Chemical Engineering Science, 1974, Vol. 29, pp. 1133-1138. Pergamon Press. Printed in Great Britain

CONTINUOUS CYCLIC ZONE ADSORPTION

DORON MEIR and RAM LAVIE Department of Chemical Engineering, Technion-Israel Institute of Technoligy, Haifa, Israel

(Receiued 4 September 1973)

Abstract-A theoretical study of a continuous version of the cyclic zone adsorption separation pro- cess, providing us with a new understanding of the parameters that determine the existence of a separation efkct and its extent is presented. A mathematical model is proposed and then solved aualyt- ically. Conditions under which a separation effect can he achieved are established, leading to the definition of dimensionless parameters that control the process. Finally, optimal operating conditions are derived and discussed.

INTRODUCTION (4) Some time ago, several articles [ 1,5] appeared de- scribing and analysing a new separation process (5) named “Cyclic Zone Adsorption”. This process, like “Parametric Pumping”l2-4, G-91, a previous (6) development of which it is indeed an extension, was based on a discrete mode of operation whereby sections of a fixed adsorbent bed are alternatingly heated and cooled. Q

The present article deals with a continuous ver- sion of “Cyclic Zone Adsorption”. A model of the process is presented and then solved analytically. This solution provides us with a new understanding of the parameters determining the existence of a separation effect and its extent.

Assumptions (l-4) are compatible with a physical system in which the fluid flow rate is sufficiently low. Assumption (5) is justifiable when the fluid consists of a dilute solution of a solute in an essen- ti@y inert solvent adsorptionwise.

The effect of pressure drop on adsorption equilibrium is negligible. Mass flux through any bed cross section is con- stant. Temperature fluctuations are limited in mag- nitude such that the effect of temperature on all physical properties, except for the equilibrium relationship, is negligible. The bed temperature at any point in time and space can be controlled at will, to desired values.

Experimental aspects and design considerations are not presented here. They will constitute the substance of a separate report.

PROCESS DE!3CRIPTION

The separation equipment consists of a fixed bed of sorbent into which a fluid having a constant sol- ute concentration yf is fed. The temperature of the bed is somehow manipulated such that it oscillates in time and space in the form of a continuously travelling wave advancing at a velocity v in the di- rection of fiow.

MATHEMATICAL MODEL

The mathematical model is based on the foilow- ing assumptions: (I) The solid and fluid phases in the bed are in local

equilibrium. (2) All bed properties are uniform over any radial

cross section. Radial gradients in velocity, con- centration and temperature are disregarded.

(3) Axial diffusion is negligible.

Under the assumptions listed above, the process is governed by the following equations which are derived from a mass balance on both phases.

(1)

J is the solute concentration in the fluid phase, (mole/cm’),

# is the solute concentration in the solid phase, (mole/g),

u is the fluid flow superficial velocity, (set-‘), h is a constant relating the physical properties of

the solid, to those of the fluid, including void fraction and density, (g/cm’),

x is the dimensionless distance along the bed (0 at entry, 1 at exit),

t is the time, (set). in a dilute system, a linear adsorption isotherm is acceptable and

4 = K(T)Y, (2)

1133

where

and

KY, is a constant, (malelg)/fmoIelcm’), AH is the heat of adsorption, (caltllmale),

R is the gas constant, (callmole “IQ, T is the temperature, (X).

Equations (1 j and (2) can be combined into

01

8.v I Wv)_O, % at (5)

where

m(T) = 1 +-M/T) (6)

is a measure of the adsorption capacity at tempera- ture 7: A feed of constant composition yi is intro- duced at x - 0. Thus one has the boundary condi- tion

Y(O. t) = 3% m

Given initial conditions, the concentration y(x, t) at any time and location in the bed can be derived by the solution of (5) with the boundary condition (7), provided that the temperature T(x, t) of the bed is explicitly known for every point in time and space.

