812
KEAL R. AMUNDSON
data show that different mechanisms are involved in gelation by acids, amnonium salts, and sodium aluminate. Gel formation with the latter reagents probably involves the formation of unionized (or slightly ionized), heavily hydrated gelatinous salts as well as copolymerization. REFERESCES
DAVIS, H. L . , A X D HAY,K. D.: J. Am. Chem. SOC. 61, 1020 (1939). (2) FLEMMISG, W . : Z . physik. Chem. 41,427 (1902). (3) HAY,H. R . : U.S. patent 2,444,774 (July 6, 1948). (4) HOLMES, H . K.: J. Phys. Chem. 22, 510 (1918). (5) HURD,C . B.: Chem. Revs. 22, 403 (1938). (6) HURD,C. B., AND BARCLAY, R . W.: J . Phys. Chem. 44,847 (19401. (7) HURD,C. B., A N D MILLER, P.S . : J. Phys. Chem. 36, 2194 (1932). (8) HURD,C. B., FREDERICK, K. J., A N D HAYXES, C. R . : J. Phys. Chem. 42, 85 (1938). (9) MERRILL, R. C . : Ind. Eng. Chem. 40, 1355 (1948). (10) PRASAD, M.:J. Indian. Chem. SOC.10, 119 (1933). (11) PRASAD, l f . , A N D HATTIAXGADI, P. B.: J. Indian. Chem. S O C . 6, 653, 893 (1929). (12) RAY,R . C., A X D GANGLILY, P. B . : J. Phys. Chem. 34,332 (1930). (13) RAY,R . C . , A X D GAFGL~LY, P. B . : J. Phys. Chem. 35, 596 (1931). (1)
AIATHEhIATICS OF ADSORPTIOK I S BEDS. I1 S E A L R . AMCSDSOS
Department of Chemical Engineering, School of Chemistry, University of Minnesota, Minneapolis 14, Minnesota Received September 16, 1949
The mathematical solution of problems of adsorption of a solute by a solid from a solution depends upon the mechanism assumed to be controlling in the bed. There is a fundamental difference in the mathematical analysis, depending upon whether one assumes that equilibrium obtains or whether one assumes a kinetic approach of some kind. In the first case the basic problem is the solution of a first-order partial differential equation. Since the definition of linearity for first-order equations is somewhat more liberal than it is for second-order equations, it derelops that solutions in the first case can be obtained for general isotherms. This problem has been considered by a great number of writers. I n the second case the problem is not quite as simple, since the equations Tvhich are to be solved are of second order and only very special cases can he solved analytically. Similarly, this problem has been considered by many writers. In a previous paper (1) the author considered one problem of the second category in full generality. It is the purpose of this paper to solve two additional problems completely. If we assume that w have an adsorbent hed of unit cross section where depths (or heights) are measured as z and if c = concentration in moles of adsorbate per unit volume of solulion
in the fluid stream,
MATHEMATICS O F ADSORPTIOE; IN BEDS. I1
n = V = (Y = 1 =
813
amount of adsorbate in bed in moles per unit volume of bed, velocity of fluid through interstices of the bed, fractional void volume of bed, and time,
then on considering an elemental thickness of bed Az we obtain the equation (see Klotz (6)) :
an V -ac+ - - ac + - - =1 O az
at
at
Previously the author considered, after Bohart and Adams (3), that the local rate of removal in the bed was governed by
-1 -an = kc(N0
- n)
a at
where NOis the saturation capacity of the bed and k is the velocity constant. In this paper two additional cases will be discussed. Case A : Case B:
-an (Y
at
-
k l c ( i ~ o- n)
- kzn
(3)
Special cases for Case A have been considered previously. It is to be noted in this case that such a kinetic assumption covers both the cases where there is a mass transfer resistance and also where adsorption itself may be controlling. Either of these conditions merely requires a redefinition of k1 and k2. Case B was first considered by Thomas (10) for ion exchange. Here \ye shall solve these problems for arbitrary initial condition of the column and for arbitrary inlet feed composition to the column. Also the term &/at n-ill not be neglected, as has often been done. Although the method of Riemann is applicable in the mathematical solution of these problems, the author prefers the use of the La Place transformation. For this method any one of several good books in the field may be consulted; however, the author recommends the book of Churchill (4).The transform operator will be denoted by L and the transform variable corresponding to time (or better, distance along the bed) as p . Therefore L[f(t)]= g ( p ) , where g ( p ) is the transform of f(t). Tables needed for finding inverse transforms are available in Churchill’s book and no further reference to these tables will be made.
