COM M UN I CAT1ON
EXISTENCE OF ASYMPTOTIC SOLUTIONS T O FIXEDBED SEPARATIONS AND EXCHANGE EQUATIONS The existence of asymptotic solutions to the differential equations which describe single-solute fixed-bed separations and exchange processes is proved for nodinear equilibria under three different sets of conditions: ( 1 ) zero longitudinal dispersion and finite mass transfer resistance, ( 2 ) equilibrium operation with finite longitudinal dispersion, and (3) finite longitudinal dispersion and finite mass transfer resistance. The use of single-solute asymptotic solutions for describing the behavior of individual species in multicomponent systems is also discussed.
I
FIXED-BED separations and exchange processes, it has long been recognized that, for certain general classes of nonlinear equilibrium distribution isotherms, constant-pattern solute concentration profiles are asymptotically approached in sufficiently long beds under either saturation or elution conditions. T h e existence and uniqueness of asymptotic solutions which predict such constant-pattern profiles have previously been proved by Rosen (5) for the case of zero longitudinal dispersion and a finite resistance to mass transfei for a single distributed solute. His development is valid for a large class of mass transfer rate functions which fulfill certain general requirements. T h e purpose of this note is to show that existence proofs similar to Rosen’s may be given for two other important cases-finite longitudinal dispersion and negliqible mass transfer resistance (equilibrium operation), and finite longitudinal dispersion and finite mass transfer resistance. I n addition we point out that all of the asymptotic solutions for single-solute systems may be used to describe the behavior of individual species in a system of n interfering solutes, if the asymptotic state of the system consists of a set of n constantpattern concentration profiles. We begin by taking the equation of continuity for singlesolute fixed-bed sorption operations
N
and rewriting it in terms of the independent variables z* and t , where t* is a distance coordinate whose origin moves with the solute front and which is defined by
where Ac = c ( - m ) - ( - ( f a ) and Aq = q ( - a ) - q ( 4 - m ) . We assume here that the interstitial moving-phase velocity, u, is not a function of z or t . By postulating the existence of a constant-pattern concentration profile, we may neglect changes of concentration with respect to time at constant I*.
T h e equation of continuity then reduces to
D- d2x
=
d2**
[+ e
4 1 - 4 (Aq/Ac) (1 - t) (Aq/Ac)
I-
4 x - Y) dz*
(3)
Here x and y are dimensionless concentrations defined as
where c, is the lesser of c ( a ) and c( - a ) , and q m is the lesser of q( m ) and q ( -a). Defined in this fashion, x and y always increase as c and q increase, respectively. Now, since d x / d z * + 0 and x + y for large ]I* 1 , Equation 3 may be integrated immediately to give
I n addition to this equation of continuity, a mass transfer rate expression is required for describing the behavior of the system. This may be written in a general fashion as
(5) or, under constant-pattern conditions, as
T h e rate function G ( x , y) will be assumed here to fulfill conditions essentially equivalent to those specific by Rosen: I n the (.Y, y ) region defined by 0 x 1 and 0 y 1, let G(x, y ) possess the following properties:
<
0. (4) (BG/bn) 0 for x = 0. (5) (bG/by) < 0.
>
VOL. 4
NO. 2
MAY
1965
233
These restrictions are realistic and reasonably general. As indicated by Rosen, these conditions on G(.x, y) permit application of the implicit function theorem (7), with the followirg results: The relation G(x, y) = 0 defines a unique and continuous function, j * ( x ) , and determines a positive constant 1 (upper limit on x ) such that y*(O) = 0 and y * ( l ) = 1, while for 0 6 x 6 1 we have 0 6 y*jx) 6 1. Therefore, for 0 6 x 6 1
~ * ’ b=) - (dG/d~)~/(dG/dy)~
(7)
The function thus defined is the equilibrium distribution isotherm, since the condition G ( x , y ) = 0 defines the equilibrium state. Case 1.
0
= 0 and Finite Mass Transfer Resistance
We now prove the existence of asymptotic solutions x ( z * ) and j ( z * ) for these conditions by a procedure similar to Rosen’s and we extend the range of applicability somewhat beyond that of Rosen’s development. First we note from Equation 4 that x = y when D = 0, so that the concentration profiles x ( z * ) and y(z*) will be identical. These profiles are readily obtained by integration of Equation 6-that is, from
z*
= (I/?)
[
dy/G(x, Y) = (I/?)
