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3. Upscaling of Adsorbing Solutes; Pore Scale
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f(c) = c and d = wR 0 (3.43)
As a result Equation (3.19) leads to the following macro-scale kinetic adsorption
relationship:
Û i = nk att c i − (1 − n) ρ s k det s i (3.44)
where the kinetic rate coefficients, k att and k det [T −1 ], are defined by
k att = SDi 0
nwR 0
(3.45a)
k det = Di 0
k i D wR 0
(3.45b)
Indeed, if we acknowledge the fact that, for a tube, porosity is unity and the
specific surface area is S = 2 R 0
, Equations (3.45) reduce to
k avg
att = 2D 0
wR 2 0
(3.46a)
k avg
det
= D 0
k i D wR 0
(3.46b)
where superscript avg stands for the averaging method. Recall that w R0 denotes
the radial position of a point in the pore where point concentration c i∣ ∣
pore
is equal to average concentration c i (see Equation 3.43). These equations agree
quite well with Equations (3.42) which were obtained through numerical averaging.
In Equations (3.42), if we neglect the dependence on the velocity and the
exponential term and approximate (0.5 + 4.5 k D
R0
) ≈ 4.5 k D
R0
, we obtain the approximation:
k att = 4Di 0
R 2 0
(3.47a)
k det = 2Di 0
k D
R 0
(3.47b)
We notice that the two equation sets, (3.46) and (3.47), are the same if w =
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