The performance of molecular sieve adsorption columns: non-isothermal systems

Chemical Engineering Science. 1975, Vol. 30, pp. X03-810.

Pergamon Press.

Printed in Great Britain

THE PERFORMANCE OF MOLECULAR SIEVE ADSORPTION COLUMNS: NON-ISOTHERMAL SYSTEMS-t D. M. RUTHVEN, D. R. GARGS and R. M. CRAWFORDO Department of Chemical Engineering, University of New Brunswick, Fredericton, New Brunswick, Canada (Received 7 August 1974) Abstract-A simple theoretical modelis used to calculatedimensionlessbreakthroughcurvesand the corresponding temperatureprofilesfor a non-isothermaladsorptioncolumn, operating under constant pattern conditions. The model

involves three parameters (1,which measures the relative rates of heat and mass transfer, b, which is a heat capacity term and A,which measures the degree of non-linearity of the equilibrium isotherm. The effects of these parameters on the form of the concentration and temperature profiles is investigated. Experimental breakthrough curves for the non-isothermal sorption of propylene, cis-2-butene and I-butene from an inert carrier gas, in a column packed with 5A molecular sieve, are analyzed and interpreted. The heat and mass transfer coefficients, calculated from the experimental curves, are shown to be consistent with independently measured values and conclusions are drawn cobceming the nature of the controlling resistances. INTRODUCTION

at an essentially constant temperature but in which,

A general solution to the problem of predicting the performance of a fixed bed adsorption column would require the simultaneous solution of the appropriate mass transfer rate equation and the differential energy balance equation, subject to the boundary conditions imposed by the equilibrium isotherm and the differential fluid phase mass balance. The complexity of this problem is such that a general solution is impracticable and the solutions which have so far been obtained are mostly limited to the special cases of either isothermal operation, in which the energy balance vanishes, or adiabatic operation in which the energy balance assumes a simplified form. In previous papers[l-31 we have considered, in some detail, the isothermal problem for a molecular sieve column. The assumption of isothermal behaviour is a fair approximation when the adsorbable components are weakly adsorbed or present only at very low concentrations but in practice one is often concerned with the sorption of strongly adsorbed species at sufficiently high concentrations for thermal effects to be important. The case of adiabatic sorption has been considered by Leavitt[4] and by Pan and Basmadjian[S, 61 among others. Industrial adsorption columns are, however, generally operated with appreciable heat loss at the wall so that conditions are intermediate between the isothermal and adiabatic cases. Some discussion of such systems has been given by Weber et a1.[7,8] but their work was limited to nearly adiabatic systems in which the column was well lagged with only a small heat loss. The present paper is concerned with systems which approach more nearly the isothermal case. As a simple idealization we consider a system in which the column wall is maintained

because of finite heat transfer resistance, there are appreciable temperature gradients in the region of the mass transfer zone. The results of a simple generalized theoretical analysis of this problem are first presented and the model is then used to interpret experimental breakthrough curves for some simple systems for which the assumptions of the theory are approximately fulfilled. THEOREllCAL

In order to simplify the theoretical analysis the following approximations which are reasonable for many practical systems are introduced. (1) It is assumed that there is plug flow with negligible axial dispersion. This is a valid approximation for a packed bed at moderate Reynolds numbers although at very low Reynolds numbers axial dispersion may become significant. (2) Pressure drop through the column is neglected and the interstitial velocity is assumed constant. (This approximation becomes invalid at high sorbate concentrations.) (3) The mass transfer rate is represented by a linear driving force equation:

$=ka(q*-q).

tThis paper, which is based on the B.Sc. thesis of Mr. R. M. Crawford, was presented at the 24th Canadian Chemical Engineering Conference, Ottawa, 23 Oct. 1974. SPresent address: Molecular Sieves Division, Union Carbide Corporation Technical Centre, Tarrytown, New York, U.S.A. PPresent address: DuPont of Canada Ltd., Kingston, Ontario, Canada.

