Chromatographic adsorption with sinusoidal input

CItemicol ~incering Printed in Great

Science. Vol.

40. NO. 11, pp. 2053-2061,

1985. 0

Britain.

CHROMATOGRAPHIC

HUGH

ADSORPTION INPUT

A. BONIFACE’

Chemical Engineering Department, (Received

and DOUGLAS

WITH

OOW-2509185 S3.00+0.00 1985. Pergamon Press Ltd.

SINUSOIDAL

M. RUTHVEN

University of New Brunswick, Fredericton, New Brunswick, Canada

4 September

1984, in revised form 4 February

1985)

Abstract-A chromatographic method based on a steady state sinusoidally varying input concentration has been developed and applied to the analysis of experimental data for some selected weakly adsorbing systems. Axial dispersion coefficients, xeolitic ditiusivities and adsorption equilibrium constants determined by this method were shown to be consistent with the values derived by other methods of analysis. The proposed method is an improvement over pulse or step input chromatography in certain systems because simpler, less accurate measurements are required from experiments and the modelling process is mathematically less complicated.

INTRODUffION

used to study adsorption processes for some time (see, e.g. van Deemter et al., 1956; Sarma and Haynes, 1974). The major disadvantage, however, is that it involves the measurement of transient concentration profiles. In the case of pulse chromatography, the exact shape of the input and output concentration peaks are critical to the analysis of an experiment and the experimental data must therefore be as accurate as possible. This requirement is exacerbated by the fact that greatest accuracy is required at points furthest from the peak mean where, because of noise and drift, the data are generally least accurate. Converting the results to the frequency domain does alleviate this difficulty somewhat (Gangwal et al., 1971), but accurate measurements are still essential. In using frequency response methods to study adsorption, advantage is taken of the steady state nature of the response, which allows compensation for drift and noise and eliminates the weighting effect mentioned above. The main difficulty is in setting up an experimental system which will provide a sufficiently pure sinusoidal concentration function at the adsorption column inlet. The method has been widely used in the study of heat transfer (see, for example, Gunn, 1970) and for the study of axial mixing (McHenry and Wilhelm, 1957; Lites and Geankoplis, 1960; Gottschlich, 1963; Chung and Wen, 1968) but in our review of the literature of this subject we encountered only a few papers in which the method had been applied to the study of mass transfer kinetics (Deisler and Wilhehn, 1953; Gunn and Pryce, 1969, Gunn and England, 1971) and no studies in which the method had been applied to a bi-porous (micropore-macropore) adsorbent. In all previous studies a double acting piston was used to provide a steady flow with a sinusoidally varying concentration. Chromatography

has been

successfully

However the development of electronic flow control valves has made it possible to achieve the required sinusoidal input in a simpler way and in the present study we used a standard Matheson mass flow controller coupled to a low frequency oscillator. MATHEMATICAL

MODEL

To represent the dynamic response of the adsorption system we used a bidisperse pore diffusion model, similar to that used by Hsu and Haynes (1981) but without external film resistance, which has been generally found small for gaseous systems. Parameters included in the model are therefore the equilibrium constant, the axial dispersion coefficient, the macropore diffusivity and the micropore diffusivity. Account was also taken of the distribution of microparticle size since in the experimental study commercial pelleted adsorbents were used. The system is described by the following set of differential equations. In the microparticles:

a24 2%

KS a4

ax2+-=-’ x ax

’

L&at

(1)

boundary conditions:

g

(0, L) = 0

(3)

q(r, t) = KCOI. t) q(x,

0) =

0.

(4)

In the macroparticles:

where:

iY2C,

2 ac,

ay2+

Y3Y

W=

-3r

D K* s

2053

-f?$

O*z

Y

= % SC, o,aL

(r, r) pt’)dr;

(5) (6)

boundary conditions:

2

+Present address: AECL, Chalk River Nuclear Labora-

tories, Chalk River, Ontario, Canada.

