with analytical curves produced by the proposed models in an attempt to simulate the column behavior. The mathematical model produced two sets of partial differential equations which were simulated on an analog computer. The resulting simulations indicated that the models based on assumptions of
perfect mixing, plug flow and isothermal linear equilibrium could be used to predict the dead time and describe the system when enough finite subdivisions are used to describe the column behavior. The simulation results, when compared with the experimental data, showed that the model gave satisfactory agreement. Values of Peclet number, dispersion coefficient, dead time and mass transfer coefficient were found to be 2.87, I.1 ft2/sec, 100 set and O-234 set-‘. These parameters characterize the column dynamic behavior.
I-% Longitudinal dispersion b”,k tlaw W
x
x=0
X
X+AX
X-L
t
Y
FIO. 1. Schematic representation of a gas-liquid chromatographic cohunn.
EXPERIMENTAL PROCEDURE The experimental studies were carried out in a packed column of 4 in. i.d. and 6-ft long copper tube. The substrate employed was Dow Corning 200 silicon oil, 10 % and 15 % by weight. Packings
t Present address: Tennessee State University, Nashville, Tennessee.
1299
cEIBNo-LIANo
LAI and J. A. ROTH
were 30/60 and 80/200 mesh Fisher Columpak with and without substrate. Figure 2 is a schematic of this experimental apparatus. Details of the apparatus are available elsewhere [3]. The experimental pulse response curves were produced by introducing an arbitrary input pulse at the column entrance. Before the input pulse was injected, the column was operated with solute free stream (carrier gas alone) until thermal equilibrium was established. All experiments were performed at room temperature. MATHEMATICALMODEL
The normal mode of gas chromatography is to sweep a mixture of components in a carrier gas through a fixed bed of an absorbent or an inert solid coated with a liquid substrate. A mathematical model was derived for a packed chromatographic column. The mathematical description of the isothermal tied bed accounting for longitudinal dispersion, forced convection and interphase mass transfer was obtained. These equations are given below. The one-dimensional partial differential equation for the gas phase is
For the corresponding phenomena phase the differential equation is
Equations (1) and (2) are based on Van Deemter’s model for gas chromatography [2]. When the interphase mass transfer is negligible, dispersion occurs only in the gas phase. This gives the following equation :
acll_ ax, ,G -’ a2 -ax at D
To simulate the experimental pulse response curves, Eqs. (l), (2) and (3) were proposed. Equations (1) and (2) represent the case where the inert solid packing with liquid substrate was used for the bed while Eq. (3) represents the case where the bed was packed with inactive packing (no liquid substrate). The modelled column was visualized as seven mixed tanks. These equations were used to describe each of the mixed tanks when simulation was made. The simulation results are shown in Fig. 3 for Eqs. (1) and (2); Fig. 4 for Eq. (3). Injected puke
in the liquid Time,
(2)
FIQ. I
Gas
Botterv
I
cylinders
FIG.2. Modified gas-liquid chromatography apparatus.
1300
set
3. Result of analog simulation.
Dynamic simulation of gas chromatogcaphic
as at the ends of the column, the visualized mixtanks have explicit incoming and out-flowing streams, the scale factors on the analog circuit are somewhat different from those for the tanks in between.
Run 6 No.3 Ethylene pure mixing
Injected pulse
column
/
-1
Simk&tign
IO
0
20
Time, FIG.
I 40
30
I 50
I 60
L
set FIG.
4. Simulation result of pure mixing. SIMULATION
The transport phenomena in the gas chromatographic column were investigated by simulation on an analog computer (Applied Dynamics Model AD-2-64-PB). The simulation response was compared with the experimental response to evaluate the model. Since the models are partial differential equations, they are approximated by converting into ordinary differential equations by the finite difference method. The following are the difference equations for the simulation: dC”
A=-
dt
5.
The packed column, well mixed-tank
models.
In preparing a system of equations for analog simulation, time and amplitude are the important factors determining the results. The real system time of this study ranged from 18 to 200 sec. The value of coefficients were computed according to the technique of mixing-tank analog and Tayaor’s first-order approximation (4). Regrouping Eqs. (4) and (5) in the order of C,, C,”and C:, the coefficients of the models were found as follows:
DL(c;-zc:+c:)-*~:-C;) Ax2
(6) (4) ..
