Rigorous calculations concerning free diffusion in several specific cases indicate that volume change on mixing effects can be appreciable in the determination of diffusion coefficients from experimental data. Procedures have been outlined which minimize the error in the diffusivity if it proves inconvenient to apply the exact equation.
T,l
=
X/2t‘/2
p
= = =
total mass density of mixture mass density of component I mass fraction of component I
pi wl
SUBSCRIPTS a
= reference to ethyl alcohol component
us
= =
m
reference to water component property evaluated a t 03 boundary = property evaluated a t - m boundary
+
Acknowledgment
0
T h e authors are indebted to R. H. Foy a n d J. A. Moffitt, Computations Research Laboratory, T h e Dow Chemical Co., for assistance in computer programming.
literature Cited (1) Boltzmann, L.? Ann. Physik Chem. W’ied. 53, 959 (1894). (2) Duda. J . L.. Vrentas, J . S., The Dow Chemical Co., Midland. Mich., unpublished data. (3) Dullien, F. A. I,., Sheinilt, 1,. i V . > Can. J . Chem. Eng. 39, 242 (1961). (4) Cricksen: J. L., Arch. Rational .Mech. .4nal. 4, 231 (1960). (5) Geddes, .A. L.. “Determination of Diffusivity,” Chap. X I I , “Physical Methods of Organic Chemistry,” Vol. 1. A. iyeissberger, ed., Interscience, New York. 1949. (6) Gosting, L. J., L 4 d ~ Protein ’ ~ ~ ~ .Chern. 1 1 , 429-554 (1956). (7) Gosting. L. J., Fujita. H., J . A m . Chem. Sac. 79, 1359 (1957). 49, 890 (8) Hammond, B. R., Stokes, R. H., ?‘runs. Faraday SOC. (1953). (9) Kirkwood, J. G., Baldwin. K. L., Dunlop. P. J., Gosting, L. J.. Kegeles, G.. J . Cheni. Phys. 33, 1505 (1960). (10) I,ainm, O., .thr,a d c t a Regiae SOC.Sci. tipsaiienszs. Ser. ZL’, 10, No. 6 (1937). (1 1) McCoiinell. A. J., “.Applications of Tensor Analysis,” Dover Publications, New York. 1957. (12) Nishijima, Y., Oster. G., J . Chem. Phys. 27, 269 (1957). (13) Nishijima, Y . , Oster. G., J . Polymer Scz. 19, 337 (1956). (14) Perry, J. H., ed.. “Chemical Engineers’ Handbook.” 3rd ed., p. 188, McGraw-Hill: Sew York. 1950. (15) Prager, S.. J . Chem. Phvs. 21, 1344 (1953). (16) Truesdell, C.. Toupin, R. A , , “The Classical Field Theories,“ “Handbuch der Physik:” Band J I I / l , p. 619 ff., S. Flugge, ed., Springer, Berlin, 1960. RECEIVED for rexsiew October 28. 1964 ACCEPTED April 22. 1965
Nomenclature
D = binary diffusion coefficient D* = approximate binary diffusion coefficient calculated from Equation 27 F,‘ = zth component of external force per unit mass acting o n component I 11’ = zth component of mass diffusion flux of component I relative to mass average velocity j1
= x component of mass diffusion flux of Component
p = TI’] = T = t
PI
= =
u‘
=
u
=
ij
117,%
= =
t
=
I
relative to volume average velocity pressure partial stress tensor of component I temperature time partial specific volume of component I zth component of mass average velocity x component of mass average velocity x component of volume average velocity zth component of diffusion velocity of component I relative to mass average velocity space coordinate in the direction of diffusion a n d flow
SLOW P A R T I C L E DIFFUSION IN ION EXCHANGE COLUMNS RA LPH S
.
C 00P E R
, University of
California, Los Alamos Scienti’c
Laboratory, Los Alamos, A’. M .
A general solution has been derived for the behavior of an ion exchange column under the conditions of particle diffusion control and irreversible equilibrium. This solution includes time dependence, and does not rely on the usual “constant pattern” assumption. A number of diffusion models were examined, including the linear and quadratic forcing term approximations and the exact solution for diffusion into spheres. The quadratic approximation was found to be in close agreement with the exact solution.