CASEA In this case the complete statement of the problem is
814
NEAL R. AMUNDSOX
c(z, t ) = co(t), where z = 0
n(z, t ) = n&),
where t
(4)
f
V
This case has been called by Thomas (11) the case of linear kinetics. Equation 4 indicates that the concentration in the feed stream can be a function of the time and can therefore vary in an arbitrary manner throughout the course of any run. Equation 5 states that the amount of adsorbate already on the column can be a function of the space variable z. From this condition we see that the solution of this system will resolve the problems of adsorption and desorption simultaneously. It follows that we are seeking two functions which will satisfy equations 1, 2, 4, and 5 . In order t o facilitate the solution we shall make the following changes of variable. Let x = z/v y = a(t
- z/V)
Then equations 1, 2, 4, and 5 become
an = klc - kzn aY c(z, t ) = co(y/a) = cl(y)
when x = 0
n(z, t ) = no(Vx) = nl(x) when y = 0 h’ote that condition 6 states that an = ay
-_axac
and hence there exists a function f(x, y) such that
df(z, y)
=
n dx - c dy
and therefore
Utilizing this fact it follows from equation 7 that
It is \vel1 known that the change of dependent variable
f(x r Y ) =
e--klz--kzu
#4x, Y)
MATHEMATICS O F ADSORPTION I N BEDS. I1
815
reduces equation 11 to
-a26 -
k 1 k 2 d x r y) = 0
ax ay
(13)
If we combine equations 10 and 12 with equations 8 and 9 we obtain: nl(z) = -kle-'12$,(x,
0)
+ e-k1*
am ax
We note that these are ordinary differential equations, linear and of the first order, whose solution can be written (14)
which equations define f(r) and g(y) and where we have set $(O, 0) = 0. This is legitimate, since f(z, y) occurs only as a derivative. It follow that we have reduced our original problem to the solution of equations 13, 14, and 15. In order to solve this system it is convenient to reduce it to ti\-o others. Consider the two problems:
u =
f(z), when y when x
=
0
=
0
0,
when y
=
0
g(y),
xhenx
=
0
u = 0,
v
=
t' =
+
Kote that the sum u u satisfies equation 13 and also equations 14 and 15, so that 4 = u 0 is the desired solution. Let L[u(x,y)] = U ( p , y). Hence
+
Li
=
F ( p ) when y
where L [ f ( x ) ]= F ( p ) . The solution of this system is
C = F ( p ) exp(klk2ylp)
=
0
816
NEAL R. AMUNDSON
It can be shown that because f(0) = 0, the inverse transform of this expression is a convolution of f’(x) and Z 0 ( 2 d / k l x y ) or u(z, Y) = [f’(t!Io(Zdkklk:y(z
- E ) ) ds
where f’(z) is the derivative of f(x) and I&) is the Bessel function of the zeroth order with imaginary argument. The solution of equation 16b can be obtained in an analogous manner by taking the transform with respect t o y. There results
Hence
In order to compute c ( z , t ) and n(z, t ) it is necessary to compute the partial derivatives of f(x, y), as shown in equation 10, and t o revert to the original variables. After some manipulation with Bessel functions the following formulae result.
MATHEhfhTICS O F ADSORPTIOS Ili BEDS. I1
817
where we have written the exponent on the left-hand side to simplify the writing. ~ ( t 1,) and n(z, t ) can be obtained from the above by substituting z = t’V and y = a(t - z / V ) and where
and
Rather than convert equations 18 and 19 into the original z and t variables it is generally easier to convert experimental data into the z and y coordinates. These formulae are rather complex but their numerical use ivould not pose too many problems, since the functions involved are well tabulated so that numerical integrations either manually or by neiv high-speed computing machines nould be relatively easy. These formulae are perfectly general and it can be shon-n that all previous solutions (see references 2 , 5 , 7, 8, and 9), assuming kinetics of the kind assumed here, are special cases of equations 18 and 19.
