concentration Co/R. On the hot half cycle, the zone effluent is composed of parts of two belts: one belt has length a / R and has concentration COR;the other has length a [ ( R- 1)/R] and concentration Co. By proper concentration averaging of the hot effluent and dividing by the concentration of t h e cold product, one obtains Equation 5 for the separation factor. If the remainder of the zone exceeds a / ( l K c ) but is less than a / ( l K H ) ,the hot effluent contains parts of three belts. The first and third belts both have concentration Co and their combined length is a [ ( R- ])/I?]; the second belt has length a / R and concentration COR.Clearly the average concentration of the hot effluent is the same for all values of CY that satisfy Equation 7. I n a similar fashion the derivation of Equation 5 can be extended to m zones. Case 3. No regular pattern. T h e separation factor u s . zone number exhibits a n oscillatory behavior if t h e remainder { 1 - N [ a / ( l K C ) a / ( l K H ) ] ] is less than a n entire cold belt or greater than an entire hot belt.
+
+
+
+
+
Discussion
The criterion for separation developed in this paper assumes linear adsorption isotherms and instantaneous local equilibrium. For certain regions of a , the separation increases from zone to zone. h computer model based on finite differences was used t o compute separation in cycling zone adsorption. Finite differencing introduces some numerical errors which are similar to the axial concentration spreading due to finite mass transfer or dispersion. It was found that for proper values of a the separation factor leveled off after a few zones. This is similar to what has been observed in parametric pumping. Ritter and Douglas (1970) have defined parametric pumping as a situation where a periodic input signal causes state variables to fluctuate, and simultaneously a time variable coefficient in the system equation is forced to be periodic. T h e advantage of having two cycling variables in parametric
pumping is that resonance call be obtained a t all the frequencies (a’s). On the other hand, a system like cycling zone adsorption with only one periodic variable produces resonance for some frequencies only. Severtheless, cycling zone adsorption, which has an advantage of nonreversal of the fluid flow, is similar to parametric pumping in the sense that a periodic time coefficient has been replaced by a periodic position coefficient, Nomenclature
K = slope of adsorption isotherm Kc R = I 1 KH t = dimensionless time T = temperature, “C z = dimensionless axial position a = dimensionless velocity, fraction of bed volume displaced 1half cycle 4f = dimensionless fluid concentration 4 8 = dimensionless solute concentration in the solid SUBSCRIPTS C = cold €I = hot
+ +
literature Cited
Pigford, R. L., Baker, B., 111, Blum, D. E., IND.ENG.CHI:M., FUNDAM. 8, 448 (1969). Ritter, A. B., Douglas, J. M., 1x0. ENG.CHEM., FUKDAX. 9, 22 ( 1970). Wilhelm, R. H., Rice, A. W., Bendelius, A. R., IND.EYG.CHEM., FIXDAM. 5 , 141 (1966). Wilhelm, R. H., Rice, A. W., Rolke, R. W.,Sweed, N. H., . IND.ENG.CHEM.,FUYDSM. 7, 337 (1968). RECEIVED for review July 13, 1970 ACCEPTED February 4, 1971 Computations were carried out at the Princeton University Computer Center, which is supported in part by National Science Foundation Grants GJ-34 and GU-3137. Financial support was provided by the National Science Foundation under Grant GK1427x1.
Cycling Zone Adsorption: Quantitative Theory and Experimental Results Burke Baker, 111,’ and Robert 1. Pigford2 Department of Chemical Engineering and Lawrence Radiation Laboratory, University of California, Berkeley, Calij. 94720
A theoretical explanation for cycling zone adsorption, a wave-propagational separation process given a qualitative explanation in an earlier article, i s presented. Experimental results confirming the theoretical predictions are also included. It i s found that in addition to accounting for the separation effect the theory predicts the effect can be amplified through the interaction of the concentration and progressing thermal waves. Possible process schemes utilizing the interaction are discussed.