The approximation differential-difference equations can be visualized as complete mix-tanks, as shown in Fig. 5 and used for the pulse response simulation. The dynamic performance of the column was then simulated by programming the analog computer, using experimental data as the initial trial-anderror values for the amplitude scale-factors. Thus, the time derivative, divided difference and concentration driving force were all formed at the same length increment of the column. The equations generated for each tank of the subdivided column are shown as Eqs. (4) and (5). However,
dC; k =--$-kc; dt
-
(7>
Pe = LE/D, and a = L/ii. These coefficients were used to determine the scale factors of Eqs. (8) through (13). The derivation of the model is based on the boundary condition such that C(X, t) approaches zero as the distance variable goes to infinity. But this condition does not apply to finite difference-differential equations. Consequently, some modification of the boundary condition is required to enable the model to 1301
CHENQ-LIANGLAI and J. A. ROTH
describe the transport phenomena on both ends of the column. To fuhill the modification, one of the possible approaches is to assume the packed column is analogous to many perfectly mixed tanks and to describe each tank by the same model. Since the model can be shown to be analogous to a series of perfectly mixed-tanks the differentialdifference equations can be derived from the “mixing-tank” system. The packed chromatographic column under consideration was approximated by 7 mixing-tanks, the maximum capability of the analog computer used. Considering Fig. 5, the gas phase of each tank has three incoming streams and three out-flowing streams. A material balance over the center tanks gives,
dCo 8,_ Ql,;
+Q+
v,
dt
Q +-pp
B
#
(8)
-(QI+Q,+QJC:~V,
dCi’/dt= (Q&i-Q&')/V;
(9)
The equations for the corresponding at the entrance of the column are dC;/dt =
(Qt-Q2)CO, -(Q,
dC:/dt
=
i&+(Q2c?,+Q&% (10)
+QdC:/&
(11)
(Q&:-Qsc:)/v,
dC;ldt = (QrC6,+QsC:)/V,
=
-(QJ7,+Q&7,>/v,
(12)
(Q&7,-
(13)
Q~c:>/v,
When the differential-difference equations (6) and (7) are compared to Eqs. (8) through (13), for the mix-tanks the systems are analogous. As an example,
dC”
-2X-
dt
QI,; V
a --E + ( Pep2
_(QI+Q~+Q+o VI3
2j3)
C;+kC;
REWLTS
phenomena
and at exit
dC:/dt
The similarity between Eqs. (6) and (8) is apparent. The sum of the flow terms (Q, + Q2+2Q,)/V,, is analogous to longitudinal dispersion, while the difference of the flow terms (Q, - Q,)V, is analogous to bulk transport and (Q, + Q,)/ VB to the mass transfer between phases. If the coefficients of the concentrations C$ C;, C: and Cp are expressed in terms of the system parameters, a correlation between the flow terms and the parameters can be obtained. At the ends of the column, the correlation differs from that for the tanks in between because of the initial and boundary conditions. With these correlations established, scaling the amplitude factors of Eqs. (8) through (13) on the circuit using experimental data for initial trialand-error values can be made. The experimental input pulse curve was generated on the analog computer by a variable Diode Function Generator [S] and used to simulate the pulse response curve.
