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3.3 Numerical upscaling of adsorbing solute transport
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Franzini, 1965]
[ ( ) ] 2 r
v(r) = 2v 1 −
R 0
(3.27)
where r is the radial coordinate, v(r) is the local fluid velocity, v is the average
flow velocity, and R 0 is the radius of the cylinder. In the Single-Tube Model,
adsorption occurs only at the wall of tube, Figure (3.1). In this study, changes
in the radius of the tube due to the adsorption process are neglected.
The mass transport in the tube is given by
Figure 3.1: Conceptual representation for the Single-Tube Model.
The velocity profile is assumed to be parabolic.
∂c
( r
) [ 2)
∂t + 2v(1 − ∂c ∂ 2
R ∂z = D c
0
∂z 2 + 1 (
∂
r ∂c )]
r ∂r ∂r
(3.28)
where D 0 [L 2 T −1 ] is molecular diffusion coefficient and z is the axial direction
along the tube.
In the tube, the wall acts as an adsorbent and therefore, the adsorption relation
must appear in the boundary condition to the differential Equation (3.28).
Adsorption to the wall may be described by the following equation which prescribes
that the diffusive mass flux to the wall is the only source of accumulation
at the wall
∂s
∂t = −D 0
∂c
∂r ∣ (3.29)
s
where s [ML −2 ] is adsorbed mass per unit area. This is just Fick’s law, Equation
(3.3). As in section 3.2, we assume that linear equilibrium adsorption
(Equation 3.4)holds at the wall, characterized by the distribution coefficient
value, k D [L]:
s = k D c| s
(3.30)
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