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D 9 p a i t m e i i i o f the N a v y
C o n t r a c t N o n r - 2 2 0 ( 2 8 )
ION EXCHANGE KINETICS A NONLINEAR D I F F U S I O N PROBLEM
F. t - ie l f fer ich and M. S. Plesset
Eng inee r i ng D i v i s i o n
C4LIFCjRN1A I b ~ 5 S l I V b J ~ E Of JECHPNOLOG'at'
Pasadena, C a l i f o r n i a
Repcirt NO. 85-7 Sspern b e r 1957
A p p r o v e d b y M S, PIesset
Office of Naval Resea rch Department of the Navy Contract ~ o n r - 2 2 0 ( 2 8 )
ION EXCHANGE KINETICS
A Nonlinear Diffusion P rob lem
F. Helfferich and M. S, P l e s s e t
Reproduction in whole o r i n pa r t is permitted for any purpose of the United S ta tes Government.
Engineering Division California Institute of Technology
Pasadena, California
Repor t No. 85-7 September , 1957
Approved by M, S. P l e s s e t
ION EXCHANGE KINETICS
A Nonlinear Diffusion Problem
by
F, Helfferich and M. S. P les se t
Summary
Ideal limiting laws a r e calculated for the kinetics of particle dif-
fusion controlled ion exchange processes involving ions s f different
mobilities between spherical ion exchanger beads of uniform size and
a well- s t i r r ed solution, The calculations a r e based on the nonlinear
Nernst-Planck equations of ionic motion, which take into account the
effect of the e lec t r ic forces (diffusion potential) within the system.
Numerical resu l t s for counter ions of equal valence and s ix different
mobility rat ios a r e presented. They were obtained by use of a digital
computer. This approach contains the well-known solution to the cor -
responding linear problem a s a limiting case. An explicit empir ical
formula approximating the numerical resu l t s is given.
1, Introduction
An ion e'xsrchange r e s in consists essentially of a three-dimensional,
cross-linked network of hydrocarbon chains carrying fixed ionic groups,
the electr ic charge of which is compensated for by mobile ions of opposite
charge ("csuater ions"'), These counter ions axe f ree to diffuse within
the res in network, Tn contact with an electrolyte solution the res in takes
up solvent and some additional mobile ions (additional counter ions, and
l'co-ions" have the same charge sign a s the fixed ionic groups). In a n
ion exchange process the counter ion species present initially i s replaced
by another species,
We consider the ion exchange between spherical ion exchange resin
beads of uniform size containing the counter ion species A and a well-
s t i r r ed solution containing the counter ion species B, During the process
ions A diffuse out of the bead and a r e replaced by an equivalent amount of
ions B.
Id has been established that such processes a r e diffusion controlled:
the ra te determining mechanism i s the interdiffusion of the two species
A and B, either within the res in particles o r in a Nernst diffusion layer
("film"] adherent to the particle surface, which is not affected by stirring, 1
Limiting law8 for fi lm controlled exchange in ideal systems have been given
previously. in this paper ideal limiting laws for particle controlled
processes a r e calculated. Part ic le control is favored by concentrated solu-
tions, large diameter and high degree of cross-linking of the beads, and
efficient s t i r r ing. 3
The driving 'tforce" for the flux of a n ionic species is , in systems
without convection, the gradient of i t s general chemical potential, the
principal elements of which a r e the concentration gradient of the species
and the gradient of the electr ic potential. Even if no external e lectr ic
field is applied, a gradient of the electr ic potential will, a s a rule, be
built up by the diffusion process (diffusion potential). The corresponding
differential equations a r e nonlinear. The interdiffusion process con-
s idered here can be described in t e r m s of one interdiffusion coefficient
which, however, is not constant but changes with the concentration ratio
of the two counter ion species present. Hence, the mathematical t rea t -
ment is that of a diffusion problem with varying diffusion coefficient,
Hitherto only the corresponding linear problem, i. e. , with constant inter-
4 diffusion coefficient, has been solved and applied to ion exchange
kine t ics. In our approach the solution to the linear problem repre-
sents a limiting case: the exchange of counter ions of equal mobility.
