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Multicomponent adsorption dynamics of polymers in a fixed bed
Pergamon
PII:
Chemical Engineering Science, Vol. 51, No. 16, pp. 4013 4024, 1996 Copyright (t) 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved S0009-2509(96)00243-6 0009 2509/96 $15.00 + 0.00
M U L T I C O M P O N E N T A D S O R P T I O N D Y N A M I C S OF P O L Y M E R S IN A FIXED BED L E O N I D K. FILIPPOV* Lehigh University, Bethlehem, Pennsylvania 18015, U.S.A.
(First received 30 October 1992; revised manuscript received and accepted 29 January 1996) Abstract--The theoretical analysis of concentration-patterns of frontal adsorption and desorption dynamics of mixtures in a fixed bed for the Langmuir adsorption isotherms is considered for the equilibrium and non-equilibrium models. For the equilibrium model, the concentration-constant-patterns and concentration-proportional-patterns may be described by using the Riemann invariants Rk and H~ invariant analytical expressionswhich are found. The methods of calculation of adsorption and desorption isotherms based on the use of outlet-concentration-distributions are developed. Adsorption and desorption isotherms of mixtures of the water soluble associative polymer for two end-groups (R := C16H33 and R := H) onto particulate surfaces of titanium dioxide are obtained. It is shown that the method developed can be used to measure the adsorption and desorption isotherms over a wide range of associative polymer concentrations. Copyright © 1996 Elsevier Science Ltd Keywords: fixed bed, adsorption, polymer mixtures, Riemann invariants
1. INTRODUCTION For the investigation of adsorption of polymers on planar and particulate surface to solid substrates, direct and indirect experimental methods have been applied (Helferrich and Klein, 1970; Lipatov and Sergeeva, 1974; Fleer and Lyklema, 1983; Hiemez, 1986; Guiochon et al., 1988). Direct methods can be divided into the following main groups: electrometric, chromatographic, and spectroscopic methods. The indirect methods leading to the thickness of the adsorbed layers are based on the use of three techniques: ellipsometry, scattering (static and dynamic light or neutrons), and hydrodynamics. In ellipsometric measurements, the thickness of the adsorbed layers on a planar surface are deduced from the changes of polarized light upon reflection. In scattering measurements, the thickness is found by static and dynamic scattering of light or neutrons. In hydrodynamics measurements, the thickness is inferred from the external displacement of the adsorbed polymer of the slipping layer between the solid and liquid phase which moves tangentially with respect to the solid surface. Guiochon and co-workers have applied computer simulation to describe the propagation of a large-concentration band profile in nonlinear liquid chromatography. The computer simulation allows us to compare experimental data and a large-concentration theoretical profile obtained for different adsorption/desorption isotherms of mixtures. However, the adsorption/desorption isotherm of mixtures may be found from the experimental data by using analytical
*Address for correspondence: Leonid Filippov, 510 Seneca Street (First floor), Bethlehem, PA 18015, U.S.A.
solutions. Therefore, in this paper we will develop in detail a method for determining the adsorption/desorption of mixtures on a particulate solid surface based on the frontal adsorption dynamics or chromatrography. This method is based upon the use of the both the experimental outlet-concentration-distributions and algebraic relations (4) for the constantpatterns, and the linear eq. (39) for the proportional patterns, when the Langmuir desorption isotherms occur. Frontal chromatography is a direct method which allows us to find the adsorption and desorption isotherms of mixtures over a wide range of concentrations of polymers. 2. ADSORPTION DYNAMICS OF MIXTURES FOR THE EQUILIBRIUM MODEL
Adsorption dynamics of multicomponent mixtures in the equilibrium case is described by the system of material balance equations for each of the component mixtures
~cm UOCm 6#f.~(C) ~t +--~-Z +
~
(1 -- a~
O,
l ~m~n Cm(0, t) = Corn= constant,
6--
~ , (1)
cm(z, 0) = c° = constant. (2)
This hyperbolic system of equations admits as solutions the following coherent frontal patterns: the constant patterns and the proportional patterns (Lax, 1957; 1971; Aris and Amundson, 1973; Rhree et al., 1986; Filippov, 1992). For the constant patterns, the concentration distribution takes the form of concentration waves traveling with a speed wi (1 ~< i ~< n) and
4013
L. K. FILIPPOV
4014
is a function of only one variable y and the discontinuous solutions have the form:
or for dimensionless concentrations in the following form
~C~ 1), y < 0 Cm(2 , l) = Cm(y ) = (C~) '
A I
y > 0
c
)
=
( F mo ) k ( K po) k C k / V
V = 1 + ~ (KObCj,
,
j
(3) y=z--wit
,
1 <~m~n
C k = c k / m a x (Cok ; C0)
W o k = (r'm)k [max(cog; cO)],
where c~ ~ and c~ - ~ are the equilibrium values of concentrations for the wave index i. The values of velocities w~ from the integration form of eqs (1) are found by (Filippov, 1989) wi
ui -
-
U (I + &i~)'
1 <~m, i<~n.
Cm(Z, t) =
g~(¢),
~ = Yi = V(t~(i 1)) ~(i 1) Vi(?(i 1)) ~ ~ ~ l,i(?(i))
~d ~,
~ = ~I ° = v~(~ "~)
~_ = z/t,
(5)
-0,
d~
~fm(C) OCk
det (fmk -- piErak) = 0,
H ( p , c) = 1 -- ~
--
1)/6 = Pi,
bj/(p* - It),
~*<~*<
--
. . .
~
vi)/(bvi) ~ ~ p--. ~*-0,
0%
H(p,c)
+ O,
d(
h(i)(c) '
j=l
?~cj
, ( )
H(p,c)
~
dCk
r~)(c) - - rtkO(C) '
1 <~m,
/~=0,
ck>0.
Therefore, for the Langmuir adsorption isotherm (8) the matrixfmk has n real, positive, and different eigenvalues Pk (1 ~< k -%
k, i ~ n
0
where r~,° (c) is the right-hand eigenvector of the matrix frog, corresponding to the eigenvalue of #i (1 -%
2.1. A n a l y s i s o f c o n c e n t r a t i o n p a t t e r n s f o r L a n g m u i r a d s o r p t i o n isotherms The Langmuir adsorption isotherms are written in the form
- - -
~
< /'/(n--l) < ~ < ~/n"
The right-hand and left-hand side eigenvectors r~i) and l~i) of the matrix fmk are found, respectively, from the equations (Arfken, 1970; Rozdhestensky and Yanenko, 1986) ,~ (i) • " .]mkrk
fro(C) = ( F m ) k ( K p ) k C k / V ,
+oc,
(7)
or in the following form dc~
- - O,
H ( ~ , c) -~ - ~:
1
=
<~*.
-- ~ bj(p* - #) 2 > O, bj > 0
H(0, c) = ~ > 0, - -
< (F,.).(G).
The function H ( p , c) is monotone, since
where E,,k is the unit matrix and/2~ is the eigenvalue of the m a t r i x f , k (1 ~< i ~< n). We write the preceding system of equations as -
bj = (F,.)j[(Kp)j]2/V 2.
are satisfied then the poles p* of the function H ( p , c) are arranged as
1 <~ m, k, i <~ n
Pi = (u
(10)
j 1
(6) (. = (ut/z
1 ~ m, k, i ~ n,
The secular eq. (10) for the Langmuir adsorption isotherms (8) may be rewritten in the following form n H(/~, c) [ I (P* - #) = 0, #* = ( F m ) j ( K p ) j / V j 1 (11)
0H(p, C) ,
1 <. k <~ n
(r,.)l (Kp), < (F,~)2(K~)2 < "
l <~ m, i <~ n
f~k -
fk(C) = f k [ m a X ( c o k ; c ° ) ] ,
If the inequalities
where ?" a), ~") are the equilibrium values of concentrations for the wave index i and Cm(O is the solution of eqs (7). When the proportional patterns of eqs (5) occur, the system of eqs (1), may be rewritten in the following matrix form (Emk -- fmk)dCk
(K°)R = (Kp)k[maX(Cok; cO)]
(4)
For the proportional patterns, the concentration distributions are in the form of expanding concentration waves and depend only on the variable n. The continuous self-similar solutions may be written as: ~-1~, ,
(9)
where (Fm), and (Kp), are the maximum amount of adsorption and the equilibrium constant for the kth component of the mixture, respectively. The eigenvalue #i of the matrixf,.k is found from the secular equation (Korn and Korn, 1968; Arfken, 1970)
fm(C(y = + c~)) - - j m ( C ( y = -- oG)) c,.(y = + oC) -- Cm(y = -- o0) '
t
l
= pirf ,
fmkl~ ') = I~l~ ~,
1 <. m, k, i % n
V = 1 + ~. ( K p ) j c j j=l
(12t (8)
where fmk is the transpose of the matrix fmk.
Multicomponent adsorption dynamics of polymers For the Langmuir adsorption isotherms (8), the preceding equtions may be written as (1/b °) (#* - pi)r~; = L (Kp); rJi), j=,
b ° = (Fm)m(Kp)mcm/V 2 [1/(Kp),,]
(13a)
(g* - / t , ) l ~ ) = L byl~i,.
Ifdcm/d~ ~ 0 (1 ~< m ~< n), then for the wave index i, according to eqs (20), we have Rk=Constant,
it:k,
Ri = variable,
i = k.
r'~'~ = boo, * - ui)
~,
t~" = (K~,)~(.*
(13b)
- ui)
1 <~k<~n.
Let us now consider the Riemann invarlants Rk (1 ~< k ~< n). By definition, for the hyperbolic systems of equations, the Riemann invariants are given by (Rozdhestensky and Yanenko, 1986)
dRi =- ~ci L l~)dc"
(15)
m 1
where ~Ckis the integrating multiplier of the differential form. For the Langmuir adsorption isotherms (8) the integrating multipliers are equal to
~:i
-/1, EEl=, b°(Kp)~(/~* -- #i)- ' ] . o , [ Z k = , bk(Kp)k(Pk - #i) 2]
-
(16)
F r o m the preceding analysis for the proportional patterns (5), the following relations are valid
(22)
since according to eqs (15)-(17) and (21), the following equality is valid dHi d~
--0,
Thus, the Riemann invariants Rk, and invariants Hi are important values which allow the calculation of concentration-distributions for the constant and proportional patterns. In fact, the discontinuous concentration-distributions, Cm(y) (1 ~< m ~ n), are described by the following algebraic system of equations, for the constant patterns (3) of the Langmuir adsorption isotherms with the wave index i: Hi = pi V 2
constant,
=
ek
=/~i
V = constant,
1 <~k, i<~n
(23)
and the continuous concentration-distributions, c,,(0 (1 ~< m ~< n), are described by the analogous algebraic system of equations for the proportional patterns (5)
Rk = Pi V = constant,
H~ =/~i V 2 = constant,
1 ~ k , i<~n. d(
V/2,
hti)(c) = --
dV
L (Kp)jr} i), j=l
1 -- hti)(c )
(17)
Thus, the Riemann invariants from eqs (15)-(17) may be obtained in the following form
Rk = l~kV, 1 ~ k <~n
(18)
The hyperbolic system of eqs (1) in the Riemann invariants has the form 8Ri #iORi g~-+~=0,
1 <~i<~n.
(19)
F o r the proportional patterns (5) from eqs (7), (15) and (17) we write
dRi
L (c3Ri~(dc,,~=(V)
d:-m=,\~Cm]\d:J
(24)
F r o m the preceding analysis, the Langmuir adsorption isotherms are valid for the following inequalities. For the proportional patterns (5):
dV
--<0, d;
dRi ~-
>0,
1 <_.i~n.
dV
-->0, dy
dRi
--<0, dy
1 <~i<~n.
(20)
where 6ik is the Schwarz-Christoffel symbol.
