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3. Upscaling of Adsorbing Solutes; Pore Scale
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where c i∣ ∣
s
denotes solute concentration in the fluid at the pore wall, and c i∣ ∣
pore
denotes the solute concentration within the pore at some distance d. We now
employ Equation (3.4) to eliminate c i∣ ∣
s
from (3.16) and substitute the result
back into (3.15)
∫
∫
Û i = − Di 0
V kD i d s i dA + Di 0
c i∣ ∣
V d pore
dA (3.17)
A fs A fs
There is of course no information on the value of c i∣ ∣
pore
. However, it is plausible
to assume that c i∣ ∣
pore
is a function of average fluid concentration, c i . This
means that we may replace c i∣ ∣
pore
by f( c i ) in Equation (3.17) and set
c i∣ ∣
pore
= f( c i ) (3.18)
Use of this approximation (Equation 3.18), recognizing the fact that c i is a
constant within the averaging volume, and use of Equation (3.4) and (3.9),
leads to the following macro-scale relationship
Û i = − Di 0
k i D d (1 − n) ρs s i + SDi 0
d f( ci ) (3.19)
where S [L −1 ] denotes the solid grain specific surface area, S = A fs /V . Substitution
of this result into the adsorbed mass balance (3.14) yields the standard
linear kinetic adsorption equation:
∂ (1 − n) ρ s s i
=
∂t Û i = nk att f( c i ) − (1 − n) ρ s k det s i (3.20)
where the kinetic rate coefficients, k att and k det [T −1 ], are defined by
and
k att = SDi 0
nd
k det = Di 0
k i D d
(3.21a)
(3.21b)
The mass balance equation for the solutes now becomes
∂nc i
∂t
+ ∇ · (nc i v i) + ∇ · (nJ i) = nr i − nk att f( c i ) + (1 − n) ρ s k det s i (3.22)
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