The exact solution
was in the form of a double infinite series, which was slow to converge in certain cases, and a single series approximation was developed which had more rapid convergence. HE differential equation describing the behavior of ion T e x c h a n g e is well known a n d has been solved under various conditions by several authors (2, 4,9 ) . T h e equation is
T h e existing solutions assume a quasi-steady state (“constant pattern”) in n h i c h the shape of the sorption wave is unchanged as it moves down the column. Vermeulen (70) developed a n empirical relation to extend these results to shallow beds o r 308
I&EC FUNDAMENTALS
early times. when the constant pattern has not yet been established. This is of practical interest in the sorption of plutonium on certain resins for which the solid diffusion rate is very low ( 7 ) . LVe have obtained solutions of Equation 3 (without the constant pattern assumption) for the case of particle diffusion control and irreversible equilibrium-Le.. E , = a constant for all G > 0. Longitudinal and film diffusions in the liquid have been assumed negligible a n d isothermal, uniform flow conditions are assumed to hold through the column. Solutions have been obtained for several models of the solid diffusion process. including linear forcing ( 3 ) , Vermeulen’s quadratic
forcing (9):and the exact equation for diffusion into spheres (G), all with constant diffusion coefficient and uniform particle size. General Method
09
.~
.ot Rt
Y (rl 0 5
-
For irreversible equilibrium the sorption wave will move down the column with a distinct front where c goes to zero. T h e front will pass a point z cm. down the column a t a time ~ ~ ( 2 ) At . that point i n the column! F \vi11 be given by the solution for the static (or batchj case with fixed surface concentration. except that it will be displaced in time by ~ ~ ( 2 )Thus. . if the static solution is
CO = f(t) Ee
I
Figure 1 . Fractional loading of resin sphere under p a r ticle diffusion control
(2)
the right side of Equation 3 is
(3) where (4)
We have included the inlet concentration, G,, Equation 3 for compactness later. If we normalize to c., Equation 3 becomes
Inversion of z ( 0 ) gives 7 f ( Z ) from Equation ID, C ( t , z ) from Equation 9, and finally the solution from Equation 8. This program can be carried through analytically for several diffusion functions, and approximately or numerically for others. We shall consider four diffusion functions for the solution of the static case.
A.
The "Early Time" Solution
explicitly in where v is constant factor. notation
This can be written in Vermeulen's
where
This is the limiting form of the exact and quadratic solutions with Y = 1.0 and 0.924, respectively, and is good to Y % 0.3. By taking Y = 5/6, the range of approximate validity is extended to Y 'v 0.7, as r a n be seen from Figure 1, Xvhich compares various Y ( t ) .
is Hall's (.5) distribution parameter. A formal solution to Equation 5a was derived by Liberman, and is given in the appendix. T h e result is
B. Linear Forcing y
=
1 -
e-kL
(16)
e-*kl)l/2
(1 7)
C. Quadratic Forcing y !where S is the unit step function, which is zero for its argument less than zero and 1.0 for its argument equal or greater than zero, and
G(t,L')
=
F[t -
Tf(Z)]
is derived from the condition that X = 0 when t = Letting T,(~)
=
(1 -
where
(9)
D. The Exact Solution T/(z).
e - n2+kl
Early l i m e Solution
we obtain
For the "early time" solution r-
; k where q ( s ) is the Laplacc transform o f f ( t ) and is also related to the transform of F ( t ) .
The motion of the front is given by
VOL. 4
NO. 3
AUGUST
1965
309
For convenience we make use of dimensionless variables defined by Vermeulen
2 1 or t > e / A k
For N,
kAz
N
= _ U
or
0,= N, - 1
@=k(,-y>
(31)
The solution of Equation 3 also is divided (Figure 4). For
0 is equal to the product N,T used by Hall. then becomes a quadratic,
Equation 21
N, 2 1
X (24)
For N,
=
1
-
N,e-'
(32)
e--B+*P-l
(33)
2 1
x
= 1 -
and the over-all solution is
The latter is the constant pattern solution, and shows no change ofform \tith time as does Equation 32. Quadratic Diffusion Solution
which is shown in Figure 2 Linear Diffusion Approximation Solution
The linear forcing term gives
For rhis case the motion of the front is divided into t\vo regions. .4t early times the sorption is relativel) slow and cannot fully deplete the solution. -4s shoun in Figure 3. the solution wave has a finite concentration at the front until t = e / A k (and t = z! h k ) . and moves with the velocity of the fluid inside the column, u,'c. Thus for N, 5 1, ZE
Tl(2)
= U
or
0, = 0
(28)
Afterward the wave moves with a different constant velocity, ub, which is the steady-state, constant-pattern velocity. U
L'b
8.0,
,
I
1
I
= ___
,
Exact Solution
The exact diffusion solution appeared difficult and in ( 7 ) approximate numerical solutions of Equation 8 were obtained by using 7/12) derived for the .'early time'' and quadratic approximation. The numerical approach could not be used, since it required b? 'bt to be kno\vn as an explicit function of I ? lvhich is not possible for the infinite series of Equation 19. Analytical solution by the methods used above proved possible, again \vith valuable aid from Liberman. For the exact solution
(29)
e + h
I
The quadratic forcing term is difficult to carry through analytically. but we used a numarical solution of Equation 3 to calculate a solution tvhich is presented in terms of the dimensionless variables in Figure 5 , The computer program used [ 7) \vas a general one \vhich allo\ved reversible equilibria, various diffusion models, time-dependent input concentration, and interaction among several ions in the resin phase. A typical case using the quadratic diffusion approximation (Figure 6) exhibits only a curved profile in contrast to the linear diffusion solution (Figure 3). X is not equal to Y. the condition X = Y being characteristic of the constant pattern case.