(26)
Let us consider the adsorption and desorption dynamics for the different Langmuir isotherms of a twocomponent mixture (n = 2). Example 1: For the adsorption dynamics (Corn > C0, m = 1, 2) and the following Langmuir desorption isotherms (9), (K°), = 2,
(r°)2 = 2,
(K°)2 = 2,
Col = C o 2 = 1, C ° = C 2 ° = 0 1, i = k (~ik= O, i ¢ k
(25)
For constant patterns (3):
( r ° ) , = 3,
--~ (}ik,
(21a) (21b)
Hi = / l i V 2 = constant
',
(14)
l~
However, for the wave index i of the Langmuir adsorption isotherms, the invariant Hi is constant, i.e.
j=,
F r o m the preceding relations, it follows that the righthand and left-hand side eigenvectors are equal to
4015
(27)
the constant patterns (3) take place for the wave indexes i = 1, 2. In these cases, according to eqs (4) and (23), we have
Co, = Co2 = 1, V = 4, R1 = 3.68, H, = 14.73,
/q = 0.9215, R2 = 0.813
/t2 = 0.2034,
/~1 = 1.5(
4016
L.K. FILIPPOV
i = l:
{
C~ lJ = 0,
R1 = 6 ,
~C1
(1)
C(21) = 1.457, H i = 14.73,
V = 2.457,
C z(1) = 1.457,
=0,
~R 2 = 0.813,
]21 = 2.44,
]22 = 0.331,
]21 = 1.5}
]21 = 2 . 4 4 ,
]22=0.331,
]22 = 0.814}
R2=0.813
Hz = 2,
V=2.457,
(28)
R1 = 6
i = 2:
C° = C° = O, V = I, ]21=6, R2=2, H2=2, Rl=6 1, 0,
CI(L,t)=CI(Z)=
]22=2,
Z > U a = 1.5, r = ~ r
1,
~>'r>ul
C2(L, t) = C2(-Q =
]22=0.814} with z = L
=1.5
C[21) = 1.457, ul > r > u2 = 0.814
(29)
O, O < ~ < u 2 where L is the length of the fixed bed. The solutions (28) and (29) are shown in Fig. 1.
The solutions (31) and (32) are shown in Fig. 2.
Example 2: F o r the a d s o r p t i o n - d e s o r p t i o n dynamics (Col > c °, Co2 < c °) and the following L a n g m u i r adsorption isotherms (9)
Example 3: F o r the desorption dynamics (Co,. < c °, m = 1, 2) and the following L a n g m u i r adsorption isotherms (9)
(F,°,)I = 4.5,
(K°)I = 2,
Col = C O =
1,
(F°)2 = 2,
(K°)2 = 1,
C ° = Co2 = 0
(30)
(F°)1=3,
the p r o p o r t i o n a l pattern (5) occurs for the wave index i = 1, and the constant pattern (3) occurs for the wave index i = 2. In these cases, according to eqs (4), (23) and (24), we write
C = O, Co2 = 1,
V = 2,
R1
R2
9,
H 1 = 18,
]2]o~= 4.5, ]22
=
(K°h =2,
Co1=Co2=0,
(r°)2=2,
C °=C2 °=1
0.5}
= 1
i=1:
{ {
C~ 1~ = 2.28,
RI = 1,
C~ 1~ = 2.28,
Re = 1 ,
C(21) = 0.433,
H1 = 6 ,
V = 6,
]2]1) =
V = 6,
]21 = 0.5,
]22 = 0.167}
0.5,
(31)
Re = 3
C(21) = 0.433,
H2=6,
]22 = 0.167,
]22 = 0.333}
R1 = 3
i = 2: C01
=
R2=2,
1,
C ° = 0,
H2=6,
V = 3,
=
=
1,
]22 = 0.666,
]22 = 0.333}
R1=3
/°, CI(L, t) = Ca(r)
]21
cc > r / > ]2]o) = 4.5
CI(T)'
#~o) >~ r > ]211j = 0.5 C] 1) = 2.28, 0.5 >~ z >/ 1'/2 = 0.333
[l,
0.333 > r >~ 0
1,
= C 2 ( L , t) = C 2 ( z )
~>r>~4.5
C2(r), ] C~2II
[o,
= 0.433,
4.5 >~ z >~ 0.5 0.5 ~> z > 0.333 0.333 > r >/0
(32)
(Kp°)2= 1,
(33)
4017
Multicomponent adsorption dynamics of polymers the proportional patterns (5) take place for the wave index i = 1 and i = 2. In these cases,
C° = C° = I, Ra
0.813,
V = 4, , , ° ) = 0 . 2 0 3 4 , H1 = 3.23,
,2=0.9215}
R z = 3.7
i=1:
{
C~1'=0.31,
R1 = 2,
C(2u=0,
H1 = 3.23,
C,"=0.31,
V = 1.62,
C(2')=0,
Re--3.71,
He=6,
, @ ) = 1.23,
,2=2-29],
(34)
3.7
R 2 =
V=1.62,
,1=1.23,
,u2=2.29}
R1=2
i=2:
Co, = Coz = O. V = 1, ,u1= 2, .2 = 6} R 2 = 6 . He =6, R1 =2 0,
o0 > r >~ #t22) = 6
CI(T), 6 >/ r >~ #t21) = 2.29 CI(L, t) = CI(r) =
C2(L, t )
C2(r)
=
=
C]1) = 0.31, 2.29 >/r i> @) = 1.29 Cl(r),
1.29 >/r >~ ,]0) = 0.203
1,
0.203 > r/> 0
0,
1.29 ~< r
C2(z)
1.29 ~> z >~ 0.203
1,
0.203 ~> r >~ 0
(35)
A
A
2.5 2.0 C1
c, 0.5i
1.5
[
I
~
Col 1.0
# ,
equilibrium case dl _ m lee:=0:1/ nons-eesquilibrl. . . .
.....
0.0 . . . . . . . . 0
1.5
b.'5 . . . . . . . . .
1.b . . . . . . . . .
0.5 c?
1.5 . . . . . . . . U1
0.0
~.0 7"
iI
. . . .
. . . .
,
a.=0.1 nnon-equlllor laa:---~):~l ° n s - e ° q ......... . . . .
I
,
. . . .
. . . .
[
. . . .
[
~ . . . .
[
. . . . .
,
. . . .
0.00 0+50 1+00 1.50 2.00 2.50 3.00 3.50 4.00 0 U2 ,U.?)
1.0 Co2
[3
~- -
I
,
,
,
,q
4+5(~) 500 ,££1° T
B
C2 i Co2
I
C2 .......
- -
O.5
0.5
equilibriurrl case a6=0.1 on-equilibrium
___
.....
C~ j
I c; 0.0
0.0
0.0
o
.
.
.
.
.
.
.
.
0.51"77"[". . . . . . . . . .1.0 ...... U2
115.. . . . . . . . U1
I
~'.0 7"
Fig. 1. (A and B) The outlet-concentration-distributionsfor adsorption isotherms (27).
The solutions (34) and (35) are shown in Fig. 3. As follows from the preceding examples, the Riemann invariants Rk, and invariants Hi remained valid for different wave indexes in cases of adsorption, desorption and adsorption~tesorption dynamics.
II
....
i .... 0.5 (1)
0.0~ U
....
U2
equilibri urn oCne_Seq uilibrium (0.=0.3)
. . . . (o.--03: , .... .0
, .... 5
, .... 20
, .... 25
cases
t .... 30
i .... 35
i .... 40
,u+1
, .... 4I~ ~+
5.0
T
Fig. 2. (A and B) The outlet-concentration-distributionsfor adsorption desorption isotherms (3l). 2.2. Analysis of solutions for proportional patterns For the fixed bed of the length L, it follows from eqs (23) and (24) that the proportional patterns (5) and the wave index i are valid relations
T
=
Hi/V 2, "r = ~ with z = L, Hi=constant,
1 ~
L. K. FILIPPOV
4018 1.0
2.5
A'
2.0 1.5 , C1 0.5
i
...... - - --
~",'-
T-I/2
equilibrium cose (%=0.1) non-equilibrium (G,=O.3) coses
1.0 0.5
.
.
.
.
.
.
t.=./
0.0 o
L0
_(o) H,1
~.--
c;
2 O) ,/2"2
_0) P1
3
4
5
/~2) Relotive Totol
3"
Fig. 4. The plots of z 1/2 vs the total concentrations of the two-component mixture for desorption isotherms (33): (S1)l = 0.78, (1,)1 = 0.66.
t
"" ~.
] C2 0.5
Concentrotion,C==Cl+C2
~
I
(0,=0,1) non-equilibrium
- --
~
(a,=0.3)
+ (Fm)E(Kp)2 - Hi - Rk]}/Ai
eoses
(39)
At = {RkE(Kph -- (Kp)2] + ( K p h × (Kp)2 [(Fm)l -- (Fm)2]} U 1/2 •
O0
o
05
03
08
43
0
~°)
' ' ' 15 ....
J8
#I ')
20
r
Fig. 3. (A and B) The outlet-concentration-distributions for desorption isotherms (33). Hence we write z 1/2 = V/(Hi)I/2,
H i = constant.
~k = Rk/V,
i # k, 1 ~ i, k ~ n (37)
#1.2 = ( f l l +fz2)/2 + [(f11 - f 2 2 ) 2 / 4 +Ji2J21] 1/2,
af.,(c) fmk
--
OCk
m,k
,
=
($1)1 = 0.78,
(36)
Thus the plot of r-1/2 vs V is a straight line for a multi-component mixture with an arbitrary number of components. In particular, for the two-component mixture (n = 2) we write using eqs (23) and (24), ~i = Hff V2,
where Cz is the total concentration of the two-component mixture, (SDi and (I,)i are the slope and intercept of the straight line o f t - 1/2 vs Cz, respectively. In particular, for the case of eqs (30)(32), as shown in Fig. 4, from eq. (39), we have ( I , h = 0.66, i = 1,
H1 =3.23,
R2=3.7.
3. NON-EQUILIBRIUM MODEL OF MULTICOMPONENT ADSORPTION DYNAMICS
Adsorption-separation process in a fixed bed is described by the system of equations of material balance in the intergranular space of the adsorber [eq. (40)] and the system of equations of kinetics [eq. (4t)] in the following form (Filippov, 1989, 1992)
1,2.
.
+ ~
= _
D~
(401
k=l
It follows from eqs (36) and (37) that the dependence
~~tq m
between concentrations C1 and Cz is linear, i.e.
L
flmk[fk(c) -- qk],
1 <~ m, k <~ n. (41)
k=l
C 1 = al C2 -I- bl al
=
-
(Kp)2E(F,.h (Kp)l
-
Equations (41) reduce to equations for the adsorption isotherm, i.e. qk = f k ( q , C2. . . . . C,), when the rates of adsorption and desorption processes are infinite
Rk]/
{(Krh [(F,,)2(K,)z - R , ] } (38)
bl = [Hi q- Rk -- (Fm)l (Kp)l - (Fm)2(Kp)2]/ {(K.h [(r.h(K,h
H~, R , = constant,
- RR]}
Let us consider the solutions of the system of equations (40) and (41) for the constant patterns (3). In this case we write
i ¢ k.
Z- 1/2 = (SI)iC z q_ (In)i,
k=lL(6wi)J
ui
C z = C 1 -~ C2, i = 1, 2
(St)~ = ( K p h (Kp)2 [(F,,h ( K p h - (F,.h(K,,)2]/A, (I.), = {[(Kph - (Kp)2]Rk + [(rmh - (Fm)2] x [ ( K p h (Kp)2 + (F.,)x(Kph
F D(I) I dAck
= u,Ac -
Therefore, for the two-component mixture, from eqs (36) and (38), we have
=
(u
-
dy
'
w~)/(~wil (42)
dAqk
"
fl,.k
dy l<~m,k<~n.
Multicomponent adsorption dynamics of polymers The Langmuir adsorption isotherms (8) admit the following transformations
(43)
4019
form
ff(1/Ao)ln(29), + oo >/y >~ Yo~ Y - Yoi = ~(1/AO)ln(29) ' Yoi >1 Y >t Ao = (26D0) -1 {c~ - [~z _ 460oflo(U~ -- l, + )]a/2} (49) A ° = (26Do) 1 {(~ -- [0~2 -- 46Doflo(Ui - l~[ )]1/2}
Therefore, according to eq. (43), the system of eqs (42) may be rewritten in the following matrix form (l) 2
6D,.kd Ack dy 2
B,.kdACk + BOkACk = O, dy Bmk = UiwiE=k
1 <~ m, k <~ n
+ ~ D,.jD}~
(44)
9(Yoi)= 1/2,
p+ = # i ( y =
+~),
P i = Pi(Y = -- ~). When Do = 0, the quasi-linear equations of the second order reduce to quasi-linear equations of the first order, and the precise analytical solution may be written in the following form: y = yo, + {(u,w,)/[Do(U,
-
~,(,,o))]}
j=l
x {lngg - [l~i(Co)/ui]ln(1 + 9/a)} ~ o = u,Dm~ - (V/Vo) ~ D.jfj~(co),
Vo = V(co)
(50)
UdCo)/Ui] Vo/AV.
a = [1 -
j=l
is the unit matrix. In the simple case, when
w h e r e E,nk
D ~ = Do = constant,
flmk = Do = constant (45)
L i ~ - [ ( A o ) - ' + ( A ° ) -~][ln(1/e)],
eq. (44) reduces to iJOodZAck
o~dAck
dy 2
dy
(51a)
+
~ ~mkACk = O, m=l
l ~ k , m<~n
(u~w, + DoDo), ~,~ = Do[U~Em~ - (Vo/V)fm~(Co)].