The linear solution holds rigorously for isotopic exchange processes ,
whereas for other cases the deviations a r e considerable,
2. Model and Simplifying Assumptions
We use the simplest model conceivable and the simplest equations
proven to be adequate for the description of ionic diffusion processes, the
Nernst-Planck equations. 11-15
Without respect to i ts actual porous s t ructure the whole res in is
t reated a s a quasi-homogeneous phase. Individual diffusion constants for
the mobile ions present a r e defined correspondingly; they can be measured
by t r ace r techniques in equilibrium systems. 1' 15-18 The assumption is
niade with exper imenta l support 15' 16' l8 that these individual diffusion
constants a r e , essent ia l ly constant fo r a given res in , i, e . , independent
of ionic composition. We make use of the ideal Einste in re la t ion between
the individual diffusion constants and the e lectrochemical mobili t ies, We
neglect any coupling of ionic fluxes o ther than by e lec t r ic fa rces . We
a s s u m e a constant concentration of fixed ionic groups throughout the res in .
Any concentration change and flux of the co-ions i s d isregarded, since
they a r e s m a l l a s compared with those of the counter ions if the concen-
t ra t ion of the solution i s not too high and that of the fixed ionic groups not
too low. l3 W e neglect changes in ionic activity coefficients and swelling
condition of the r e s i n a s well a s the effect of gradients aP swelling p re s -
s u r e . Under the s imple boundary condition used here , the magnitude of
the selectivity coefficient i s i r re levant . l9 We r e s t r i c t ou r se lves to
r a d i a l diffusion, i. e . , spher ical symmetry. We a s s u m e that interdif-
fusion within the r e s i n beads i s the r a t e determining mechanism through-
out the process . Th i s l as t assumption i s not s t r i c t ly valid; i t leads to
exchange fluxes that a r e infinite initially, whe reas the possible diffusion
r a t e through the f i lm is finite. F o r a very sho r t init ial period f i lm
diffusion m u s t be the r a t e controlling step.
The assumptions and simplification l is ted above a r e those usually
made in the formulation of a model fo r ion exchange kinetics. 5- 10
We do not neglect the effect of the e l ec t r i c potential gradient on the
ionic f luxes; in th is r e spec t our approach dif fers f r o m previous theories.
3 , Formulation of the Problem
Under the assumptions mentioned, the fluxes of the two counter ion
species A and B a r e given by the Nernst-Planck equations 11-14
O B = - D B r ad CB -I zBCB (F/./RT) grad rg] , I lb )
where @ is the flux, 2) i s the individual diffusion constant, G i s the
molar concentration, z i s the electrochemical valence, is the
Faraday constant, R is the gas constant, T i s the absolute temperature,
i s the e lec t r ic potential, and the subscripts A and B refer to the
counter ion species ,
Electroneutrali ty requires that the total equivalent concentration of
counter ions is constant throughout the bead, since the concentrations of
both the fixed ionic groups and the co-ions a r e constant; thus
The absence of an electr ic cur rent in combination with (2) gives the con-
dition
By use of (2) and (31, Eqs. ( l a ) and ( lb) may be combined to give
2 D ~ D ~ ( ~ ~ ~ ~ + .BeB)
2 grad CA . D A A A z Z c + DBz BCB
%
'She quantity in brackets may be termed the interdiffusion coefficient
D~~ of the process , Its value depends not only on the individual dif-
fusion constants DA and Dg; it i s also a function of the concentration
rat io cA/cB and therefore a function of the time and space coordinates.
The dependence of the interdiffusion coefficient on ionic composition i s
shown in Fig. 1 for various values of the rat io D ~ / D * . F o r CA 6( CB9
DAB assumes the value DA, and for CB ( CA the value DB. The ion
present in smal le r concentration always has the stronger influence on
the interdiffusion coefficient. 33 l9 This simple rule i s a consequence of
of the fact that in (1) the concentration of the species en ters in the elec-
t r i c term. It is also physically evident since the e lec t r ic field ac ts on
every ion present and thus causes a large transference of the species
present in high concentration, whereas the flux of the species present in
low concentration is hardly affected.
