Kinetics of Adsorption - American Chemical Society

2 Kinetics of Adsorption J. M. SMITH University of California, Davis, Calif.

The

effects of physical

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adsorption models into

on

are presented

account

interpreting

transport

porous in

solids by which

designing

observed

often of major importance for

liquid

systems.

-particle transfer

The

processes

are

in adsorption diffusivities

correctly

kinetics, which

in are

particularly

describe

is commonly

the mass transfer

or

processes

even for gaseous

More than a single rate coefficient represent

taken

equipment

Intraparticle

are complex,

overall

Quantitative

these effects can be

adsorption

data.

on the

discussed.

intra-

adsorbates. necessary

in the interior

of

to the

adsorbent.

A d s o r p t i o n is n o r m a l l y t h o u g h t of as the process b y w h i c h a m o l e c u l e or a t o m i n a fluid is a t t a c h e d to a s o l i d surface, a n d it is i m p l i e d that the m o l e c u l e ( o r a t o m ) is i n the same l o c a t i o n as the site. K i n e t i c s of s u c h processes is c o n c e r n e d w i t h force fields b e t w e e n sites a n d molecules a n d forms a n i m p o r t a n t area of surface c h e m i s t r y . H o w e v e r , i n this p a p e r b o t h a w i d e r a n d m o r e r e s t r i c t e d v i e w w i l l b e t a k e n of a d s o r p t i o n k i n e t i c s i n that emphasis w i l l b e p u t o n the so-called p h y s i c a l processes t h a t m u s t a c c o m p a n y a d s o r p t i o n , i f the o v e r a l l process is to c o n t i n u e . I n p a r t i c u l a r the k i n d of k i n e t i c s discussed w i l l b e that necessary to e x p l a i n the p e r f o r m a n c e of, or to d e s i g n a n apparatus for, s e p a r a t i n g or r e m o v i n g c o m ponents i n a fluid stream. T h e use of the t e r m p h y s i c a l to d i s t i n g u i s h other steps f r o m the process at the site, w h i l e c o m m o n , is unfortunate. T h e so-called p h y s i c a l processes of diffusion m a y i n v o l v e m e c h a n i s m s best d e s c r i b e d b y c h e m i c a l means. process.

F o r example, surface diffusion o n the p o r e w a l l s is a n a c t i v a t e d O n t h e other h a n d , the a d s o r p t i o n itself at a surface site m a y

i n v o l v e s u c h w e a k forces that a p h y s i c a l rather t h a n c h e m i c a l b o n d best characterizes the mechanics. W h a t is sought is a d i s t i n c t i o n b e t w e e n the a d s o r p t i o n step at a fixed l o c a t i o n a n d processes b y w h i c h the adsorbent is transferred to the site. T h e d i s t i n g u i s h i n g feature is w h e t h e r the process i n v o l v e s the m o v e m e n t of t h e adsorbate.

H e n c e , a m o r e correct c h a r a c -

8

In Adsorption From Aqueous Solution; Weber, W., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1968.

2.

S M I T H

Kinetics

of

9

Adsorption

t e r i z a t i o n w o u l d b e space or p o i n t processes. A p p l i c a t i o n s of a d s o r p t i o n r e q u i r e transfer of the adsorbate b e t w e e n a b u l k fluid a n d a site o n the s o l i d surface. H e n c e , the space processes, of necessity, do occur. A u s e f u l g e o m e t r i c a r r a n g e m e n t for a d s o r p t i o n is a n assembly of s o l i d p a r t i c l e s , p e r m e a b l e to the adsorbate, w i t h the a d s o r b a t e - c o n t a i n i n g t h r o u g h the i n t e r p a r t i c l e v o i d s p a c e — t h e c o n v e n t i o n a l

fluid

fixed-bed

flowing

absorber.

B a s i c features of b o t h types of processes h a v e b e e n t a k e n i n t o account i n u s i n g this f o r m of a d s o r p t i o n a p p a r a t u s .

R e c o g n i t i o n that the rate

depends u p o n the n u m b e r of sites has necessitated p e r m e a b l e particles so that i n t e r i o r sites are p o t e n t i a l l y usable. R e a l i z a t i o n t h a t the transfer distance m u s t not b e excessive f r o m b u l k fluid to site, or i n t e r i o r sites Downloaded by TUFTS UNIV on December 2, 2014 | http://pubs.acs.org Publication Date: June 1, 1968 | doi: 10.1021/ba-1968-0079.ch002

w i l l b e valueless, l e a d to the use of s m a l l particles. A s i n t i m a t e d , the i m p o r t a n t p o i n t is not w h e t h e r space processes are i n v o l v e d , b u t w h e t h e r t h e y affect the rate of a d s o r p t i o n . T h e results of a d s o r p t i o n are m e a s u r e d i n terms of concentrations of the adsorbate i n the b u l k

fluid;

tions o n the s o l i d surfaces are not u s e f u l i n the d e s i g n or e v a l u a t i o n tasks.

S u c h concentrations

concentra performance

are u s u a l l y u n k n o w n .

W h a t is

n e e d e d are rate equations w h i c h express the k i n e t i c s of a d s o r p t i o n i n terms of properties of b u l k fluid adjacent to the s o l i d p a r t i c l e — s p a c e or g l o b a l k i n e t i c s i n contrast to p o i n t k i n e t i c s , w h i c h express the rate i n terms of properties at the a d s o r p t i o n site. Space processes are significant to the extent that v a r i a b l e s s u c h as fluid v e l o c i t y a n d p a r t i c l e size affect the g l o b a l k i n e t i c s equations. F r o m this p o i n t of v i e w the p r o p e r a p p r o a c h to d e s i g n of a d s o r p t i o n e q u i p m e n t is a two-step

procedure:

A . E s t a b l i s h the p o i n t k i n e t i c s a n d e q u i l i b r i u m i s o t h e r m for the a d s o r p t i o n step at the site, a n d the rate coefficients for the v a r i o u s space processes. T h e s e are t h e t w o types of d a t a r e q u i r e d for d e s i g n . T o some extent estimates m a y b e m a d e for the coefficients for the space processes. B . B y a p p l y i n g the c o n s e r v a t i o n equations (mass, e n e r g y ) a n d the d a t a f r o m A , c a l c u l a t e the c o n c e n t r a t i o n as a f u n c t i o n of t i m e a n d p o s i t i o n i n the a d s o r p t i o n apparatus. It is d e s i r a b l e to list the sequence of space a n d p o i n t steps w h i c h together constitute g l o b a l a d s o r p t i o n .

