An Equilibrium Theory of the Parametric Pump SIR: Pigford, Baker, and Blum (1969) presented a n equilibrium theory of the parametric pump, which was more recently generalized by Aris (1969). The theory shows a good agreement with experimental results for a reasonable range of concentration. Such a linear equilibrium model, however, will be open to question for very large concentrations, as Aris (1969) pointed out, and the deviations in Figure 6 appear due to the nonlinear nature of the equilibrium rather than to finite mass transfer resistance. The nonlinear equilibrium is also considered to have the prime effect that makes the separation factors limited. The effect of the nonlinear Langmuir-type adsorption isotherm and the approach toward a limiting separation were observed by Wilhelm, Rice, Rolke, and Sweed (1968). Using the authors’ notation, we put v = (1 -
(1)
€)Ps/ePF
g=y+vx
(2)
(3) where the equilibrium relation, f, satisfies the conditions fu
> 0, fuu < 0, and f r < 0
(4)
At the end of every half cycle the solute is redistributed in the following manner (constant g) : Yh
-
Yc = -vlf(Th,Yh)
- f(Tc,~c)l
heat of adsorption per mole of solute. Equation 6 is then reduced to
- Yc =
+
-Y[(Th - TC)fT(T*,Y*)
(Yh
+
Yc
+
from which Equation 8 follows simply. Incidentally, %-eobtain lim
yh/yc
%1-
YS-0
Y ( T~ Tc)N dK* 1 vNK* dT
+
(10)
which may be applied in the lower part of the column for large n. The discussion can be better visualized in the f vs. y diagram, for each cycle is characterized by a pair of constant g lines that are of constant slope, - Y - I (cf. Equation 5 ) . Figure 1 clearly elucidates that the linear equilibrium model may lead to a significant e;ror for large values of n and thus the nonlinear nature of the equilibrium must be taken into account. According to the mathematical theory of quasilinear equations (Rhee, 1968), the concentration boundary in the cold half cycle tends to be sharp (shock), whereas the one in the heating half cycle becomes diffuse (simple wave). As ( y T ) % increases, the head of the diffuse boundary is accelerated faster than the sharp boundary, since their propagation speeds are given by the reciprocals of
(5) at
00-dz = ($)T
where subscripts h and c denote the heating and cooling half cycles, respectively. It is clear that the larger the difference, the more effective the parametric pumping becomes. By applying the law of the mean, Equation 5 can be rewritten in the form Yh
dK* y* V(Th - Tc)N - dT Y c (1 K * Y * ) ~ vNK*
=
1
+
Y
(z)
- Yc)fu(T*,Y*)]
or
where
Tc
< T* < Th, Y c < Y* < Y h
(7)
c
0
I n the upper part of the column both y c and Y h increase as the number of cycles, n, becomes large. Since it follows on physical grounds that
( a ) Linear
Y
equilibrium
lim f ~ ( T , y ) l = y 0 Y-m
we obtain
which shows the approach toward the limiting separation. Suppose that the equilibrium is described by the Langmuir adsorption isotherm
7
0 ( b ) Langmuir adsorption isotherm
Figure 1.
in which N and KO are constant and AH(