F o r the t reatment of time dependent processes the equation of con-
tinuity must be introduced
In previous work on time dependent processes , only the limiting
l inear case has been considered, to which (4) and (5) reduce when a
constant interdiffusion coefficient DAB is assumed. Analytical solutions
to the linear problem have been derived for a number of systems including
those with spherical symmetry with various boundary conditions. 4, 20
However, Eq. (4) and Fig. 1 show that the assumption of a constant inter-
diffusion coefficient holds only i f the individual diffusion constants DA and
D a r e equal, a s is the case for example in isotopic exchange processes. B
This paper d e a l s with the general, nonlinear problem and will t reat
s p e ~ i f i c a l l y sys tems with spherical symmetry. Upon substitution of (4)
into (5) an expression i s obtained which can be written conveniently in
dimensionless form with the introduction of the following dimensionless
var iables and parameters:
a s z D / z D - 1 ; A A B B b = aA/zB - 1
where t is time, r is the radial coordinate, and ro is the radius of the
beads. The resulting equation
was derived in a preliminary note. The fraction qA of the species A
s t i l l present in the sphere a t the dimensionless t ime T is expressed in
t e r m s of the solution y ( p , .c ) of Eq. (7) a s follows:
The ion exchange ra te , deiined as the decrease with time of the amount
QA of the species A (in moles) present in the spheres, may be written a s
where V is the total volume of ion exchanger used.
If the problem is limited to the exchange of counter ions of equal
valence (zA = zB)$ SO that b = 0. the general equation (7) may be
simplified by the iniroduction of the new variable
to be come
which is a convenient form for numerical integration. The resu l t s
reported a t the present time will be for this case. Determination of the
solutions of Eq, (7) for non-zero values of b a r e planned for a la ter time.
We use the simple s t initial and boundary conditions possible, assuming
that initially the beads contain counter ions A only and the solution counter
ions B only, and that no concentration changes occur in the solution,
The boundary condition (12b) corresponds to infinite solution volume o r
to continuous renewal of the solution, The selection of initial and bound-
a r y conditions other than (12) would require the introduction of further
parameters which, for the numerical evaluation we car r ied out, would
have to be chosen a rb i t r a r i ly and have led to resu l t s applicable in special
cases only. In contrast , by using the conditions (121, more general l imit-
ing laws a r e obtained, which can be approached fair ly closely by experi-
mental techniques.
Even under the simple conditions (12) a n analytical solution to the
Eqs. (7) o r (11) was not obtained, We have evaluated (11) numerically for
six values of the ratio D ~ / D ~ of the individual diffusion constants:
The result; a r e given in section (5).
4. Calculation Procedure
F o r the numerical evaluation, (11) was approximated by the finite
difference form
- R- ( P ) [u(P, 7 - u(P-AP,.I)]
where
g(p, = exp - 4 ~ 3 T If A 7 / ( A P ) ~ ;
By use of (13), U(P, z + h 7 ) can be calculated f rom u ( p + ~ p , T ),
u(pI 1~ ), and u ( p - 4 , T ). F o r the calculation the initial condition and the
boundary conditions for p = 0 and p = ]I a r e required. The initial
condition is obtained f rom (12a) and (10):
\
The boundary conditions a t the bead surface ( P = 1) i s given by (12b) and
It can be shown that the gradients of y and u vanish a t the center of the
sphere:
( a u / a P I P ' 0 = o .
Hence the condition a t p = 0 becomes
By choice of Ap an upper limit is given for A T by the stability
condition f o r the numerical integration. 22 In case of (131, this condition
The function u(p, 7 ) was calculated numerically f rom (13) and (14)
in accord with the requirement (15).
F o r the numerical integration corresponding to (a), which leads to
the fraction qA st i l l present in the sphere, S impsonfs rule was used:
where
The dimensionless ra te , -dqA/d r, was approximated by
The ca l cuh t ion s t a r t s out with a step' function, Therefore, a very
fine subdivision of space and t ime is requi red initially, The init ial
spacing of p used was Ap = l/640, The spacing of AT was
selected i n accordance with the stability condition (15). As the cal-
culation proceeded, Ap was increased stepwise to 1/320, 1/160,
1/80. 1/40 ( a t about qA = 0 . 9 ) , 1/20 ( a t about qA = 0.6), and 1/10 (at
qA 4 0.1). AT was increased correspondingly. The estimated max-
4- irnurn e r r o r in qA is t 0.01; the e r r o r i n dqA/d.r i s about - 2%.