The process is based on a deliberate periodic per- turbation in space and time of the bed temperature. Say

T(x, t) = T, i-AT COS?W(X - ut) @I

where

Tti is the average operating temperature, (“IQ AT is the amplitude of the periodic perturbation,

FKI, t’ is the velocity of the thermal wave, (sec.-‘), w is the frequency of the oscillation, (dimen-

sionless length-‘).

ThUS,

m(T)=l+hK(T)=l

-t- kIL exp i

AH R (T, -I- AT CDS 2W(X - ut)) I (9)

For small deviations AT, this can be approximated

where

= m(T,)[i -A cos 2nw(x - at)], W>

Being interested in the asymptotic periodic solu- tion, one can do without initial conditions and the problem is now fully defined. To fmd such a sohr- tion, it is useful to change the independent variables t0

,7=20(x - ut),

0=t

Equation (IO) can thus be expressed as

(12)

m(T) = p(z) = m(T,)[I --A CDS ml, (13)

a function of z alone. Expressing Eq. (S) in terms of the new independent variables. one has

Thus,

Using the method of characteristics, Eq. (14) is equivalent to the pair of ordinary diierential equa- tions

_- = &&$!Ldr. dz u-pu P dw

( 1 yv z!

(15)

Or, written separately,

$&dz =2wd0

(ISa)

Consider frrst the particular case where A = 0, that is the process is operated at a constant temperature

Continuous cyclic zone adsorption 1135

Then,

EL(z) = m(Z) = constant,

and Eq. (5) takes the form

(16)

In an isothermal bed, one has a concentration wave advancing at a velocity

w(X) =u m(T,)’ (17)

called henceforth the isothermal concentration wave velocity.

In their paper on cyclic zone adsorption, Baker and Pigford [ l] predict amplification of a separation effect when the thermal wave is made to pass the bed at a velocity adjusted to that of the (isothermal) concentration wave. Let us define a tuning ratio at an average temperature T, as

(18)

The tuning ratio r is the ratio of the velocity of the thermal wave to that of the isothermal concen- tration wave at a temperature T,.

As it turns out, the solutions to our system are best expressed in terms of dimensionless parame- ters 3 and C, depending on the tuning ratio r and the amplitude A of the oscillation in adsorption capacity:

C=20(A2rZ-(1-r)*~‘“=2w~l-r(~S’-l~”’. (20)

Integration of Eq. (Isa) along the characteristics (15b), yields the following solutions for distinct ranges of the dimensionless parameter B.

(sinrrz)sinh{rrCx}-coszz-L -’ B 1 > IBl>L

(21a)

(B i-cos 7rZ)COS{?rCX}-~B2- ljln

(sin ~2) sin {7&x} - cos ~2 - $ 1 -I, IBl~f, w-7)

$=2 (1tcosTz)x [

( 1 -i- [ 2?ro(l- r)x -ctg I 7 Ml 2 -1 , B = 1, (21~)

$=2 (l-cosTrz)x [

, B=-1, (21d)

Evidently, the concentration profiles y(x, t) in the bed and in particular, the outlet composition ~(1, t) follow a pattern depending on the value of B. A significant separation effect is only obtained in the case IB I > 1, a condition expressed equivalently in terms of r and A as

1 1 -----<‘<l-A’ l+A (22)

meaning that in order to obtain a significant separa- tion effect, the process should be tuned at a ratio constrained by inequality (22).

Figures 1 and 2 depict respectively a typical con- centration profile in the bed at some fixed time and outlet composition as a function of time. Fig. 3 is a topographical or contour map of the concentration as a function of time and space. It can be described as waves of solute moving in the direction of flow at regular intervals, with peaks increasing in height in that same direction. Thus, one can get for a suitable choice of process parameters satisfying inequality (221, an effluent of composition ~(1. t), consisting most of the time of an essentially solute free stream, interspersed at regular time intervals with pulses of a concentrated solute (see Fig. 2).

FEATWJZS OF IN-T

Let us now consider some features of interest in the solution described above.