The solutions of the preceding system of equations may be found in the following form: ck(y) = ck(y =
Li ~- (wi/Do) {[1 - #i(y = - oc)/ui] - 1 + [1 - #dY -- + ov)/u~] -1} [In(i/e)].
#dY=
.q(y= + o C ) = 0 ,
(47)
0<~9(y)~
where 9(Y) is an unknown function. In this case the system of eqs (46) transforms into the following quasi-linear equation of the second order ;~dg(y) dy
-oc)
WhenDmk#0,
D ~ vsO
(l <~ m, k <~ n)
u max (F,,)k bmk --
D~)k=Do
(Lu)
(1 ~ < m , k ~ n )
(48) ao-
Vo=V(co),
6 D~)kmax (Fm)k ,
(LDmR)
WhenDmk=D0,
AV=V °-Vo,
+~).
In the general case, when [3rag¢ O, D~)k 4= 0 (1 ~< m, k ~< n), the estimation of effects of dispersive factors (adsorption kinetics and axial dispersion) may be realized only by numerical integration of the system of eqs (40) and (41) by a computer. The effects of dispersive factors take into account the following dimensionless parameters (Filippov, 1989):
amk --
+ f l o [ U i - Pi(Co)/(l + g(y)A ~--~)]g(y)=O
(51b)
According to the conditions of existence of the constant patterns, these patterns take place if the following inequalities are valid (Lax, 1957, 1971; Rhree et al., 1986; Filippov, 1992)
+ o~) + g O ' )
x [Ck(Y = T Or) -- c d y = _+ OC)],
6Ood2g() ,) dy 2
A o > 0 , A° > 0
&
(46) c~ =
The solutions (49) and (50) describe the concentration distribution in the fixed bed and may be used to calculate the mass transfer zone. The mass transfer zone, Ls, for the wave index i, obtained from the solutions (49) and (50) when g(y = +_ Go) = e,, are, respectively, equal to
V °=V(c°).
Strictly speaking, the solution of the quasi-linear eq. (48) may be found by a numerical method using a computer. However, the approximate analytical solution may be written in the following
u max (F,,)k (LDo)
,
b0
6 Do max (Fm)k (Lu)
. (52)
The system of eqs (40) and (41) for the non-equilibrium model transforms into eqs (1) for the equilibrium model when ~21mk =
aO =
bmk = ao = 0.
(53)
4020
L.K. FILIPPOV
Below we will analyze the numerical solutions of the system of eqs (4) and (41) for different values ao (ao = 0.1 and 0.3) and bo = 0.05. The numerical solutions for the adsorption isotherms (27), (30), and (33) are shown in Figs 1-3, respectively. As it follows from these solutions, the numerical solutions approach uniformly to the solutions of the equilibrium model (1), when the value of the parameter ao decreases and approaches zero. These results make it possible to find adsorption isotherms using analytical solutions of eqs (1) for the equilibrium model and experimental outlet-concentration-distributions. The numerical solutions have been analyzed earlier using a computer by Guiochon and co-workers. However, only analytical solutions allow us to solve the reverse problem, i.e. to find parameters of adsorption and desorption isotherms of mixtures. Therefore, below we will use the analytical solutions in the form of eqs (4) and (39). 3.1. C a l c u l a t i o n o f p a r a m e t e r s o f a d s o r p t i o n isotherms To calculate adsorption isotherms we have to use the algebraic eqs (4). For a two-component mixture (n = 2), when the first component is strongly adsorbed, we write wi = u/(1 + 6ui),
i = 1,
Ui =fl(Col, C02)/C01 ,
e0 = co = 0
For the the convex adsorption isotherms (in particular Langmuir adsorption isotherms), the outletconcentration-distribution of polymers depends on only one variable t*
t* =
e 2 ( L , t) = e2(t*),
t -- L/wi,
i
=
I
Fig. 5. The flow diagram for the study of adsorption and desorption dynamics in the fixed bed. 1: twin micropumps, 2a: pure solvent in syringe, 2b: solvent plus adsorbate in syringe, 3: flow loop, 4: thermostat, 5: fixed bed, 6: filters, 7: DRF detector, 8: flowmeter, 9: computer interface.
and t o is the time needed for the removal of fluid through outlet and inlet capillaries. The retention times can be found by using the experimental outletconcentration-distributions, for both the empty fixed bed and the fixed bed filled with the adsorbent particles, and the following expressions /'fOiled =
[1 -- c ( g , t)exp" filled/C(L, t) . . . . . . p. filled] dt
1.
(58)
To exclude the effects of outlet and inlet tubes (capillaries), as shown in Fig. 5, we measured the two outlet-concentration-distributions for the empty fixed bed and the fixed bed filled with the adsorbent particles. According to eqs (54) and (55), the retention t i m e s , tfilled and tempty , for these cases are equal to
te0mpty =
It follows from eqs (56)-(58) that the polymer concentration on the particulate surface of the adsorbent equals
o o A t ~ texp. filled -- texp. empty
fl(CO1, C02) = Tl(eOl, Co2)
702(c01, c02) = y02(c~ ~) = 0, d21))
+ 6 D ' 2 ( e o l , £o2)
-- ~2(e] 1, = O, C(21))3/(Co2 -- C(21l)},
[-1 -- c ( L , t)exp" empty/e(L, t) . . . . . . p. empty] dt.
7ol(e01, Co2 ) = Col [(Atuo)/(AL) - tr],
trilled = t o + ( L / u ) [ l + 671 (col, Co2)/Col ],
tnlled ~--- t O + ( L / u ) { l
'
(54)
ul = [J2(Col, c02) -- f2(c~11) = O, c02)]/(Co2 -- c~1)).
C l ( L , t) = C l ( t * ),
I
+ (c02 - c~21))[(Atuo)/(AL) - a]
(56)
or equals f z l ( C o l , Co2) = 72(Col, Co2)
701(Col, Co2) = Cox [(Atuw)/(AVo) - a],
tempty = t 0 -~- a ( L / u )
since the linear velocity of the fluid in the empty fixed bed is Uo = u / a
(57)
A t = tOp. filled -- tOp.
empty
yol(C01, Co2) = yo2(C~ 1) = 0, c ~ ' ) + (Co2 -
c~')[(atuw)/(AVo)
-
a]
4021
Multicomponent adsorption dynamics of polymers since the volume velocity of the fluid, Uw, and the volume of adsorbent in the fixed bed, Vo, are equal to Uw = uoS,
i = 1, using to eqs (24), we write Hi(Col, Co2) = HI(C~t1)= 0, c~21)),
Vo = Wo/[(1 - a)po]
where W0 and Po are the weight and the density of the adsorbent in the fixed bed, respectively; S is an area of a cross-section of the fixed bed. In most cases porosity of a is equal to 0.4. If the polymer concentration in a solution, Coa, is in [%wt], then the polymer concentration on the adsorbent surface, 7k(c0), is °]k(co) [/~mol/g] = 106Cok[°/oWt][(A tUw)/ (3Vo) - a ] / M W ,
z = (ut/L - 1)/3. For a two-component mixture under
k=l,2.
It follows from the preceding equation that the amount of polymer adsorption, F,,(c), is given by Fro(Co)[/~mol] = Wo[g]ym(Co) [/~mol/g] (61)
R2(co1, Co2) = R2(c]1) = 0, H 1 = #1V2,
((21))
R 2 =/12V
(62)
/~1,2 = ( f l l +/22)/2 +_ [ ( f l l --f22)2/4 + f , 2 J 2 , ] 1'2, f~k = Of.,(C)/eCk.
TO calculate four parameters of the Langmuir desorption isotherms [(F.,h, (Fro)2, ( K p h , and (Kp)z], we have to find from the experimental plot o f t 1/2 vs the total mixture concentration, C~, values of (S1h, (1.)1 and c~11. According to eqs (39), the values of (S1h and ( l . h depend on four parameters of the Langmuir desorption isotherms. Thus, from these two algebraic equations and two algebraic eqs (62), four parameters of the Langmuir desorption isotherms [(F,,h, (Fro)2, (Kph, and (Kp)2] can be found.
F,,(Co) [/2mol/2 ] = y,,(Co) [l~mol/g]/S ° [mE/g] 4. EXPERIMENTAL
where So is the specific surface area of the adsorbent per gram. From the experimental outlet-concentration-distributions, the concentrations, ct21) and At, for different values of Col and Co2 are found. The adsorptions of Fx(COl, Co2) and F2(col, Co2) are calculated by expressions (61). It should be noted that expression (61) can be used only for calculations of the convex domain of adsorption isotherms. These expressions are invalid for the concave domain of adsorption isotherms. For the calculation of the concave domain of adsorption isotherms (or convex desorption isotherms) we need to use eq. (7) or (36) in the case of Langmuir adsorption isotherms. 3.2. Calculation of parameters of desorption isotherms To calculate the parameters of the Langmuir desorption isotherms the Riemann invariants Rk, and invariants Hi and the linear plot of z- 1/2 vs the total concentrations of mixtures, Fz, may be used. Strictly speaking, eqs (24) and (39) are valid only for the equilibrium model. Therefore, it is reasonable to compare outlet-concentration-distributions for the equilibrium and non-equilibrium models. For the Langmuir desorption isotherms (33), the system of eqs (40) and (41) were integrated numerically by a computer. The dependencies of r 1/z on the total concentrations of mixture, Cs, for the equilibrium and non-equilibrium model, are shown in Fig. 4. If the parameter ao is reduced, these dependencies approach a straight line, according to eq. (39) for the equilibrium model. Thus, the slope and intercept of the straight line, ($I)1, (1,)1 and also the equilibrium concentration of a weak component, c~21~, can be found from the outlet-concentration-distribution for the total concentration of the two-component mixture, Cz(z), since
The model polymer is a non-ionic polyurethane polymer based on ethylene oxide. The polymer's structure consists of linear water-soluble poly(oxyethylene) (PEO) backbones with an average molecular weight of 51,000 with hydroxyl, dodecyl linear alkyl end-groups. The structure of the model associative polymer is R - O - ( D I - P E O ) 6 - D I - O - R [ R is C16H33 or H, DI is isophorone diisocyanate, and PEO is with a normal molecular weight of 8200]. The polydispersity indices of 1.06 for the polymer with (R := C16H33) end-group and of 1.04 for the polymer with ( R : = H) end-group, respectively, were determined by gel permeation chromatography. We have been investigating the adsorbent TiO2(T315 500 Fisher Scientific Co.). The particle size and shape, surface topography, chemical compositions, and bonding of the titanium dioxide has been characterized using a combination of X-ray photoelectron spectroscopy (ESCA or XPS) and transmission and scanning electron microscopy (TEM and SEM). X-ray photoelectron spectra were collected with a Perkin-Elmer LS-5000 ESCA instrument, using a magnesium anode operating at 400 W, and a 35 eV pass energy for the analyzer. Samples of these particle substrates were compacted onto double-sided adhesive tape. Empirical sensitivity factors were used to convert peak intensities into atomic percentage. For particle size analysis, TEM was used. Samples were secured on an SEM stub with 3 m adhesive transfer tape and examined with a JOEL 6300F field emission (FE)-gun SEM at 1 kV accelerating voltage which is low enough to eliminate the need for sputtered conductive coatings. Particle size distribution was obtained from the photomicrographs using a MOP-3 image analyzer interfaced with a Zenith PC. The results of quantitative analysis of the photomicrographs show that the titanium dioxide particles
4022
L. K. FILIPPOV
have an average diameter of 178 nm and polydispersity index of 1.45. The chemical composition in the unit atom percent at the surface of titanium dioxide (Fisher Scientific Co.) is Ti(18%), 0(55%), C(25%), and O/Ti(2.6%, contaminant oxygen subtracted C20). The only significant component, this composition, is a 1% phosphorous signal from phosphate. Titanium dioxide had > 1% sodium, and some had 1% nitrogen. The carbon concentration is in the normal range found on powdered inorganic materials. Curve fitting the carbon 1s peak into three components (C-C, C-O and C = O) gives approximately C50 stoichiometry. Subtracting the oxygen associated with the contaminant, the remaining oxygen-to-metal ratio shown in the last column was calculated. This ratio is too high by about the same factor, perhaps because the calibration factor was determined on a large, fiat plate. The value of the specific surface per gram of grains of the adsorbent (titanium dioxide) is measured by MONOSORB (Quantochrome Co.) and is calculated by the BET method (Hiemez, 1986). The specific surface area of titanium dioxide is equal to 13.5 m2/g. The solvent was distilled-deionized (DDI) water. Solutions were made fresh daily. Adsorption and desorption dynamics were being investigated in the fixed beds as shown in Fig. 5. The concentration of polymers in DDI water for outlet adsorption and desorption distributions was determined by differential refractometer (DRF) KNAUER. In our case, the fixed bed having a 6-mm interior diameter is enclosed between inlet and outlet tubes, which are located in a thermostat jacket.