This form of equation (or the equivalent expression in terms of fluid phase concentration) is correct when external fluid film resistance is dominant but if the mass transfer rate is controlled by intraparticle diffusion then the mass transfer coefficient ka must be considered as a lumped parameter. This has been discussed by Glueckauf[9] who showed that, for linear systems ka = 15D/r*. Equivalent relationships for non-linear systems have been recently developed [ lo]. (4) The mass transfer coefficient ka is assumed to be independent of temperature. This is a fair assumption if external film resistance is rate controlling but for intraparticle diffusion control it is a severe approximation.

803

D. M. RUTHVEN et al.

804

(5) The temperature is assumed to be uniform across any section of the column with negligible temperature difference between gas and pellet but, in the region of the adsorption zone, a significant temperature difference between gas and column wall. This implies a high effective bed thermal conductivity and a high rate of heat transfer between gas and pellets with all the thermal resistance, characterized by an overall heat transfer coefficient h, at the inner surface of the tube wall. The assumption that the temperature difference between gas and wall is significantly greater than the temperature difference between gas and pellet can be approximately justified if similarity of i factors is assumed. However the radial temperature difference between the centre. of the bed and the wall is often comparable with the temperature difference at the wall so the assumption of a uniform temperature across the bed may be a severe approximation. (6) It is assumed that heat transfer at the external surface of the column wall is sufficiently rapid to maintain the wall at a uniform temperature. This is a good approximation for a thermostatted experimental column and under certain conditions it may be approximately true for an unlagged industrial column. (7) We assume that the equilibrium isotherm may be represented by a Langmuir equation: q*_ -_qs

bc

1 t bc

balance equation i-6

aq’

( >

$tv$t

-y-

x=0

(7)

reduces simply to [ 111:

03) In dimensionless form the mass transfer rate equation (Eqn. 1) may be written: (9) where T = kat, IJ = q/q0 and $I* = q*/qo. Eliminating $ and $I* between Eqs. (2), (8) and (9) we obtain:

d4

W-4)

iG=YjTipG

(10)

For moderate differences in temperature Eq. (3) may be approximated by:

(11) (2) so that, with dimensionless temperature defined by:

with the temperature dependence of the equilibrium constant b given by the usual van’t Hoff relationship: b = b. exp (-AHIRT) (AH independent of sorbate concentration).

(3)

With these approximations the heat balance for a differential element of the column may be written: (I-r)(-AH)~=ogc~~~t(l-~)p’c~~~

e’g+To) we have 1 ee -=-= bco boco where A = q&5.

(4)

The dimensionless rate equation (Eq. 10)thus becomes: (14) For a reasonable length of column and with a favourable equilibrium isotherm, constant pattern behaviour is established; that is both the temperature and concentration fronts move at a uniform velocity with no further change in shape. Under these conditions:

while the heat balance (Eq. 6) may be written:

gf=pg+ae

(15)

where lpc, t(l-E)P’C:,

so that the heat balance becomes:

(16) ’ =

qb(l - c)R(+?*

and t!y(T-

To).

(6)

In the constant pattern limit the differential fluid mass

4h ’ = kaqbd(l- e)R(&?*’

(17)

805

The performanceof molecularsieve adsorptioncolumns: non-isothermalsystems The generalized constant pattern breakthrough curve (4 vs 7) and the corresponding dimensionless temperature curve (fl vs T) may thus be obtained, in terms of the parameters a, /3 and A, from the simultaneous solution of Eqs. (14) and (15) with the initial condition C#= 0, fl = 0, d4/d7 = 0. For negative values of /l the temperature front leads the concentration front and under these conditions it proved simpler to start the integration from the final condition C#I = 1, 0 = 0, d+/dT = 0. Solutions were obtained numerically, for various values of the parameters, using a fourth order Runge-Kutta scheme[l2] and the results of these calculations are shown in Figs. l-3. In the isothermal case (a + 00)an analytical solution is readily obtained since Eq. (1) may be integrated directly with Eqs. (2) and (8) to give[ll]: 72-7,=fln[~(~)]tln(~).

0.6

-

0.6. 9 0.4.

