Adsorption - Introduction to Carbon Capture and Sequestration - page 247

Note that if a CO 2 molecule gets adsorbed, it accumulates in the
control volume. If ρ A is the density of the adsorbent (kg/m 3 ) then the total
amount of adsorbent (kg) in our control volume dV is:

( )

m =−ερ 1 dV

ads

A

We can now compute the accumulation of CO 2 due to adsorption in the
adsorbent in our control volume:

( )

∆=σ× 1 ε ρ

A dxdydz

CO CO

2 2

As the adsorption is the only contribution to the accumulation, we have:
( )

d ∆ / dxdydz

∂ρ d σ CO

CO CO

( )

2

2

= = 1 − ε ρ

2

,

∂ t dt dt

A

using Henry's law:

∂ρ dp

CO CO

( )

2

=−ερ 1 H

2

CO

∂ t dt

A 2

Now we can write the mass balance as:

dp dp dp

1 ε u

CO CO CO

( )

2

=

2

+ 1 − ε ρ H

2

CO

RT dt RT dz dt

A 2

or:

dp dp

1 ε u

CO CO

( )

−−ερ 1 H

2

=

2

CO

RT dt RT dz

A 2

If we assume N 2 does not adsorb, we have the equation that we need to
solve to compute the breakthrough curve, with the boundary condition,
p CO 2 ( t ,0) = p CO 2 , fl ue . To fi nd the solution, we guess that this solution looks
like:

() ( )

pz t , =Θ z − t ,

CO

2

where Θ = Θ ( y ) is an unknown function with argument y = ( z - Bt ). Given
Θ , we can write:

dp dp

CO CO

2

=Θ ′ and

2

=− Θ B ′ ,

dz dt