I n a recent article by Pigford, Baker, and Blum (1969b) a qualitative description was presented of a cyclic fixed-bed separation process given the name “cycling zone adsorption.” The first objective of this article will be to explain the process theoretically. This leads to certain predictions of ways of Present address, Shell Oil Co., Deer Park, Texas. To whom correspondence should be sent.
introducing heat into the bed that will increase the concentration change. Finally, experimental evidence of the predictions is offered. Consider the process shown in Figure 1, in which a fluid having a constant solute concentration, yf, is passed through a bed of solid particles. The temperature of the bed is cyclically altered, either b y heating and cooling the walls, as in Ind. Eng. Chern. Fundam., Vol. 10, No. 2, 1971
283
0
3=at
Tc @o~ack~e;bsorbant
Q I"T H
Heating h a l f -cycle
(1 - ~)~sAHada 9. (4) [PSCdl - E ) P,Cf€I b t
+
where CY is the fraction of the packed bed outside the adsorbent particles and E is t h e fraction by volume of immobile fluid inside the particles. The interface between the surface of the solid and the fluid inside the solid particles is considered to be a t equilibrium and is represented by an expression of the type
Yn
'h
+ PfCIEI(1 - a) (Ts- T ) +
h*%
-4
[P,C,(1
Cooling half-cycle
(5)
P = dTs,c*)
I
Yf
I---F
uA
U
h'
A
F i r s t half - c y c l e
Second h a l f - c y c l e
Figure 1. Single-zone operation of cycling zone adsorber. The standing thermal wave mode is indicated b y (a) above; (b) below illustrates the progressive thermal wave mode.
Figure l a (standing thermal wave), or by using a pre-exchanger to heat and cool the input stream, as in Figure l b (traveling thermal wave). As explained in the earlier article, cyclic changes in the effluent concentration are obtained; these changes can be amplified by using a series of such columns. Mathematical Model
The equations governing the process are those for a nonisothermal, fixed-bed system which are derived from mass and heat balances over each phase. Disregarding radial gradients in velocity, concentration, and temperature, the balances for a packed column through which a solution containing a single adsorbate is flowing yield (Baker, 1969) A. The solute balance on the fluid and solid phases
a c (1 -+-bt
CY
CY)€
bc* bt
+ Ps (1 - ff)U CY
E)
2
Any effect of temperature on the parameters has been neglected except in the equilibrium relationship. The assumption of over-all linear driving-force relationships for the mass transfer h e e d and Wilhelm, 1969) may not be precisely true in some cases and may need to be replaced b y the corresponding radial intra-particle diffusion terms. For fast transfer, however, the simpler expressions are sufficient. I n the equations the stationary phase is treated in two parts: the adsorbent itself (concentration = q ) and the immobile fluid in the pores (concentration = c*), assumed to be in local equilibrium with the adsorbent. Since the separation effect in the process depends on a cyclic alteration in the phase equilibrium rather than on any rate differences, the approach taken in this study was first to design a good process based on the equilibrium theory, assuming that rates of exchange are very high, and then to develop the mass and heat relationships for the general case in which rates may be important. If it is assumed t h a t at any point in the column the solid and fluid phases are in equilibrium, Le., c = c* and Ts = T , and that axial dispersion and heat of adsorption effects may be neglected, a n analytic solution to the simplified equations is possible. This is presented below. (The Freundlich isotherm p = d(c*)'; 0
< IC 6
1
(6)
will be used later in eq 5 since i t will apply to the experimental system discussed below.) The equations, which have been simplified using the above assumptions, become
+
at where Ulh
= u/(l
+ (1 - ff)[P,Cs(1 -
E)
f PfCfEl/PfC,ff]
is the velocity at which a thermal wave advances through the column. Assuming that the temperature of the column is changed by changing the temperature of the feed stream, the boundary conditions are
c=co;z=O,t>O
C. The solute balance on the solid phase
T