T h i s is not a n e w c o n c e p t a n d

s u c h descriptions h a v e f r e q u e n t l y b e e n p r e s e n t e d

(1,

14),

particularly

for fluid reactions o n porous catalyst particles. T h e first space process, a x i a l d i s p e r s i o n , is not a p a r t of the. sequence, b u t it does affect

the

observed

Its

k i n e t i c s , a n d is l o g i c a l l y c o n s i d e r e d

significance d e p e n d s E /(2R)v. x

as a space process.

u p o n the r e c i p r o c a l of the a x i a l P e c l e t

number,

T h e s e q u e n t i a l steps a r e :

( 1 ) E x t e r n a l transfer: transfer of the absorbate f r o m b u l k f l u i d to outer surface of p a r t i c l e b y m o l e c u l a r a n d c o n v e c t i v e diffusion.


10

ADSORPTION

F R O M

AQUEOUS

SOLUTION

( 2 ) I n t e r n a l transfer: transfer of adsorbate f r o m p a r t i c l e surface to i n t e r i o r site; b y d i f f u s i o n i n the v o i d space of the pores, b y surface m i g r a t i o n o n the p o r e surface, or b y v o l u m e diffusion, for e x a m p l e , i n the holes i n the c h e m i c a l structure of the s o l i d phase. ( 3 ) A d s o r p t i o n o n the i n t e r i o r site. F o r aqueous solutions s u c h a d s o r p t i o n is u s u a l l y classified b y either i o n i c t y p e c h e m i c a l steps, or b y a r e l a t i v e l y loose a d s o r p t i o n w i t h l o w heat of a d s o r p t i o n , that is b y p h y s i c a l a d s o r p t i o n . I n either case the process has a h i g h r a t e constant so that this step u s u a l l y does not influence the g l o b a l k i n e t i c s .


( 4 ) I n r e p l a c e m e n t or r e a c t i o n operations w h e r e a p r o d u c t is p r o d u c e d , the same steps as ( 2 ) a n d ( 3 ) o c c u r for t h e p r o d u c t i n the reverse direction. W h i l e the treatment so f a r has not c o n s i d e r e d the effect of t e m p e r a t u r e o n g l o b a l k i n e t i c s , heats of a d s o r p t i o n or c h e m i c a l r e a c t i o n c a n b e significant.

T e m p e r a t u r e gradients are g e n e r a l l y less for l i q u i d systems

—e.g., aqueous s o l u t i o n s — t h a n for gaseous since the heat c a p a c i t y

(c p) v

of the l i q u i d s t r e a m is a n o r d e r of m a g n i t u d e h i g h e r t h a n t h a t for gases. H o w e v e r , to present a m o r e c o m p l e t e a p p r o a c h to the p r o b l e m , the steps b y w h i c h the h e a t of a d s o r p t i o n is t r a n s f e r r e d to the f l o w i n g stream are: 1.

H e a t release c a u s e d b y a d s o r p t i o n at t h e i n t e r i o r site.

2.

I n t e r n a l transfer: transfer of energy to t h e outer surface of the s o l i d p a r t i c l e . T h i s is c o m m o n l y t r e a t e d as t h o u g h the p a r t i c l e w a s h o m o g e n e o u s w i t h a single effective t h e r m a l c o n d u c t i v i t y . 3. E x t e r n a l transfer: transfer of energy f r o m t h e surface of the p a r t i c l e i n t o the f l u i d stream. T h e properties of flowing fluids are s u c h that t h e resistance to heat transfer c a n b e l a r g e r t h a n t h a t for mass transfer, so t h a t a n e g l i g i b l e c o n c e n t r a t i o n difference m a y exist b e t w e e n b u l k fluid a n d p a r t i c l e surface a n d yet the c o r r e s p o n d i n g t e m p e r a t u r e difference w i l l b e significant. 4. A x i a l d i s p e r s i o n of energy a l o n g the fluid stream. T h e i m p o r t a n c e of this process is d e t e r m i n e d b y t h e r e c i p r o c a l of the P e c l e t n u m b e r for heat transfer, k /(2R) C . x

Conservation

P

t

Equations

If the processes just d e s c r i b e d are a s s u m e d to c h a r a c t e r i z e t h e t r a n s fer of mass a n d energy i n a fixed-bed adsorber, t h e c o n s e r v a t i o n p r i n c i p l e s m a y b e a p p l i e d to t h e m to describe the t e m p e r a t u r e a n d c o n c e n t r a t i o n as a f u n c t i o n of t i m e a n d p o s i t i o n . P r e s e n t i n g the equations for a

fixed-bed

geometry has the a d v a n t a g e of i n c l u d i n g also equations, as s p e c i a l cases, for transient a d s o r p t i o n i n single particles or groups of particles i n b a t c h systems. I n aqueous systems, as i n others, some of the steps i n t h e transfer processes h a v e r e l a t i v e l y h i g h rate coefficients.

S u c h steps m a y b e t r e a t e d

as o c c u r r i n g at near e q u i l i b r i u m so t h a t the d r i v i n g force, either a c o n -


2.

Kinetics

S M I T H

of

11

Adsorption

c e n t r a t i o n or t e m p e r a t u r e difference, approaches zero. T h e s i m p l i f i c a t i o n i n the treatment u n d e r those c o n d i t i o n s c a n b e r e a d i l y seen f r o m the equations. L e t us suppose that a single species is a d s o r b e d f r o m a n o t h e r w i s e i n e r t stream flowing t h r o u g h a fixed b e d of p o r o u s p a r t i c l e s ( e x t e r n a l p o r o s i t y == a, i n t e r n a l p o r o s i t y =

/?).

