984
LEONLAPIDUS
AND
NEALR. AMTJNDSON
other factors such as chloride ion concentration than it is a t higher pH. (3) Cobalt is adsorbed to a greater extent than barium under comparable conditions, the ratio between the two being as high as four in some cases. In experiment B a t pH 7.50, 49.7y0 of the cobalt was adsorbed, while in experiment D at pH 7.51, 13.2y0 of the barium was adsorbed. In experiment H with the quantity of adsorbent 4 X 10W6gram atom, the adsorption of cobalt was 44.7%, while in experiment J with this same quantity of adsorbent the adsorption of barium was 10.9%. Despite this difference in per cent. adsorbed, barium and cobalt which occur together in low concentrations in fission products cannot be separated from one another to give samples of high radioactive purity, since the amount of barium adsorbed, while considerably less than the amount of cobalt adsorbed, is still quite large. (4) The relative extent of adsorption of cobalt and barium is the reverse of that reported by Iiress-
Vol. 50'
man and Kitchener for the adsorption of these ions on a phenol sulfonic ac.id resin.3 They indicate that the extent of adsorption of cobalt is determined by the ion association of cobalt ions and a ((primary determining factor, which may or may not be ionic size." In the case of adsorption on hydrous ferric oxide, it is believed that a more significant factor is the weakly basic character of cobalt, that is, the hydrolysis of cobalt ion is a controlling factor in its adsorption by the very weakly acidic adsorbent, hydrous ferric oxide. Cobalt and barium ions are adsorbed on hydrous ferric oxide in the reverse order of the basicities of their hydroxides. The support of this research by the Development Fund of The Ohio State University and grants received through the Graduate School from the Research Foundation are gratefully acknowledged by the authors. (3) T. R. E. ICressrnnii %nd .I. A. Tiitcliener. J. Chern. Soc., 1201 (1949).
MATHEMATICS OF ADSORPTION I N BEDS. VI. THE EFFECT OF LONGITUDINAL DIFFUSION I N ION EXCHANGE AND CHROMATOGRAPHIC COLUMNS' BY LEONLAP ID US^ AND NEALR. AMUNDSON Universitg of Minnesota, Minneapolis id, Aiiiinmola Received January I , 1966
The effect of longitudinal diffusion in chromatographic and ion eschnnge columns is considered. Calculations made under t h e assumption of pointwise local equilibrium show t h a t sharp boundaries are smoothed out, thus casting some doubt on the column method for determining isotherms. The problem in which the local rate of removal follows a first order kinetic law is alao solved and this solution-is a new one.
Of the many factors which may determine the dynamic behavior of an adsorption column, the effect of diffusion has received the least attention. The diffusional effects may manifest themselves, in general, in three ways under the most simple assumptions. If there is a resistance to mass transfer between the fluid and solid, it is usually assumed that this is a diffusional phenomenon. If the particles used as adsorbent are of some size and if the whole particle is to be used effectively, it is necessary that the adsorbate diffuse through the fluid in the intraparticle volume before adsorption can take place inside the particle. These two effects have already been considered by the writersa3 The third effect, that of longitudinal diffusion in the interparticle fluid, was considered partially by Glueckauf, Barker and IGtt,' and it is the purpose of this paper to generalize and extend this work. Longitudinal diffusion causes a flow of adsorbate which is superimposed upon the convective flow of fluid through the column. Physically, it is clear that diffusion will tend to smear out sharp adsorption (1) Presented a t the XIIth International Congress of Pure and Applied Chemistry, New York, N. Y., Sept. 13, 1951. (2) Forrestal Research Center, Chemicsl Kinetics Division, Princeton University, Princeton, New Jersey (3) P. R. Kaeten, L. Lapidus and N. R. Amundson, Trrxe JOURNAL, 56, 883 (1952). (4) E. Glueokauf, K. H. Barker and 0.P. Kitt, Discussiona of the Faraday Soc., No,7 , 199 (1949).
