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3. Upscaling of Adsorbing Solutes; Pore Scale
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then the kinetic process will be important. This condition can be represented
by the dimensionless number σ D
σ D = ρs (1 − n)vK i D
SD i 0
(3.26)
Thus, if σ D ≥ 1, then kinetic effects are important. Note that if the flow
velocity is very small, then the kinetic effects become negligible. In the limiting
case of no flow, as is the case in batch experiments, the equilibrium relationship
(3.24) applies with no approximation. Thus, batch experiments can be used to
obtain the macro-scale distribution coefficient.
In the following section, we will perform numerical experiments to explore the
assumptions leading to Equations (3.24) and also find an approximate value
for d in Equation (3.16).
3.3 Numerical upscaling of adsorbing solute transport
Perhaps the simplest step in upscaling is to replace the three-dimensional flow
and concentration fields within the pore (or a tube) by 1D fields, whereby velocity
and concentration are averaged over the pore cross section. As mentioned
in the Introduction, this upscaling has been considered for homogeneous reactions
as well as dissolution. Here we treat the upscaling of adsorptive solute
transport.
To analyze the scale dependence of adsorption process, we have developed
two models: a) a Single-Tube Model in order to simulate details of transport
within a pore, and b) an equivalent upscaled 1D model for the cross-sectionallyaveraged
concentration. These models allow us to investigate some of the assumptions
made in our upscaling approach and also to verify results of that
approach.
3.3.1 Flow and transport at pore scale (Single-Tube Model)
Consider a long single tube with a constant circular cross section with radius
R 0 . We assume fully developed, steady-state, laminar flow in the tube
(Poiseuille flow) so that the velocity distribution is given by [Daugherty and
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