Fig. 1. Concentration profile.

D~RON MEIR and RAM LAVIE 1136

A=02 r i I-10

w;30 Y’O ,667

Fig. 2. Effluent composition.

Firstly, one is certainly interested to know when to expect a concentration peak at the outlet and its duration. To gather this information, we differen- tiate y with respect to z and thus find that in the

direction z, y reaches extreme values at

* = iarctg c

zi 2w(r - l)tgh@Zx)

+ 2n,

n=0,1,2,3 ,... (23)

At any location x in the bed and in particular at the outlet x = 1, peaks of concentrated solute appear at time intervals of

This was derived by inserting (12) in (23) for two consecutive values of z: and solving for (t,,., - t,) with x = constant. It is interesting to note that ac- cording to (24), the time interval r depends only on the thermal wave velocity v and the frequency w

Continuous cyclic zone adsorption 1137

but not for instance on the flow rate u. Thus, the sampling timer need only be adjusted to the temper- ature.

The crest values of the concentration at a loca- tion x are derived by inserting (23) in (21a). Thus one gets

y = [(l + g2(x)y2rg(x)]*, (25)

where

B &I = cB2_ lj,n sinb ;Cx ( >

. CW

The plus sign in the r.h.s. of Eq. (25) corresponds to y&x) = Max[y(x, t)], the maximum value, while

the minus sign in this same expression corresponds to y,;,(x) = Minly(x, t)], the minimum value of the

concentration ; at a location x in the bed. The width T+(X) of a concentration peak, defined

as the time interval during which

at a given location x, can be evaluated by equating y(x, t) in (21a) to yf and solving for t. When this is done, it is found that S(x), the portion of a cycle with Y(x, t) > yf is

In particular, one has at the outlet,

One other feature of interest concerns the dis- tribution of the solute in the effluent between the lean and rich portions of the cycle. From (251, at any location x one has

-~--[(1+g’(n))-g?(X)]*=1. (29) YmdX) Y,“(X) _

Yf Yf

That is, the average concentration at any loca- tion, and in particular at the outlet, equals the geometric average of the maximal and minimal val- ues at that location.

OPTIMAL OPERATING CONDITIONS

Up to this point, it has only been established that separation will be obtained when the process

parameters satisfy an inequality constraint (22). One may now ask if, within this constraint,some conditions would provide “better” separation than others.

One likely objective will be to maximize the peak value of the effluent concentration or,

vc$x {Ynrax(l) = Nl + g*(l)‘!* + gm17. (30) , .w

According to (29) this will also minimize the lowest value of the solute concentration in the lean portion of the cycle.

From (261, (191, (20)

g(l) =2wd4r ( 2 ) - sinh s = rwAu2 -

“4 (2n + l)!

= TuArg {rr*o*[A'r'- (1- r>“l} (2n+l)! . (31) n 4

A, 0, r and C are all positive. y,,.( 1) increases with g(1). g(1) increases with A and with o indefinitely. Within the range defined in (22), C is a symmetric function of r with zeros at the two extremes, ri = l/l + A and rz = l/l -A. It reaches a maximum at the center of that interval rc = l/l - A2. As can be seen in (3 l), given A and w, g( 1) can only increase with r for

1 1 -<r5------ 1-l-A l-A?‘ (32)

However, when r > l/l -A’, there are conflicting trends in g(l) due to r and to C. Therefore, there may exist a local maximum in g(1) within the interval

1 1 1_A2++---- 1-A’ (33)

and the global maximum of g (1) certainly lies in this range.

It may be concluded that the best separation in the sense of (30) will be obtained for as high a value as practical for both the frequency w and the amp- litude A of the oscillations, while the optimal tun- ing ratio r must be searched for (fine tuning) within the interval (33).

DISCUSSIONS AND CONCLUSIONS

The theoretical study of a continuous version of cyclic zone adsorption carried out here has estab- lished conditions under which a separation effect is achieved and amplified. This has contributed to our

CES VOL. 29 NO. -5-F

1138 DORON MEZR and RAM LAVIE

understanding of the parameters controlling the process and the definition of relevant dimensionless numbers.