B
The simulation results have been replotted in Figs. 3 and 4 showing both experimental and predicted curves. A summary of the dead time and system parameters is given in Tables 1 and 2. Other experimental runs not presented here are available elsewhere [3]. The gas phase flow with no interphase mass transfer is described by Eq. (3). Experiments were made by pulsing ethylene passing through the column packed with inert solid packing alone (i.e. with no liquid substrate). This simulation was accomplished by setting the pots associated with k, the interphase mass transfer coefficient, to zero. As shown in Fig. 4, both responses approximated Gaussian curves. Thus, representation of the axial dispersion phenomena by Eq. (3) for the gas phase is satisfactory. The simulation dead time was found to be 35 sec. An experiment including interphase mass transfer was made by using silicon oil as liquid substrate on the inert solid packing and normal butane as the solute gas. Equations (1) and (2) were used for the simulation. In Fig. 3, the pulse response curve produced by various models are compared with the experimental curve from Run 11, No. 15. This model produced a tailing curve which tended to converge slowly toward the Gaussian curve as the
1302
Dynamic simulation of gas chromatographic column
mass transfer rate approached zero. The simulation dead time was 100 sec. If the interphase mass transfer rate approaches infinity, instantaneous equilibrium is reached. The model Eqs. (1) and (2) also gave a Gaussian curve. Therefore, the mass transfer process to and from the stationary phase can be explained in terms of random walk with resulting Gaussian distribution. The so-called instantaneous equilibrium is assumed but is not reached at high flow rate in practical problems. Accordingly, a term for the finite rate of mass transfer must be retained in the model to describe the system. A consequence of the finite rate of interphase mass transfer is the factor causing the elution curve to tail. TABLE1. EXPERIMENTAL OPERATING CONDITIONS Run 11, No. 15
helium-n-butane
Temperature
at 22°C
Injection time
060 set helium at 4.70 psi normal butane at 5.20 psi loomA helium 68GO cm3/min
Flow rate n-butane 26.00 cm3/min
TABLE2. SIMULATION RESULTS Dead time
Pe
DI,
Model I Run 11, No. 15 1OO~OOsec3.00
1.00
Experimental
k
M
1176Osec
Experimental
4O.w set
Model II Run6,Nos.2,3
35GOsec 9-00
0.87
a/p = a (a/PejF) -(a/2jl)
= 6
(14) (15)
klM=d
(16)
k=c
(17)
where a, b, c and dare the values of the amplitudescale factors on the simulation circuit. M is readily obtained from Eq. (16). Values of Peclet number and k are obtained from Eqs. (15) and (17). For example, from the simulation data of Run 11, No 2, a=7*00, Pe=2*50, k=80040 and M=1*24x 104. These values are consistent with those obtaind by the frequency response method using digital computer techniques [3]. CONCLUSIONS
Injection pressure
Current used
M can be obtained. The equations of the scale coefficients for obtaining the parameters are
0*2200
-
12x 103
-
From the correlations of Eqs. (6) and (7) to Eqs. (8) and (9) and the amplitude factor resulting from simulation, the values of the three parameters characterizing axial dispersion (the Peclet number), gas-liquid mass exchange (the mass transfer coefficient k), and gas-liquid equilibrium constant,
The mathematical model and pulse response technique proposed in this study is useful in the study of packed beds under the conditions of simultaneous axial dispersion and gas-stationary liquid mass transfer. The model tested here is a generalization of the dispersion model used to study mixing pattern in the fluid phase of packed column. The pulse response curve simulation estimated the dead time of the system. The equations for gas-liquid chromatography previously reported by KIZULEMANS [2] have been experimentally verified. Equation (3) excludes the gas-liquid chromatography process. This model sufficiently described single phase fluid dispersion in an inert packed column. The two models, one with and one without chromatography, produced pulse response curves which had a dead time after an input pulse was initiated. The observed dead times were greater than those obtained by analog simulation. This resulted from insuthcient column increments used in the approximation solution of the partial differential equation describing column behavior. However, these models gave a good simulation after a time delay. Simulation dead time was 100 set when the packed column was approximated by subdivision into seven parts. Since the
1303
ckiJINO-LUIW
LAI and J. A. ROTH
simulation is good (except the dead time) for the computer simulation and control applications the models presented here are adequate to describe gas-liquid chromatography. Although the values of the system parameters can be obtained by the model simulation, this simulation cannot mathematically define the precise physical events which contribute to dispersion and mixing. This problem remains unsolved and still requires rigorous mathematical description.