bands formed in the column and this is substantiated by the calculations. Certainly longitudinal diffusion will be a small effect on the total flow of adsorbate through the column. This brings up, however, a rather interesting conflict on the basic character of chromatographic analysis. The early theory of chromatography assumed that equilibrium between adsorbent and adsorbate solution was immediately established. Later work showed that this required an extremely small rate of flow through the column, in which case the effect of diffusion becomes more pronounced. On the other hand, if flow rates are high the equilibrium theory becomes questionable and some rate process must be used for the local adsorption mechanism. It is assumed here that a column of u i i t cross sectional area is packed with a finely divided adsorbent such that the interparticle volume is filled with solvent or solution. The column may have an initial adsorbate content on the adsorbent as well as in the interparticle volume. At time zero a solution, whose concentration may vary with the time, is admitted to the column. I t is desired to know the concentration of the i~it~erparticle solution and the adsorbate content of the adsorbent at any time and at any position in the bed. Let c = concentration of adsorbate in the .fluid stream, moles per unit volume of solution
NOV.,
1952 EFFECT O F IIONGITUDINATAn I F F U S I O N
I N I O N EXCH.4NGE AND CHROMATOGR4PHIC COTjUMNS
n = amount of adsorbate on the adsorbent, moles per unit volume of bed as packed V = velocity of fluid through interstices of the bed ' z = distance variable along the bed D = diffusion coefficient of the adsorbate in solution in the bed co(t)' = concentration of solution admitted to the bed. Ci(Z) = initial concentration of solution in the interparticle volume in the bed ni(x) = initial adsorbate concentration on the adsorbent a = fractional void volume in the bed Then on analyzing an elemental length of bed Ax, t,he equation D -dzc
= 8 -dc
a22
ax
at
a
a
two special cases will be treat,ed here
+
n = klc kz an/& = klc - kzn
(2) (2')
The first implies that equilibrium is established at each point in the bed while the second assumes the rate of adsorption is finite. The second also contains in it the special case of mass transfer a t the particle surface in which the equilibrium is linear as assumed by Hougen and Marshsl1,j i.e. an/& = kr(c
- c*)
where kr is a mass transfer coefficient and n = hc*. The additional relations n = ni(z)
>0 = 0, z > 0
when z = 0, t
(3)
when t
(4)
describe, respectively, the inlet fluid and the initial condition of the bed. The problem as stated above is the most general one with the assumptions used here. One assumption which is generally not valid but which has been tacit in practically all previous work concerns the character of the hydrodynamic velocity profile. If the velocity profile is not uniform one would not expect the concentration profile to be uniform. Hence equation 1 is in error in not only neglecting the hydrodynamic variations but also the consequent concentration effects. Equilibrium Case.-If pointwise equilibrium is established in the column a t each point, it is necessary to solve equations 1, 2, 3 and 4. This is possible since the problem may be easily reduced to problems which have already been solved. Details of the so1ut)ion are given in the appendix. The solution is
( 5 ) 0. Hougeri and (1947).
Special cases of equat,ion 6 may be obtained with ease. Suppose, for example, that co(t) and c i ( x ) are constants, co and ci, respectively, then straightforward manipulation of the integrals produces
+ -dc + -1-dn
results. It is necessary to make some assumption concerning the mechanism of adsorption, that is, the local relation between c and n. Although it would be desirable to use a general relation of the form
c = co(t) c = ci(2)
986
W. R. hlarshall, Chem. Eng. P r o g . , 43, 197
where Vta = v and
where erf and erfc are the error and complementary error functions, respectively. From equations 8 and 9 it is seen that the concentration ratio is dependent upon two variables, V / D and v / y a . In Fig. 1 equation 8 is plottJed as concentration ratio c/co versus v for x = 50 and with values of V / D = 1, 2 , 10 and 100 for y a = 5 and y a = 25. Curves A, B, C and D are for decreasing diffusivity with fixed velocity or for increasing velocity and fixed diffusivity. Hence as the velocity decreases through the bed the sharp brealrthrough point which would normally be expected tends to be smeared out. Curves E, F, G are for a smaller value of ya and illustrate the effect of a lower capacity adsorbent. In this case the effect of diffusion is not as pronounced.