T h e internal diffusivity D

is

c

b a s e d u p o n the t o t a l area p e r p e n d i c u l a r to the r a d i a l d i r e c t i o n i n the p a r t i c l e so that for non-porous particles the o n l y c h a n g e w o u l d b e /? =

1.

T h e equations are w r i t t e n for s p h e r i c a l particles. A n u m b e r of a s s u m p tions are m a d e , s u c h as diffusion o n l y i n the r a d i a l d i r e c t i o n , b u t these are e v i d e n t f r o m the f o r m of the equations a n d n e e d not b e

described

i n d e t a i l . It s h o u l d b e m e n t i o n e d that a l i n e a r f o r m is a s s u m e d for the Downloaded by TUFTS UNIV on December 2, 2014 | http://pubs.acs.org Publication Date: June 1, 1968 | doi: 10.1021/ba-1968-0079.ch002

point adsorption kinetics ( E q u a t i o n 4 ) .

D o i n g this results i n l i n e a r

equations i n c o n c e n t r a t i o n w h i c h m a y b e s o l v e d b y s t a n d a r d m a t h e m a t i c a l methods.

E q u a t i o n 4 is a s i m p l i f i e d f o r m of the L a n g m u i r ex

pression, a p p l i c a b l e for gaseous systems w h e n the extent of a d s o r p t i o n is l o w . F o r i o n exchange i n aqueous systems E q u a t i o n 4 w o u l d c o r r e s p o n d to first order, r e v e r s i b l e k i n e t i c s at the a d s o r p t i o n site. C o n s e r v a t i o n of M a s s . I f c represents the c o n c e n t r a t i o n of a d sorbent i n t h e b u l k fluid stream, c the i n t r a p a r t i c l e c o n c e n t r a t i o n , a n d Cads the c o n c e n t r a t i o n of a d s o r b e d c o m p o n e n t , mass c o n s e r v a t i o n i n the fluid phase r e q u i r e s A

a

SZ

&Z

2

$t

V

and w i t h i n the particle

P

8r

2

2 8Cj _ 7 ~Sr

8Ci _ ~&t

pp S c ~(3 ' 8*

a d s

(2)

T h e last t e r m i n E q u a t i o n 1 represents the g l o b a l k i n e t i c s , expressed as the rate of a d s o r p t i o n p e r u n i t v o l u m e of b e d . i n terms of concentrations i n the

fluid

It cannot b e w r i t t e n

phase—the

equations m u s t

be

s o l v e d to d o t h i s — b u t it m a y b e expressed q u a n t i t a t i v e l y i n terms of Ci b y w r i t i n g a n expression for the diffusion r a t e : (3) T h e p o i n t k i n e t i c s , i n terms of a first order, r e v e r s i b l e a d s o r p t i o n rate is 8t

(Ci -

c

a d B

/K)

(4)

If other k i n e t i c s represent the p o i n t process, the p r o b l e m is s t i l l defined as l o n g as R

v

c a n b e w r i t t e n i n terms of concentrations at the a d s o r p t i o n

site, a l t h o u g h the s o l u t i o n w i l l b e m o r e difficult.


12

ADSORPTION

F R O M

AQUEOUS

SOLUTION

T h e e x t e r n a l a n d i n t e r n a l concentrations are r e l a t e d b y Step 1 o f the sequence:

M S ) , - . - * - "

(

5

)

R e t u r n n o w t o the first p a r t of the two-step p r o c e d u r e f o r c h a r a c t e r i z i n g a n a d s o r p t i o n c o l u m n . O u r m o d e l represented b y E q u a t i o n s 1, 2, 3, 4, a n d 5 shows that t h e f o l l o w i n g space a n d p o i n t rate

coefficients

define the p r o b l e m : Space coefficients: E , k , D x


Point coefficient:

t

c

fc

adg

R e l i a b l e values f o r a l l these coefficients w o u l d b e r e q u i r e d f o r situations w h e r e a l l the steps are significant. C o n s e r v a t i o n o f E n e r g y . T h e c o n s e r v a t i o n e q u a t i o n f o r energy i n the f l u i d phase is s i m i l a r to E q u a t i o n 1:

w h e r e Q is t h e rate o f heat transfer f r o m p a r t i c l e to fluid phase p e r u n i t v

v o l u m e ; t h a t is

T h e energy b a l a n c e w i t h i n the p a r t i c l e i n c l u d e s the heat of a d s o r p t i o n , Q

p

p e r u n i t v o l u m e o f p a r t i c l e ; t h a t is,

where Q= v

(-AH)flp= (-AH)fc

a d s / 0 p

(

- c

C i

a d g

/K)

(9)

T h e e q u a t i o n r e l a t i n g T i a n d T is analogous t o E q u a t i o n 5, o r (10)

••htiT-TJ T e m p e r a t u r e has a strong influence o n k

&da

a n d sometimes o n D . c

I f the specific step d e s c r i b e d b y these t w o coefficients has a significant effect o n the k i n e t i c s o f a d s o r p t i o n , t h e n energy c o n s e r v a t i o n equations m a y h a v e t o b e i n c l u d e d i n the analysis t o e s t a b l i s h the t e m p e r a t u r e t o every point i n the bed.

I n these circumstances t h e a d d i t i o n a l space

coefficients, k , h , a n d k m u s t b e a d d e d to the p r e v i o u s list. x

t

coefficient analogous to k

e

&as

A point

is not i n c l u d e d , because it has b e e n a s s u m e d

t h a t the t e m p e r a t u r e b e h a v i o r of the p a r t i c l e c a n b e f a i t h f u l l y r e p r e s e n t e d b y assuming a homogenous material.

W i t h this s i m p l i f i c a t i o n i t is n o t


2.

S M I T H

Kinetics

of

13

Adsorption

necessary to i n t r o d u c e different temperatures w i t h i n the p a r t i c l e for s o l i d a n d p o r o u s regions. T o i l l u s t r a t e the n a t u r e of t e m p e r a t u r e effects for aqueous systems, c o n s i d e r a n i o n exchange b e d .