G concentration of solute gas in liquid
CL c:,
ii2
Q,, Q,,
NOTA-DON C,
C:, C+,, C,
phase, mole/unit volume concentration of solute gas in liquid CT phase for difference equation over tanks 1, 7 and central tank, mole/ unit volume DL dispersion coefficient, ft’/sec k mass transfer coefficient, time- ’ column length, ft distribution coefficient, C,/C, axial Peclet number, Lii/D= iit volume flow rate, ft3/sec t time, set ii average interstitial velocity of moving gas, ft/sec VI volume of mixture, ft3 AX axial distance increment, ft a system time, set B dimensionless, Ax/L
concentration of solute gas in gas phase, mole/unit volume concentration of solute gas in gas phase, for difference equation, mole/unit volume
v,,
&l%RENCEs CLEMENTSW. C. JR,, Ph.D. Thesis, Vanderbilt University, June 1963. KEULEMANS A. I. M., Gas Chromatography, Reinhold 1959. LAI C. L., Dynamic Analysis of Gas Chromatography, Ph.D. Thesis, Vanderbilt University, January 1966. MICKLEY H. S., SHERWOODT. K. and REED C. E., Applied Mathematics in Chemical Engineering, McGraw-Hill ROGERSA. E. and CONNOLLY T. W., Analog Computer in Engineering Design, McGraw-Hill 1960. R&surr~&-Le comportement dynamique d’une colonne chromatographique gaz-liquide a et& etudi6 af?n de developper des methodes pour l’estimation de la reponse d’impulsions et du temps mort. Une m&ode mathematique d&rivant le comportement de la colonne a et& proposee et les courbes de reponse. qui en resultent compare-es au comportement de la colonne experimentale. Cette etude indique que l’on pourrait utiliser les mod&s bases sur l’assomption d’un melange parfait et d’un courant a obturation pour determiner le temps mort qui pourrait reprt%enter le temps de residence du gaz solute il travers la colonne, si l’on emploie suffisamment de subdivisions pour la simulation. Le modele de Van Deemter a ete v&f%. experimentalement et d&it de facon adequate la colonne chromatographique gaz-liquide au cours de la simulation du contr6le par ordinateur du systeme. Les valeurs des parametres du systeme qui caracterisent le comportement dynamique ont et& obtenues a partir de la simulation. La simulation de la courbe de reponse des impulsions est unique pour l’estimation du temps mort, ainsi que la courbe de reponse transitoire pour le temps en &at stable. Znsammenfassnng-Das dynamische Verhalten einer chromatographischen Gasfliissigkeits~ule wurde untersucht, urn Methoden zur Absch&ung der Impulsantwort und der Totziet zu entwickeln. Es wurde ein mathematisches Model1 erortert, das das Verhalten der S%ule beschreibt, und die sich ergebenden Antwortkurven wurden mit dem Verhalten der Versuchsslule verglichen. Diese Untersuchung wies darauf hin, dass die auf Annahme perfekter Mischung und Pfropfenstriimung basierenden Modelle zur Vorhersage der Totziet verwendet werden konnten, die die Verweilszeit des in der Lijsung befindlichen Gases in der S&de darstellen kormte, falls genug Unterteilungen fur die Nachbildung verwendet werden. Van Deemters Modell wurde auf dem Versuchswege bestatigt. D&es Modell beschreibt die chromatographische Gasfltlssigkeitss&ule bei Nachbildung des Systems durch einen ProteBrechner ausreichend. Die Werte der System-Parameter, die das dynamische Verhalten kennzeichnen, wurden durch die Nachbildung gewonnen. Die Nachbildung der Impulsantwortkurve ist auf die Schiitzung beschriinkt, ebenso wie die Ubergangsfunktion fti die konstante Totziet.
1304
1963.
Vol. 22, pp. 1299-1304.PergamonPress Ltd., Oxford. Printed in Great Britain.
Dynamic simulation of gas chromatographic column CHENG-LIANGLNt and J. A. ROTH VanderbiltUniversity,Nashville,Tennessee (Received 25 August 1966; accepted 7 March 1967) Abstract-The dynamic behavior of a gas-liquid chromatographic column was investigated to develop methods for estimation of the pulse response and dead time. A mathematical model describing the column behavior was proposed and the resulting response curves compared with the experimental column behavior. This study indicated that the models based on assumptions of perfect mixingand plug flow could be used to predict the dead time which could represent the residence time of the solute gas through the column if enough subdivisions were used for the simulation. Van Deemter’s model was verified experimentally and adequately describes the gas-liquid chromatographic column in process control computer simulation of the system. The values of the system parameters characterizing the dynamic behavior were obtained from the simulation. The pulse response curve simulation is unique for the estimation of dead time just as is the transient response curve for the steady-state time.