The calculations were c a r r i e d out o n the "Datatronl' digital com-
puter (manufactured by Electrodata, Inc. , Pasadena, California) in the
computing cen te r a t the California Institute of Technology. Floating
point numbers were used. The total t ime required was about 35 hours.
5. Resul ts and Discussion
The calculation yields the dimensionle s s functions u ( ~ , 7 ), qA( s ),
and -dqA( T ) / d l . The f i r s t i s readily converted to y (p. .r ), which
gives the rad ia l concentration profiles of the species A. In tabulating
and plotting the resu l t s , we have followed general practice and given
the f ract ional at tainment of equilibrium, ~ ( s ), r a the r than qA(7 ).
Under the init ial and boundary conditions (121, this quantity is given by
F ( 7 ) = 1 - q A ( 7 ) (18)
The representat ion of the data i s kept in dimensionless form, since the
actual values of the concentrations, the fractional attainment of equilib-
r iurn, and the r a t e s a t a given time t depend a lso on D A# ro, C, and V.
They can be obtained frqm the tables and graphs by substituting numcri-
ca l values for these quantities and using ( 6 ) and (9).
Table I gives the fractional attainment of equlibriurn, F ( T ), and
the dimensionless rate , -dq / d q l for the various cases . Figure 2 A
shows a plot of F ve r sus T and includes the solution to the linear ease
4 (DA = DB) which is well known to be
2 2 cup(-n a T) . (19) F ( T ) = f - - mi! n = ~ n
It should be emphasized that the time coordinate T ~ o n t a i n s the
quantity DA; hence, these resu l t s can be used fo r d i rec t comparison of
different processes only if the ion A initially present in the beads is the
same in a l l cases . F igure 2 shows that the curves for different rat ios
D ~ / D ~ can not be transformed into one another by a linear t ransforma-
tion of the time. If the ion B is s b w e r , the process i s slower, but not
by a constant factor, because the interdiffusion coefficient increases with
decreasing concentration of A, On the other hand, i f B is fas te r , the
process i s faster , but the interdiffusion coefficient decreases during the
process.
I t is of in te res t to compare the r a t e s for forward and backward
exchange of two counter ion species. In Fig, 3 we have plotted the f rac-
-I- tional attainment of equilibrium for the exchange processes H against
+ -I- 13' and L i against H , assuming a ratio D ~ / D ~ ~ = 10. F o r direct
comparison we have chosen D t / r2 a s the dimensionless time H o
coordinate in both cases , i t is seen that the process is faster if H', the
fas te r ion, *is initially present in the resin, The half t imes of exchange
differ by about the factpr 2, the time required for 90% exchange by about
Che factor 3. According to previous theoiies, which assume a constant
interdiffusion coefficient, the two curves should coincide.
The radial concentrakion profiles of the species A for 25%, 50%,
75%, and 90% exchange a r e shown in Fig, 4. T h e profiles for the linear
case (D* = ~ g ) given by 4
2 2 sin (nnp) exp (-n n T ) (20) T P , = 1 n
a r e included. The shape of the profiles depends strongly on the ratio
DA/Dg If the ian present in the sphere initially is much the faster one,
a comparatively sharp boundary moves in toward the center of the bead
(I?ig. 4a). If the ion present in the bead initially is much the slower one,
the boundary is diffuse and the process reaches the center rapidly
(Fig. 4b). Again, the explanation is straightforward. In the f i r s t case
the concentration of the faster ion is sma l l near the surface and large
near the center of the bead. Hence, the interdiffusion coefficient
decreases toward the center. Therefore the outer shells of the bead a r e
exhausted rapidly, whereas the exchange near the center remains slow,
In the second case, the opposite holds, The interdiffusion coefficient
increases toward the canter ; therefore the exhaustion is more uniform.
Previous theories lead to one se t of curves for a l l ca ses (Fig. 4g).