+(1

(0, t) = 0

(7)

HUGH

2054

CJR,

I) = C(z,

A.

BONIFACE

and DOUGLAS M.

(8)

I)

w

C,(Y, 0) = 0. In the column: 3(1 - 0,)N

ac,

a2c,

v

i?Z2

-D,az+

where:

RD,

0, ac,

(10)

=- D, at

N=

(13)

C(z, 0) = 0.

(14)

For simplicity these equations have been written for a system in which the average concentration is zero. Since the equations are linear, it can be shown that the solution for a system in which there is a finite mean concentration will be precisely the same (relative to that mean) as the solution for the zero mean case. Assuming a log-normal distribution of microparticle diameters, the solution in Laplace form is riven bv: _

1

%(A--1) Z [

dZ=l+-

(15)

4Y&

(16)

V

3(1 - @,)I& R=v

X

* s

exp[-z*]

-cc.

r*

(@R coth (BR)-

(ar

1)

(17)

coth (UT)- 1) dz

(18)

a* E--zK

(1%

D, r = Texp[J2oz]

(20)

and the frequency domain solution is found by substituting jw for s. The above equations can be separated into real and imaginary parts: g(L,jw) 0

= exp[Re+jIm].

(21)

response theory (see, e.g. Using frequency Coughanowr and Koppel, 1970), by setting w equal to f, the time domain solution at steady state is:

&

(L., f) = exp[ - Re] sin(ft

E = e,+(i

(12)

Co = C(0, t) = C, sin(ft)

g(L,s)=exp

(24) where:

C(o0, t) = 0

V

particular interest are where the dominant attenuating parameter is either axial dispersion or micropore diffusion. In the case where axial dispersion is dominant, the equations for the amplitude ratio and phase lag simplify to: LB, -ln(A)=-vTE2f2 (23)

(11)

boundary conditions:

7=8Ds+

RUTHVEN

-

Im)

(22)

where exp[ - Re] is the amplitude ratio (A) and Im is the phase lag (4). Thus an explicit relationship between the adsorption and system parameters and the column response to a sinusoidal input concentration function is available. We can further investigate some simplified cases. Two physical situations which are of

-e8,)ey+(i

-e,)(i

-e,)K,. (25)

The following simplification, made for the case when micropore diffusion dominates, is based on a Taylor series approximation for the coth term taken to the fourth order. Consequently, it is of use only under limited conditions. -m(A)

=

&Gf*

(26) (27)

where: G = (1 -f&)(1

-0,)K~

exp[Za’J.

(28)

In each case, if -In(A) were plotted against f*, a straight line would indicate the applicability of the assumptions and the slope of the line would yield the mass transport parameter. Note that the amplitude ratio equations produced here are almost identical to certain terms in the equation for the second moment obtained using the same model (Boniface and Ruthven, 1985). This has been noted in more detail by others (e.g. Boersma-Klein and Moulijn, 1979). EXPERIMENTAL

PROCEDURE

experimental apparatus is shown schematically in Fig. 1. A Varian model 3700 gas chromatograph was used as the basis of the system. The helium carrier gas flow was controlled by a Matheson model 6240 mass flow controller (FCl) calibrated in the range 15300 ml/ruin. Argon, oxygen or nitrogen sample gas was added to the carrier gas at the head of the column (60 x 0.77 cm i.d.) which was packed with 40-50 mesh particles. This flow was also regulated by a Matheson model 6240 mass flow controller (FC2), but calibrated in the range 0.1-5 ml/min. A modification was made here so that the control set point could be adjusted externally by means of a small applied voltage. In this way, the low frequency oscillator (HP 3300A function generator) was used to vary the flow of sample gas sinusoidally. A mean sample flow of 1.0 ml/ruin was used and an amplitude of 0.5 ml/min set so that the input sample flow always varied between 0.75 ml/min and 1.25 ml/min. Before connecting the oscillator, the controller response to a step input from 0 to 1.0 ml/min was examined. Internal adjustments to the feedback loop were made to optimize the response. Three standard carrier flows were selected: 30,60 and 150 ml/min. This The

