Absorption - Introduction to Carbon Capture and Sequestration - page 203

We can make a Taylor expansion of the sin h and cos h :

( )

3

Lk ′

Lk ′ − Lk ′

e − e

( )

sin hkL ′ = = Lk ′ + +

2 3!

( )

2

Lk ′

Lk ′ − Lk ′

( )

e + e

cos hkL ′ = = 1 + +

2 2!

If we keep only the linear terms, we get the equation for a reaction-limited
fl ux:

Dk

( )

D

( )

CO CO

()

0 ∗ 0 ∗

j 0 =

2

c − c =

2

c − c

CO CO CO

CO CO

L

2

Lk ′

2 2

2 2

We get the results of the previous section back again; the chemical reac-
tion is too slow to have any effect.
The other limit is if the chemical reaction is fast compared to
diffusion:

kc

rB

kL ′ = L 1

D

CO

2

In this limit, we have the equation for diffusion-limited fl ux:

( )

cos hkL ′

()

0 0

j 0 ≈ D k c ′ ≈ D k c c

( )

CO CO CO rB

CO CO

2 2 2

2

sin hkL ′

2

We see that if the chemical reactions dominate, we get a typical term
related to the square root of the diffusion constant times the reaction rate
constant. In the derivation of this expression, we assume that the reac-
tion is irreversible, so we cannot relate this result to the equilibrium
concentration.
For amine solutions the reaction rate is often the limiting step, in
which case we have as a driving force the difference in CO 2 concentra-
tion and the concentration at the interface:

D

( ) ( )

CO

()

∗ ∗

2

j 0 = c − c = k c − c

CO CO , i CO eff CO , i CO

L

2 2 2 2 2