0.2

-

(18)

For large values of (Ythe numerically generated curves agreed well with Eq. (18) confhming the validity of the numerical scheme. From Fig. 1 it may be seen that as Q decreases, approaching the adiabatic case, the temperature rise becomes increasingly large and there is a corresponding increase in the spread of the breakthrough curve. Thus, for a system in which (Yis small, a column designed on the assumption of isothermal operation would give premature breakthrough. The effect of the parameter /3 is illustrated in Fig. 2. This parameter controls the relative positions of the temperature and concentration fronts and the shape of the temperature curve and has only a minor influence on the spread of the breakthrough curve. A negative value of p means that the heat capacity of the gas is greater than

-12

-8

-4

0

4

6

12

Fig. 2. Theoretical concentration temperature curves showing the effect of fi (a = 0.05, A = 0.67).

06 + I 2 3 4 5

0.4

0.2

I.0

0167 0.287 0,445 0.667 0,800

0.8 1.6

0.6 +

I ,2

0.4 .9

0.8

I.6

-24

-16

-8

0

8

I6

24

T-h

Fll. 3. Theoreticalconcentrationand temperaturecurvesshowing the effect of A ((I = 0.05, fi = 0).

Fig. 1. Theoretical concentration and temperature curves showing the effect of (1(fl = 0, A = 0.67). CES Vol. 30, No. 6-C

that of the sieve so that the temperature front leads the concentration front while a positive value of B corresponds to the reverse situation. The effect of A, the non-linearity parameter, is shown in Fig. 3. For given values of a and r9, as A decreases, approaching a linear isotherm, the spread of the constant pattern breakthrough curve increases, while the maximum temperature rise decreases.

806

D. M. RLITHVEN et

al.

(16) becomes:

EXPERIMENTAL

Breakthrough curves and the corresponding temperature curves were measured experimentally for sorption of propylene, I-butene and cis-2-butene from an inert carrier in a column packed with $’ Linde 5A molecular sieve pellets. The apparatus and procedure were essentially the same as in the previously reported isothermal studies of Garg and Ruthven[21 except that an additional potentiometric recorder was incorporated to monitor continuously the thermocouple outputs. The column was enclosed in a

circulatingair thermostatwhichmaintainedthe temperatureof the column wall essentially constant. The existence of a constant pattern type of temperature front in which the shape of the temperature-time curve is independent of axial position was confirmed in preliminary studies with the helium-butane system[l3]. However, for the main part of the present study olefinic sorbates were used since these have a somewhat higher heat of sorption than the corresponding paraffins leading to more pronounced nonisothermal effects. Details of the column and adsorbent are given in Table I while the conditions of the experimental runs are summarized in Table 2. For comparison with the theory one requires simultaneous measurements of both the temperature and concentration at the column outlet. It is not possible to obtain a reliable measurement of the temperature at the end of the column so the thermocouple had to be placed about three inches into the bed. Attempts to determine the time of the experimental temperature curve relative to the breakthrough curve by using another thermocouple in the exit gas stream were not entirely successful, probably because of the significant time delay associated with the response of the thermocouples, which were enclosed in a thermowell within the column.

(20)

The heat capacity of the adsorbent (p’c: = 0.234 cal/crn3) was taken from Breck[l6] while the values of AH were obtained from the gravimetric equilibrium data[l4]. (C,&, AH = - 10 kcallmole; 1-butene and cis-2-butene, AH = - 13k&mole). The values of a and the corresponding overall heat and mass transfer coefficients (h and ka) are more difficult to estimate a priori. A trial and error procedure was therefore employed to derive these parameters from the experimental breakthrough curves. For each run a series of curves was computed for the known values of A and fl and a range of CYvalues. The value of a which gave the correct maximum temperature rise was then selected and the value of ka was found by matching the experimental time for lo-90 per cent breakthrough to the dimensionless theoretical time. The corresponding value of h was then obtained from Eq. (17). Values of the parameters and the experimental heat and mass transfer coefficients are summarized in Table 2. In some of the earlier experiments

Table 1. Details of column and adsorbent 91.5 cm

Packed length:

3.81 cm

i.d. Cd)

Linde 5A sieve - l/8" pellets

Adsorbent

0.38

Bed Voidage (E)

730 gin

Total "t. Of adsorbent

0.8

w

0.33 1.8 x 10-4 cm 0.23 cm -3 1.14 gm. cm

FP rz p'p

0.206 Cal. w-1

c' P

is taken from Breck(16' and is slightly =P (2) . larger than the value used previously by Garg and Ruthven