CASEB This case is a mathematical generalization of a problem previously considered by the author; honever, it involves a different physical assumption since it assumes that adsorbed adsorbate exerts a back pressure. Thomas (10, l l ) , in his treatment of ion exchange, solved a special case of n-hat follows. The system t o be solved is ac + 1an Ti.-ac + 0 (1) at at (Y at
~ ( t t,) = c o ( t )
n-hen i = 0
(4)
n(z, t ) = n&)
when t
(5)
= t’T’
Thomas has called this case adsorption with Langmuir-type kinetics, since in the limit anfat = 0 and therefore
n=
kl s oc k2 klC ~
+
We proceed, in the solution, as in the first case, by making the same changes of variable. A function j ( z , y) can once again be defined such that
Although this is a nonlinear equation it can be linearized by the substitution:
818
NEAL R . AMUNDSON
Equation 20 then reduces to
a2@ - klk?h’O+(X, y) = o
axay and equations 4 and 5 to
These equations can be solved as ordinary differential equations whose solutions are, after defining +(O, 0) = 1, @ (0, Y) = exp
I’
@(z,0 )
l
=
exp
+ k2l dv G(y) nl(E) + S o l kl dE F(x) =
[klcl(d
[-
=
(22)
(23)
which equations define G(y) and F(x). S o t e that G(0) = 1 = F(0). Hence the problem to be solved is similar to that under Case A; however, it must be handled in a slightly different manner. Consider the tn-0 problems:
u(0, y) = G(y) when x u ( z , 0) = 1
=
0
when y = 0
when
0, Y) = 0 v(x, 0)
=
2 =
0
F ( x ) - 1 when y = 0
+
S o t e that u v satisfies all the conditions that $I satisfies. Splitting this problem up into two in such a peculiar fashion is desirable to preserve the continuity at the origin. If the method of Case A is used, u and v are discontinuous a t the origin and some care must be exercised. Let L[u(s,y)] = U ( r , p ) , L[v(x,y)] = V ( q , y), L[G(y)] = P ( p ) , L[F(x)] = &(a); then equations 24a and 24b become respectively
U
= P ( p ) when
x
=
0
819
M4THEhCATICS OF ADSORPTION Ih’ BEDS. I1
and
dV qG
- kikzNoV = 0
V = &(a)
- -1 P
whose solutions are
u
= p ( p ) exp
when y
=
0
t y )
The inverse transforms of these can be written after some manipulation as:
and
where
?iow to calculate c and n in terms of z and y variables we need ax
1 a+ k1 ax
af&ay kl
ir;
a
f
4
7
a
-
1
+
(25)
and 1 a+ 1
4
(26)
The derivatives a+/& and a+/ay can be calculated in a manner analogous to that in Case A ; hence c(z, t ) and n(r, t ) can be obtained after reversion to the
820
NEAL R. AMUNDSON
old variables. Again, it v-ould probably be better to convert the experimental data to z and y variables if these formulae n-ere to be used. I t can be shown that previous solutions (see references 1, 3, 10, and 11) reduce to equations 25 and 26. These formulae are cumbersome and their use would be enhanced by some reduction to simpler form. However, since the problem solved was a general one, no simplification can be expected and yet retain the generality. Experimental work is now being carried on in these laboratories, and it is hoped that verification of these results n.ill be forthcoming. This solution, so far as the author is aware, has not appeared in the literature in connection with adsorption, although, as mentioned, Thomas considered a somewhat simpler problem for ion exchange. SUMMARY
General equations are developed for adsorption in beds of adsorbent which cover simultaneously adsorption and desorption. The bed may have an arbitrary distribution of adsorbate adsorbed on it, and the inlet solution concentration may have an arbitrary dependence upon the time. These equations are developed for two different cases depending upon the mechanisms assumed to be controlling in the bed. The first is that originally used by Bohart and Adams (3), while the second is that which Thomas (IO) called Langmuir kinetics. No simplifying approximations are made; hence the solution is mathematically exact for the cases considered. This paper was prepared while the author Tvas on a Summer Research Fellowship from the Graduate School of the University of Minnesota. REFEREXCES (1) AMUKDSOIC, S . R . : J. Phys. b. Colloid Chem. 62, 1153 (1948). (2) ASZELIUS,A , : Z. angew. Math. Mech. 6, 291 (1926). (3) BOHART, G. S., ASD . 4 ~ . 4 m ,E. Q , : J. Am. Chem. SOC.42, 543 (1920). (4) CHURCHILL, R . V . : Modern Operational Mathematics i n Engineering. McGraw-Hill Book Company, Inc., S e w York (1944). ( 5 ) FURICAS, C. C.: U. S. Bur. Mines Bull. Xo. 361 (1932). (6) KLOTZ,I . $ 5 . : Chem. Revs. 39, 241 (1946). (7) NUSSELT, W.: Z. Ver. deut. Ing. 71, 85 (1927). (8) SCHUMASK, T . E . W.:J. Franklin Inst. 208, 405 (1929). (9) THIELE,E. W . : Ind. Eng. Chem. 38, 646 (1946). (10) THOMAS, H. C . : J. Am. Chem. SOC.66, 1664 (1944). H . C.: Ann. S . Y. Acad. Sei. 49, 161 (1918). (11) THOMAS,