I
,
,
,
,
1
,
4
03 02
oi-
l
20
10
0
2
NP
Figure 2. Time-dependent column behavior for irreversible equilibrium with particle diffusion control Early time diffusion approximation with
310
IBEC FUNDAMENTALS
Y
= 5/6
30
40
(CMI
Figure 3. Typical calculated form of solution concentration profile for Pu in ion exchange column, assuming linear diffusion forcing term Profiles for various times fabeled b y time in seconds sec.-l
k = A = 5
1----F04
gives
3.6 3.8 34
t
-1
I T h e motion of the front is
30 28
26 24
This function has a second-order pole at s = 0 and an infinite series of simple poles on the negative real axis where
22
0
2 0
tan noi
=
nbt
(39)
Evaluating the residues at these poles gives the front as vo + + Ak A u
z(e) =
2 -
3
C -pairs
ueoiz ~~
As,
T h e S t are easily evaluated from Equation 39 and are approximately given by 06
-
04
st
* - 4 k (i
+ +)'
(41)
02
"
I
0
I
1
12
06
0.4
I 6
1
1
2 0
1
I
2.4
1
28
Use the dimensionless notation and the equation for the front (Figure 7) is
N,
Figure 4. Time-dependent column behavior for irreversible equilibrium wih particle diffusion control
T o integrate Equation 8 directly it is necessary to have O f [which is equivalent to 71(z)] as an explicit function of N, (which is equivalent to z). I n dimensionless notation the solution is
Linear dlffusion approximation 1 similar to quasi-steady-state solutlon Result for N,
2.
Taking the Laplace transform of Equation 34 and using the identity *cotin$ =
1
-
9
+ E@_29-_n2
o=&
=
s
)(:
-
25
NTJ
e-nZ+eS,
d ~ !ewei(Np') ,
n=l
(43)
(35)
1
1-
with
X(O,N,)
(36)
NP
Figure 5. Time-dependent column behavior for irreversible equilibrium with particle diffusion control Results determined for quadratic diffusion approximation b y numerical calculatlon o f column behavior, in good agreement with exact solution
Evans noted that by changing the variable to O fwith the use of Equation 42 to obtain dN,/dO,, the integral could be done analytically giving
Figure 6. Typical calculated forms of concentration profiles for Pu in ion exchange column, assuming quadratic diffusion approximation At early times c/c, not equol to c/c, Proflles for various times labeled by time in seconds k = lo-' ret.-'
A = 1 VOL. 4
NO. 3
AUGUST 1965
311
Discussion and Conclusions
3.2
'/
I
NP Figure 7. Exact solution for time-dependent column behavior for irreversible equilibrium with particle diffusion control
T h e time-dependent equation for a n ion exchange column with solid particle diffusion control and irreversible equilibrium has been solved in general, and specifically for several diffusion models. I n addition to the exact solution for diffusion into spheres, several diffusion approximations were examined. as they are more widely used because of their mathematical simplicity. T h e '.early time'' approximation is a limiting form of the exact solution and the quadratic approximation and yields a very simple analytic result. The Jvidely used linear approximation has a n analytic solution with two phases. I n the first phase the concentration wave moves with the velocity of the fluid and the form of the break-through curve varies with time. T h e second phase is equivalent to the constant pattern solution with the sorption wave moving at the steady-state velocity. T h e quadratic approximation gives a much better fit to the exact diffusion solution, a n d is stili well suited for use in analytic or numerical calculations. Glueckauf (3) in a comparison of various approximations found that for irreversible equilibrium and constant pattern conditions the quadratic approximation was the best of several proposed. O u r results confirm this and extend it to the initial transient state. T h e 0, N, diagrams allow simple physical interpretation. O n them a breakthrough curve is a vertical line a t the appropriatevalue of N,? and thesolution concentration throughout the column a t any fixed time is represented by a line with intercept on the 0 axis of kt and a slope of -€,'A. Appendix.