=
To
+ ( T H- T o ) S p ( d ) ;
2 =
0, t
>0
i.e., the inlet temperature is cycled in a square wave of frequency W / ~ T cyclesjmin. The solution to eq 8 is
D. The energy balance on the solid phase 284
Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
T - To
=
( T H - To)sQ(wt -
W Z / U ~ ~ )
(9)
representing the propagation of temperature through the bed as a traveling wave of velocity ~ [ h If. the column is heated and cooled through the walls, this natural thermal velocity is overridden; then a standing temperature n a v e is produced and, effectively, u t h a The occurrence of the independent variables in the combination t - z / u t h suggests a change to the nen- independent variables I
7
= t
-
z/uth;
[ =
Using these variables eq 7 becomes 1 - a
e+-
"0
1-a a
COLD
311
t
11
HOT 2 n
HOT
411 51T COLD
YO Figure 2. Characteristic lines used in equilibrium theory for linear isotherms-single zone, square-wave cycling
with the boundary and forcing conditions C = C o ; { = o , T > o
T
To
=
+ ( T H- T ~ ) s ~ ( u.tT=) ;0 ,
T
linear Isotherm
>0
Using the method of characteristics as in Pigford, et al. (1969a), eq 10 is equivalent to the pair of ordinary equations
dl
-
dr
=
I n the event k = 1, ie., if t,he Freundlich isotherm is straight, u, i,: concentration-independent. Then eq 13 reduces to
-
ci _ -
Uc,t-l-'
tbc,i-'
ci-1
- UZh-' - UZh-l
(16)
Since in nearly all adsorption systems solute is rejected from t-he solid when the system temperature is raised, u c C < u c H and Solutions to eq l l a are lines representing the r - { characteristics. Solutions to eq l l b , yielding c as a function of T , are valid only along these characteristics. If the temperature is constant ( d T / b r = 0), the solution to eq l l b i i
(12)
Thus, c reniains constant along a characteristic, except when the temperature changes. When the temperature is switched, the concentration changes can be expressed by 1-a [I+_
6
- (o/uZh)](ct 1-CY ~~
a
- cz-1) (1
-
+
E)PS(Q~
-
qz-1)
=
0
(13)
from eq 12, where i - 1 refers to conditions just before the switch and i to conditions afterward. The concentration velocity, u,, is the speed with which small changes in conceiitration move along the column. I n general it is concentration and temperature dependent and is given by
cC/CH
=
Q
c
-
0.10 -
.-
-
z
Feed c o m p o s i t i o n
L
0.05
0 0
800 Time
Figure 12.
,
I600
sec
Comparison of standing and traveling square-wave forcing using two zones
to the high temperature. Thus it never gets shifted to higher concentration. T o obtain this shift, the temperature in the second zone would have to be changed sooner in each cycle. Nevertheless the beneficial effect of multiple zoning is evident from the separation factors. The following studies were made with larger particles of 2 0 4 0 mesh carbon than those represented in Table 11. The prediction of a n optimum cycling frequency for the two-
zone case was confirmed. The prediction of an optimum in the phase lag between the temperature forcing of the two columns was also substantiated. Baker (1969) showed that the optimum phase lags giving the maximum in aav and a p k are not exactly a t 180", as expected from the linearized theory, but approximate the values found experimentally. Finally, Figure 12 shows the use of traveling-wave forcing in the dual-zone process. Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
291
Discussion and Conclusions
It has been demonstrated theoretically and on a laboratory scale that the cycling zone adsorption process is feasible. Significant separations are predicted mathematically using the equilibrium theory, a t least when the isotherms are not too strongly curved. Significant but by no means complete separations were observed. Particularly evident was the predicted increase in separation power gained by using multiple zones in series. Furthermore, the cyclic transfer of thermal energy in and out of the zones, which is essential to the process, can be accomplished very efficiently in liquid-solid systems by the use of the natural thermal, traveling-wave properties of the zones. Certain difficulties are evident. The small shift in the equilibrium isotherm for moderate temperature changes