Theoretically, there is no limit to the degree of separation that can be achieved through an edu- cated choice of design and operation parameters. In practice however, there are important limitations that should be taken into consideration.

Firstly, it is clear that in practice, the degree of separation will be limited because such high con- centration gradients as predicted for the peaks are not very probable. One would expect the peaks to be smeared by diffusion (that was neglected in the assumptions) even at rest. but certainly under flow conditions, however slow.

Secondly, dilute concentrations were assumed in the model. This assumption may not be valid at the peaks, again limiting the practically attainable de- gree of separation.

Thirdly, the bed temperature can certainly not be controlled in practice at will to “any” desired value. In particular, there are limits to the frequency and amplitude of the temperature oscillations that can be produced. However, these can be compensated by proper design: Frequency limitations can be compensated by increased length of the bed. As for the amplitude limitations, since what really counts is the amplitude of the oscilIations in adsorption capacity m(T), careful attention to the properties of the adsorbent mater&I and the choice of the av- erage operating temperature can be of great help.

In practice, when the oscillations in bed tempera- ture must be induced externally through some transfer process, there will be a strong inverse reia- tionship between the frequency and the amplitude. Thus, they can not be made both arbitrarily large and a compromise is necessary.

There are indiations, from the results of simula- tions where the assumptions of the mathematical model were relaxed, that those assumptions are not as restrictive as they seem.

ii

m n r t

I(

NOTATION

function defined in (26) constant relating the physical properties of

the solid to those of the fluid, &m3 adsorption capacity defined in (4) integer tuning ratio, defined in (18) time, set fluid flow superficial velocity, set-’

velocity of the therma wave, sec..’ velocity of the isothermal concentration

wave, defined in (171, see-’ dimensionless distance along the bed solute concentration in the fluid phase,

mole/cm’ feed composition, mole/cm’ peak compostition, mole/cm3 base composition, mole/cm3 coordinate defined in (12) amplitude of the oscillation in adsorption

capacity, defined in (1X) dimensionless number defined in (19) dimensionless nymber defined in (20) heat of adsorption, callmole slope of the adsorption isotherm,

(molelg)l(molelcm”) gas constant, c&mole “K temperature, “K average operation temperature, “K ampIitude of the temperature oscillation, “K dimensionless number defined in (31) portion of a cycle during which y > yf solute concentration in the solid phase,

mole/g adsorption capacity in terms of coordinate z,

defined in (13) frequency of oscillation, number of wave

length in the total bed length coordinate defined in (12) time interval between consecutive peaks, set time interval during which y > yf, set

FtEFERENCFS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Baker B. and Pigford R. L., Ind. Engng Chem., Fun- dam. 1971 10 283. Gregory R. A. and Sweed N. H.. Chem. Engng J. 1970 1 207. Gupta R. and Sweed N. H., Ind. Engng Chem., Fun- dam. 1971 10 28t). Pigford R. I..., Baker B. and Blum D. E., Xnd. Engng Ckem., Fundam. 196518 144. Pigford R. L., Baker B. and Blnm D. E., ind. Engng Chem.. Fwuiatn 1969 8 848. Sweed.N. H. and Wilhelm R. A,, Ind. Engng Chem., Fundam. 1969 8 221. Wilhelm R. H., Rice A. W. and Bendelius A. R., Ind. Engng Chem. 1966 5 141. Wilhelm R. H. and Sweed N. If., Science 1%8 X59 522. Wilhelm R. H., Rice A. W., Roike R. and Sweed N. H.. Tnd. E~gug Chew., Fundam. 1968 7 337. Zhukhovitskii A. A., Gas Ckromatogruphy, 3rd Sym- posium, Edinburgh, (R. P. W. Scott, Ed). Butterworth Scientific Publications, London 1960.