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m

o v e

d f

r o

m

t

h

e

_g

a s.

I f

t

h

e c o nta

m i

n a nt c

h

a

l l

e n

g

e co n c e ntr at

i

on

i

s r e

d

u ce

d

_,

t

h

e a

d

s o r

b

ent w

i l l

r e

l

e a s e c o nta

m i

n a nt *

In a c a r

b

o n

b

e

d

, t

h

e u

p

str e a

m p

o rt

i

o n

l

o a

d

s

f

ir st,

p

o s s

i b l y

r e a c

h i

n

g

e

q

u

il i b

r

i

u

m

w

h i l

e t

h

e

d

ow n str e a

m

c a r

_b

o n

i

s st

i l l

n e a r

l

_y e

_{m p}

t

_y

.

T h

e s

h

ape o

f

t

h i

s c o n c e ntr at

i

o n

wa v e

f

r o nt

i

n a n a

d

s o r

_p

t

i

o n

b

e

d d

e

_p

e n

d

s o n c a r

b

o n a n

d

c o nta

m

i

n a nt

_p

r o

_p

e rt

i

e s, c o n c e ntr atio n , a n

d b

e

d f l

owr ate.

W

h

e n t

h

e c o nta

m

i

n a nt

p

e n etr ate s t

h

e

b

e

d

_, c a

l l

e

d

b

r e a

k

t

h

r o u

_{g h}

, t

h

e

d

own str e a

m

c on c e ntr at

i

on

m

a

y

c

h

a n

g

e r a

p i d l y f

r o

m

v e r

y l

o

w

to t

h

e

c

h

a

l l

e n

_g

e co n c e ntr at

i

o n .

^ *

T

e st r e s u

l

ts can

b

e

_{g i}

ven a s

b

r e a

k

t

h

r o u

_{g h}

t

i

m

es.

E

q u

i l i b

r

i

u

m

A d

s o r pt

i

on

M

o

d

e

l

s.

A

n u

m b

e r o

f m

at

h

e

m

at

i

c a

l m

o

d

e

l

s

h

a v e

b

e e n

d

e v e

l

o

_p

e

d

o v e r t

h

e

_y

e a r s to

d

e s c r

i b

e e

_q

u

il i b

r

i

u

m

a

d

s o r

_p

t

i

o n a s o c c u r s

i

n c a r

b

o n

b

e

d

s, in c

l

u

di

n

g i

s ot

h

e r

m m

ode

l

s s u c

h

a s t

h

e

L

a n

g m

u

i

r,

F

r e u n

d l i

c

h

, B

E T

(

B

r u n a u e r

-E

n

m

i et t-

T

_e

l l

_{e r}

)

,

G i b b

s, a n

d P

o

l

a n

y i

-

_D

u

b i

n

i

n

i

sot

h

e r

m

s.

'

T h

e e

_q

u

i l i b

r

i

u

m

a

d

s o r

_p

t

i

o n

con c e ntr at

i

on,

i

s

k

nown as an a

d

sor

p

t

i

o n

i

sot

h

er

m

.

A f

r e

q

ue nt

l y

u se

d i

s o t

h

e r

m i

s t

h

at

d

er

i

v e

d b y D

u

b i

n

in

, w

h

o us e

d Po

l

an

y i

'

s conce

_p

t o

f

a

d

s o r

_p

t

i

o n to

d

e v e

l

o

_p

a

se

m i

e

m p i

r

i

c a

l f

u n ct

i

o n

d

e sc r

i b i

n

_g

t

h

e v o

l

u

m

e o

f

t

h

e a

d

sor

b

e

d

s

_p

ace as a

f

u nc t

i

o n o

f

t

h

e a

d

sor

_p

t

i

o n

_p

o te nt

i

a

l

.

T h

e

f

u n ct

i

o n

is a

p p

r o

p

r

ia

te

f

o r a

d

s o r

p

t

i

o n

in

w

h ic

h

c a

p i l l

ar

y

c o n

d

e n s at

i

o n

is

o c c u r r

i

n

_g

, a n

d i

s

k

n own as t

he

D

u

b i

nin -R_ad

u sh

k

e v

i

c

h

(

D

-

R

)

_e

q

u at

io n

.