If the external d i f f u s i o n step controls the

g l o b a l k i n e t i c s , for e x a m p l e as d e s c r i b e d b y S c h l o g l a n d H e l f f e r i c h ( 1 2 ) , s m a l l t e m p e r a t u r e differences i n the b e d are not l i k e l y to b e significant, since n e i t h e r fc

ads

or D

c

are necessary to describe the g l o b a l k i n e t i c s . O n

the other h a n d i f i n t e r n a l diffusion has a n effect o n the rate of a d s o r p t i o n , D

c

is i m p o r t a n t a n d the k i n e t i c s w i l l c h a n g e w i t h t e m p e r a t u r e l e v e l .

H o w e v e r , the necessity of i n c l u d i n g energy c o n s e r v a t i o n equations

will


d e p e n d i n a d d i t i o n u p o n the heat c a p a c i t y characteristics of the system. A s m e n t i o n e d , the h i g h heat c a p a c i t y of l i q u i d streams makes i t m u c h less l i k e l y that i m p o r t a n t t e m p e r a t u r e v a r i a t i o n s , for e x a m p l e f l u i d a n d s o l i d , w i l l d e v e l o p i n aqueous systems.

between

H e n c e , for aqueous

a d s o r p t i o n it appears that a n i s o t h e r m a l treatment u s i n g o n l y the mass c o n s e r v a t i o n equations w i l l u s u a l l y b e satisfactory.

F u r t h e r analysis of

space a n d p o i n t processes w i l l b e b a s e d u p o n i s o t h e r m a l o p e r a t i o n .

Importance of Space Processes A recent analysis (4)

of k i n e t i c d a t a for the a d s o r p t i o n of h y d r o

c a r b o n gases o n s i l i c a g e l illustrates the r e l a t i v e influence of the three s e q u e n t i a l steps o u t l i n e d earlier as w e l l as the significance of a x i a l d i s p e r s i o n . T h e d a t a w e r e o b t a i n e d at 5 0 ° C . for a h i g h surface area g e l (832 sq. m e t e r s / g r a m ) , a n d are of a c h r o m a t o g r a p h i c t y p e ; t h a t is, c o n c e n t r a t i o n — t i m e s curves w e r e o b s e r v e d i n the effluent gas f r o m a n a d s o r p t i o n b e d i n response to a square w a v e i n p u t to the b e d .

R a t e coeffi

cients w e r e e v a l u a t e d b y c o m p a r i n g the c h r o m a t o g r a p h i c c u r v e w i t h the s o l u t i o n of E q u a t i o n s 1, 2, 3, 4, a n d 5. T h e i m p o r t a n c e of a x i a l d i s p e r s i o n d e p e n d s u p o n the p a r t i c l e d i a m e t e r a n d v e l o c i t y of the f l u i d stream, t h a t is, the v a r i a b l e s i n the P e c l e t n u m b e r . I n the c h r o m a t o g r a p h i c analysis the effect is best v i e w e d as a p l o t of the s e c o n d m o m e n t of the effluent p e a k vs. the r e c i p r o c a l of the square of the v e l o c i t y .

F o r certain operating

c o n d i t i o n s s u c h graphs are straight lines. A n i l l u s t r a t i o n is F i g u r e 1, i n w h i c h d a t a for t h e r e d u c e d s e c o n d m o m e n t of p r o p a n e are s h o w n for three p a r t i c l e sizes. T h e p a r t of the t o t a l resistance associated w i t h a x i a l d i s p e r s i o n at a n y v e l o c i t y v is e q u a l to the v a l u e of the o r d i n a t e at less its v a l u e at 1/vr =

1/v

2

0. T h e f r a c t i o n of the resistance associated w i t h

a x i a l d i s p e r s i o n is this difference i n ordinates d i v i d e d b y the o r d i n a t e at 1/v . 2

F o r the c o n d i t i o n s of F i g u r e 1, a x i a l d i s p e r s i o n is o b s e r v e d

b e a significant factor for a l l velocities at t h e smallest p a r t i c l e size.

to

How

ever, for the particles of 0.50 m m . r a d i u s at h i g h velocities, the effect is s m a l l , b u t h a r d l y n e g l i g i b l e at the highest velocities s t u d i e d .


14

ADSORPTION

F R O M

AQUEOUS

SOLUTION


60

l0 /v2,(min/cm) 6

Figure

1.

2

Chromatography of propane (50°C.) [^-(t\ /12y\/2(xMonl/v*

and dependence of

A

O R — 0.11 mm. A R = 0.39 mm. • R — 0.50 mm. T h e three s e q u e n t i a l steps are a d d i t i v e so that t h e i r i m p o r t a n c e c a n b e s h o w n as percentage figures. T a b l e I shows results at 5 0 ° C . for three hydrocarbons.

T h e effect o f a x i a l d i s p e r s i o n has b e e n e l i m i n a t e d f r o m

these d a t a . A s p a r t i c l e size increases the effect o f i n c r e a s i n g diffusion p a t h l e n g t h is c l e a r l y seen.

E x t e r n a l diffusion resistance also

increases

w i t h p a r t i c l e d i a m e t e r b u t is not a v e r y i m p o r t a n t factor i n a n y case. C o m p a r i s o n o f the results for different h y d r o c a r b o n s shows t h e effect o f fcads-

T h e l o w e r the m o l e c u l a r w e i g h t the slower the rate o f p h y s i c a l

adsorption.

H e n c e , f o r ethane

a d s o r p t i o n step accounts

a n d small particle diameter t h e local

f o r n e a r l y h a l f of t h e t o t a l resistance. F o r

s m a l l e r p a r t i c l e sizes t h e t r e n d continues u n t i l t h e p o i n t process c a n control

global

kinetics.

T h i s is i l l u s t r a t e d later w i t h

some d a t a o n

c h e m i s o r p t i o n , g e n e r a l l y a slower process t h a n p h y s i c a l a d s o r p t i o n . F o r aqueous solutions i n v o l v i n g r a p i d a d s o r p t i o n o r r e a c t i o n a t the site, f o r example, i o n exchange processes, g l o b a l kinetics are n o r m a l l y c o n t r o l l e d b y external a n d i n t e r n a l diffusion.