5. RESULTS AND DISCUSSION
We developed the above method based on the frontal adsorption dynamics of a mixture in the fixed bed to find adsorption and desorption isotherms in mixtures of water soluble polymers with a polymer structure R O - ( D I - P E O ) 6 - D I - O - R for two endgroups R : = C16H33 and R : = H. Let us consider the adsorption isotherms for mixtures of these polymers. The adsorption isotherms were calculated by using expressions (54)-(61) from the experimental outlet-concentration-distributions for two lengths (L = 0.9 and 6.3 cm) of the fixed bed over a wide range of polymer concentrations (from 0.03 to 3 g/kg). The experimental adsorption isotherms are shown in Figs 6 and 7. The adsorption isotherm for the individual water soluble associative polymer ( R : = C I 6 H 3 3 ) is almost linear for the polymer concentrations 0 ~< c01 ~< 1.1 g/kg and is of the Langmuir type for the polymer concentration 1.2 g/kg ~< col ~< 3 g/kg. For the mixture of polymers of R : = C 1 6 H 3 3 and R : = H , the linear domain of adsorption isotherms reduces. The linear domain of adsorption isotherm of associative polymer is 0~
result may be explained by the arrangement of polymer chains into adsorption layers. The adsorption isotherms of the polymer with the end-group (R:= H) are the Langmuir type for the polymer concentrations 0 ~< c02 ~< 3 g/kg. The maximum amount of adsorption (Fm)l and equilibrium constant (Kp)l for the associative polymer (R := C16H33 ) are almost three times more than the maximum amount of adsorption (Fro)2, and equilibrium constant (Kp)2 for the polymer (R := H) due to the end-group R := C16H33. Let us now consider the desorption isotherms for mixtures of these polymers. The desorption isotherms were calculated using the expressions (39) and (62) from the experimental outlet-concentration-distributions for the two lengths (L = 0.9 and 6.3 cm). The experimental plots of z- 1/2 vs are C~ are shown in Fig. 8. "E ~.5
L ~
Polymers(c,:R=C,sH~~ ~.0
O00
Relative Concentratlon,C1=cl/col
Fig. 6. The adsorption and desorption isotherms for the polymer with the end-group (R := C16H33) for the different concentrations of the polymer with the end-group (R := H). Adsorption (solid curves): (@E~) c2 = 0, (A--A)c2 = 1 g/kg, ([~-E]) c2 = 3 mg/kg; desorption (dashed curves): (0---0) c2 =0, (A A) c2 = l g/kg, (m---m) c2 =3mg/kg; col = Coz = 3 g/kgg (Fm)l = 0.172pmol/m 2, (Kp)1 = 1.16kg/g, (Kp) 2 = 0.4 kg/g.
~E
3.0
Polymers(c2:R=H;c , : R = C , 6 H y
E ~2
2.0.
1.0
Retatlve Ooncentration, C2=cJCo2 Fig. 7. The adsorption and desorption isotherms for the polymer with the end-group (R := H) for differentconcentrations of the polymer with the end-group (R:= C16H33): Adsorption (solid curves): (()-O) cl = 0, (A---A)cl = 1 g/kg, ([2~E]) Cl = 3 mg/kg, desorption (dashed curves): (o----o) cl = 0, (A---A) cl = 1 g/kg, (11 II) c~ = 3 mg/kg; %1 = c02 = 3 g/kg; (Fro)2= 0.05 #mol/mz, (Kv)2 = 0.4 kg/g, (Kp)l = 1.16 kg/g.
Multicomponent adsorption dynamics of polymers It follows from these results that the experimental plots approach a straight line when the length of the fixed bed, L, increases. If L ~ 0% then, according to eqs (52) and (53), the experimental plot of r -x/2 vs C~ can be described by the equilibrium model. For the equilibrium model, according to eq. (39), the straight line AB corresponds to the proportional pattern (5) for the wave index i = 1, and the straight line CD corresponds to the proportional pattern (5) for the wave index i = 2. It follows from the experimental data, shown in Fig. 8, that desorption isotherms of mixtures of polymers with the endgroups ( R : = C 1 6 H 3 3 ) and R : = H are described by Langmuir desorption isotherms. The dimensionless parameters of these isotherms calculated by the above-mentioned method are: ( F ° h = 38.7,
(K°)I = 3.48,
1.25 Polymers(cl:R=C~6Haa; ...... ....
theoretical equilibrium caae ,,,,~, 1~--0.9 . . . . perimentol
l , ~
""
0.75 T-1/2 0.50
0.25
~C --
0.00 0.0
--
.....
Ci'-
~(I.)l(intercept ) -
l, Ji . . . . . . . . . C(~)0.5
Relotive
Totol
J ......... 1.0
i ....... 1.5
2.0
Concentrotion,Ce=Cl+C2
Fig. 8. The experimental and theoretical outlet-concentration-distributions for a two-component mixture of polymers with the end-groups (R:= C 16H33 ) for desorption isotherms (63): (St), = 0.361, (I,)1 = 0.277, (F°m)l= 38.7, (K~)I = 3.48, (r°~)2 = 11.25, and (K°)2 = 1.2.
C°=C °=l,
Riemann invariants Rk, and invariants Hi for the Langmuir adsorption isotherms are found. The linear dependence between concentrations of two-component mixtures for the constant and proportional patterns are shown. We proved that the plot of z-1/2 (z is
c° = c ° = 3 g / k g .
For the equilibrium model the proportional patterns (5) occur for the wave indexes i = 1, 2. For these patterns, according to eqs (5) and (37) we write C o = C o = 1, 5.56,
j' _ d , ¢
(63)
Coa=Co2=0,
R1
c2:R=H)
1.00
(F°)2 = 11.25,
(K°)2 = 1.2
4023
V = 5.68,
Hi = 31.8,
#]o, = 0.986,
[-/2 = 10.06]
I
R 2 = 57.14
i=1:
{ {
C{a) = 0.39,
Ct21)= 0 ,
RI = 13.5,
H1 = 31.8,
Cp ) = 0 . 3 9 ,
Re=57.1,
Ct21)=0, H 2 = 134.7,
V = 2.36, R 2 =
#]1) = 5.72,
#2
=
24.2}
57.14
V=2.36,
#1 =5.72,
#2
=
(64)
24.2}
R1 = 13.5
i = 2: Coa = Co2 = 0, V = 1, #1 = 13.5, #2 = 134.7"( 134.7, H i = 134.7, R1 = 13.5
R2 =
0,
oo > r ~> #t22) = 134.7
Cl(z),
134.7 >~ z ~> #t21) = 24.2
CI(L, t) = Cl(z) =
C 2 ( L , t) -~
C2(z ) =
Cp ) = 0.39, 24.2 >/r >i u]~) = 5.72
CI(z),
5.72 >~ r >1 #~o) = 0.986
l,
0.986 ~> r/> 0
0,
5.72 ~< z
C2(z),
5.72/> z >/0.986
1,
0.986 >~ z >~ 0.
It follows from eqs (64) and (65) that for the experimental desorption isotherms (63) the Riemann invariants R~, and invariants Hi are constant for the wave indexes i = 1, 2, according to eq. (24).
6. C O N C L U S I O N S
For the equilibrium model of multicomponent adsorption dynamics, analytical expressions of the
(65)
the dimensionless time) vs the total concentration of the mixture is linear for the proportional patterns. By using the slope and intercept from these linear dependencies, we developed the method of calculating the parameters of the Langmuir desorption isotherms. The developed method, based on the frontal adsorption dynamics in the fixed bed, may be applied usefully to find adsorption and desorption isotherms
4024
L. K. FILIPPOV
of mixtures over a wide range of polymer concentrations onto particulate surfaces of adsorbents. NOTATION cm D~k
fro(c) (Kp)k L Li MW n q,, t u wi z i
concentration of the ruth component in an external fluid, g/kg matrix of diffusion coefficient (m = k) and mutual diffusion (m # k) in an intergranular space of the porous media of the adsorber, cmZ/s equations of adsorption isotherms for the mth component mixture the equilibrium constant for the kth component mixture, kg/g length of the fixed bed, cm width of the mass transfer zone, cm molecular weight of polymer, g/mol number of mixture components concentration of the ruth component in an adsorbed state, g/kg time, s linear velocity, cm/s wave velocity, cm/s distance from the bed entrance, cm wave index
Greek letters matrix of kinetic coefficients of mass exchange 1/s (F,.)k the maximum amount of adsorption for the kth component mixture,/~mol/m 2 tr porosity
[Jmk
REFERENCES
Arfken, G., 1970, Mathematical Methodsjbr Physicists. Academic Press, New York.
Aris, R. and Amundson, N. R., 1973, Mathematical Methods in Chemical Engineering. Prentice Hall, Englewood Cliffs, NJ. Filippov, L. K., 1989, Theoretical basis of separation processes and dynamics of adsorption of multicomponent mixtures. Chem. Engng Sci. 44, 575-582. Filippov, L. K., 1992, Coherent and incoherent frontal patterns for multicomponent dynamics of adsorption. I. Adsorption dynamics for convex mixture isotherms of adsorption. Chem. Engng Sci. 47, 1199-1210. Fleer, G. J. and J. Lyklema, J., 1983, Adsorption of polymers, in Adsorption from Solution at the Solid/Liquid Interface (Edited by G. D. Parfitt and C. H. Rochester). Academic Press, New York. Golshan-Shirazi, S., Gholdbane, S. and Guiochon, G., 1988a, Comparison between experimental and theoretical band profile in nonlinear liquid chromatography with a pure mobile phase. Anal. Chem. 60, 2630-2634. Golshan-Shirazi, S. and Guiochon, G., 1988b, Comparison between experimental and theoretical band profile in nonlinear liquid chromatography with a binary mobile phase. Anal. Chem. 60, 2634 2641. Guiochon, G., Golshan-Shirazi, S. and Jaulmes, A., 1988, Computer simulation of the propagation of a large-concentration band in liquid chromatography. Anal. Chem. 60, 1856-- 1866. Helferrich, F. and Klein, G., 1970, Multicomponent Chromatography. Theory of Interference. Marcel Dekker, New York. Hiemez, P. C., 1986, Principles of Colloid and Surfaces Chemistry. Marcel Dekker, New York. Korn, G. A. and Korn, T. M., 1968, Mathematics Handbook .for Scientists and Engineers. McGraw-Hill, New York. Lax, P., 1957, Hyperbolic systems of conservation laws. Comm. Pure Appl. Math. 10, 537-566. Lax, P., 1971, Shock Waves and Entropy in Contribution to Nonlinear Function Analysis. University of Wisconsin Press, Madison, WI. Lipatov, Yu. S. and Sergeeva, L. M., 1974, Adsorption of Polymers. Wiley, New York. Rhree, H. K., Aris, R. and Amundson, N. R., 1986, First Order Partial Differential Equations. Prentice-Hall, Englewood Cliffs, NJ. Rozdhdestensky, B. L. and Yanenko, N. N., 1986, Systems of Quasi-Linear Equations. Nauka, Moscow.