*

I

t

h

a s

be e n

f

o u n

d

to

f i

t

d

ata we

l l f

or

m

an

_y

n o n

_p

o

l

ar or w e a

k l y p

o

l

a r a

d

s o r

b

ate s

b

e

l

ow

t

h

e

ir

cr

i

t

i

ca

l

t e

m p

er atu re.

T h

e

D

-

R

_e

q

u at

i

o n

h

a s

b

e e n w

i d

e

l

_y u s e

d fo r

d

ata c o r r e

la

t

io n

f

or a

d

s o r

_p

t

i

o n o

f

V

O C

s on a ct

iv a

te

d

car

b

o n.

T h

e e

q

u at

io n

i

s :

w

h

e re:

W

, = a

d

s o r

p

t

io n

s

p

a c e o

f

t

h

e c a r

b

on

f

or t

h

e v a

p

o r , c

m

V

g,

W

q =

m

a x

i

m

u

m

a

d

s o r

p

t

i

on s

p

ace

f

or t

h

e car

b

o n, c

m

V

g

,

k

= c a r

b

o n s tr u ctu ra

l

c o n sta nt,

m

o

le

^

/

c a

l

^

e = a

d

s o r

_p

t

io n

_p

o te nt

i

a

l f

o r t

he

c a r

b

on,

R T l

n

(

P/ P

J

, ca

l /

m

o

l

e,

R

=

_g

a s c o n sta nt,

T

= a

b

s o

l

ute te

m p

e r atu r e,

P

= e

_q

u

i l i b

r

iu

m

_p

a rt

ia

l p

r e s s u r e o

f

c o nta

m i

n a nt,

P, = s atu r at

i

o n v a

p

o r

p

r e s s u r e

f

o r c o nta

m

in a nt at

T

,

P

= a

f f i

nit_y c o e

f fi

c

ie n

t

f

o r c o nta

m i

n a nt c o

m

_pa r e

d

to a r efe r e n c e c o

m p

o n e nt

f

o r w

h i

c

h

W

_q an

d k h

a ve

b

een

d

ete r

m

i

n e

d

. =

P

_c

(. c s .)

/P

c(r efe r» c c);

K

=

(

d

l

'

- ₁

)

M

}

/

{ (

_T

1

^ +

2

₎

_p

_,

T

_|

= r e

fr

a ct

i

v e

i

n

d

e x at t

h

e s o

d i

u

m D

wa v e

l

e n

_g

t

h

M

=

_m

_o

_l

_{e c u}

_l

_{ar w} e

i g h

t

P

j

= _{c o nt a}

_m

_i

_na nt

l i q

u

i d d

e n s

it

_y

,

g /

c

m

^

.

D

y na

m

i

c

R

e sp ons e.

S i

n c e t

h

e e x

i

t c o n c e ntr at

i

on e

q

u a

l

s t

h

e

in

l

et con c e ntr at

i

on

at e

_q

uil

i b

riu

m

, t

h

e d

y

n a

m i

c, non

-_e

q

ui l

i b

r

i

u

m p

e r

f

o r

m

a n c e o

f

a n a

d

so r

b

e r

i

s o

f p

r

i

m

a r

_y

i

nte r e st.

W

_^ a n

d k

a r e

f

u n c tio n s o

f

t

h

e c a r

bo n

, n ot o

f

t

h

e v a

p

o r.

T h

e u s e o

f

t

h

e D

-

_R

e

_q

u at

i

o n

b

e

_gi

n s

b y

a c c u

m

u

l

at

i

n

_{g k i}

n et

ic

a

_{ds o r}

_p

t

i

o n c a

_p

a c

i

t

_{y d}

ata

f

o r t

he

c a r

b

o n o

f

i

nte r e st

in

t

h

e

f

o r

m

o

f

m

ass o

f

r e

f

e r e n c e c onta

m i

n a nt

_p

e r

m

a ss o

f

c a r

b

on ,

W

^ , at

e

_q

u

i l i b

r

iu

m

w

i

th t

h

e c onta

m i

n a nt at sev e r a

l p

ar t

i

a

l p

r e s s u r e s a n

d

a s

in

_{g l}

e

te

m

_p

e r atu r e.