This value of

RESULTS AND DISCUSSION

Equilibrium isotherms for all three sorbates have been determined gravimetrically[l4,15] and the equilibrium points calculated from overall mass balances in the present series of experiments showed satisfactory agreement with the gravimetric data. In order to compare the experimental and theoretical temperature and concentration curves, values of the three parameters A, (Yand fi are required. Values of A were found by fitting the gravimetric isotherms, over the appropriate concentration range, to a Langmuir equation (h = bcJ(l+ ho)). The value of /3 is also easy to estimate since o/v’ P 1 and, from a straightforward mass balance we have: 0 qb _1-c -=2)’ co ( l >

(19)

For an inert carrier gas c, = 2.5R callmole so that eqn.

the temperature curves were not recorded. In order to estimate the mass transfer coefficients for these runs a mean value of h was used and the corresponding estimated values of a and AT,, are given in parentheses. Representative plots showing the comparison between experimental and theoretical concentration curves are given in Figs. 4 and 5. The absolute time scale for the experimental concentration curve is known directly from the recorder output but the precise location of the experimental temperature curve is not known. Furthermore, the assumption of constant pattern behaviour in the development of the theoretical model means that the time scale of the theoretical curves is not absolute, although the relative times of the temperature and concentration curves are fixed. The dimensionless time scales of the theoretical curves were converted to real time using the values of ka given in Table 2 and the time scales of the theoretical concentration curves were then located to give

4.05 2.05 2.08 I.05 2.05 1.12 2.26

Cis-2-Butene Cis-2-Butene Cis-2-Butene Cis-2-Butene Cis-2-Butene Cis-2-Butene Cis-2-Butene

21 28 29 30 31 32 33 40.4 27.0 58.0 72.0 51.0 25.3 25.3

54 26.2 17.8 23.8

Tll

44 38 47 21 30 18 22 15.5 26 -

50 50 50 50 50 50 50 50 125 155 155 092 0.89 0.88 0.79 0.76 0.55 0.70

0.9 0.9 0.92 0.89

0.75 0.85 0.9 0.85 0.87

(15) 26 38 19 47

:i 50 50 50 50 34 34 85 50 50 50 50

A 0.85 0.85 0.85 0.88 0.88 0.85 0.85 0.85 0.9

DegC

AT-

(24) (24) (24) (32) (32) (24) (24) (24) I::;

50 50

Deg.C

The range of Reynolds numbers (Re,) corresponding to these runs is 3-30. *The carrier gas was He in all runs except 9 and 10 for which the carrier was Ar.

2.12 2.4 5.85 2.03

21 21 37 28.2 16.7 23.4 11.7 46.6 23.4 51.4 61.0 24.8 19.9 19.2 42.3

2.18 2.02 2.54 4.01 4.37 2.08 2.09 2.12 199 2.05 2.03 2.03 4.98 2.03 4.03

V

cm.sec-’

%

1-Butene I-Butene 1-Butene I-Butene

Sorbate

18 19 21 22

2 4 5 7 8 9* lo* 13 14 I5 16 17 23 24 25

Run

to49

t 0.05 -0.12 -0.09 - 0.24 -0.06 -0.28

-0.12

t 046

-0.11 - 0.05

-0.21 t 0.01

to46

-0.2 -0.21 -0.14 0 t 0.03 - 0.20 -0.2 -0.2 - 0.22 - 0.22 -0.16 -0.2

B

0.13 0.25 0.19 0.19 0.18 (0.29) (0.13)

0.04 0.06 0.04 0.22

;:;; (0.15) (0.15) (0.3) 0.16 0.11 0.30 0.06

(0.15) (0.15) (0.2)

(0.2) (@2) (0.2)

a

Non-Isothermal

Table 2. Summary of experimental runs and model parameters

4.04 2.0 2.42 2.68 4.0 4.0 7.5

10 8.2 13.3 3.3

4.6 5.4 5.2 8.0 7.0 3.0 2.9 6.4 4.4 6.2 6.8 5.3 7.9 3.8 13.1

*

,

’

ka x 10’ set-’

Model

(3.2)