T h i s s i t u a t i o n corresponds

closely t o

b u t a n e a d s o r p t i o n o n the larger p a r t i c l e size i n T a b l e I . T h e shape o f b r e a k t h r o u g h curves p r o v i d e s a q u a l i t a t i v e d e s c r i p t i o n of the c o n t r o l l i n g processes i n a n a d s o r p t i o n b e d .

E q u a t i o n s 1, 2, 3, 4,

a n d 5 w i t h proper boundary a n d initial conditions can b e compared w i t h


2.

S M I T H

Kinetics

15

of Adsorption

m e a s u r e d b r e a k t h r o u g h curves to evaluate q u a n t i t a t i v e l y the resistances chosen to describe the process.

T h i s is c o n v e n i e n t l y i l l u s t r a t e d w i t h d a t a

(9, J O ) for the a d s o r p t i o n of n i t r o g e n o n 9 6 % s i l i c a glass ( V y c o r ) at l i q u i d nitrogen temperature.

F o r this p h y s i c a l a d s o r p t i o n , it is expected t h a t

either e x t e r n a l or i n t e r n a l d i f f u s i o n w o u l d c o n t r o l the k i n e t i c s . F i g u r e 2 shows o b s e r v e d b r e a k t h r o u g h curves for t w o p a r t i c l e sizes a n d also p r e d i c t e d curves b a s e d u p o n e x t e r n a l diffusion c o n t r o l l i n g the date.

The

p r e d i c t e d curves d o not s h o w a l a r g e e n o u g h effect of p a r t i c l e d i a m e t e r to fit the o b s e r v e d d a t a .

F i g u r e 3 shows the same t y p e of d a t a for f o u r

p a r t i c l e sizes a n d p r e d i c t e d curves b a s e d u p o n i n t e r n a l d i f f u s i o n c o n t r o l l i n g t h e k i n e t i c s ( w i t h a fixed v a l u e for D ) . c

T h i s m o d e l fits the d a t a


rather w e l l . Table I.

Contributions to the Second Central Adsorption on Silica Gel at 5 0 ° C . Ethane

Moment—

Propane

Butane R = 0.11mm

R = 0.50mm

R = 0.11mm

R = 0.50mm

R = 0.11mm

44.5

4.0

37.7

3.0

16.0

1.0

54.8

95.0

61.5

95.9

81.4

95.1

1.1

2.6

3.1

R = 0.50mm

A Adsorption Resistance, % B Intraparticle Diffusion Resistance, % C

External Diffusion Resistance, %

1.0

F o r c h e m i s o r p t i o n the p o i n t a d s o r p t i o n process is m o r e l i k e l y to b e significant. T h e p e r t i n e n t v a r i a b l e to investigate i n this case is t e m p e r a ture level.

I n F i g u r e 4 b r e a k t h r o u g h curves are i l l u s t r a t e d for e t h y l

alcohol on silica gel ( I I ) .

T h e effect of t e m p e r a t u r e is v e r y strong. F o r

the smallest size (0.0099 c m . r a d i u s ) at 1 5 5 ° C . the c u r v e is n e a r l y v e r t i c a l a n d corresponds to a v e r y r a p i d rate. T h e fact that the curves for the same t e m p e r a t u r e b u t different sizes are not the same suggests

that

i n t e r n a l or external d i f f u s i o n resistance also is i m p o r t a n t . A m a t h e m a t i c a l analysis of the curves gives e v i d e n c e

that a m o d e l b a s e d u p o n

point

a d s o r p t i o n a n d i n t e r n a l diffusion satisfactorily represents the d a t a . F i g u r e 5 shows that at 9 0 ° C ,

the b r e a k t h r o u g h curves for the t w o

smallest

particles are essentially the same, i n d i c a t i n g that p o i n t a d s o r p t i o n controls the o v e r a l l k i n e t i c s . I n this s i t u a t i o n the rate is p r o p o r t i o n a l to the t o t a l


16

A D S O R P T I O N

F R O M

A Q U E O U S

S O L U T I O N

n u m b e r o f sites o r t o the mass o f adsorbent, regardless o f p a r t i c l e size. A t 1 5 5 ° C . ( F i g u r e 6 ) the rate constant for the p o i n t process has b e c o m e so m u c h larger that the b r e a k t h r o u g h curves v a r y w i t h p a r t i c l e size f o r the w h o l e r a n g e o f p a r t i c l e s . 1.0

— — 3 ^ 0

a=0.00 / *

\


N

= 0.0793 cm a = 0.0463 a = 0.0151

0.6 0.5

0.2

//pi

/A

—

Infinite Rate Line, t = !23 sec. s

fix

50

Figure 2.

100

Breakthrough

150 TIME, t, sec.

curves for adsorption controlling

200

250

300

of nitrogen—external

diffusion

Calculated from Equation 32 — O — Experimental data I n aqueous systems i n v o l v i n g electrolytes, the r a p i d p o i n t process h a d l e d t o e q u i l i b r i u m treatments ( 1 7 ) , e v e n for space steps.

Consider

a b l e i n s i g h t into the d e s i g n o f a d s o r p t i o n beds c a n b e g a i n e d b y t r e a t i n g b o t h external a n d i n t e r n a l diffusion steps, as w e l l as p o i n t a d s o r p t i o n , as o c c u r r i n g a t e q u i l i b r i u m . T h i s p r o c e d u r e has b e e n p a r t i c u l a r l y h e l p f u l for m u l t i c o m p o n e n t systems (2, 5, 9, 15, 18). W i t h respect t o k i n e t i c s it has l o n g b e e n r e c o g n i z e d

(1, 3, 6, 7, 16) that d i f f u s i o n steps c o n t r o l

the kinetics o f i o n exchange processes. O f p a r t i c u l a r interest is the effect of the electric g r a d i e n t , i n d u c e d b y the c o n c e n t r a t i o n g r a d i e n t , o n i n t r a p a r t i c l e mass transfer. H e l f f e r i c h a n d Plesset (6,7) h a v e u s e d the N e r n s t P l a n c k e q u a t i o n f o r the flux, w h i c h i n c l u d e s the c o n t r i b u t i o n a t t r i b u t e d to electric gradient, i n c o n s e r v a t i o n equations for s p h e r i c a l particles a n d for slab geometry. process,

A s s u m i n g this i n t e r n a l d i f f u s i o n step controls t h e

t h e n o n - l i n e a r equations

were

integrated numerically. T h e


2.