Chromatographic adsorption

2055

Argon

-7 “1

GCColumn i

OxYg

1 I

Time

Fig. 2. Adsorption FC PT TCO -

Flow Controller Procsurc, Trmnsducor ThormaI Conductivity

RV ” Oatmotor

-

Rogutoting Volvo Shut-off Valve

Fig. 1. Schematic diagram of adsorption

apparatus.

meant that the concentration of sample gas was dependant on the carrier flow, but as long as the adsorption was linear, this would not affect the results. The advantage was that the sample flow controller response could be optimized for one flow. The thermal conductivity detector (TCD) output was connected to an HP 59313A analog/digital converter and an HP 83 microcomputer was used to analyse the signal. Initially, the carrier flow and column pressure were measured and recorded. The carrier gas velocity could thus be calculated. From the output signal and the input signal, which was obtained from the sample gas flow transducer, the amplitude ratio and phase lag response of the adsorption column were determined in the following manner. By measuring the input and output signals of the system over at least one full cycle, a maximum and minimum value of each was obtained. Taking differences, amplitudes of the two signals were obtained and an amplitude ratio could be calculated. Although the measurements were not converted to concentrations, a true amplitude ratio could be calculated with one further piece of information. Input and output signals measured at steady state for two different (but constant) sample gas flows provided the required relationship between thetwo signals for a situation where the input and output concentrations must have been the same. To determine the exact frequency of the sample concentration function and the phase lag between the inlet and outlet concentrations, the times when the signals crossed predetermined reference voltages (for a rising concentration) were recorded (see Fig. 2). This was done for three full cycles and the results averaged over those three cycles. Calibration of the system instrumentation and control was done with a “null column”-the standard column was removed and a very short connection made directly between inlet and outlet. The phase lag indicated that the inlet and detector dead volumes were negligible. However the amplitude ratio varied with

+

column input and output signals.

frequency, and in fact varied in direct proportion to frequency but was independent of carrier gas flow and sample gas type. The actual relationship of amplitude ratio to frequency was determined and the resulting calibration curve gave the detector response correction factor at any given frequency. Dividing the observed amplitude ratio in each experiment by the appropriate factor resulted in the true amplitude ratio. To apply the model, some method was needed to compare the amplitude ratio and phase lag calculated from the model with experimental values and adjust the model parameters until agreement was reached. In most cases, where only a single parameter was unknown, the simplest method was adequate (e.g. trial and error or Newton’s method) using a least squares fit test. For estimating transport properties, the phase lag was of little use and the amplitude ratio was used exclusively. The phase lag was used to determine the adsorption equilibrium constant. RESULTS

AND

DISCUSSION

Several different adsorption systems were selected for study based on three criteria: the systems had been studied before and were known to be well behaved, i.e. linear adsorption. the adsorption equilibrium constant was low so that relatively high frequencies and consequently short times were required in the experiments. a range of microparticle diffusivities was required to test the limitations of the method. Table 1 lists the molecular sieves that were studied and the physical properties of the packed columns. Ar-NaX The Ar-NaX system was selected on the basis of our earlier study (Boniface and Ruthven, 1985), as an example of a system in which micropore diffusional resistance should be negligible. Attenuation of the signal should therefore be due entirely to the combined effects of axial dispersion and macropore diffusion. This was born out by the experimental results which are summarized in Table 2. When the results are plotted as - In (A) against f’ (see Fig 3), three straight

2056

A. BOUIFACE

HUGH

and DOUGLASMRCJTHVEN

Table 1. Column Sieve

Column

voidage 6.