INTRODUCTION To PREDICTthe influence of process variables upon a gas chromatographic column, mathematical models must be developed which simulate the dynamic behavior. Uses of such models are the determination of the length of dead time, the development of transfer functions for process analysis and control and the simulation of interrelated processes in a dynamic computer control system. Gas chromatography involves a homogeneous volatile mixture of n-components to be separated. The mixture is injected into a carrier gas stream which passes through a packed column on which the n-components are separated. Figure 1 illustrates a typical section of a gas-liquid chromatographic column. The mathematical and experimental investigation of the dynamic behavior .of the column is reported. Experimental arbitrary pulse response data were obtained on a packed chromatographic column with helium as carrier gas and normal butane or ethylene as solute gas. The resulting response curves were compared with analytical curves produced by the proposed models in an attempt to simulate the column behavior. The mathematical model produced two sets of partial differential equations which were simulated on an analog computer. The resulting simulations indicated that the models based on assumptions of
perfect mixing, plug flow and isothermal linear equilibrium could be used to predict the dead time and describe the system when enough finite subdivisions are used to describe the column behavior. The simulation results, when compared with the experimental data, showed that the model gave satisfactory agreement. Values of Peclet number, dispersion coefficient, dead time and mass transfer coefficient were found to be 2.87, I.1 ft2/sec, 100 set and O-234 set-‘. These parameters characterize the column dynamic behavior.
I-% Longitudinal dispersion b”,k tlaw W
x
x=0
X
X+AX
X-L
t
Y
FIO. 1. Schematic representation of a gas-liquid chromatographic cohunn.
EXPERIMENTAL PROCEDURE The experimental studies were carried out in a packed column of 4 in. i.d. and 6-ft long copper tube. The substrate employed was Dow Corning 200 silicon oil, 10 % and 15 % by weight. Packings
t Present address: Tennessee State University, Nashville, Tennessee.
1299
cEIBNo-LIANo
LAI and J. A. ROTH
were 30/60 and 80/200 mesh Fisher Columpak with and without substrate. Figure 2 is a schematic of this experimental apparatus. Details of the apparatus are available elsewhere [3]. The experimental pulse response curves were produced by introducing an arbitrary input pulse at the column entrance. Before the input pulse was injected, the column was operated with solute free stream (carrier gas alone) until thermal equilibrium was established. All experiments were performed at room temperature. MATHEMATICALMODEL
The normal mode of gas chromatography is to sweep a mixture of components in a carrier gas through a fixed bed of an absorbent or an inert solid coated with a liquid substrate. A mathematical model was derived for a packed chromatographic column. The mathematical description of the isothermal tied bed accounting for longitudinal dispersion, forced convection and interphase mass transfer was obtained. These equations are given below. The one-dimensional partial differential equation for the gas phase is
For the corresponding phenomena phase the differential equation is
Equations (1) and (2) are based on Van Deemter’s model for gas chromatography [2]. When the interphase mass transfer is negligible, dispersion occurs only in the gas phase. This gives the following equation :
acll_ ax, ,G -’ a2 -ax at D
To simulate the experimental pulse response curves, Eqs. (l), (2) and (3) were proposed. Equations (1) and (2) represent the case where the inert solid packing with liquid substrate was used for the bed while Eq. (3) represents the case where the bed was packed with inactive packing (no liquid substrate). The modelled column was visualized as seven mixed tanks. These equations were used to describe each of the mixed tanks when simulation was made. The simulation results are shown in Fig. 3 for Eqs. (1) and (2); Fig. 4 for Eq. (3). Injected puke
in the liquid Time,
(2)
FIQ. I
Gas
Botterv
I
cylinders
FIG.2. Modified gas-liquid chromatography apparatus.
1300
set
3. Result of analog simulation.
Dynamic simulation of gas chromatogcaphic
as at the ends of the column, the visualized mixtanks have explicit incoming and out-flowing streams, the scale factors on the analog circuit are somewhat different from those for the tanks in between.