PII:
Chemical Engineering Science, Vol. 51, No. 16, pp. 4013 4024, 1996 Copyright (t) 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved S0009-2509(96)00243-6 0009 2509/96 $15.00 + 0.00
M U L T I C O M P O N E N T A D S O R P T I O N D Y N A M I C S OF P O L Y M E R S IN A FIXED BED L E O N I D K. FILIPPOV* Lehigh University, Bethlehem, Pennsylvania 18015, U.S.A.
(First received 30 October 1992; revised manuscript received and accepted 29 January 1996) Abstract--The theoretical analysis of concentration-patterns of frontal adsorption and desorption dynamics of mixtures in a fixed bed for the Langmuir adsorption isotherms is considered for the equilibrium and non-equilibrium models. For the equilibrium model, the concentration-constant-patterns and concentration-proportional-patterns may be described by using the Riemann invariants Rk and H~ invariant analytical expressionswhich are found. The methods of calculation of adsorption and desorption isotherms based on the use of outlet-concentration-distributions are developed. Adsorption and desorption isotherms of mixtures of the water soluble associative polymer for two end-groups (R := C16H33 and R := H) onto particulate surfaces of titanium dioxide are obtained. It is shown that the method developed can be used to measure the adsorption and desorption isotherms over a wide range of associative polymer concentrations. Copyright © 1996 Elsevier Science Ltd Keywords: fixed bed, adsorption, polymer mixtures, Riemann invariants
1. INTRODUCTION For the investigation of adsorption of polymers on planar and particulate surface to solid substrates, direct and indirect experimental methods have been applied (Helferrich and Klein, 1970; Lipatov and Sergeeva, 1974; Fleer and Lyklema, 1983; Hiemez, 1986; Guiochon et al., 1988). Direct methods can be divided into the following main groups: electrometric, chromatographic, and spectroscopic methods. The indirect methods leading to the thickness of the adsorbed layers are based on the use of three techniques: ellipsometry, scattering (static and dynamic light or neutrons), and hydrodynamics. In ellipsometric measurements, the thickness of the adsorbed layers on a planar surface are deduced from the changes of polarized light upon reflection. In scattering measurements, the thickness is found by static and dynamic scattering of light or neutrons. In hydrodynamics measurements, the thickness is inferred from the external displacement of the adsorbed polymer of the slipping layer between the solid and liquid phase which moves tangentially with respect to the solid surface. Guiochon and co-workers have applied computer simulation to describe the propagation of a large-concentration band profile in nonlinear liquid chromatography. The computer simulation allows us to compare experimental data and a large-concentration theoretical profile obtained for different adsorption/desorption isotherms of mixtures. However, the adsorption/desorption isotherm of mixtures may be found from the experimental data by using analytical
*Address for correspondence: Leonid Filippov, 510 Seneca Street (First floor), Bethlehem, PA 18015, U.S.A.
solutions. Therefore, in this paper we will develop in detail a method for determining the adsorption/desorption of mixtures on a particulate solid surface based on the frontal adsorption dynamics or chromatrography. This method is based upon the use of the both the experimental outlet-concentration-distributions and algebraic relations (4) for the constantpatterns, and the linear eq. (39) for the proportional patterns, when the Langmuir desorption isotherms occur. Frontal chromatography is a direct method which allows us to find the adsorption and desorption isotherms of mixtures over a wide range of concentrations of polymers. 2. ADSORPTION DYNAMICS OF MIXTURES FOR THE EQUILIBRIUM MODEL
Adsorption dynamics of multicomponent mixtures in the equilibrium case is described by the system of material balance equations for each of the component mixtures
~cm UOCm 6#f.~(C) ~t +--~-Z +
~
(1 -- a~
O,
l ~m~n Cm(0, t) = Corn= constant,
6--
~ , (1)
cm(z, 0) = c° = constant. (2)
This hyperbolic system of equations admits as solutions the following coherent frontal patterns: the constant patterns and the proportional patterns (Lax, 1957; 1971; Aris and Amundson, 1973; Rhree et al., 1986; Filippov, 1992). For the constant patterns, the concentration distribution takes the form of concentration waves traveling with a speed wi (1 ~< i ~< n) and
4013
L. K. FILIPPOV
4014
is a function of only one variable y and the discontinuous solutions have the form:
or for dimensionless concentrations in the following form
~C~ 1), y < 0 Cm(2 , l) = Cm(y ) = (C~) '
A I
y > 0
c
)
=
( F mo ) k ( K po) k C k / V
V = 1 + ~ (KObCj,
,
j
(3) y=z--wit
,
1 <~m~n
C k = c k / m a x (Cok ; C0)
W o k = (r'm)k [max(cog; cO)],
where c~ ~ and c~ - ~ are the equilibrium values of concentrations for the wave index i. The values of velocities w~ from the integration form of eqs (1) are found by (Filippov, 1989) wi
ui -
-
U (I + &i~)'
1 <~m, i<~n.
Cm(Z, t) =
g~(¢),
~ = Yi = V(t~(i 1)) ~(i 1) Vi(?(i 1)) ~ ~ ~ l,i(?(i))
~d ~,
~ = ~I ° = v~(~ "~)
~_ = z/t,
(5)
-0,
d~
~fm(C) OCk
det (fmk -- piErak) = 0,
H ( p , c) = 1 -- ~
--
1)/6 = Pi,
bj/(p* - It),
~*<~*<
--
. . .
~
vi)/(bvi) ~ ~ p--. ~*-0,
0%
H(p,c)
+ O,
d(
h(i)(c) '
j=l
?~cj
, ( )
H(p,c)
~
dCk
r~)(c) - - rtkO(C) '
1 <~m,
/~=0,
ck>0.
Therefore, for the Langmuir adsorption isotherm (8) the matrixfmk has n real, positive, and different eigenvalues Pk (1 ~< k -%
k, i ~ n
0
where r~,° (c) is the right-hand eigenvector of the matrix frog, corresponding to the eigenvalue of #i (1 -%
2.1. A n a l y s i s o f c o n c e n t r a t i o n p a t t e r n s f o r L a n g m u i r a d s o r p t i o n isotherms The Langmuir adsorption isotherms are written in the form
- - -
~
< /'/(n--l) < ~ < ~/n"
The right-hand and left-hand side eigenvectors r~i) and l~i) of the matrix fmk are found, respectively, from the equations (Arfken, 1970; Rozdhestensky and Yanenko, 1986) ,~ (i) • " .]mkrk
fro(C) = ( F m ) k ( K p ) k C k / V ,
+oc,
(7)
or in the following form dc~
- - O,
H ( ~ , c) -~ - ~:
1
=
<~*.
-- ~ bj(p* - #) 2 > O, bj > 0
H(0, c) = ~ > 0, - -
< (F,.).(G).
The function H ( p , c) is monotone, since
where E,,k is the unit matrix and/2~ is the eigenvalue of the m a t r i x f , k (1 ~< i ~< n). We write the preceding system of equations as -
bj = (F,.)j[(Kp)j]2/V 2.
are satisfied then the poles p* of the function H ( p , c) are arranged as
1 <~ m, k, i <~ n
Pi = (u
(10)
j 1
(6) (. = (ut/z
1 ~ m, k, i ~ n,
The secular eq. (10) for the Langmuir adsorption isotherms (8) may be rewritten in the following form n H(/~, c) [ I (P* - #) = 0, #* = ( F m ) j ( K p ) j / V j 1 (11)
0H(p, C) ,
1 <. k <~ n
(r,.)l (Kp), < (F,~)2(K~)2 < "
l <~ m, i <~ n
f~k -
fk(C) = f k [ m a X ( c o k ; c ° ) ] ,
If the inequalities
where ?" a), ~") are the equilibrium values of concentrations for the wave index i and Cm(O is the solution of eqs (7). When the proportional patterns of eqs (5) occur, the system of eqs (1), may be rewritten in the following matrix form (Emk -- fmk)dCk
(K°)R = (Kp)k[maX(Cok; cO)]
(4)
For the proportional patterns, the concentration distributions are in the form of expanding concentration waves and depend only on the variable n. The continuous self-similar solutions may be written as: ~-1~, ,
(9)
where (Fm), and (Kp), are the maximum amount of adsorption and the equilibrium constant for the kth component of the mixture, respectively. The eigenvalue #i of the matrixf,.k is found from the secular equation (Korn and Korn, 1968; Arfken, 1970)
fm(C(y = + c~)) - - j m ( C ( y = -- oG)) c,.(y = + oC) -- Cm(y = -- o0) '
t
l
= pirf ,
fmkl~ ') = I~l~ ~,
1 <. m, k, i % n
V = 1 + ~. ( K p ) j c j j=l
(12t (8)
where fmk is the transpose of the matrix fmk.
Multicomponent adsorption dynamics of polymers For the Langmuir adsorption isotherms (8), the preceding equtions may be written as (1/b °) (#* - pi)r~; = L (Kp); rJi), j=,
b ° = (Fm)m(Kp)mcm/V 2 [1/(Kp),,]
(13a)
(g* - / t , ) l ~ ) = L byl~i,.
Ifdcm/d~ ~ 0 (1 ~< m ~< n), then for the wave index i, according to eqs (20), we have Rk=Constant,
it:k,
Ri = variable,
i = k.
r'~'~ = boo, * - ui)
~,
t~" = (K~,)~(.*
(13b)
- ui)
1 <~k<~n.
Let us now consider the Riemann invarlants Rk (1 ~< k ~< n). By definition, for the hyperbolic systems of equations, the Riemann invariants are given by (Rozdhestensky and Yanenko, 1986)
dRi =- ~ci L l~)dc"
(15)
m 1
where ~Ckis the integrating multiplier of the differential form. For the Langmuir adsorption isotherms (8) the integrating multipliers are equal to
~:i
-/1, EEl=, b°(Kp)~(/~* -- #i)- ' ] . o , [ Z k = , bk(Kp)k(Pk - #i) 2]
-
(16)
F r o m the preceding analysis for the proportional patterns (5), the following relations are valid
(22)
since according to eqs (15)-(17) and (21), the following equality is valid dHi d~
--0,
Thus, the Riemann invariants Rk, and invariants Hi are important values which allow the calculation of concentration-distributions for the constant and proportional patterns. In fact, the discontinuous concentration-distributions, Cm(y) (1 ~< m ~ n), are described by the following algebraic system of equations, for the constant patterns (3) of the Langmuir adsorption isotherms with the wave index i: Hi = pi V 2
constant,
=
ek
=/~i
V = constant,
1 <~k, i<~n
(23)
and the continuous concentration-distributions, c,,(0 (1 ~< m ~< n), are described by the analogous algebraic system of equations for the proportional patterns (5)
Rk = Pi V = constant,
H~ =/~i V 2 = constant,
1 ~ k , i<~n. d(
V/2,
hti)(c) = --
dV
L (Kp)jr} i), j=l
1 -- hti)(c )
(17)
Thus, the Riemann invariants from eqs (15)-(17) may be obtained in the following form
Rk = l~kV, 1 ~ k <~n
(18)
The hyperbolic system of eqs (1) in the Riemann invariants has the form 8Ri #iORi g~-+~=0,
1 <~i<~n.
(19)
F o r the proportional patterns (5) from eqs (7), (15) and (17) we write
dRi
L (c3Ri~(dc,,~=(V)
d:-m=,\~Cm]\d:J
(24)
F r o m the preceding analysis, the Langmuir adsorption isotherms are valid for the following inequalities. For the proportional patterns (5):
dV
--<0, d;
dRi ~-
>0,
1 <_.i~n.
dV
-->0, dy
dRi
--<0, dy
1 <~i<~n.
(20)
where 6ik is the Schwarz-Christoffel symbol.