3.6 3.2 3.11 3.22 2.8

344 4.24 4.85 6.25

4.21 4.81 5.6 4.25

4.76 (mean)

h x 10’ cahcm-* set-‘.deg-’

2.6 1.4 1.6 1.8 2.5 2.1 3.0

3.6 3.6 4.8 2.6

3.1 3.8 3.6 4.7 4.2 2.2 2.0 4.7 3.0 3.9 5.1 4.0 4.5 3.1 6.0

Isothermal Eqn. 18 ka x 10’ set-’

3

808

D. M. RUTHVEN et al.

AT

Fig. 4. Comparison of theoretical and experimental concentration and temperature curves (Run 18,2% I-butene at SOT).

-50

06.

-40 AT -30

55

60

65

70 1,

75

80

85

tin

tion curve. These trends were indeed shown by the experimental curves. The values of the heat and mass transfer coefficients calculated from the experimental curves also appear to be reasonable thus providing further confhmation of the validity of the theoretical analysis. Heat transfer in a packed bed has been extensively studied by Yagi and Kunii[I6]. It was shown that at low Reynolds numbers (Re < -30) both the wall heat transfer coefficient and the effective thermal conductivity of the bed become essentially independent of fluid velocity. The values of the overall coefficient h calculated from the experimental curves show no significant trend with gas velocity and the values for propylene and I-butene are essentially the same (mean value =4*7x IOV4Cal/cm’ sec. deg.). The values for cis-butene are somewhat smaller but again there is no significant concentration or velocity dependence. The numerical values of h are within the range of values which may be estimated from Yagi and Kunii’s results by considering the combined effects of heat transfer resistance at the wall and the effective thermal conductivity of the bed. The range of reported values for heat transfer coefficients in packed beds is however quite large and more detailed analysis is therefore probably not justified. There are three resistances which may contribute to the overall mass transfer resistance: the external fluid film resistance, the macropore diffusional resistance of the molecular sieve pellet and the micropore resistance of the zeolite crystals. Under the conditions of the present study all three resistances are significant although the external resistance is generally smaller than the intraparticle resistances. The external film coefficient, based on a driving force expressed in terms of sorbate concentration in the zeolite crystals, as in Eq. (l), is related to the gas film coefficient by:

Fig. 5. Comparison of theoretical and experimental concentration and temperature curves (Run 27,4% cis-butene, SOT).

coincidence with the experimental breakthrough curves at the @J= 0-I and 4 = 0.9 points. The absolute times of the experimental temperature curves were adjusted to make the times of the maxima coincident with the maxima of the theoretical curves. The theoretical curves give a fairly good representation of the experimental curves but since two of the four parameters required to calculate the theoretical curves (a and ka) were obtained directly by matching the theoretical and experimental curves, such agreement is not perhaps surprising. The qualitative effect of variables such as the sorbate concentration is, however, correctly predicted by the theory. For example, it may be seen from Eq. (20) that, as the sorbate concentration increases, /3 increases in a positive sense. For the present sorbates the change from negative to positive values occurs at about 4 per cent. The theory therefore predicts that as the sorbate concentration increases the temperature curve should occur progressively later, relative to the concentration curve and that for concentrations greater than about 4 per cent the temperature curve will lag behind the concentra-

(21) where co/q0is obtained directly from the equilibrium data and the film coefficient (kc) may be estimated from the correlation of Petrovic and Thodos [ k,/o

(for

> 3)

(22)

and macropore coefficients (k,a and be estimated from available Both micropore diffusivity and the effective macropore diffusivity so, systems with highly film resistance becomes increasingly the individual coefficients are with experimental over-all coefficients that in runs and IO (C&-Ar) the mass transfer resistance is macropore diffusion and the macropore coefficients for these runs are with experimental The macropore coefficients are larger for the systems with a helium carrier because the higher molecular diffusivity. (At 50°C Dc,H~+= The

The performance of molecular sieve adsorption columns: non-isothermal systems

809

Table 3. Comparison of theoretical and experimental mass transfer coefficients Theoretical kfa

kza

x 103

9-28

CjHg-Ar

6-8

l-C4H6-He

Experimental

kpa

(SK-')

csec-1) CQi6-He

Values

x lo3

ka x 103

x lo3

(SW-11

I set-1)