Run 6 No.3 Ethylene pure mixing
Injected pulse
column
/
-1
Simk&tign
IO
0
20
Time, FIG.
I 40
30
I 50
I 60
L
set FIG.
4. Simulation result of pure mixing. SIMULATION
The transport phenomena in the gas chromatographic column were investigated by simulation on an analog computer (Applied Dynamics Model AD-2-64-PB). The simulation response was compared with the experimental response to evaluate the model. Since the models are partial differential equations, they are approximated by converting into ordinary differential equations by the finite difference method. The following are the difference equations for the simulation: dC”
A=-
dt
5.
The packed column, well mixed-tank
models.
In preparing a system of equations for analog simulation, time and amplitude are the important factors determining the results. The real system time of this study ranged from 18 to 200 sec. The value of coefficients were computed according to the technique of mixing-tank analog and Tayaor’s first-order approximation (4). Regrouping Eqs. (4) and (5) in the order of C,, C,”and C:, the coefficients of the models were found as follows:
DL(c;-zc:+c:)-*~:-C;) Ax2
(6) (4) ..
The approximation differential-difference equations can be visualized as complete mix-tanks, as shown in Fig. 5 and used for the pulse response simulation. The dynamic performance of the column was then simulated by programming the analog computer, using experimental data as the initial trial-anderror values for the amplitude scale-factors. Thus, the time derivative, divided difference and concentration driving force were all formed at the same length increment of the column. The equations generated for each tank of the subdivided column are shown as Eqs. (4) and (5). However,
dC; k =--$-kc; dt
-
(7>
Pe = LE/D, and a = L/ii. These coefficients were used to determine the scale factors of Eqs. (8) through (13). The derivation of the model is based on the boundary condition such that C(X, t) approaches zero as the distance variable goes to infinity. But this condition does not apply to finite difference-differential equations. Consequently, some modification of the boundary condition is required to enable the model to 1301
CHENQ-LIANGLAI and J. A. ROTH
describe the transport phenomena on both ends of the column. To fuhill the modification, one of the possible approaches is to assume the packed column is analogous to many perfectly mixed tanks and to describe each tank by the same model. Since the model can be shown to be analogous to a series of perfectly mixed-tanks the differentialdifference equations can be derived from the “mixing-tank” system. The packed chromatographic column under consideration was approximated by 7 mixing-tanks, the maximum capability of the analog computer used. Considering Fig. 5, the gas phase of each tank has three incoming streams and three out-flowing streams. A material balance over the center tanks gives,
dCo 8,_ Ql,;
+Q+
v,
dt
Q +-pp
B
#
(8)
-(QI+Q,+QJC:~V,
dCi’/dt= (Q&i-Q&')/V;
(9)
The equations for the corresponding at the entrance of the column are dC;/dt =
(Qt-Q2)CO, -(Q,
dC:/dt
=
i&+(Q2c?,+Q&% (10)
+QdC:/&
(11)
(Q&:-Qsc:)/v,
dC;ldt = (QrC6,+QsC:)/V,
=
-(QJ7,+Q&7,>/v,
(12)
(Q&7,-
(13)
Q~c:>/v,
When the differential-difference equations (6) and (7) are compared to Eqs. (8) through (13), for the mix-tanks the systems are analogous. As an example,
dC”
-2X-
dt
QI,; V
a --E + ( Pep2
_(QI+Q~+Q+o VI3
2j3)
C;+kC;
REWLTS
phenomena
and at exit
dC:/dt
The similarity between Eqs. (6) and (8) is apparent. The sum of the flow terms (Q, + Q2+2Q,)/V,, is analogous to longitudinal dispersion, while the difference of the flow terms (Q, - Q,)V, is analogous to bulk transport and (Q, + Q,)/ VB to the mass transfer between phases. If the coefficients of the concentrations C$ C;, C: and Cp are expressed in terms of the system parameters, a correlation between the flow terms and the parameters can be obtained. At the ends of the column, the correlation differs from that for the tanks in between because of the initial and boundary conditions. With these correlations established, scaling the amplitude factors of Eqs. (8) through (13) on the circuit using experimental data for initial trialand-error values can be made. The experimental input pulse curve was generated on the analog computer by a variable Diode Function Generator [S] and used to simulate the pulse response curve.