(26)
Let us consider the adsorption and desorption dynamics for the different Langmuir isotherms of a twocomponent mixture (n = 2). Example 1: For the adsorption dynamics (Corn > C0, m = 1, 2) and the following Langmuir desorption isotherms (9), (K°), = 2,
(r°)2 = 2,
(K°)2 = 2,
Col = C o 2 = 1, C ° = C 2 ° = 0 1, i = k (~ik= O, i ¢ k
(25)
For constant patterns (3):
( r ° ) , = 3,
--~ (}ik,
(21a) (21b)
Hi = / l i V 2 = constant
',
(14)
l~
However, for the wave index i of the Langmuir adsorption isotherms, the invariant Hi is constant, i.e.
j=,
F r o m the preceding relations, it follows that the righthand and left-hand side eigenvectors are equal to
4015
(27)
the constant patterns (3) take place for the wave indexes i = 1, 2. In these cases, according to eqs (4) and (23), we have
Co, = Co2 = 1, V = 4, R1 = 3.68, H, = 14.73,
/q = 0.9215, R2 = 0.813
/t2 = 0.2034,
/~1 = 1.5(
4016
L.K. FILIPPOV
i = l:
{
C~ lJ = 0,
R1 = 6 ,
~C1
(1)
C(21) = 1.457, H i = 14.73,
V = 2.457,
C z(1) = 1.457,
=0,
~R 2 = 0.813,
]21 = 2.44,
]22 = 0.331,
]21 = 1.5}
]21 = 2 . 4 4 ,
]22=0.331,
]22 = 0.814}
R2=0.813
Hz = 2,
V=2.457,
(28)
R1 = 6
i = 2:
C° = C° = O, V = I, ]21=6, R2=2, H2=2, Rl=6 1, 0,
CI(L,t)=CI(Z)=
]22=2,
Z > U a = 1.5, r = ~ r
1,
~>'r>ul
C2(L, t) = C2(-Q =
]22=0.814} with z = L
=1.5
C[21) = 1.457, ul > r > u2 = 0.814
(29)
O, O < ~ < u 2 where L is the length of the fixed bed. The solutions (28) and (29) are shown in Fig. 1.
The solutions (31) and (32) are shown in Fig. 2.
Example 2: F o r the a d s o r p t i o n - d e s o r p t i o n dynamics (Col > c °, Co2 < c °) and the following L a n g m u i r adsorption isotherms (9)
Example 3: F o r the desorption dynamics (Co,. < c °, m = 1, 2) and the following L a n g m u i r adsorption isotherms (9)
(F,°,)I = 4.5,
(K°)I = 2,
Col = C O =
1,
(F°)2 = 2,
(K°)2 = 1,
C ° = Co2 = 0
(30)
(F°)1=3,
the p r o p o r t i o n a l pattern (5) occurs for the wave index i = 1, and the constant pattern (3) occurs for the wave index i = 2. In these cases, according to eqs (4), (23) and (24), we write
C = O, Co2 = 1,
V = 2,
R1
R2
9,
H 1 = 18,
]2]o~= 4.5, ]22
=
(K°h =2,
Co1=Co2=0,
(r°)2=2,
C °=C2 °=1
0.5}
= 1
i=1:
{ {
C~ 1~ = 2.28,
RI = 1,
C~ 1~ = 2.28,
Re = 1 ,
C(21) = 0.433,
H1 = 6 ,
V = 6,
]2]1) =
V = 6,
]21 = 0.5,
]22 = 0.167}
0.5,
(31)
Re = 3
C(21) = 0.433,
H2=6,
]22 = 0.167,
]22 = 0.333}
R1 = 3
i = 2: C01
=
R2=2,
1,
C ° = 0,
H2=6,
V = 3,
=
=
1,
]22 = 0.666,
]22 = 0.333}
R1=3
/°, CI(L, t) = Ca(r)
]21
cc > r / > ]2]o) = 4.5
CI(T)'
#~o) >~ r > ]211j = 0.5 C] 1) = 2.28, 0.5 >~ z >/ 1'/2 = 0.333
[l,
0.333 > r >~ 0
1,
= C 2 ( L , t) = C 2 ( z )
~>r>~4.5
C2(r), ] C~2II
[o,
= 0.433,
4.5 >~ z >~ 0.5 0.5 ~> z > 0.333 0.333 > r >/0
(32)
(Kp°)2= 1,
(33)
4017
Multicomponent adsorption dynamics of polymers the proportional patterns (5) take place for the wave index i = 1 and i = 2. In these cases,
C° = C° = I, Ra
0.813,
V = 4, , , ° ) = 0 . 2 0 3 4 , H1 = 3.23,
,2=0.9215}
R z = 3.7
i=1:
{
C~1'=0.31,
R1 = 2,
C(2u=0,
H1 = 3.23,
C,"=0.31,
V = 1.62,
C(2')=0,
Re--3.71,
He=6,
, @ ) = 1.23,
,2=2-29],
(34)
3.7
R 2 =
V=1.62,
,1=1.23,
,u2=2.29}
R1=2
i=2:
Co, = Coz = O. V = 1, ,u1= 2, .2 = 6} R 2 = 6 . He =6, R1 =2 0,
o0 > r >~ #t22) = 6
CI(T), 6 >/ r >~ #t21) = 2.29 CI(L, t) = CI(r) =
C2(L, t )
C2(r)
=
=
C]1) = 0.31, 2.29 >/r i> @) = 1.29 Cl(r),
1.29 >/r >~ ,]0) = 0.203
1,
0.203 > r/> 0
0,
1.29 ~< r
C2(z)
1.29 ~> z >~ 0.203
1,
0.203 ~> r >~ 0
(35)
A
A
2.5 2.0 C1
c, 0.5i
1.5
[
I
~
Col 1.0
# ,
equilibrium case dl _ m lee:=0:1/ nons-eesquilibrl. . . .
.....
0.0 . . . . . . . . 0
1.5
b.'5 . . . . . . . . .
1.b . . . . . . . . .
0.5 c?
1.5 . . . . . . . . U1
0.0
~.0 7"
iI
. . . .
. . . .
,
a.=0.1 nnon-equlllor laa:---~):~l ° n s - e ° q ......... . . . .
I
,
. . . .
. . . .
[
. . . .
[
~ . . . .
[
. . . . .
,
. . . .
0.00 0+50 1+00 1.50 2.00 2.50 3.00 3.50 4.00 0 U2 ,U.?)
1.0 Co2
[3
~- -
I
,
,
,
,q
4+5(~) 500 ,££1° T
B
C2 i Co2
I
C2 .......
- -
O.5
0.5
equilibriurrl case a6=0.1 on-equilibrium
___
.....
C~ j
I c; 0.0
0.0
0.0
o
.
.
.
.
.
.
.
.
0.51"77"[". . . . . . . . . .1.0 ...... U2
115.. . . . . . . . U1
I
~'.0 7"
Fig. 1. (A and B) The outlet-concentration-distributionsfor adsorption isotherms (27).
The solutions (34) and (35) are shown in Fig. 3. As follows from the preceding examples, the Riemann invariants Rk, and invariants Hi remained valid for different wave indexes in cases of adsorption, desorption and adsorption~tesorption dynamics.
II
....
i .... 0.5 (1)
0.0~ U
....
U2
equilibri urn oCne_Seq uilibrium (0.=0.3)
. . . . (o.--03: , .... .0
, .... 5
, .... 20
, .... 25
cases
t .... 30
i .... 35
i .... 40
,u+1
, .... 4I~ ~+
5.0
T
Fig. 2. (A and B) The outlet-concentration-distributionsfor adsorption desorption isotherms (3l). 2.2. Analysis of solutions for proportional patterns For the fixed bed of the length L, it follows from eqs (23) and (24) that the proportional patterns (5) and the wave index i are valid relations
T
=
Hi/V 2, "r = ~ with z = L, Hi=constant,
1 ~
L. K. FILIPPOV
4018 1.0
2.5
A'
2.0 1.5 , C1 0.5
i
...... - - --
~",'-
T-I/2
equilibrium cose (%=0.1) non-equilibrium (G,=O.3) coses
1.0 0.5
.
.
.
.
.
.
t.=./
0.0 o
L0
_(o) H,1
~.--
c;
2 O) ,/2"2
_0) P1
3
4
5
/~2) Relotive Totol
3"
Fig. 4. The plots of z 1/2 vs the total concentrations of the two-component mixture for desorption isotherms (33): (S1)l = 0.78, (1,)1 = 0.66.
t
"" ~.
] C2 0.5
Concentrotion,C==Cl+C2
~
I
(0,=0,1) non-equilibrium
- --
~
(a,=0.3)
+ (Fm)E(Kp)2 - Hi - Rk]}/Ai
eoses
(39)
At = {RkE(Kph -- (Kp)2] + ( K p h × (Kp)2 [(Fm)l -- (Fm)2]} U 1/2 •
O0
o
05
03
08
43
0
~°)
' ' ' 15 ....
J8
#I ')
20
r
Fig. 3. (A and B) The outlet-concentration-distributions for desorption isotherms (33). Hence we write z 1/2 = V/(Hi)I/2,
H i = constant.
~k = Rk/V,
i # k, 1 ~ i, k ~ n (37)
#1.2 = ( f l l +fz2)/2 + [(f11 - f 2 2 ) 2 / 4 +Ji2J21] 1/2,
af.,(c) fmk
--
OCk
m,k
,
=
($1)1 = 0.78,
(36)
Thus the plot of r-1/2 vs V is a straight line for a multi-component mixture with an arbitrary number of components. In particular, for the two-component mixture (n = 2) we write using eqs (23) and (24), ~i = Hff V2,
where Cz is the total concentration of the two-component mixture, (SDi and (I,)i are the slope and intercept of the straight line o f t - 1/2 vs Cz, respectively. In particular, for the case of eqs (30)(32), as shown in Fig. 4, from eq. (39), we have ( I , h = 0.66, i = 1,
H1 =3.23,
R2=3.7.
3. NON-EQUILIBRIUM MODEL OF MULTICOMPONENT ADSORPTION DYNAMICS
Adsorption-separation process in a fixed bed is described by the system of equations of material balance in the intergranular space of the adsorber [eq. (40)] and the system of equations of kinetics [eq. (4t)] in the following form (Filippov, 1989, 1992)
1,2.
.
+ ~
= _
D~
(401
k=l
It follows from eqs (36) and (37) that the dependence
~~tq m
between concentrations C1 and Cz is linear, i.e.
L
flmk[fk(c) -- qk],
1 <~ m, k <~ n. (41)
k=l
C 1 = al C2 -I- bl al
=
-
(Kp)2E(F,.h (Kp)l
-
Equations (41) reduce to equations for the adsorption isotherm, i.e. qk = f k ( q , C2. . . . . C,), when the rates of adsorption and desorption processes are infinite
Rk]/
{(Krh [(F,,)2(K,)z - R , ] } (38)
bl = [Hi q- Rk -- (Fm)l (Kp)l - (Fm)2(Kp)2]/ {(K.h [(r.h(K,h
H~, R , = constant,
- RR]}
Let us consider the solutions of the system of equations (40) and (41) for the constant patterns (3). In this case we write
i ¢ k.
Z- 1/2 = (SI)iC z q_ (In)i,
k=lL(6wi)J
ui
C z = C 1 -~ C2, i = 1, 2
(St)~ = ( K p h (Kp)2 [(F,,h ( K p h - (F,.h(K,,)2]/A, (I.), = {[(Kph - (Kp)2]Rk + [(rmh - (Fm)2] x [ ( K p h (Kp)2 + (F.,)x(Kph
F D(I) I dAck
= u,Ac -
Therefore, for the two-component mixture, from eqs (36) and (38), we have
=
(u
-
dy
'
w~)/(~wil (42)
dAqk
"
fl,.k
dy l<~m,k<~n.
Multicomponent adsorption dynamics of polymers The Langmuir adsorption isotherms (8) admit the following transformations
(43)
4019
form
ff(1/Ao)ln(29), + oo >/y >~ Yo~ Y - Yoi = ~(1/AO)ln(29) ' Yoi >1 Y >t Ao = (26D0) -1 {c~ - [~z _ 460oflo(U~ -- l, + )]a/2} (49) A ° = (26Do) 1 {(~ -- [0~2 -- 46Doflo(Ui - l~[ )]1/2}
Therefore, according to eq. (43), the system of eqs (42) may be rewritten in the following matrix form (l) 2
6D,.kd Ack dy 2
B,.kdACk + BOkACk = O, dy Bmk = UiwiE=k
1 <~ m, k <~ n
+ ~ D,.jD}~
(44)
9(Yoi)= 1/2,
p+ = # i ( y =
+~),
P i = Pi(Y = -- ~). When Do = 0, the quasi-linear equations of the second order reduce to quasi-linear equations of the first order, and the precise analytical solution may be written in the following form: y = yo, + {(u,w,)/[Do(U,
-
~,(,,o))]}
j=l
x {lngg - [l~i(Co)/ui]ln(1 + 9/a)} ~ o = u,Dm~ - (V/Vo) ~ D.jfj~(co),
Vo = V(co)
(50)
UdCo)/Ui] Vo/AV.
a = [1 -
j=l
is the unit matrix. In the simple case, when
w h e r e E,nk
D ~ = Do = constant,
flmk = Do = constant (45)
L i ~ - [ ( A o ) - ' + ( A ° ) -~][ln(1/e)],
eq. (44) reduces to iJOodZAck
o~dAck
dy 2
dy
(51a)
+
~ ~mkACk = O, m=l
l ~ k , m<~n
(u~w, + DoDo), ~,~ = Do[U~Em~ - (Vo/V)fm~(Co)].
