The efficiency of the counter was estimated to be about 8% using the method described by Welch, et al. (1967). As
expected, this is less than values reported for large counters (Welch, et al., 1967).
3
r
Acknowledgment
2
John Root gave valuable advice about handling and counting problems with radioactive tracers and provided space for the work. Also the use of electronic equipment associated with AEC Contract AT-(04-3)-34, Agreement 158, is gratefully acknowledged.
b z -I
f
1
Y
170
im
I
I
190
200
I
210
220
TIME (SEC)
Figure 3. Comparison of responses of counter and TC cell; TC cell after counter
essentially identical curves. Therefore, the difference between the curves in Figure 2 is due to dispersion in the T C cell and not in the counter. Hence, the counter design shown in Figure 1 should be useful for studying transient phenomena in flow reactors, using carbon-14 isotopes.
literature Cited
Lee, J. K., Lee, E. K. C., Musgrave, B., Tang, Y. N., Root, J. W., Rowland, F. S., Anal. Chem. 34, 741 (1962). Padberg, G., Smith, J. &I., J . Catal. 12, 172 (1968). Schneider, P., Smith, J. AT., A.Z.Ch.E. J . 14, 762 (1968). Welch, M., W'hithnell, R., Wolf, A. P., Anal. Chem. 39, 273 (1967). Wolfgang, R., Rowland, F. S., Anal. Chem. 30, 903 (1958). RECEIVED for review October 29, 1971 ACCEPTEDMay 1.5, 1972 Thanks are due to the Convenio between the University of Chile and the University of California for providing fellowship funds.
COMMUNICATIONS
Equilibrium Theory of the Parametric Pump. Effect of Boundary Conditions The separation possible in a cyclic parametric pump process i s shown to be strongly affected by the amount of mixing in the end compartments. Using an equilibrium model, algebraic solutions are found for the bands of high concentration which would develop if mixing were prevented. A two-column back-to-back operation i s proposed to minimize mixing and the expected performance of ihis device i s calculated. The separation factor for a repetitively cycled closed system i s shown to be of the order (constant)n, after n cycles if the end compartments are well mixed. However, if mixing i s prevented, the separation factor for the band of highest concentration i s of the order (constani)2n-' for a single column and of the order for the double column.
Pigford, Baker, and Blum (1969), Aris (1969), and Gregory and Sweed (19iO) have analyzed cyclical adsorption and desorption operations by means of an equilibrium model. These analyses, along with the computational solutions obtained by Sweed and Rilhelm (1969) and Rolke and Wilhelm (1969), all consider that the fluid m-hich emerges from the column during oiie half cycle is perfectly mixed in a reservoir before being returned to the rolumn during the next half cycle. The purpose of this communication is to examine the effect of alternative end conditions on the predicted performance of the column and to consider back-to-back or close-coupled operation of two columns whose adsorption-desorption cycles are 180' out of phase.
Sweed and Wilhelm (1969) comment that, in their experiments, a small portion of the effluent fluid was contained in the connecting tubing between column and reservoir and therefore was not mixed with the major portion. Under some circumstances it might be desirable to temporarily store all of the effluent fluid during one half cycle in a tube, or in a heat exchanger, or in a section of inert packing, and displace it by means of a n immiscible drive fluid. The idealization of this arrangement is a plug flow model of the reservoir sections, in which each element of fluid leaving the active section of column retains its spatial relationship to the rest of the fluid in the reservoir. KOmixing takes place and the last element of fluid to leave the column during one half cycle Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
415
example shown in the figure the concentrations of each band and the fraction of the total stream a t each of these concentrations (in brackets) are given by the series
for n 2 2
I
t
C
H
t H
I
t
C
H
1 C
( k = 1, 2, , . . n - l), where n is the number of cycles. If the column is operated in this way for n cycles and the effluent is then well mixed, the average concentration is given by
3
2
I
(2)
(3) Figure 1. Single column with plug flow ends C
H
C
H
C
H
t
I
t
I
t
I
The general solution given by Aris (1969) for mixing after each displacement is
+
1
I
t
I
f
I
H
C
H
C
H
C
2
I
3
Figure 2. Double column, no mixing in center region
becomes the first element to re-enter the column during the subsequent half cycle. Figure 1 shows the characteristic lines for three cycles in a system chosen for its algebraic simplicity. I n Figure 1 we assume that all of the fluid leaving the column during one half cycle is returned to the column during the next half cycle (no dead volume), and that a solute front entering the column during one half cycle does not emerge from the other end of the column during the same half cycle (no breakthrough). The convention of previous authors is followed, that the fluid flows upward during the first half cycle and that desorption from the stationary phase takes place a t this time. The system is closed and the concentration of the component of interest becomes progressively depleted a t the bottom of the column. A linear sorption isotherm is assumed. At each temperature swing from cold to hot the equilibrium concentration changes by the ratio ( 1 b ) / ( l - b). In the figure yn denotes yo[(l b ) / ( l - b)]" where yo is the initial concentration; positive indices denote enrichment and negative indices depletion. At the bottom of the column the concentration leaving during the nth cycle is given by
+
+
[Note: the term q [ ( l b ) / ( l - b ) ] is incorrectly written q [ ( l - b)/(l b ) ] in Aris (1969)l. For the special case illustrated in Figure 1,the concentration front entering the column a t the start of the upward displacement half cycle just reaches the top of the column a t the end of this half cycle. I n the notation of Aris (1969) this corresponds to (q = 0, p = 1) in eq 4, which then becomes identical with (3). I n the more general case, where the front may take any number of cycles to emerge ( p = any integer) and/or may only partially emerge during a given cycle (0 6 p < l), the concentration of each band and the total fraction of the effluent in each band are given by the following series (here d = n - p - 1).