Na-X Na mordenite 4A

Table 2. Adsorption

Interstitial velocity u (ems-‘) 11.20 11.20 11.20 11.20 11.20 11.20 4.78 4.78 4.78 4.78 4.78 4.78 2.42 2.42 2.42 2.42

0.41 0.40 0.38

of argon in Na-X

Frequency

Amplitude ratio A

Porosity

packing parameters

radius

standard deviation Q

(mm) 0.17 0.27 0.13

2.34 3.10 2.54

0.81 0.82 0.83

R

0.33 0.31 0.32

An approximate a priori estimate

at 30°C Phase lag

coefficient expression

o.m44

0.0296 0.0606 0.1230 0.2504 0.3781 0.5022 0.02% 0.06% 0.1230 0.1882

may be derived (see, for example, D,

1.00 0.99 0.98 0.94 0.90 0.82 0.99 0.96 0.89 0.60 0.33 0.14 0.97 0.83 0.44 0.16

Log radius

crystalline radius i (ctm)

(racii-‘) 0.0296 0.0605 0.1230 0.2501 0.3774

Mean

Macroparticle

*,

26.0 25.9 25.6 25.7 25.5 25.4 60.9 61.1 61.1 60.7 60.2 59.8 120.6 119.0 119.0 115.7

= DJt& = 0.7 D,,, + vR.

known

(29)

The values of D, calculated from this expression are also given in Table 3 and it is evident that there is remarkably close agreement with the measured values. The adsorption equilibrium constant (K,), calculated from the phase lag, was found to be 3.51 which is also consistent with previous results obtained by pulse chromatography (Boniface, 1983).

lines are obtained_ The values of the axial dispersion coefficient calculated from these slopes are given in Table 3. The third set of dispersion values is obtained when an estimate of the macropore diffusivity is made based on known gas phase and Knudsen diffusivity values, and these results agree with previous results somewhat more closely (Boniface, 1983).

Ar and O2 in Na mordenite These systems were selected, from our earlier studies, as examples of systems in which intracrystalline diffusional resistance may be expected to be small but still measurable in a chromatographic system. The experimental results confirm that this is indeed true. The amplitude ratios for argon are plotted in Fig. 5 and the mass transfer and equilibrium parameters derived by matching the model to the experimental response are summarized in Table 4. In order to avoid calculating more than one diffusion coefficient from the experimental response we set the axial dispersion coefficient equal to the value measured experimentally at the same velocity for the Ar-NaX system, since the difference in molecular diffusivities between Ar-He

2

1.5

.15

.1 f’

Fig. 3. -In(A)

of the dispersion

from the well Ruthven, 1984):

Grad=/.‘>

vs f * for argon in Na-X.

.a

.25

Chromatographic

2057

adsorption

Table 3. Axial dispersion of argon in Na-X Interstitial velocity Y (ems-‘)

Axial dispersion coefficient Dz --In(A)

direct application

vs f2

(D,/R’ 11.20 4.78 2.42

.2

0

I

0

0.305 0.245 0.213

.I

= 00)

.3 f

of model = 25 s- ‘) 0.260 0.241 0.218

0.317 0.253 0.221

.2

(0,/R’

(cm’ s- ‘) from:

.4

[ D,/R= from eq. (29)i 0.275 0.23 0.22

.5

.6


Fig. 4. Experimental and theoretical amplitude ratios for argon in Na-X at 30°C. D, values used to calculate the theoretical lines are given in Table 3.

1-

.8

.5 A <-, .4

.2

IJL cl

Fig. 5. Experimental and theoretical amplitude ratios for argon in Na mordenite at 30°C. D, values used to calculate the theoretical lines are shown.

2058

HUGH

Table 4. Adsorbent

Sample gas Ar

Na-X

A.

BONIFACE

and DOUGLAS

M.