The solutions of the preceding system of equations may be found in the following form: ck(y) = ck(y =
Li ~- (wi/Do) {[1 - #i(y = - oc)/ui] - 1 + [1 - #dY -- + ov)/u~] -1} [In(i/e)].
#dY=
.q(y= + o C ) = 0 ,
(47)
0<~9(y)~
where 9(Y) is an unknown function. In this case the system of eqs (46) transforms into the following quasi-linear equation of the second order ;~dg(y) dy
-oc)
WhenDmk#0,
D ~ vsO
(l <~ m, k <~ n)
u max (F,,)k bmk --
D~)k=Do
(Lu)
(1 ~ < m , k ~ n )
(48) ao-
Vo=V(co),
6 D~)kmax (Fm)k ,
(LDmR)
WhenDmk=D0,
AV=V °-Vo,
+~).
In the general case, when [3rag¢ O, D~)k 4= 0 (1 ~< m, k ~< n), the estimation of effects of dispersive factors (adsorption kinetics and axial dispersion) may be realized only by numerical integration of the system of eqs (40) and (41) by a computer. The effects of dispersive factors take into account the following dimensionless parameters (Filippov, 1989):
amk --
+ f l o [ U i - Pi(Co)/(l + g(y)A ~--~)]g(y)=O
(51b)
According to the conditions of existence of the constant patterns, these patterns take place if the following inequalities are valid (Lax, 1957, 1971; Rhree et al., 1986; Filippov, 1992)
+ o~) + g O ' )
x [Ck(Y = T Or) -- c d y = _+ OC)],
6Ood2g() ,) dy 2
A o > 0 , A° > 0
&
(46) c~ =
The solutions (49) and (50) describe the concentration distribution in the fixed bed and may be used to calculate the mass transfer zone. The mass transfer zone, Ls, for the wave index i, obtained from the solutions (49) and (50) when g(y = +_ Go) = e,, are, respectively, equal to
V °=V(c°).
Strictly speaking, the solution of the quasi-linear eq. (48) may be found by a numerical method using a computer. However, the approximate analytical solution may be written in the following
u max (F,,)k (LDo)
,
b0
6 Do max (Fm)k (Lu)
. (52)
The system of eqs (40) and (41) for the non-equilibrium model transforms into eqs (1) for the equilibrium model when ~21mk =
aO =
bmk = ao = 0.
(53)
4020
L.K. FILIPPOV
Below we will analyze the numerical solutions of the system of eqs (4) and (41) for different values ao (ao = 0.1 and 0.3) and bo = 0.05. The numerical solutions for the adsorption isotherms (27), (30), and (33) are shown in Figs 1-3, respectively. As it follows from these solutions, the numerical solutions approach uniformly to the solutions of the equilibrium model (1), when the value of the parameter ao decreases and approaches zero. These results make it possible to find adsorption isotherms using analytical solutions of eqs (1) for the equilibrium model and experimental outlet-concentration-distributions. The numerical solutions have been analyzed earlier using a computer by Guiochon and co-workers. However, only analytical solutions allow us to solve the reverse problem, i.e. to find parameters of adsorption and desorption isotherms of mixtures. Therefore, below we will use the analytical solutions in the form of eqs (4) and (39). 3.1. C a l c u l a t i o n o f p a r a m e t e r s o f a d s o r p t i o n isotherms To calculate adsorption isotherms we have to use the algebraic eqs (4). For a two-component mixture (n = 2), when the first component is strongly adsorbed, we write wi = u/(1 + 6ui),
i = 1,
Ui =fl(Col, C02)/C01 ,
e0 = co = 0
For the the convex adsorption isotherms (in particular Langmuir adsorption isotherms), the outletconcentration-distribution of polymers depends on only one variable t*
t* =
e 2 ( L , t) = e2(t*),
t -- L/wi,
i
=
I
Fig. 5. The flow diagram for the study of adsorption and desorption dynamics in the fixed bed. 1: twin micropumps, 2a: pure solvent in syringe, 2b: solvent plus adsorbate in syringe, 3: flow loop, 4: thermostat, 5: fixed bed, 6: filters, 7: DRF detector, 8: flowmeter, 9: computer interface.
and t o is the time needed for the removal of fluid through outlet and inlet capillaries. The retention times can be found by using the experimental outletconcentration-distributions, for both the empty fixed bed and the fixed bed filled with the adsorbent particles, and the following expressions /'fOiled =
[1 -- c ( g , t)exp" filled/C(L, t) . . . . . . p. filled] dt
1.
(58)
To exclude the effects of outlet and inlet tubes (capillaries), as shown in Fig. 5, we measured the two outlet-concentration-distributions for the empty fixed bed and the fixed bed filled with the adsorbent particles. According to eqs (54) and (55), the retention t i m e s , tfilled and tempty , for these cases are equal to
te0mpty =
It follows from eqs (56)-(58) that the polymer concentration on the particulate surface of the adsorbent equals
o o A t ~ texp. filled -- texp. empty
fl(CO1, C02) = Tl(eOl, Co2)
702(c01, c02) = y02(c~ ~) = 0, d21))
+ 6 D ' 2 ( e o l , £o2)
-- ~2(e] 1, = O, C(21))3/(Co2 -- C(21l)},
[-1 -- c ( L , t)exp" empty/e(L, t) . . . . . . p. empty] dt.
7ol(e01, Co2 ) = Col [(Atuo)/(AL) - tr],
trilled = t o + ( L / u ) [ l + 671 (col, Co2)/Col ],
tnlled ~--- t O + ( L / u ) { l
'
(54)
ul = [J2(Col, c02) -- f2(c~11) = O, c02)]/(Co2 -- c~1)).
C l ( L , t) = C l ( t * ),
I
+ (c02 - c~21))[(Atuo)/(AL) - a]
(56)
or equals f z l ( C o l , Co2) = 72(Col, Co2)
701(Col, Co2) = Cox [(Atuw)/(AVo) - a],
tempty = t 0 -~- a ( L / u )
since the linear velocity of the fluid in the empty fixed bed is Uo = u / a
(57)
A t = tOp. filled -- tOp.
empty
yol(C01, Co2) = yo2(C~ 1) = 0, c ~ ' ) + (Co2 -
c~')[(atuw)/(AVo)
-
a]
4021
Multicomponent adsorption dynamics of polymers since the volume velocity of the fluid, Uw, and the volume of adsorbent in the fixed bed, Vo, are equal to Uw = uoS,
i = 1, using to eqs (24), we write Hi(Col, Co2) = HI(C~t1)= 0, c~21)),
Vo = Wo/[(1 - a)po]
where W0 and Po are the weight and the density of the adsorbent in the fixed bed, respectively; S is an area of a cross-section of the fixed bed. In most cases porosity of a is equal to 0.4. If the polymer concentration in a solution, Coa, is in [%wt], then the polymer concentration on the adsorbent surface, 7k(c0), is °]k(co) [/~mol/g] = 106Cok[°/oWt][(A tUw)/ (3Vo) - a ] / M W ,
z = (ut/L - 1)/3. For a two-component mixture under
k=l,2.
It follows from the preceding equation that the amount of polymer adsorption, F,,(c), is given by Fro(Co)[/~mol] = Wo[g]ym(Co) [/~mol/g] (61)
R2(co1, Co2) = R2(c]1) = 0, H 1 = #1V2,
((21))
R 2 =/12V
(62)
/~1,2 = ( f l l +/22)/2 +_ [ ( f l l --f22)2/4 + f , 2 J 2 , ] 1'2, f~k = Of.,(C)/eCk.
TO calculate four parameters of the Langmuir desorption isotherms [(F.,h, (Fro)2, ( K p h , and (Kp)z], we have to find from the experimental plot o f t 1/2 vs the total mixture concentration, C~, values of (S1h, (1.)1 and c~11. According to eqs (39), the values of (S1h and ( l . h depend on four parameters of the Langmuir desorption isotherms. Thus, from these two algebraic equations and two algebraic eqs (62), four parameters of the Langmuir desorption isotherms [(F,,h, (Fro)2, (Kph, and (Kp)2] can be found.
F,,(Co) [/2mol/2 ] = y,,(Co) [l~mol/g]/S ° [mE/g] 4. EXPERIMENTAL
where So is the specific surface area of the adsorbent per gram. From the experimental outlet-concentration-distributions, the concentrations, ct21) and At, for different values of Col and Co2 are found. The adsorptions of Fx(COl, Co2) and F2(col, Co2) are calculated by expressions (61). It should be noted that expression (61) can be used only for calculations of the convex domain of adsorption isotherms. These expressions are invalid for the concave domain of adsorption isotherms. For the calculation of the concave domain of adsorption isotherms (or convex desorption isotherms) we need to use eq. (7) or (36) in the case of Langmuir adsorption isotherms. 3.2. Calculation of parameters of desorption isotherms To calculate the parameters of the Langmuir desorption isotherms the Riemann invariants Rk, and invariants Hi and the linear plot of z- 1/2 vs the total concentrations of mixtures, Fz, may be used. Strictly speaking, eqs (24) and (39) are valid only for the equilibrium model. Therefore, it is reasonable to compare outlet-concentration-distributions for the equilibrium and non-equilibrium models. For the Langmuir desorption isotherms (33), the system of eqs (40) and (41) were integrated numerically by a computer. The dependencies of r 1/z on the total concentrations of mixture, Cs, for the equilibrium and non-equilibrium model, are shown in Fig. 4. If the parameter ao is reduced, these dependencies approach a straight line, according to eq. (39) for the equilibrium model. Thus, the slope and intercept of the straight line, ($I)1, (1,)1 and also the equilibrium concentration of a weak component, c~21~, can be found from the outlet-concentration-distribution for the total concentration of the two-component mixture, Cz(z), since
The model polymer is a non-ionic polyurethane polymer based on ethylene oxide. The polymer's structure consists of linear water-soluble poly(oxyethylene) (PEO) backbones with an average molecular weight of 51,000 with hydroxyl, dodecyl linear alkyl end-groups. The structure of the model associative polymer is R - O - ( D I - P E O ) 6 - D I - O - R [ R is C16H33 or H, DI is isophorone diisocyanate, and PEO is with a normal molecular weight of 8200]. The polydispersity indices of 1.06 for the polymer with (R := C16H33) end-group and of 1.04 for the polymer with ( R : = H) end-group, respectively, were determined by gel permeation chromatography. We have been investigating the adsorbent TiO2(T315 500 Fisher Scientific Co.). The particle size and shape, surface topography, chemical compositions, and bonding of the titanium dioxide has been characterized using a combination of X-ray photoelectron spectroscopy (ESCA or XPS) and transmission and scanning electron microscopy (TEM and SEM). X-ray photoelectron spectra were collected with a Perkin-Elmer LS-5000 ESCA instrument, using a magnesium anode operating at 400 W, and a 35 eV pass energy for the analyzer. Samples of these particle substrates were compacted onto double-sided adhesive tape. Empirical sensitivity factors were used to convert peak intensities into atomic percentage. For particle size analysis, TEM was used. Samples were secured on an SEM stub with 3 m adhesive transfer tape and examined with a JOEL 6300F field emission (FE)-gun SEM at 1 kV accelerating voltage which is low enough to eliminate the need for sputtered conductive coatings. Particle size distribution was obtained from the photomicrographs using a MOP-3 image analyzer interfaced with a Zenith PC. The results of quantitative analysis of the photomicrographs show that the titanium dioxide particles
4022
L. K. FILIPPOV
have an average diameter of 178 nm and polydispersity index of 1.45. The chemical composition in the unit atom percent at the surface of titanium dioxide (Fisher Scientific Co.) is Ti(18%), 0(55%), C(25%), and O/Ti(2.6%, contaminant oxygen subtracted C20). The only significant component, this composition, is a 1% phosphorous signal from phosphate. Titanium dioxide had > 1% sodium, and some had 1% nitrogen. The carbon concentration is in the normal range found on powdered inorganic materials. Curve fitting the carbon 1s peak into three components (C-C, C-O and C = O) gives approximately C50 stoichiometry. Subtracting the oxygen associated with the contaminant, the remaining oxygen-to-metal ratio shown in the last column was calculated. This ratio is too high by about the same factor, perhaps because the calibration factor was determined on a large, fiat plate. The value of the specific surface per gram of grains of the adsorbent (titanium dioxide) is measured by MONOSORB (Quantochrome Co.) and is calculated by the BET method (Hiemez, 1986). The specific surface area of titanium dioxide is equal to 13.5 m2/g. The solvent was distilled-deionized (DDI) water. Solutions were made fresh daily. Adsorption and desorption dynamics were being investigated in the fixed beds as shown in Fig. 5. The concentration of polymers in DDI water for outlet adsorption and desorption distributions was determined by differential refractometer (DRF) KNAUER. In our case, the fixed bed having a 6-mm interior diameter is enclosed between inlet and outlet tubes, which are located in a thermostat jacket.