6-16

s-14

15-19 16

3

2-2.5

14-24

14-18

7-16

3-13

12-31

1.3-2.6

4-13

2-4

cis-2-C4H6 5ooc 125'C

19

15O'C

9-16

The

ranges

19 24-35

of values for each

estimation

of K,a were

of kpa

macropore

mechanism

with

system.

diffusion

to the

The

taken

and DCIHsHei= 0.53 cm* set-‘.) For these systems (with CJ-L and l-G&) all three mass transfer resistances are significant and no single process is rate controlling. The partial contribution of external film resistance is illustrated in Fig. 6 in which the experimental mass transfer coefficients for the runs with 2 per cent CJHs in He are plotted against fluid velocity. The resulting curve is lower in absolute value and shows a less pronounced dependence on fluid velocity compared with the theoretical curve, calculated from Eqs. (22) and (23), assuming external film control. Under comparable conditions the overall mass transfer coefficient for cis-2-CdHs is much smaller than for l-CJ-b and this difference is attributable to the difference in the zeolitic diffusivities since, for these species, both the film and macropore coefficients are essentially the same. The critical diameter of the cis-2-C.,Hs molecule is, however, larger and this leads to a higher activation energy for zeolitic diffusion[191 and, at low temperatures, a relatively small value of k,a. At 50°C the experimental overall coefficients for cis-2-butene are essentially independent of fluid velocity and close to the theoretically estimated micropore coefficients, which are much smaller than either film or macropore coefficients. Due to the high

Fig. 6. Velocity dependence of mass transfer coefficients for runs with 2% propylene in helium at 50°C(--Eq. (22);-experimental).

ref. (19).

assumed

factor

0.134 cm* set-’

range

of experimental

diffusivities

from

was

a tortuosity

4-7.5

6-10

correspond

conditions

4

7

of

used

In

to occur

in the

the calculation by a molecular

3.0.

activation energy the micropore coefficient increases very rapidly with temperature so that at the higher temperatures the mass transfer rate becomes controlled mainly by macropore resistance. For all three sorbates the mass transfer coefficients calculated from the non-isothermal model are in approximate agreement with the values estimated from independent diffusivity data. However, more detailed analysis is probably not justified on account of the nature of the approximations involved in the mathematical model. CONCLUSIONS

Although several important approximations are involved the theoretical model appears to provide a very satisfactory representation of the behaviour of the experimental systems. When there is a significant temperature rise the spread of the breakthrough curve becomes much greater than would be expected for an isothermal system with the same mass transfer resistance. The magnitude of this effect is illustrated in Table 2 in which the experimental mass transfer coefficients calculated from the non-isothermal model are compared with the values calculated from Eq. (18) on the assumption of isothermal behaviour. As is to be expected the “apparent” mass transfer coefficients calculated from Eq. (18) are all smaller than the coefficients calculated from the nonisothermal model, the ratio ranging from a modest 1.2for runs with a relatively small temperature rise (e.g. runs 16, 24, 22) to a factor of nearly three for runs 18 and 21 in which the temperature rise exceeds 40°C. The practical implication of this is that if a column is designed on the basis of a mass transfer zone length (equivalent to the spread of the breakthrough curve) calculated using correct estimates of the mass transfer coefficient but assuming isothermal behaviour, then, if the system is in fact non-isothermal, premature breakthrough will occur. Such difficulties may also occur in the scale-up from a laboratory column to a larger unit. The column diameter occurs in the denominator of the parameter a so that in going from a small diameter experimental column to a larger practical column the value of a will decrease. Non-isothermal effects will therefore be more pronounced in the larger diameter column leading to a longer

810

D.