RUIIWEN

Summary of equilibrium constants and diffusivities u (ems-‘)

KS

11.20

3.51

0.26

0.0077 0.0077

(cm%‘)

D

(cm+)

DZC

(cmzs-1)

Ar

4.78 2.42

3.51

0.24 0.22

Na Na Na Na

Ar Ar ::

12.50 5.38 13.10 5.51

9.84 9.84 10.13 10.13

0.26 0.24 0.26 0.24

-

8.1 x 9.2 x 1.9 x 1.2 x

Ar Ar 02 02 N* %

4.95 2.40 4.95 2.50 4.90 2.36

2.62 2.62 2.76 2.76 9.52 9.52

0.24 0.22 0.24 0.22 0.24 0.22

-

2.5 x 2.2 x 10.2 x 9.0 x 3.2 x 3.2 x

mordenite mordenite mordenite mordenite

4A 4A 4A 4A 4A 4A

and O,-He is small. Using realistic values for D,, the macropore diffusion term was found to have only a minor effect on the predicted response so this term was neglected and the best fit values of D, were then determined by trial and error assuming only axial dispersion and micropore resistance. The values of D, for Ar (8-9 x 10es cm2 s-l), found in this way, are in reasonably good agreement with the values obtained previously for the same system by pulse chromatography (11-14 x 10m8 cm2 s-i). The best fit amplitude ratio curves are shown in Fig. 5. The error term given in Table 4 may be used as an indicator of how well (or how poorly) the theory is able to model the system behaviour. There is evidently little difference between the best and worst cases. Ar, O2 and N2 in 4A Diffusion of these gases in 4A zeolite is quite slow so that these systems represent the other extreme in which

-

60

-

Na-X

Error ( x 106)

1;:

lo-* 10-e 10-6 10-e

85 17 240 220

lo-” 10-12 lo- a0 lo- I0 lo-‘= lo- I2

74 290 390 100 150 130

micropore resistance is very high. The variation of the amplitude ratio with frequency, for the Ar4A system, is shown in Fig. 6 together with the theoretical curves calculated using the best fit values of D,, determined in the same way as for the Na mordenite systems. The three theoretical curves shown (for the 2.4 cm s- ’ velocity case) indicate the sensitivity to small changes in the micropore diffusivity. The phase lag data for the 02-4A system are presented in Fig. 7. The agreement between theory and experiment is seen to be good and this was typical of all the systems studied. In contrast to the situation with an Na-X sieve in which the phase lag is highly sensitive to the equilibrium constant, in the case of the 4A sieve the phase lag is quite insensitive to changes in the equilibrium constant. The simple explanation is that the adsorption rate is so small (because of the low micropore diffusivity) that the system is approaching the zero adsorption situation.

1

.8

.6 A <-, .4

2.2xlcr'"sn%

.i

P q

c

tl~mds, 4.95 2.40

2.9xlo-'+sn4. E

Fig. 6. Experimental and theoretical amplitude ratios for argon in 4A at 30°C. D, values used to calculate the theoretical lines are shown.

Chromatographic

adsorption

2059 v (cm/s) 4.95

2.50

A 0

Fig. 7. Experimentaland theoreticalphase lags for oxygen in 4A at 30°C. D, values used to calculate the theoretical lines are shown.

Diffusion in 4A is sufficiently slow that the intracrystalline diffusivities for these systems may be determined directly from uptake rate measurements. Comparative data obtained from gravimetric and volumetric uptake rate studies, as well as from one pulse chromatographic study are summarized in Table 5. The adsorption equilibrium constants (KS) determined in the present study are based on total microparticle (solid) volume and since the adsorbents used contain about 20% inert clay binder the values may be expected to be about 80% of the crystal based equilibrium constants (K,) determined from uptake studies. In fact the values of K, are about 60% of the previously reported K, values and this may be considered as acceptable agreement. However, the intracrystalline diffusivities derived from the present study are much lower than the values derived from the uptake rate measurements. Yucel and Ruthven (1980) showed that the intracrystalline diffusivity is very sensitive to the initial

Table 5. Comparison

of chrotnatograpbic

This study

Ar

KS

DC(cm2s-‘)

2.6

9 x lo-‘”

02

2.76

3.5 x lo-“’