S M I T H

Kinetics

of

Adsorption

17

results s h o w that the electric effect depends u p o n the r a t i o of the d i f f u sivities o f the c o u n t e r - m o v i n g ions D / D , a n d t o a s l i g h t extent u p o n A

B

the r a t i o o f electric charges of the t w o ions. N o r m a l l y i n t r a p a r t i c l e d i f f u sion has b e e n t r e a t e d i n s u c h systems u s i n g F i c k ' s l a w w i t h a single concentration-independent

d i f f u s i v i t y f o r a b i n a r y system.

N e r n s t - P l a n c k flux r e l a t i o n s h i p , t h e rate of exchange increase w h e n the faster i o n ( f o r e x a m p l e H i n the exchange r e s i n i n i t i a l l y .

+

i n the H - L i +

W h e n the r a t i o D / D A

Using the

is p r e d i c t e d t o =

B

system) is

+

10 the t i m e

r e q u i r e d for 9 0 % exchange differed b y a factor of three f r o m that b a s e d u p o n the u s u a l F i c k ' s l a w result.


1.0

h / / 11 0.8

0.6 a = 0.0793 cm 0.5

0 = 0.0463^ a = 0.0151 ^>

0.4

//

Vf

0.2

V

"a = 0.0099 ^ a = 0. or Infinite Rate Line

1

TIME, t, sec.

Figure 3.

Breakthrough

curves for adsorption controlling

of nitrogen—internal

diffusion

o -o- o o Experimental data !— Computed results, Equation 25

N

Helfferich

(4)

tested t h e essential features of t h e N e r n s t - P l a n c k

c o n c e p t b y e x p e r i m e n t a l measurements w i t h the H - N a +

+

ion pair.

Diffu-

sivities of e a c h i o n i n the exchanger w e r e e v a l u a t e d f r o m i n d e p e n d e n t c o n d u c t i v i t y measurements. T h e s e diffusivities w e r e t h e n u s e d t o p r e d i c t exchange

rates a n d c o n c e n t r a t i o n profiles w i t h i n t h e r e s i n u s i n g t h e

p r e v i o u s l y d e v e l o p e d t h e o r y ( 6 , 7 ) . T h e agreement w a s g o o d , a n d i n p a r t i c u l a r the e x p e r i m e n t a l d a t a c o n f i r m e d that the rate of exchange w a s larger w h e n the i o n of greatest m o b i l i t y ( H ) w a s o r i g i n a l l y i n the e x +

changer a n d N a w a s i n s o l u t i o n . +


18

ADSORPTION

1.0

6=406 e =8.89 R=0.054b (cm)!/ i/

/ 1

1 4 I3I°C Jr /i I55°C/ /1 1 / i / 1 / i / 1 / i / 1 / 1 R=Q009J/ i / (cm) 1/ If s I55°C p I3I°C 0.5


Figure

10

/

4. Breakthrough

!

1 !

i y '90.5° C i A

II0°C

i

SOLUTION

-II0°C

f

15

AQUEOUS

0=37.8 1 1 1

6=188

0.5

i

F R O M

20 25 TIME,0(min.)

30

curves for adsorption

35

!

40

45

50

of ethyl alcohol on silica gel

1.0

0.90

,30

1.00

CORRECTED TIME,0

O

Figure 5.

Converted

breakthrough curves for adsorption silica gel at 90.5°C.

of ethyl alcohol on

Expt. data: particle radii o 0.0793 cm. jo' 0.0540 cm. 6 0.0311 cm. \x 0.0151 cm. -o- 0.0099 cm. If F i c k ' s e q u a t i o n is u s e d a n d t h e diffusivities o f t h e i o n p a i r D B A

a n d D A are a s s u m e d to b e e q u a l , as for b i n a r y gas diffusivities, p r e d i c t e d B

exchange rates are t h e same, regardless o f d i r e c t i o n o f diffusion. T h u s , the n e w e r theory t a k i n g i n t o a c c o u n t t h e electric p o t e n t i a l is c l e a r l y a n i m p r o v e m e n t over t h e F i c k s l a w a p p r o a c h .

However, if an empirical

v i e w o f this s i m p l e t h e o r y is u s e d , a l l o w i n g e a c h d i f f u s i v i t y t o assume a


2.

Kinetics

S M I T H

of

Adsorption

19

v a l u e d e t e r m i n e d b y t h e d a t a , exchange rates c a n b e a d e q u a t e l y r e p r e sented. H e r i n g a n d Bliss ( 8 ) s t u d i e d t h e exchange rates o f six i o n p a i r s o n D o w e x 5 0 W a n d f o u n d that t h e f r a c t i o n a l c o n v e r s i o n vs. t i m e results c o u l d b e r e p r e s e n t e d w i t h e q u a l a c c u r a c y b y either theory.

However,

the e m p i r i c a l l y e s t a b l i s h e d values f o r t h e diffusivities w e r e a n o r d e r of m a g n i t u d e different f o r t h e c o u n t e r d i f f u s i n g ions a n d p r o b a b l y h a d little p h y s i c a l m e a n i n g . h = 0.00 4.76 0 9.5lO

X =6090

23.8k

0.8


v

s

o

Nv

J / /< -i-T""""" , I P

N

IO

\\I;/J&e

o 0.6 0.5

^47.5 = h 95.0

0.4

/y/\\ # * 111 v

0.2

t i l l

/ /II 1 / '' / / 1I I \ 1 / /

0.60

0.70

0.80

0.90

Figure 6. Converted

I.I 0

1.00

CORRECTED TIME, 8

1.20

1.30

0

breakthrough curves for adsorption silica gel at 155°C.

of ethyl alcohol on

Expt. data: particle radii o 0.0793 cm. ,o' 0.0540 cm. (J) 0.0311 cm. X 0.0151 cm. -o- 0.0099 cm. Methods of Evaluating