M. RWHVEN et al.

mass transfer zone and a more diffuse breakthrough curve. In systems

t T TO v 0’ w

NOTATION

;

bo C CO

CP

c: d D

h -AH k k rc, h

kz

m 4 q* 40 4s 4’

rp rz R Re, SC

ratio of external area to pellet volume, cm-’ Langmuir equilibrium constant, cm3.mole-’ constant in Eq. (3), cm3.mole-’ gas phase concentration of sorbate, mole.cm-’ sorbate concentration in feed, mole.cm-3 heat capacity of gas, cal.g-‘.deg-’ heat capacity of adsorbent, cal.g-‘.deg-’ column diameter, cm diffusion coefficient, cm2.sec-’ overall heat transfer coefficient at column wall, cal.cm-‘.sec.deg heat of adsorption, caLmole_’ overall mass transfer coefficient defined by Eq. (l), cm.sec-’ gas film mass transfer coefficient based on gas phase concentration driving force, cm.sec-’ gas film mass transfer coefficient based on zeolitic concentration driving force, cm.sec-’ equivalent film coefficient for macropore diffusion based on zeolitic concentration driving force, cm.sec-’ equivalent film coefficient for zeolitic (micropore) diffusion based on zeolitic concentration driving force, cm.sec-’ ratio of column voidage to crystal volume = E/W(l -q)(l -e) sorbate concentration in zeolite crystal, mole.cm-’ equilibrium sorbate concentration in zeolite crystal, mole.cm-’ crystal based sorbate concentration in equilibrium with feed gas concentration co,mole.cm-’ constant in Langmuir equation (Eq. 2), mole.cm-3 sorbate concentration based on pellet volume (q’ = q/w(l- Ed)), mole.cm-3 pellet based sorbate concentration in equilibrium with feed gas concentration, mole.cm-’ equivalent radius of adsorbent pellet, cm equivalent radius of zeolite crystal, cm gas constant, caLmole_‘.deg-’ Reynolds number based on superficial velocity = 2eptlrpIcL Schmidt number = p/pD,

y z

time, set temperature, “K wall temperature (constant), “K linear gas velocity in bed, cm.sec-’ linear velocity of constant pattern front, cm.sec-’ fraction of zeolite crystals in molecular sieve pellet mole fraction of sorbate in feed gas axial distance, cm

Greek symbols

parameter defined by Eq. (17) parameter defined by Eq. (16) voidage of packed column porosity of molecular sieve pellet density of gas phase, g.cmm3 density of adsorbent, g.cm-’ dimensionless gas phase concentration (c/co) dimensionless adsorbed phase concentration (4 140) dimensionless time parameter (kat) dimensionless temperature defined by Eq. (12) non-linearity parameter = q$q# = bco( 1+ bc&’

ltlm%RENm

1. Garg D. R. and RuthvenD. M., Chem. Engng Sci. 197328 791, 799. 2. Garg D. R. and RuthvenD. hf., Chem. Engng Sci. 197429 571. 3. Garg D. R. and Ruthven D. M., Chem. EngngSci., in press. 4. Leavitt F. W., Chem. Engng Progr. I%2 58 54. 5. Pan C. Y. and Basmadjian D., Chem. Engng Sci. 196722 285. 6. Pan C. Y. and Basmadjian D., Chem. EngngSci. 197025 1653; Ibid. 197126 45. 7. Meyer 0. and Weber T. W., A.J.Ch.E. fl. 196713 457. 8. Lee R. G. and Weber T. W., Can. I: Chem. Engng 1%947 60. 9. Glueckauf E., Trans. Faraday Sot. 195551 1540. 10. Ruthven D. M. and Garg D. R., A.J.Ch. JI., 197521 200. 11. Vermeulen T., Ado. in Chem. Engng. Vol. II (1958), 147. 12. Lapidus L., Digital Computation for Engineers, p. 113. McGraw-Hill. New York. 1%2. 13. Garg D. R., Ph.D. Thesis, University of New Brunswick (1972). 14. Derrah R. I., Ph.D. Thesis, University of New Brunswick (1973). 15. Derrab R. I., Loughlin K. F. and Ruthven D. M., J. Chem. Sot. Faradav Trans. I 197268 1947. 16. Breck 6. W., aolite Molecular Sieves, p. 753. WileyInterscience, New York, 1973. 17. Yagi S. and Kunii D., A.I.Ch.E. 8. 19606 97. 18. Petrovic L. J. and Thodos G., Ind. Eng. Chem. Fund. 19687 274. 19. Ruthven D. M., Derrah R. I. and Loughlin K. F., Can. J. Chem. 197351 3514.