" '

A

ssu

m i

n

_g

that t

h

e

b

u

l k l i q

u

i d d

e n s

i

t

_y

can

b

e u se

d

to re

_p

re s e nt t

h

e

v o

lu

m

e o c c u_p

i

e

d b

_y t

h

e a

d

s o r

be n

t o n t

h

e a

d

s o r

b

at e,

W

^

is

co n v e rte

d

to

W

^

b

y

d i

v

i d

in

g

b

_y

_P

c-

A p

lot o

f l

n

(

W

_y

)

v e r s u s e

^

y i

el ds a str a

i g

_ht l in e of s

lo

_p

e -

k

₍

b

_{e c a u s e}

t

h

e r e

fe r e n c e

c o

m p

o u n

d

)

a n

d i

nte rce

_p

t

l

n

₍

W

_o

₎

.

O

n c e

W

_{q a n}

d k h

ave

b

een

d

e te r

m

i

ne

d

_,

W

^

(

a n

d h

e n c e

W

^, t

h

e e

q

u

i l i b ri

u

m

m

a s s a

d

sor

b

e

d

)

c a n

be

co

m p

ute

d f

o r a n

y

c o nta

m i

n a nt a t a n

_y

c o n c e ntr at

io

n a

ft

e r c o

m p

ut

i

n

_g

p

a n

_d

e

f

o r t

h

at c onta

m

i

nant.

A l

te

rn

at

i

v e

l y

, e

q

u

i h b

r

iu

m d

ata at o n e c o n c e ntr at

i

o n o n a

p

rev

i

ous

l y

c

h

ar ac te

ri

z e

d

c ar

b

o n ca n

b

e u s e

d

to c o

m p

ute

P

a n

d

e.

T h

e

D

-

R

e

_q

u at

io n

h

a s

b

e e n use

d

su c c e s s

f

u

_{l l y d}

own t o a

_{p p}

r o x

i

m

ate

_{l y 10 0}

p p m

. " *

I

t s use

f

u

ln

ess at

l

owe r c on c e ntr at

i

o n s, o r

f

or e xtr a

p

o

l

at

i

o n

f

r o

m

p p

m

to

p

p

b

l

e v e

l

s,

h

a s n ot

b

e e n r e

p

o rte

d

.

T h

e s

h

a

p

e o

f

t

h

e

i

so t

h

e r

m

c u r v es a s t

h

e

y

a

p p

r o a c

h

z e ro

c o nce ntr at

i

on

is

n ot

k

n own, ' ^

a n

d

u n e x

_p

e c te

d

c

h

a n

_g

e s

i

n

_b

e

h

a v

_{io r c o u}

_{l d}

c a use

s

i g

n

i f

i c a nt e r r o r s.

H

owe v e r

, t

h

e e

q

u at

i

o n wa s

d

e

ri

v e

d f

o r

m i

c r o

p

o r o u s c a r

b

on a n

d

m

a

_{y b}

e v al

i d

at

in

d

o or c o n c e nt rat

i

on s.

A d d i

t

i

o n a

l p h

e n o me n a

m

u st

b

e c o n s

i d

e r e

d

w

h

e n cor re

l

at

in

_{g d}

ata

f

r o

m

r e a

l

,

d y

n a

m i

c a

d

s o r

_p

t

io n

b

e

d

s.

A

n a n a

l y

s

is o

f

t

h

e

b

r e a

k

t

hr o u

gh d y

n a

m

i

c s o

f

a n a

ds o r

b

e nt

f i l

ter c a n

b

e rat

h

e r c o

m p l i

c ate

d

.