Space and Point Resistances

A c o m m o n m e t h o d of assessing t h e r e l a t i v e i m p o r t a n c e o f i n t e r n a l diffusion a n d p o i n t a d s o r p t i o n resistances is t o measure, as a f u n c t i o n o f t i m e , t h e u p t a k e o f adsorbent f r o m a s o l u t i o n c o n t a i n i n g s o l i d particles. B a t c h d a t a o f this t y p e t a k e n at different temperatures a n d p a r t i c l e sizes c a n u s u a l l y b e a n a l y z e d so as t o e s t a b l i s h t h e i m p o r t a n c e o f i n t e r n a l resistances. H o w e v e r , some types o f d i f f u s i o n h a v e r e l a t i v e l y h i g h a c t i v a t i o n energies so that t h e separation is c o m p l e x . A l s o , i n such methods care m u s t b e t a k e n t o ensure r a p i d m o t i o n o f t h e fluid w i t h respect t o t h e particles, f o r e x a m p l e b y s t i r r i n g , i n o r d e r t o e l i m i n a t e e x t e r n a l d i f f u s i o n


20

ADSORPTION

F R O M

AQUEOUS

SOLUTION

resistance. T h i s is p a r t i c u l a r l y i m p o r t a n t i f the results of b a t c h m e a s u r e ments are to b e u s e d i n d e s i g n i n g a

fixed-bed

adsorber, because e x t e r n a l

d i f f u s i o n resistance is difficult to m a i n t a i n the same i n b a t c h a n d

flow

systems. T w o relatively n e w methods have been developed d i r e c t l y the p e r f o r m a n c e of a d s o r p t i o n beds.

w h i c h analyze

B o t h are b a s e d u p o n c o m

p a r i n g solutions of E q u a t i o n s 1, 2, 3, 4, a n d 5 w i t h e x p e r i m e n t a l d a t a a n d differ o n l y b y the i n i t i a l c o n d i t i o n a p p l i e d to the b e d .

I n t h e one the

i n p u t is a step f u n c t i o n of adsorbent so that b r e a k t h r o u g h curves, s u c h as s h o w n i n F i g u r e s 2, 3, 5, a n d 6 represent the c o n c e n t r a t i o n i n the


effluent f r o m the b e d .

I n t h e other the i n p u t f u n c t i o n is a square w a v e ,

c o r r e s p o n d i n g to c h r o m a t o g r a p h i c t e c h n i q u e s .

I n this case the first a n d

s e c o n d m o m e n t s c a n b e c o n v e n i e n t l y a n a l y z e d for the rate coefficients of space a n d p o i n t processes. T h e m e t h o d is n o w l i m i t e d to l i n e a r a d s o r p t i o n isotherms, b u t f r e q u e n t l y e x p e r i m e n t a l c o n d i t i o n s c a n b e chosen so t h a t this r e s t r i c t i o n is f u l f i l l e d .

A d s o r p t i o n rate coefficients

so

determined

s h o u l d b e v a l i d over a n y c o n c e n t r a t i o n range, w h e t h e r the i s o t h e r m is l i n e a r or not, as l o n g as the coefficients are i n d e p e n d e n t of c o m p o s i t i o n . T h e first m o m e n t , m e a s u r i n g the center of g r a v i t y of the effluent peak, c a n b e r e l a t e d to the e q u i l i b r i u m constant for the p o i n t a d s o r p t i o n step, a n d the s e c o n d

a n d h i g h e r m o m e n t s to the rate coefficients

for a x i a l

d i s p e r s i o n , external a n d i n t e r n a l diffusion, a n d p o i n t a d s o r p t i o n .

Nor

m a l l y i t is not possible to evaluate m o m e n t s h i g h e r t h a n the second w i t h e n o u g h p r e c i s i o n to b e u s e f u l . H o w e v e r , b y m a k i n g measurements

for

different velocities a n d p a r t i c l e sizes, u n d e r c a r e f u l l y chosen c o n d i t i o n s , sufficient d a t a are o b t a i n a b l e to evaluate the space a n d p o i n t parameters. T h e m e t h o d has several a d v a n t a g e s : 1. R a t e coefficients are d e t e r m i n e d i n a n a d s o r p t i o n b e d , t h a t is for the same geometry that t h e y w o u l d b e u s e d for i n d e s i g n purposes. 2. E q u i l i b r i u m a n d rate coefficients o u s l y i n the same apparatus. 3.

can be established simultane

T h e a p p a r a t u s e s s i m p l e a n d d a t a are o b t a i n e d r a p i d l y .

Summary T h e i n t e n t of this p a p e r is to p o i n t out that p h y s i c a l or space processes, w h i c h u s u a l l y influence a n d f r e q u e n t l y c o n t r o l k i n e t i c s of a d s o r p t i o n i n aqueous systems, c a n b e represented effectively b y q u a n t i tative m o d e l s . T h e rate coefficients i n s u c h m o d e l s are m o r e m e a n i n g f u l t h a n those associated w i t h schemes w h i c h d o not r e c o g n i z e space p r o c esses. P u b l i s h e d reports h a v e f r e q u e n t l y a n a l y z e d d a t a b y a c h e m i c a l m o d e l , b u t i n s u c h instances the " r e a c t i o n r a t e " constants are f o u n d to


2.

S M I T H

Kinetics

of

21

Adsorption

be f u n c t i o n s of p a r t i c l e size or fluid flow rate, as p o i n t e d o u t b y G r i f f i n and Dranoff ( 3 ) . T h e a p p l i c a t i o n of a d s o r p t i o n k i n e t i c s is u s u a l l y t h e d e s i g n of p r o c esses f o r r e m o v i n g o r s e p a r a t i n g components f r o m a fluid s t r e a m ; a process for w h i c h t h e fixed-bed adsorber is w e l l s u i t e d . R e c e n t l y , analysis m e t h ods h a v e b e e n d e v e l o p e d b y w h i c h i t is possible to evaluate t h e r e l a t i v e i m p o r t a n c e of space a n d p o i n t processes, a n d o b t a i n n u m e r i c a l values of rate coefficients, f r o m measurements

o n laboratory-scale

fixed

beds.