N2

9.5

3.4 x 10-13

pretreatment of the adsorbent and is generally much smaller for pelleted material than for unaggregated zeolite crystals, probably reflecting partial blinding of the surface by binder and/or the effect of severe hydrothermal pretreatment during the pelletization process (see Kondis and Dranoff, 1971). However, the diffusivities obtained here are significantly smaller than even the smallest of the previously reported values. It is possible that the low intracrystalline diffusivities for the present sample arise simply from particularly severe hydrothermal pretreatment during the pelletization of this particular batch of material. On the other hand it has been rather clearly shown (Balow et al., 1982) that the effect of hydrothermal treatment is to reduce the diffusivity of the outer layers of the crystal far more than the centre. In the sinusoidal chromatographic method, with a slowly diffusing sorbate, only the outer surface of the crystal is penetrated on the time scale of the sine wave, so if the crystal is not homogeneous, the observed diffusivity

and gravimetric data for sorption in 4A zeolite at 303 K previous data

K,

4.6

D&m’s-1)

8x lo-”

c or p

Method

Reference

C

grav* chrom.

Ruthven and Derrah (1975) Sarma and Haynes (1974)

4.0

10-t’

4.2

3.6 x 1O-9

C

grav.

Ruthven and Derrah

3.8 x 1.2 x 3.8 x 1.3 x 2.8 x

C

grav. grav. grav. grav.

Ruthven and Derrah (1975) Yucei and Ruthven (1980) Yucel and Ruthven (1980) Yucel and Ruthven (1980) Habgood (1958)

14.9 17.9

lo-” 10-g 1O-‘o lo- I0 lo-‘r

P

C

c P P

vol.

D, = D,/K.. c = crystals. p = pellets, K. is based on total solid volume, including crystals plus binder. K, on crystal volume basis.

(1975)

2060

HUGH

A.

BONIFACE

and DOUGLAS

values will reflect the diffusivity in the outermost layers. By contrast the apparent diffusivity in an uptake experiment will reflect some kind of average value of the diffusivity throughout both the interior and the surface and this value may therefore be considerably higher. CONCLUSlONS

The results of this limited study suggest that the use of sinusoidal input is a practically useful alternative to conventional pulse chromatography for the study of adsorption kinetics and column dynamics. While the adsorption equilibrium constant may be more easily determined by the pulse method, the frequency response method can be used to determine mass transport parameters more easily and over a wider range of values. The method is most useful for weakly adsorbed species since when the equilibrium constant is large (strongly adsorbed species), very low frequencies and therefore very high detector sensitivity and stability would be required. The present limited study has demonstrated that with weakly adsorbed species it is possible to measure intracrystalline diffusional time constants spanning a range of about six orders of magnitude. The technique is perhaps most valuable for the study of relatively fast-diffusing systems such as oxygen in Na mordenite (DC/T2 - Zs- ‘) which cannot be easily studied by conventional uptake rate measurements_ (The maximum value of DC/f which can be reliably measured in an uptake rate experiment is about 10-zs-l.) A major difficulty in the application of this method, which is also encountered in other chromatographic methods, is that the results are not unique and there are generally several combinations of parameters that yield virtually the same amplitude ratio/frequency or phase lag/frequency relationships. The only way to handle this difficulty is by independent determination of a range of reasonable or possible parameter values. This may sometimes be achieved simply by varying system parameters such as particle size or flow velocity. When applied to the slow diffusing 4A systems the method appears to yield diffusivity values representative more of the crystal surface than of the interior. The present results are only preliminary and more detailed experimental studies using a variety of different experimental methods to study the same adsorbent samples would be needed to clarify the behaviour of the 4A systems. Nevertheless it appears that the frequency response method could prove valuable in elucidating the complex differences in diffusivity which have been observed between pellets and unaggregated crystals and in examining whether or not these differences are due to the creation of a low diffusivity region near the crystal surface_ Acknowledgement-We wish to acknowledge the contribution of Dr. J. J. C. Picot of U. N. B. in suggesting the application of frequency response measurements to the measurement of intracrystalline diffusivities.