Y - VOLUME OF SOLUTION Fig. 1.-Plot showing the effect of longit.udina1 diffusion for an infinite coluinii in which equilibriuni is estahlishecl locally. Initial adsorbate concentration is zero. Influent Concentration is co.
D
It is well known that the solution for no diffusion, = 10, is
986
1,EON IAAPIDUSAND N E A L
and hence the graph corresponding to Fig. 1 is a step function, the steps occurring at v = 250 and 1250. The graphs in Fig. 1 are seen to converge to these step functions. A second special case of some interest is that one in which the feed solution contains a pulse in composition. It is supposed the concentration is co from zero time to time t, after which it jumps t o coo and remains so after t. Let ci be zero for convenience only. Equation 5 reduces to
+ (coo - rg) H ( Q- V ) , V
0 , when t = 0, x > 0
= c,(t) c = u(z) n = n&) This system seems to be a new one so that its solut,ion will be discussed in some detail. The solution will be obtained by the method of the Laplace transformation. The definition of the transforms will be t,aken as
L[c(z,L)]=
Lm
e - p t c(z,t)
=
P( s)G(z,s) ds
where the Green’s function G(z,s) is defined by
dt = h ( z , p ) = h
L[n(z,l)l = N ( z , p ) = N The transforms of equations 1 and 2 are
1
It is obvious that the solution for y2 is 2/2
=
ho(p)e-ZdiJ
vz
Since h = y e””, it is clear that the inverse transforms of and yz may be taken directly. Now y~ may be writ,ten in the form e -243 ,-m 112 kzho(p) pkz + VI
p N -’ ni(z) = k l h - k?N Elimination of N between these two equations gives
+
m2
Consider the quantity e-sdB
+
(6) R. V. Churchill, “Modern Operational Mathematics in Engineering,” McGraw-Hill Bnok Co., Inc., New York, N. Y., 1944, p. 117. (7) Reference 6, p. 109.
P kz The inverse of this function is not available directly from tables and its inversion by means of the inversion integral (8) Kamke, “Differentialgleichungen,” Chelsea Puhlishing Co., New
York, N. Y., 1948, p. 607,
LEONLAPIDUS AND NEAL R. AMTJNDRON
988
is beset with difficulties. Howcvcr, by using seldom uscd operational form the answer may be obtained. The Laplace transform of the function
In order to find the invcrse transform of ~ J I it is necessary to consider the inverse P(s)G(z,s). P(s)may be written
with X and Y defined in Equations 16 and 17. Considor the function
where JUis the zero-urder Bessel function, is
-9(P 1 P
Vol. 56
+i)
where $ ( p ) is the tmnsform of ,f(t). (Sco rofcrence 9, t,he eight,h formula on t,he pagc rtnd Iiote also t,he difference in notation.) From t.his it, is a simple matter to show that the transform of
with 9 independent of p . The inverse of this function may be found by the method used on U S by taking $ ( p ) to IF e -qdPTd whose inverse is
-
a
Hence the inverse of equation 18, omitt.ing for the moment t.hr factor p X Y , is given by equat,ion 15. Now H(o,q) = 0 so that the invcrse of R is L-L(R) = X"(l,g) YH(1,p) witaht,he prime denot.ing differentiation with respect t,o I . The quant,it,y 9 used herr ncedx some e1ucidat)ion. From t,he definit,ion of t,he Green's functionG(z,s) the following may be written
+
If nnc chooses for + ( p )
+
+(p) = e-kd/PTd
t>hcn
,-.die
Therefore if one chooses
-a = - B
+ kaA + k?, = + klke
01
sinh figs = !(e-d'B(Z--a) 2
- e-dz(rfs))
Therefore in taking the inverse transform above, q must be chosen successively as
b =z / a then the inverse of Q is L-I(&) = F ( l ) , defined in Equation
.., .
1 I
hatthe inverse of may be written as a but these details will be omitted. It may be rigorously shown that the solutions obtained convolution, as in equation 12. in this paper converge to the solutions obtained from the (9) X . W. MrImhlan and Pierre Hitnibert, hfem. des Science equations when the diffusion is neglected, as D + 0. This M a f h . , 100, 12 (19-11). has some mat,hemat,ical interest. The inverse of + ( p ) is
+