(5)
+
Since no characteristic lines cross the bottom of the column in this example there are no breakthroughs of fronts during any downward displacement cycles and hence the concentration of the fluid leaving the column remains constant during these periods. Thus it makes no difference whether or not this stream is mixed before returning it to the column. At the top of the column there are concentration changes during the displacement and these successively emerging fronts may either be assumed t o retain their identity or be mixed together before being returned t o the column. For the 416 Ind. Eng. Chem. Fundom., Vol. 1 1 , No. 3, 1972
If all of the components are mixed after n cycles this series reduces to eq 4 above. Thus if the column is to be characterized by a separation factor based on the ratio of the average concentration a t the top to the average concentration a t the bottom then it makes no difference whether or not there is mixing in the ends. The separation factor in this case is of the order [(l b)/ ( 1 - b)]". If, however, the highest concentration band can be removed after n unmixed cycles, then the separation factor for this portion will be
+
Figure 2 shows a possible physical arrangement that would minimize mixing. Two column sections are operated back-toback in a close-coupled arrangement so that effluent leaving
[
(1
(,2k-d
- q)
(s) (&9k],
Y2k-d+2
(er]- }, --
[4 (*b)
(k = 1 , 3 , - - - - - d - 3) yd-2[(1 Y2n-k
-q)
[q(*b)($:y-l],---
(+b)(&iy-l],Yd-l
(Tb) 1,- - -
[
1
$b
(k = 2p
b
2"-k
y2n-1
[(s)'"-'], [2b (
- 1, - - - - 5, 3) (*b)
{y2h-d+2[q
y2k-d
(&:)k],
k =d
[
l + b
(k
=
2n
(k = 2 , 4 ,
- - - 2p)
")"I}
('
- q) l+b 2b l+b -
for d
---
>1
the fluid flowing from the high concentration end of one column to the high concentration end of the other column is
(1
l + b
(7)
- 1, - - - 3, 1 )
The concentration a t the outer ends of the column is given by
-9 +Q
(s)2} (&3d+1] (11)
These analytical results have been extended by numerical studies in which the assumption of plug flow within the column is relaxed. (Local equilibrium between phases is still assumed.) Here it is found that mixing in the end reservoirs always lowers the average separation factor. Nomenclature
b
["
l + b '_")2n-k],
- 2, d - 4,- - - - 2 , O )
the top of the lower column enters immediately into the bottom of the upper column. The adsorption and desorption cycles for the two columns are 180' out of phase and SO timed that the high concentration region occurs in the middle. The outer ends of the two columns can either terminate in piston and cylinder reservoirs, as in the conventional single column, or they can be connected so that fluid leaving the bottom of the dual column system is pumped directly into the top for one half cycle and vice oersa. The latter arrangement will be assumed for this example. The model is for a closed system in which no mixing occurs. In the central, high concentration region in Figure 2, the concentration of each band and the fraction of the total displacement accounted for by each band are given by the series
yZk-2n
I f b
Y2n-k
= constant in equilibrium relationship
n
=
p
= no. of cycles for characteristic to traverse column
no. of cycles
p = fraction of characteristic emerging j = index denoting concentration band k = index denoting concentration band yo = initial concentration yn E yo[(l b ) / ( l - b)]" enriched concentration y-" E yo[(l - b)/(l b ) 1" depleted concentration CY = separation factor
+
+
SUBSCRIPTS B = bottom of single column E = outer ends of double column I = inner ends of double column T = top of single column literature Cited
The highest concentration band thus experiences a separation factor of
Aris, R., IND. ENG.CHEM.,FUNDAM. 8, 603 (1969). Gregory, R. D., Sweed, N. H., Chem. Eng. J. 1,207 (1970). Pigford, R. L., Baker, B., Blum, D. E., IND. ENG.CHEM.,FUNDAM. 8, 144 (1969).
CYn
=
(:_+
Rolke, R. W., Wilhelm, R. H., IND.ENG.CHEM.,FUNDAM. 8, (9)
3'"-l
while the separation factor for the averaged concentration a t the center is of the order of [(l b ) / ( l - b)]*". The more general solution has the same features as this special case and can be expressed by the series shown in eq 10 (here d = 2 ( n - p ) ) . The average concentration of
+
235 (1969).
Sweed, N. H., Wilhelm, R. H., IND.ENG.CHEM.,FI;WDAM. 8, 221 (1969).
DONALD W. THORIPSOX* BRUCE D. BOWEN
University of British Columbia Vancouver 8, British Columbia, Canada RECEIVED for review August 9, 1971 ACCEPTEDMay 19, 1972
Ind. Eng. Chem. Fundom., Vol. 1 1 , No. 3, 1972
417