B
The simulation results have been replotted in Figs. 3 and 4 showing both experimental and predicted curves. A summary of the dead time and system parameters is given in Tables 1 and 2. Other experimental runs not presented here are available elsewhere [3]. The gas phase flow with no interphase mass transfer is described by Eq. (3). Experiments were made by pulsing ethylene passing through the column packed with inert solid packing alone (i.e. with no liquid substrate). This simulation was accomplished by setting the pots associated with k, the interphase mass transfer coefficient, to zero. As shown in Fig. 4, both responses approximated Gaussian curves. Thus, representation of the axial dispersion phenomena by Eq. (3) for the gas phase is satisfactory. The simulation dead time was found to be 35 sec. An experiment including interphase mass transfer was made by using silicon oil as liquid substrate on the inert solid packing and normal butane as the solute gas. Equations (1) and (2) were used for the simulation. In Fig. 3, the pulse response curve produced by various models are compared with the experimental curve from Run 11, No. 15. This model produced a tailing curve which tended to converge slowly toward the Gaussian curve as the
1302
Dynamic simulation of gas chromatographic column
mass transfer rate approached zero. The simulation dead time was 100 sec. If the interphase mass transfer rate approaches infinity, instantaneous equilibrium is reached. The model Eqs. (1) and (2) also gave a Gaussian curve. Therefore, the mass transfer process to and from the stationary phase can be explained in terms of random walk with resulting Gaussian distribution. The so-called instantaneous equilibrium is assumed but is not reached at high flow rate in practical problems. Accordingly, a term for the finite rate of mass transfer must be retained in the model to describe the system. A consequence of the finite rate of interphase mass transfer is the factor causing the elution curve to tail. TABLE1. EXPERIMENTAL OPERATING CONDITIONS Run 11, No. 15
helium-n-butane
Temperature
at 22°C
Injection time
060 set helium at 4.70 psi normal butane at 5.20 psi loomA helium 68GO cm3/min
Flow rate n-butane 26.00 cm3/min
TABLE2. SIMULATION RESULTS Dead time
Pe
DI,
Model I Run 11, No. 15 1OO~OOsec3.00
1.00
Experimental
k
M
1176Osec
Experimental
4O.w set
Model II Run6,Nos.2,3
35GOsec 9-00
0.87
a/p = a (a/PejF) -(a/2jl)
= 6
(14) (15)
klM=d
(16)
k=c
(17)
where a, b, c and dare the values of the amplitudescale factors on the simulation circuit. M is readily obtained from Eq. (16). Values of Peclet number and k are obtained from Eqs. (15) and (17). For example, from the simulation data of Run 11, No 2, a=7*00, Pe=2*50, k=80040 and M=1*24x 104. These values are consistent with those obtaind by the frequency response method using digital computer techniques [3]. CONCLUSIONS
Injection pressure
Current used
M can be obtained. The equations of the scale coefficients for obtaining the parameters are
0*2200
-
12x 103
-
From the correlations of Eqs. (6) and (7) to Eqs. (8) and (9) and the amplitude factor resulting from simulation, the values of the three parameters characterizing axial dispersion (the Peclet number), gas-liquid mass exchange (the mass transfer coefficient k), and gas-liquid equilibrium constant,
The mathematical model and pulse response technique proposed in this study is useful in the study of packed beds under the conditions of simultaneous axial dispersion and gas-stationary liquid mass transfer. The model tested here is a generalization of the dispersion model used to study mixing pattern in the fluid phase of packed column. The pulse response curve simulation estimated the dead time of the system. The equations for gas-liquid chromatography previously reported by KIZULEMANS [2] have been experimentally verified. Equation (3) excludes the gas-liquid chromatography process. This model sufficiently described single phase fluid dispersion in an inert packed column. The two models, one with and one without chromatography, produced pulse response curves which had a dead time after an input pulse was initiated. The observed dead times were greater than those obtained by analog simulation. This resulted from insuthcient column increments used in the approximation solution of the partial differential equation describing column behavior. However, these models gave a good simulation after a time delay. Simulation dead time was 100 set when the packed column was approximated by subdivision into seven parts. Since the
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LAI and J. A. ROTH
simulation is good (except the dead time) for the computer simulation and control applications the models presented here are adequate to describe gas-liquid chromatography. Although the values of the system parameters can be obtained by the model simulation, this simulation cannot mathematically define the precise physical events which contribute to dispersion and mixing. This problem remains unsolved and still requires rigorous mathematical description.
