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3.3 Numerical upscaling of adsorbing solute transport
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from the Single-Tube Model to the solution of the 1D transport equation (Equation
(3.32)) with no adsorption (i.e., k att = kdet ∗ = 0). Figure (3.3) shows the
resulting graph, where Pe is plotted as a function of pore-scale Peclet number
(P e p ). In the same figure, the Taylor dispersion formula (Equation 3.38) [Taylor,
1953, Aris, 1999] for the upscaled Peclet number in a tube is also plotted.
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P e = 1 + P e p
P e p 48
(3.38)
The excellent agreement indicates the accuracy of our numerical code in capturing
the transport within the tube and also shows that the Taylor assumption
is valid for this problem. We use Equation (3.38) to obtain the upscaled Peclet
number (P e) in the simulations of adsorbing solute in the next section.
Figure 3.3: Graph of P e versus P e p. The points are the result of fitting
of the breakthrough curve of average concentration (with κ = 0) to
Equation (3.32) to find P e. The solid line shows the Taylor formula,
Equation (3.38).
3.3.4 Upscaled adsorption parameters (k ∗ att and k ∗ det )
As we employ the Taylor formula to estimate the upscaled Peclet number,
there are only two upscaled parameters, katt ∗ and kdet ∗ , left to be determined as
a function of pore-scale parameters
k ∗ att = f(P e p , κ) (3.39a)
k ∗ det = f(P e p , κ) (3.39b)
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