F o r practical purposes, an explicit expression for F ( .c ) which
approximafes the numerical resu l t s will be useful. The relation
~ ( 2 ) = 2 - exp Cn ( f l ( a ) e t f 2 ( a ) z 2 + f 3 ( a ) 'z .~)J] 1/2
where a =: D ~ / D ~ and the coefficients f a), f2( a ) , and f 3 ( a ) a r e
given by
t f i t s a l l the numerical resu l t s within a n e r r o r of - 6%. I t should hold
equally well for intermediate values of the rat io D ~ / D ~ . Equation (21)
was developed a s an extension of a simpler but l e s s accurate approxi-
mation given by vermeulenZ3 for the linear case.
We wish to emphasize again that the resu l t s represent ideal limiting
laws only, f rom which the behavior of actual sys tems may be expected to
show more o r l e s s pronounced deviations. However, the comparison with
the solution fo r the linear case demonstrates conclusively that the effect
of the e lec t r ic potential included in ou r approach is an essent ial feature of
the process which no theory can omit.
4. Acknowledgments
We a r e indebted to Drs . J. Franklin, C. B. Ray, and R. Nathan
fox helpful suggestions concerning the numerical integration. We also
wish to express our gratitude to Mr . D. Brouillette and Miss Z. Lindberg
for their valuable assis tance in the computations.
FRACTIONAL ATTAINMENT OF EQUILIBRIUM, ~ ( t ) , AND DIMEI\TSIONLESS
RIZTE, - aqA(*)/a+ , FOR SIX D ~ W I T VALUES OF THE RATIO D ~ / D ~
TABLE I ( ~ o n t ' d , )
FRACTIONAL ATTAJXMENT OF EQUILIBRIUM, F ( * ) , AND DIMEXVSIONLESS
RATE, - dqA( r ) i d s , FOR SIX DIFFERWT VALUES OF 'PHE RRTIO ~ ~ 1 %
TABLE I (~ont'd.) %
FRACTIONAL ATTAUNMENT OF EQUILIBRIUM, F( * ) , AND DLMENSIOIEZSS
RATE, - dqA( ( )/ d z , FOR SIX D~~ VALUES OF THE RATIO ~ ~ 1 %
TABLE I (C!ontfd,) 1
FRACTIONAL ATTAINMENT OF EQUILIBRIUM, F( ) , AND DIMENSIONLESS
RATE, - dqA( l )/ d~ FOR SIX DIFFEBEXQ VALUES OF THE RATIO D ~ / D ~
a) ~ ~ 1 % - 1/2
TABLE I (cont9d.) x
FRACTIONAL ATTAINMENT OF EQUILIBRIUM, F(Z ) , AND DIMENSIONLESS
UTE, -aql((% ) / a~ , FOR SM ~rn- VALUES OF TIE WTO D ~ / D ~
FRACTIONAL ATTAIJSTMENT OF EQUILIBRIUM, F(*E' ), AND DLMENSIONLESS
RATE, - aqA(-z )/ , FOR SIX DIFFERENT VALUES OF THE RATIO D ~ / D ~
EQUIVALENT FRACTION OF B EQUIVALENT FRACTION OF B
Fig. 1 . Dependence of the interdiffusion coefficient DAB on the ionic composition of the exchanger, calculated from Eq. (4), for the interdiffusion of ions of equal valence (left) and of a uni- valent and a bivalent ion (right). The different curves cor- respond to different ratios D * / D ~ .
CH-pA
O 0.2 u .4 u
$"1& b""; V
.*-+ 2 2% 2 , ZJ+\ 4 +h X IT.; <
0) 2 ; f.9; U &I-* *v c d * k+?* e - c W CrXJ d.2
1
(a) DA /DB=IO
W -I a (el DA/DB= 2 > - 1.0 3 0 W
0.5
0 0 0.5 1.0 0 0 . 5 1.0
RADIAL COORDINATE p '
RADIAL COORDINATE p
RADIAL COORDINATE p
Fig. 4. Concentration profiles of the species A in the bead for 2570, 5070, 7570, and 90% exchange (F = 0 . 2 5 , 0 . 5 0 , 0 . 7 5 , and 0 .90) for seven different rat ios DA/DB. The profiles for the linear case (DA = DB) were calculated f rom Eq. ( 2 0 ) .
<
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