T h

e

f i

rs t re

q

u

ir e

m

e nt

f

o r r e

m

ov a

l

o

f g

a s

-p h

a s e

c o nta

m i

n a nt s

i

s t o

_g

et t

h

e

m

n e a r t

h

e a

d

s o r

b

e nt su r

f

a ce.

T hi

s r e

q

u

i

r es tr a n s

p

o rt

b

ot

h

f

r o

m

t

h

e

b

u

l k g

a s to t

he

r e

_{g i}

o n o

f

t

h

e a

d

s o r

be n

t a n

d f

r o

m

t

h

e

b

o u n

d

a r

_{y l}

a

_y

e r

i

n to t

h

e

p

o r e s.

T

r a n s

p

o rt ste

p

s a r e o

f

te n t

h

e r ate

-

_{l i m i}

_t

_in

g

ste

_p

s

i

n t

h

e o v e r a

l l

s o r

_p

t

i

o n

_p

r o c e ss.

C h

e

m i

c a

l

r e ac t

i

o n s

m

a

_y

o c c u r at t

h

e su r

f

a c e.

A

r e a

l fl

ow s

y

ste

m i

s

f

u rt

h

e r

c o

m p l i

c ate

d b y

n o n-

_{i d}

e a

l f l

ow co n

d i

t

io n

s a n

d

a

d

so r

be r

b yp

as s.

T h

ese

l

at te r t wo

f

a ct or s c a n o v e r s

h

a

d

ow a

d

s o r

_p

t

i

o n e

_q

u

i l i b ri

u

m

a n d

k i

n et

i

cs

i

n s o

m

e c a s e s.

U

n

d

e r c e rta

in

s

i m pl i fy i

n

_g

a s s u

m p

t

i

o n s—a s

in

_{g l}

e-_{c o}

m p

on e nt

l i

n e a r

i

s ot

h

e r

m

,

i

s ot

h

e r

m

a

l

c on

d i

t

i

o n s, a n

d

n o a x

i

a

l d i

s

p

er s

i

on—an a

l y

t

i

ca

l

ex

p

r e s s

i

on s su c

h

as t

h

e

M

e c

k l

e n

b

e r

_g

,

R

o s e n, a n

d

W

he e

l

e r e

q

u at

io n s

h

a ve

be e n

de v e

lo

p

e

d

t o

d

es c

ri b

e t

h

e

b

r e a

k

t

hr

o u

_{g h}

cu r ve o

f

a n a

d

so r

b

e nt

f i l

ter. "

T h

ese e

_q

u at

i

o ns

_p

e r

m

i

t e st

i

m

ates o

f

c o nta

m i

n a nt r e

m

o v a

l

e

f f

i c

ie n c

_y

a n

d

c a

_p

a c

i

t

_y

w

i

t

h

o ut

_p

e r

fo r

m i

n

_g

t

h

e e x

_p

e r

i

m

e nta

l

te sts

o v e r t

h

e e nt

ir e

b

r e a

k

t

h

r o u

_{g h}

r a n

_g

e.

F

o r t

h

e

p

r e s e nt

p

u r

p

o s e,

b

r e a

k

t

h

r o u

g h

e x

p

r e s s

io n

s

that a

_{p p}

l

_y

in the lo

w

co n c e ntr atio n r a n

_g

e a r e

_p

a rtic ula rl

_y

u s efu

l

.

T h

e

m

o

d i fi

e

d

W

h

ee

le r e

_q

u at

io n

i

s r e

l

a t

iv e

_{l y}

succ e ss

f

u

l in

e x tr a

_p

o

l

at

i

n

_g

t

h

e

p

e r

f

o r

m

a n c e o

f

r e s

_p

u

-ato r

f

i

l

te rs. ' " "

I

t

w

a s

d

e

ri

v e

d f

r o

m

a

m

a s s c o nt

in u

i

t

_y

e

_q

ua t

i

o n

a c ros s a n a

d

so r

b

e nt

b

e

d

. T

he

a

d

s o r

p

t

i

o n r ate

k i

n et

i

c s a r e c o n s

i d

e r e

d p

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