G concentration of solute gas in liquid
CL c:,
ii2
Q,, Q,,
NOTA-DON C,
C:, C+,, C,
phase, mole/unit volume concentration of solute gas in liquid CT phase for difference equation over tanks 1, 7 and central tank, mole/ unit volume DL dispersion coefficient, ft’/sec k mass transfer coefficient, time- ’ column length, ft distribution coefficient, C,/C, axial Peclet number, Lii/D= iit volume flow rate, ft3/sec t time, set ii average interstitial velocity of moving gas, ft/sec VI volume of mixture, ft3 AX axial distance increment, ft a system time, set B dimensionless, Ax/L
concentration of solute gas in gas phase, mole/unit volume concentration of solute gas in gas phase, for difference equation, mole/unit volume
v,,
&l%RENCEs CLEMENTSW. C. JR,, Ph.D. Thesis, Vanderbilt University, June 1963. KEULEMANS A. I. M., Gas Chromatography, Reinhold 1959. LAI C. L., Dynamic Analysis of Gas Chromatography, Ph.D. Thesis, Vanderbilt University, January 1966. MICKLEY H. S., SHERWOODT. K. and REED C. E., Applied Mathematics in Chemical Engineering, McGraw-Hill ROGERSA. E. and CONNOLLY T. W., Analog Computer in Engineering Design, McGraw-Hill 1960. R&surr~&-Le comportement dynamique d’une colonne chromatographique gaz-liquide a et& etudi6 af?n de developper des methodes pour l’estimation de la reponse d’impulsions et du temps mort. Une m&ode mathematique d&rivant le comportement de la colonne a et& proposee et les courbes de reponse. qui en resultent compare-es au comportement de la colonne experimentale. Cette etude indique que l’on pourrait utiliser les mod&s bases sur l’assomption d’un melange parfait et d’un courant a obturation pour determiner le temps mort qui pourrait reprt%enter le temps de residence du gaz solute il travers la colonne, si l’on emploie suffisamment de subdivisions pour la simulation. Le modele de Van Deemter a ete v&f%. experimentalement et d&it de facon adequate la colonne chromatographique gaz-liquide au cours de la simulation du contr6le par ordinateur du systeme. Les valeurs des parametres du systeme qui caracterisent le comportement dynamique ont et& obtenues a partir de la simulation. La simulation de la courbe de reponse des impulsions est unique pour l’estimation du temps mort, ainsi que la courbe de reponse transitoire pour le temps en &at stable. Znsammenfassnng-Das dynamische Verhalten einer chromatographischen Gasfliissigkeits~ule wurde untersucht, urn Methoden zur Absch&ung der Impulsantwort und der Totziet zu entwickeln. Es wurde ein mathematisches Model1 erortert, das das Verhalten der S%ule beschreibt, und die sich ergebenden Antwortkurven wurden mit dem Verhalten der Versuchsslule verglichen. Diese Untersuchung wies darauf hin, dass die auf Annahme perfekter Mischung und Pfropfenstriimung basierenden Modelle zur Vorhersage der Totziet verwendet werden konnten, die die Verweilszeit des in der Lijsung befindlichen Gases in der S&de darstellen kormte, falls genug Unterteilungen fur die Nachbildung verwendet werden. Van Deemters Modell wurde auf dem Versuchswege bestatigt. D&es Modell beschreibt die chromatographische Gasfltlssigkeitss&ule bei Nachbildung des Systems durch einen ProteBrechner ausreichend. Die Werte der System-Parameter, die das dynamische Verhalten kennzeichnen, wurden durch die Nachbildung gewonnen. Die Nachbildung der Impulsantwortkurve ist auf die Schiitzung beschriinkt, ebenso wie die Ubergangsfunktion fti die konstante Totziet.
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1963.