T h i s a p p r o a c h facilitates the d e s i g n of large scale e q u i p m e n t , a n d i n t h e case o f t h e c h r o m a t o g r a p h i c m e t h o d , possesses other advantages

as a


means of e v a l u a t i n g rate coefficients. F i n a l l y , i n t r a p a r t i c l e diffusion appears to b e a n i m p o r t a n t factor i n a d s o r p t i o n k i n e t i c s for m a n y types of systems.

I n the past i t has b e e n

c u s t o m a r y to define s u c h mass transfer q u a n t i t a t i v e l y i n terms of a n effective d i f f u s i v i t y . H o w e v e r , e v e n i n g a s - s o l i d systems m o r e t h a n one process c a n b e i n v o l v e d for porous

particles.

Thus, two-dimensional

m i g r a t i o n o n t h e p o r e surface, surface diffusion, is a p o t e n t i a l c o n t r i b u t i o n . L i q u i d systems a p p e a r to b e m o r e c o m p l e x , a n d , w i t h electrolytes, i t has b e e n s h o w n that t h e electric p o t e n t i a l i n d u c e d b y c o u n t e r - d i f f u s i n g ions s h o u l d b e t a k e n i n t o account. A r e a l i s t i c d e s c r i p t i o n of i n t r a p a r t i c l e mass transfer i n s u c h cases requires m o r e t h a n a single rate coefficient f o r a b i n a r y system. Symbols C c

Specific heat; C for fluid; C for p a r t i c l e ; c a l . / ( g r a m ) ( ° K . ) C o n c e n t r a t i o n of a d s o r b a b l e gas i n t h e i n t e r p a r t i c l e space, mole/ml. C o n c e n t r a t i o n of a d s o r b a b l e gas i n t h e i n t r a p a r t i c l e space, mole/ml. C o n c e n t r a t i o n of a d s o r b e d gas p e r u n i t w e i g h t of adsorbent, mole/gram E f f e c t i v e i n t r a p a r t i c l e d i f f u s i o n coefficient, c m . / s e c . E x t e r n a l heat transfer coefficient, c a l . / (sec.) ( c m . ) ( ° K . ) E f f e c t i v e a x i a l d i s p e r s i o n coefficient, b a s e d u p o n t o t a l crosss e c t i o n a l area of b e d , c m . / s e c . H e a t of a d s o r p t i o n , c a l . / m o l e I n t r a p a r t i c l e effectivity t h e r m a l c o n d u c t i v i t y , c a l . / ( c m . ) (sec.) ( ° K . ) A x i a l effective t h e r m a l c o n d u c t i v i t y of b e d , c a l . / ( c m . ) (sec.) ( ° K . ) A d s o r p t i o n rate constant, m l . / ( g r a m ) (sec.) E x t e r n a l mass transfer coefficient, c m . / s e c . A d s o r p t i o n e q u i l i b r i u m constant, m l . / g r a m Radius coordinate i n spherical particle, c m . R a d i u s of p a r t i c l e , c m . f

Ci Cads D ht E

p

2

c

2

x

2

AH k e

k

x

fcads k K r R t


22

ADSORPTION

Rt, R P Q t T

AQUEOUS

SOLUTION

G l o b a l a d s o r p t i o n rate, m o l e / (sec.) ( m l . ) P o i n t a d s o r p t i o n rate, m o l e / (sec.) ( m l . ) P o i n t heat release i n p a r t i c l e , c a l . / (sec.) ( m l . ) G l o b a l heat release c a l . / (sec.) ( m l . ) T i m e , sec. A b s o l u t e t e m p e r a t u r e i n fluid stream; T i = t e m p e r a t u r e w i t h i n p a r t i c l e , °K. L i n e a r v e l o c i t y of fluid i n t h e i n t e r p a r t i c l e space, c m . / s e c . A x i a l coordinate i n the b e d , c m . Interparticle v o i d fraction Intraparticle v o i d fraction D e n s i t y ; p f o r fluid; p f o r p a r t i c l e , g r a m / m l .

p

P

v

v z a p p


F R O M

t

v

Acknowledgment The

financial

assistance

of t h e A m e r i c a n C h e m i c a l Society, P R F

G r a n t N o . 1633, is g r a t e f u l l y a c k n o w l e d g e d . Literature

Cited

(1) B o y d , G . E., Adamson, A . W . , Myers, L. S. J r . , J. A m . Chem. Soc. 69, 2836 (1947). (2) Glueckauf, E . , Proc. Roy. Soc. (London) A 1 8 6 , 35 (1946). (3) Griffin, R. P . , Dranoff, J . S., A.I.Ch.E. J. 9, 283 (1963). (4) Helfferich, F . G . ,J.Phys. Chem. 66, 39 (1962). (5) Helfferich, F . G., Ind. Eng. Chem. Fundamentals 6, 362 (1967). (6) Helfferich, F . G., Plesset, M. G., J. Chem. Phys. 28, 418 (1958). (7) Ibid., 2 9 , 1 0 6 4 (1958). (8) H e r i n g , B . , Bliss, H., A.I.Ch.E. J. 9, 495 (1963). (9) K l e i n , G., Tondeur, D . , Vermeulen, T . , Ind. Eng. Chem. Fundamentals 6, 339 (1967). (10) Masamune, S., Smith, J. M., A.I.Ch.E. J. 10, 246 (1964). (11) Ibid., 1 1 , 4 1 (1965). (12) Schlogl, R., Helfferich, F., J. Chem. Phys. 26, 5 (1957). (13) Schneider, P., Smith, J . M., A.I.Ch.E. J. 14, 762 (1968). (14) Smith, J. M., " C h e m i c a l Engineering Kinetics," M c G r a w - H i l l Book C o . Inc., N e w York, 1956. (15) Tondeur, D . , K l e i n , G., Ind. Eng. Chem. Fundamentals 6, 35 (1967). (16) Vassiliou, B . , Dranoff, J . S., A.I.Ch.E. J. 8, 248 (1962). (17) Vermeulen, T., "Advances i n Chemical Engineering," V o l . 2, Academic Press, N e w York, 1958. (18) Walter, J . E., J. Chem. Phys. 13, 299 (1945). RECEIVED

October 26, 1967.


Kinetics of Adsorption - American Chemical Society

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