Solution of Column Equation
Calculated from Equation 44
u
= V/€
This has the formal solution
X(t,z) = S (i T h e results (Figure 7) are very close to those based on the quadratic difhsion approximation. I n many cases the summation o n i required very many terms (-lo3) for convergence and a computer program was developed to evaluate X . An approximation which is amenable to hand computation can be based on Equation 43. by assuming Of to be quadratic in N,. Let
O f = A2Np2
(45)
Thz integral I in Equation 43 becomes
= S
(1
-
-):
-
L' l
i>
-
s,': (I
dt'
G
G(t',z')
-
x
c z ' ,z ' )
(A3)
where S is the unit step function. We must know T ~ ( z )in order to determine function G. We get i,(z) from the condition that X = 0 a t the wave front, when t = i f ( z ) ; then,
0 = 1 -
lt:
G [ r j ( z ) - --, - "
2'
]
(A4)
But? 'This integral has been discussed a n d tabulated in the literatiire (8) in terms of a function F,,(y) which is less than l .O for all
Y. ~ ( y )= e-y*
eYzdY
(47)
With this, Equation 4 3 can be written
X(O,N,)
=
S
(T)
('46)
-
By letting
U
312
l&EC
FUNDAMENTALS
Acknowledgment
Equation .44becomes u =
dz‘F [O(z) - O ( z ’ ) ]
or
‘ I h e author is indebted to D. B. James who raised the problem, to D. Liberman who was instrumental in its mathematical solution, and to J. Evans for performing the numerical computation Nomenclature
Csing L,aplace transforms on Equation A9 gives
Then let R
=
0 - O’, so that
T h e integral on the right of Equation A13 is the transforrn of F ( R ) . or equivalently the transform of F ( t ) of Equation 4 . Now let
e { F ( t ) \ = Q(J) Since F ( t ) is a derivative of a functionf(t),
Equation A13 then becomes
Any consistent set of units can be applied. tions are given.
Defining e q u a -
45) concentration in solution = concentration in resin phase c2 = equilibrium resin concentration D = diffusion coefficient in resin phase k = a diffusion parameter (Equation 15) N, = a dimensionless variable (Equation 22) q ( s ) , Q(s) = Laplace transforms (Equation 12) K = resin particle radius s = Laplace transform variable t = time u = fluid velocity in column = u,/e u = fluid velocity outside of column ub = constant pattern sorption wave velocity X = normalized solution concentration Y = normalized resin concentration = a variable (Equation 47) y z = distance along column E = void fraction 0: = a dimensionless variable (Equation 23) f = indicates a functional dependence F = indicates a functional dependence G = indicates a functional dependence s = unit step function = a variable (Equation 10) 0 Lk = distribution parameter (Equation 7) v = a constant (Equation 13) = arrival time of wave front 7, = a variable (Equation 36) C$J 4 = a constant = 6 = Dirac delta function A
= a constant (Equation
c
=
or literature Cited
-u = Asq(s)s6:(.’(e’)] S Hence,
Thus, given the static solution f ( t ) , one can. in principle. determine its transform q ( s ) , and with Equation ,419 determine z(i3). This must be inverted to get O(z) and the T ~ ( Z )determined as given by Equation ’47. With T,(L) function G can be determined and the solution obtained from Equation A3.
(1) Cooper, R. S., James, D. B., Los Alamos Scientific Laboratory Kept. LA-3046 (July 1964). (2) Glueckauf, E., Trans. Furnduy Soc. 51, 34 (1955). (3) Zbid.; p. 1540. (4) Glurckauf, E., Coats, J.: J . Chem. SOC. 1947, 1315. (5) Hall: K. R., Acrivos, A , Eagleton, L. C., Vermeulen, T.. IND.ENG.CHEM.FUNUAMESTALS. to be published. (6) ,Helfferich. F. G., “Ion Exchange.” p. 260, McGraw-Hill, h e w York, 1962. (7) James, D. B., Cooper, R. S.? Los Alarnos Scientific Laboratory, Kept. LA-2838 (March 1965). (8) Miller, LY. L., Gordon, A . R., .I. Phys. Chem. 35, 2760 (1931). (9) Vermeulen, T., Adztun. Chrm. Eng. 2, 148 (1958). (10) Vernieulen, T., Znd. En!. Chem. 45, 1664 (1953). RECEIVED for review November 9, 1964 ACCEPTED .4pril 30, 1965 Work done under the auspices of the U. S. Atomic Energy Commission.
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