5. RESULTS AND DISCUSSION
We developed the above method based on the frontal adsorption dynamics of a mixture in the fixed bed to find adsorption and desorption isotherms in mixtures of water soluble polymers with a polymer structure R O - ( D I - P E O ) 6 - D I - O - R for two endgroups R : = C16H33 and R : = H. Let us consider the adsorption isotherms for mixtures of these polymers. The adsorption isotherms were calculated by using expressions (54)-(61) from the experimental outlet-concentration-distributions for two lengths (L = 0.9 and 6.3 cm) of the fixed bed over a wide range of polymer concentrations (from 0.03 to 3 g/kg). The experimental adsorption isotherms are shown in Figs 6 and 7. The adsorption isotherm for the individual water soluble associative polymer ( R : = C I 6 H 3 3 ) is almost linear for the polymer concentrations 0 ~< c01 ~< 1.1 g/kg and is of the Langmuir type for the polymer concentration 1.2 g/kg ~< col ~< 3 g/kg. For the mixture of polymers of R : = C 1 6 H 3 3 and R : = H , the linear domain of adsorption isotherms reduces. The linear domain of adsorption isotherm of associative polymer is 0~
result may be explained by the arrangement of polymer chains into adsorption layers. The adsorption isotherms of the polymer with the end-group (R:= H) are the Langmuir type for the polymer concentrations 0 ~< c02 ~< 3 g/kg. The maximum amount of adsorption (Fm)l and equilibrium constant (Kp)l for the associative polymer (R := C16H33 ) are almost three times more than the maximum amount of adsorption (Fro)2, and equilibrium constant (Kp)2 for the polymer (R := H) due to the end-group R := C16H33. Let us now consider the desorption isotherms for mixtures of these polymers. The desorption isotherms were calculated using the expressions (39) and (62) from the experimental outlet-concentration-distributions for the two lengths (L = 0.9 and 6.3 cm). The experimental plots of z- 1/2 vs are C~ are shown in Fig. 8. "E ~.5
L ~
Polymers(c,:R=C,sH~~ ~.0
O00
Relative Concentratlon,C1=cl/col
Fig. 6. The adsorption and desorption isotherms for the polymer with the end-group (R := C16H33) for the different concentrations of the polymer with the end-group (R := H). Adsorption (solid curves): (@E~) c2 = 0, (A--A)c2 = 1 g/kg, ([~-E]) c2 = 3 mg/kg; desorption (dashed curves): (0---0) c2 =0, (A A) c2 = l g/kg, (m---m) c2 =3mg/kg; col = Coz = 3 g/kgg (Fm)l = 0.172pmol/m 2, (Kp)1 = 1.16kg/g, (Kp) 2 = 0.4 kg/g.
~E
3.0
Polymers(c2:R=H;c , : R = C , 6 H y
E ~2
2.0.
1.0
Retatlve Ooncentration, C2=cJCo2 Fig. 7. The adsorption and desorption isotherms for the polymer with the end-group (R := H) for differentconcentrations of the polymer with the end-group (R:= C16H33): Adsorption (solid curves): (()-O) cl = 0, (A---A)cl = 1 g/kg, ([2~E]) Cl = 3 mg/kg, desorption (dashed curves): (o----o) cl = 0, (A---A) cl = 1 g/kg, (11 II) c~ = 3 mg/kg; %1 = c02 = 3 g/kg; (Fro)2= 0.05 #mol/mz, (Kv)2 = 0.4 kg/g, (Kp)l = 1.16 kg/g.
Multicomponent adsorption dynamics of polymers It follows from these results that the experimental plots approach a straight line when the length of the fixed bed, L, increases. If L ~ 0% then, according to eqs (52) and (53), the experimental plot of r -x/2 vs C~ can be described by the equilibrium model. For the equilibrium model, according to eq. (39), the straight line AB corresponds to the proportional pattern (5) for the wave index i = 1, and the straight line CD corresponds to the proportional pattern (5) for the wave index i = 2. It follows from the experimental data, shown in Fig. 8, that desorption isotherms of mixtures of polymers with the endgroups ( R : = C 1 6 H 3 3 ) and R : = H are described by Langmuir desorption isotherms. The dimensionless parameters of these isotherms calculated by the above-mentioned method are: ( F ° h = 38.7,
(K°)I = 3.48,
1.25 Polymers(cl:R=C~6Haa; ...... ....
theoretical equilibrium caae ,,,,~, 1~--0.9 . . . . perimentol
l , ~
""
0.75 T-1/2 0.50
0.25
~C --
0.00 0.0
--
.....
Ci'-
~(I.)l(intercept ) -
l, Ji . . . . . . . . . C(~)0.5
Relotive
Totol
J ......... 1.0
i ....... 1.5
2.0
Concentrotion,Ce=Cl+C2
Fig. 8. The experimental and theoretical outlet-concentration-distributions for a two-component mixture of polymers with the end-groups (R:= C 16H33 ) for desorption isotherms (63): (St), = 0.361, (I,)1 = 0.277, (F°m)l= 38.7, (K~)I = 3.48, (r°~)2 = 11.25, and (K°)2 = 1.2.
C°=C °=l,
Riemann invariants Rk, and invariants Hi for the Langmuir adsorption isotherms are found. The linear dependence between concentrations of two-component mixtures for the constant and proportional patterns are shown. We proved that the plot of z-1/2 (z is
c° = c ° = 3 g / k g .
For the equilibrium model the proportional patterns (5) occur for the wave indexes i = 1, 2. For these patterns, according to eqs (5) and (37) we write C o = C o = 1, 5.56,
j' _ d , ¢
(63)
Coa=Co2=0,
R1
c2:R=H)
1.00
(F°)2 = 11.25,
(K°)2 = 1.2
4023
V = 5.68,
Hi = 31.8,
#]o, = 0.986,
[-/2 = 10.06]
I
R 2 = 57.14
i=1:
{ {
C{a) = 0.39,
Ct21)= 0 ,
RI = 13.5,
H1 = 31.8,
Cp ) = 0 . 3 9 ,
Re=57.1,
Ct21)=0, H 2 = 134.7,
V = 2.36, R 2 =
#]1) = 5.72,
#2
=
24.2}
57.14
V=2.36,
#1 =5.72,
#2
=
(64)
24.2}
R1 = 13.5
i = 2: Coa = Co2 = 0, V = 1, #1 = 13.5, #2 = 134.7"( 134.7, H i = 134.7, R1 = 13.5
R2 =
0,
oo > r ~> #t22) = 134.7
Cl(z),
134.7 >~ z ~> #t21) = 24.2
CI(L, t) = Cl(z) =
C 2 ( L , t) -~
C2(z ) =
Cp ) = 0.39, 24.2 >/r >i u]~) = 5.72
CI(z),
5.72 >~ r >1 #~o) = 0.986
l,
0.986 ~> r/> 0
0,
5.72 ~< z
C2(z),
5.72/> z >/0.986
1,
0.986 >~ z >~ 0.
It follows from eqs (64) and (65) that for the experimental desorption isotherms (63) the Riemann invariants R~, and invariants Hi are constant for the wave indexes i = 1, 2, according to eq. (24).
6. C O N C L U S I O N S
For the equilibrium model of multicomponent adsorption dynamics, analytical expressions of the
(65)
the dimensionless time) vs the total concentration of the mixture is linear for the proportional patterns. By using the slope and intercept from these linear dependencies, we developed the method of calculating the parameters of the Langmuir desorption isotherms. The developed method, based on the frontal adsorption dynamics in the fixed bed, may be applied usefully to find adsorption and desorption isotherms
4024
L. K. FILIPPOV
of mixtures over a wide range of polymer concentrations onto particulate surfaces of adsorbents. NOTATION cm D~k
fro(c) (Kp)k L Li MW n q,, t u wi z i
concentration of the ruth component in an external fluid, g/kg matrix of diffusion coefficient (m = k) and mutual diffusion (m # k) in an intergranular space of the porous media of the adsorber, cmZ/s equations of adsorption isotherms for the mth component mixture the equilibrium constant for the kth component mixture, kg/g length of the fixed bed, cm width of the mass transfer zone, cm molecular weight of polymer, g/mol number of mixture components concentration of the ruth component in an adsorbed state, g/kg time, s linear velocity, cm/s wave velocity, cm/s distance from the bed entrance, cm wave index
Greek letters matrix of kinetic coefficients of mass exchange 1/s (F,.)k the maximum amount of adsorption for the kth component mixture,/~mol/m 2 tr porosity
[Jmk
REFERENCES
Arfken, G., 1970, Mathematical Methodsjbr Physicists. Academic Press, New York.
Aris, R. and Amundson, N. R., 1973, Mathematical Methods in Chemical Engineering. Prentice Hall, Englewood Cliffs, NJ. Filippov, L. K., 1989, Theoretical basis of separation processes and dynamics of adsorption of multicomponent mixtures. Chem. Engng Sci. 44, 575-582. Filippov, L. K., 1992, Coherent and incoherent frontal patterns for multicomponent dynamics of adsorption. I. Adsorption dynamics for convex mixture isotherms of adsorption. Chem. Engng Sci. 47, 1199-1210. Fleer, G. J. and J. Lyklema, J., 1983, Adsorption of polymers, in Adsorption from Solution at the Solid/Liquid Interface (Edited by G. D. Parfitt and C. H. Rochester). Academic Press, New York. Golshan-Shirazi, S., Gholdbane, S. and Guiochon, G., 1988a, Comparison between experimental and theoretical band profile in nonlinear liquid chromatography with a pure mobile phase. Anal. Chem. 60, 2630-2634. Golshan-Shirazi, S. and Guiochon, G., 1988b, Comparison between experimental and theoretical band profile in nonlinear liquid chromatography with a binary mobile phase. Anal. Chem. 60, 2634 2641. Guiochon, G., Golshan-Shirazi, S. and Jaulmes, A., 1988, Computer simulation of the propagation of a large-concentration band in liquid chromatography. Anal. Chem. 60, 1856-- 1866. Helferrich, F. and Klein, G., 1970, Multicomponent Chromatography. Theory of Interference. Marcel Dekker, New York. Hiemez, P. C., 1986, Principles of Colloid and Surfaces Chemistry. Marcel Dekker, New York. Korn, G. A. and Korn, T. M., 1968, Mathematics Handbook .for Scientists and Engineers. McGraw-Hill, New York. Lax, P., 1957, Hyperbolic systems of conservation laws. Comm. Pure Appl. Math. 10, 537-566. Lax, P., 1971, Shock Waves and Entropy in Contribution to Nonlinear Function Analysis. University of Wisconsin Press, Madison, WI. Lipatov, Yu. S. and Sergeeva, L. M., 1974, Adsorption of Polymers. Wiley, New York. Rhree, H. K., Aris, R. and Amundson, N. R., 1986, First Order Partial Differential Equations. Prentice-Hall, Englewood Cliffs, NJ. Rozdhdestensky, B. L. and Yanenko, N. N., 1986, Systems of Quasi-Linear Equations. Nauka, Moscow.