M.

RUTHVEN

NOTATION

A C

/” Im

j KC K,

N p(r) 4 r 7 R Re S t U V

W x Y ; u w

amplitude ratio gas phase concentration, MLm3 diffusivity, L2T ’ frequency of sinusoid, T- ’ imaginary part of Fourier transform (-1) dimensionless Henry’s law constant based on zeolite crystal volume dimensionless Henry’s law constant based on total solid (microparticle) volume including binder flux defined in eq. (ll), MLp2T-’ probability function of radius adsorbed phase concentration in microparticles, ML--3 microparticle radius, L mean microparticle radius, L macroparticle radius, L real part of Fourier transform Laplace operator, T - ’ time, T interstitial velocity, LT-. ’ superficial velocity, LT - ’ term defined by eq. (6), ML-“T-l microparticle dimension (radial), L macroparticle dimension (radial), L distance along column, L phase lag log-normal crystal radius standard deviation frequency, T - ’

Subscripts x microparticle (zeolite crystal) quantity

Y J

macroparticle quantity chromatographic column quantity

Note. The axial dispersion coefficient (D, ) used here is related to the dispersion coefficient (DL) used by Shah and Ruthven (1977) and by Ruthven (1984) -ording to the relation D, = @,D,. The microparticle diffusivities (D,) used here and in our earlier paper (F5oniface and Ruthven, 1985) are based on a hypothetical fluid phase concentration and are related to the intracrystalline diffusivity (0,) used in earlier studies by 0, = K,D,. 0, is the quantity which is measured directly in an uptake rate experiment. In comparing with other work note that Hsu and Haynes (1981) use 0, while Sarma and Haynes (1974)

use D,. REFERENCES

Boersma-Klein W. and Moulijn .J. A., 1979, Evaluation in

time domain of mass transfer parameters from chromatographic peaks. Chem. Engn@ Sci. 34 959. Bonifacc H. A., 1983, Separation of argon from air by the use of zeolites. Ph.D. thesis, University of New Brunswick. Boniface H. A. and Ruthven D. M., 1985, The use of higher moments to extract transuort data from ChrOIDatOPraDhic adsorption experiments. them. Engng Sci. 40 1401. _ Chung S. F. and Wen C. Y.. 1968, Longitudinal dispersion of liquids unpacked and fluid&d bed& A.Z.Ch.E. 3. 14 857. Coughanowr D. R. and Koppel L. B., 1974 Process Systems

Chromatographic Arr&sis end Control, Part V. p. 211. McGraw-Hill, New York. Deisler P. F. Jr. and Wilhelm R. H., 1953, Diffusion in beds of porous solids, measurement by frequency response techniques. Ind. Engng Chem. 45 1219. Gangwal S. K.. Hudgins R. R., Bryson A. W. and Silveston P. L.. 1971, Interpretation of chromatographic peaks by Fourier analysis. Can. J. Cheat. Engng 49 113. Gottschlich T. F., 1963, Axial dispersion in packed beds. A.Z.Ch.E. J. 9 88. Gunn D. J., 1970, Transient and frequency response of particles and beds of particles. Chem. Engng Sci. 25 53. Gunn D. J. and Pryce C., Dispersion in packed beds. Trans. I. Chem. E. 47 341. Gunn D. J. and Enaland R.. 1971. Disnersion and diffusion in beds of porous -particles. Chekt. E&ng Sci. 26 1413. Habgood H. W., 1958, The kinetics of molecular sieve action, sorption of nitrogen-methane mixtures by Linde molecular sieve 4A. Can. J. C&m. 36 1384. Hsu L.-K. P. and Haynes H. W. Jr., 198 1, Effective diffusivity by the gas chromatography technique. Analysis and application to measurement of diffusion of various hydrocarbons in zeolite Nay. A.1.Ch.E. J. 27 81.

adsorption

2061

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