indicates that several zones will be required for significant separation. This means additional thermal requirements. Internal reuse of the heat is possible; the feasibility of a n interstage lieat regenerator has been discussed by Baker (1969). The experimental results and perturbation theory predictions (Baker, 1969) indicate that the low mass-transfer rates reduce the separation from the equilibrium theory predictions. This means that for these intra-particle diffusion-controlled systems, one must operate a t low flow rates or with fine particles. The presence of axial dispersion, caused by the nonlinear isotherms and finite transfer rates, may require recycle arrangements. One may want to collect the effluent only while the high and low peaks are emerging, returning the rest to the column feed a t the proper time in the cycle. The practicality of imposing traveling waves on this type of system remains to be demonstrated, although i t has been done in analytical gas chromatography. If it should not prove to be feasible for thermal waves, it should be kept in mind that the effect would occur for any intensive variable that can shift the solid-fluid equilibrium. Certain such variables, for example electrical or magnetic fields or fluid pressure, may be easier to use as traveling waves, thus amplifying a small separation effect. Cycling zone adsorption is an attractive process which may economically perform the intermediate scale separation of material which now must be separated chromatographically. Acknowledgments
TVe thank the Pittsburgh Activated Carbon Co. for donating the carbon for this study. Studies supported by AEC Contract R-7405-eng-48. Additional support through NASA Predoctoral Traineeship ASG(T)-117-2 and a Shell Companies Foundation Fellowship was provided to one of the authors. Nomenclature (dimensionless unless specified)
A
=
D.w
=
292 Ind.
solid-fluid equilibrium distribution parameter, units depend on function mass molecular diffusivity, cm2/min
Eng.
Chem. Fundom., Vol. 10, No. 2, 1971
DT
thermal molecular diffusivity, cmZ/min eddy axial dispersion, cm2/min differential heat of adsorption, cal/mole gas constant, cal/(g-mole)(OC) temperature, “ C length of bed, cni surface area/bed volume ratio, cm-1 fluid concentration, moles/l. fluid concentration in equilibrium with solid-phase concentration, moles/]. = heat transfer coefficient, cal/(cm2)(min) (“C) = exponent in Freundlich isotherm = solid-phase concentration, moles/kg of d r y solid = time, min = wave velocity, cm/min = interstitial fluid velocity, cm/min = c/co, concentration in fractions of feed concentration = axial distance, cm
= = AHads = R = T = = Zo = a C = = c*
ED
h k y
t u V
y 2
GREEKLETTERS CY = interparticle void fraction CYpk = ? / h pk/?/l,pkJ peak separation factor a a v = (yh)/(y cycle-averaged separation factor e = intra-particle void fraction ps = structural density of solid, kg/cma = (t - z ) / u t n , time, niin 7 W = frequency, radians/min [ = distance, cm l ) J
SUBSCRIPTS w = refers to tube wall surrounding the system C = cold C = concentration-related value j” = feed condition H = hot h = high i 1 = conditions after a concentration characteristic crosses a temperature boundary i = conditions before a concentration characteristic crosses a temperature boundary; if i = hot, i - 1 = cold, and m e versa 1 = low S = solid th = thermal
+
SUPERSCRIPTS C = cold H = hot literature Cited
Baker, B., 111, Ph.D. Dissertation, University of California, Lawrence Radiation Laboratory, Berkeley, 1969. Goldetein. F. R . S.. Proc. Rou. Soc.. Ser. A 219. 151 11953). OhlinelR: W.,DeFord, D. D”., A n d . Chem. 35; 227 (1963). Pigford, R. L., Baker, B. 111, Blum, D. E., IND.ENG.CHEM., FUNDAM 8, .144 (1969a); 8, 848 (1969b). Sweed. N . H.. Wilhelm. R . H., IKD. E N GCHEM., . FUNDAM. 8,221 (1969). Tudge, A. P., Can. J . Phys. 40, 557 (1962). Zhukhovitskii, A. A., Zololareva, 0. V., Sokolov, V. A., Turkeltaub, K.ll.,Dokl. Akad. Nauk SSSR 77, 453 (1951). Zhukhovitskii, A. A,, “Gas Chromatography,” Third Symposium, Edinburgh, R . P. W. Scott, Ed., Butterworth Scientific Publications, London, 1960. RECEIVED for review August 17, 1970 ACCEPTED January 25, 1971