s,:
dx/G(x,
Y)
(8)
where xo and j o are those (identical) values of x and y, respectively, a t which z* = 0. Since x = y over 0 6 x 6 1 for a constant-pattern front when D = 0, and since x # y* in 0 < x < 1 because of the nonlinearity of the system, it follows that 2 # y* over 0 < x < 1-that is, the rate function G(x, y) must be nonzero for this interval. Consequently, the integrand l/G(x, y) in Equation 8 is finite for the region 0 < x < 1. A solution x ( L * ) , equal to y ( z * ) , therefore exists throughout this interval. It remains then to prove that a solution also exists a t x = 0 and a t x = 1 [at which points 1/G(x, y) is infinite] and to show that the over-all solution for 0 < x < 1 is an asymptotic solution. We write
dG(x, Y ) / ~ Y = (aG/dy),
+ (dG/dx),(dx/dy)
or, using Equation 7 , ~ G ( xy)ld’ ,
W/ayy),[1
-y*’
(x)(dxldy)l
(9)
Then, after noting that dx/dy = 1 for this case, we substitute Equation 9 into 8 to obtain
wheref(x, Y ) = [(dG/dy), (1
- r*’)l-’
A. Convex Isotherms: y*“ < 0. Let us consider first the specific class of nonlinear isotherms y * ( x ) for which y*” < 0 over the closed x interval [0, 11. Let 2 be that value of x a t which y*’ = 1 (only one such point exists). Consider then the interval 0 6 x < 2 . Clearly the function f ( x , y) is positive and is bounded over this interval, since by previous specifications (dG/dy), < 0 and by our choice of the x interval and specification of the behavior of j*”, (1 - y*’) < 0. The fact that f ( x , y ) is bounded permits application of the integral mean value theorem (7) to Equation 10 By this theorem there must exist an f, with 0 < f < f , such that for any x in the interval 0 6 x < f 234
I&EC FUNDAMENTALS
where f is the value of f ( x , y ) at x = 2 . that
It therefore follows
may be written for any x in the interval 0 6 x 0) z* fm . For the interval f < x 6 1 we find by the same procedure that a solution exists throughout 2 < x 6 1 (and at x = 1 in particular) and that as x 1, L* + --oo (since now f < 0). This behavior corresponds to the satisfaction of the boundary conditions for a saturation process as t +. m . B. Concave Isotherms: y*” > 0. For y * ” > 0 over the closed x interval [0, 11 we find by the above procedure that a solution x ( z * ) , and thus a solution y ( z * ) , exists everywhere in [0, 11 and that, as x + 1, z* + +a, and as Y + 0, z* -, -a. This behavior corresponds to the satisfaction of the boundary conditions for an elution process as t -, a , I t should be stressed that, for a particular solute, an asymptotic constant-pattern solution will exist either, for a saturation process or for an elution process, but not for both, depending on the nature ofy*(x) in [O? 11. C. Special Types of Isotherms. The conditions on C(x, y ) listed above exclude convex isotherms whose slope y*’ becomes infinite at x = 0-e.g., convex Freundlich isotherms, or concave isotherms for whichy*’ becomes infinite a t x = 1. Inasmuch as isotherms of these types have often found considerable use in practice for the representation of equilibrium data, an investigation of the extension of the above proof to these cases was made. It was found that: (a) for convex isotherms with infinite slope at x = 0, a solution exists everywhere in 0 6 x 6 1, but that, as x -,0, L* -,some finite positive number, instead of f - m , and ( b ) for concave isotherms with infinite slope at x = 1, a solution likewise exists for 0 6 x 6 1, but that, as x -P 1, z* + some finite positive number, instead of f a . However, as before, we still find that t* + -a as x -P 1 in the convex case, and z* -P --m as x +. 0 in the concave case. The behavior in these special cases, therefore, still represents the satisfaction of the boundary conditions, at t =-m, for an elution process (for the concave isotherm) or for a saturation process (for the convex isotherm). Although we will not consider these special situations further, it can easily be shown that the above conclusions apply also in cases I1 and 111.
-
-
Case II.
Negligible Mass Transfer Resistance and D
>0
Here y = y*, so that Equation 4 may be written as z* = ( I / @
lI
d x / ( x - y*)
(11)
Since for nonlinear isotherms the function ( x - ,q*) is nonzero over 0 < x < 1, the integrand l/(x - ,*) in Equation 11 must be finite for 0 < Y < 1. Hence, a solution t ( z * ) as well as the corresponding solution for 3. exists for this region. It, therefore, remains only to show that a solution exists also at x = 0 and Y = 1 and that the over-all solution for 0 6 x 6 1 is an asymptotic solution. Substitution of d(x - I*) = (1 - y*’) dx into Equation 11 yields
z* =
s’-”
(lip)
d(x - y * ) / [ ( l
- y*’)
( x - y*)]
(12)
( x --Y*)o
Clearly, if Z is that value of x at whichy*’ = 1, then the function (1 - y*’)-’ is bounded over 0 6 x < 2 and 2 < x 1. As previously shown, this fact implies (by application of the integral mean value theorem) that a solution x(r*), and thus a solution ,y(r*), exists at x = 0 and x = 1. An examination of the behavior of these solutions as x + 0 and as x -+ 1 shows that, for t = m , they satisfy the boundary conditions for a saturation process if y * ( x ) is convex and for an elution process if y * ( x ) is concave.
0
From our postulate of a constant-pattern front, and the given conditions of nonzero longitudinal dispersion and a finite mass transfer resistance, we know that G(x, y ) # 0 and x f j over 0 < x < 1 and that solutions x ( z * ) and y ( r * ) must exist for the region 0 < x < 1. In particular, we know from Equation 4 that either x > y or y > x throughout 0 < x < I.
We substitute d ( x - y) = (1 - dy/dx)dx into Equation 4 to obtain
4. - Y ) / [ ( l - d y / d x ) ( x
z* = (l/P)
- r)l
(13)
x -Y)o
Then we take the ratio of Equations 4 and 6 to give
4 l d x = ( r / P ) G ( x ,Y ) / ( x - Y )
(14)
Now, over 0 < x < 1, ( x - y) is nonzero and of constant sign. Thus, by Equation 14 the quantity dyldx must be nonzero and of constant sign over 0 < x < 1 . This fact implies that the function y ( x ) is either uniformly convex or uniformly concave over the region 0 x 1. Hence, no inflection points can occur in this interval and dy/dx equals unity a t only one point, which we may call 2 . The function (1 - dy/dx)-’ x 1. This is thus unbounded only a t the single point in 0 fact is sufficient to permit us to prove by application of the integral mean value theorem to Equation 13 that solutions x(z*) and y ( z * ) exist at x = 0 and x = 1 . Furthermore, it may readily be shown that these solutions satisfy, a t t = a , the boundary conditions for a saturation or elution process, depending on whether y * ( x ) is convex or concave, respectively, in [0, 11. ,411 of the solutions presented above confirm our original postulate of constant-pattern behavior-that is, they demonstrate that x and y can be expressed entirely as functions of the single distance variable z* and independently of time, provided that t is very large.
<
