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12 0. Modelling Processes in Porous Media with Differential Equations

(1) (2)
with D := τD η + D mech, Q η := Q η + Q η .
Because the mass flux consists of qc η , a part due to forced convection,and
(1) (2)
of J +J , a part that corresponds to a generalized Fick’s law, an equa-
tion like (0.29) is called a convection-diffusion equation. Accordingly, for
the part with first spatial derivatives like ∇·(qc η )the term convective part
is used, and for the part with second spatial derivatives like −∇·(ΘD∇c η )
the term diffusive part is used. If the first term determines the character of
the solution, the equation is called convection-dominated. The occurrence
of such a situation is measured by the quantity Pe, the global P´eclet num-
ber, that has the form Pe = q L/ ΘD [- ]. Here L is a characteristic
length of the domain Ω. The extreme case of purely convective transport
results in a conservation equation of first order. Since the common mod-
els for the dispersion matrix lead to a bound for Pe, the reduction to the
purely convective transport is not reasonable. However, we have to take
convection-dominated problems into consideration.
Likewise, we speak of diffusive parts in (0.17) and (0.20) and of (nonlin-
ear) diffusive and convective parts in (0.21) and (0.25). Also, the multiphase
transport equation can be formulated as a nonlinear convection-diffusion
equation by use of (0.24) (see Exercise 0.2), where convection often dom-
inates. If the production rate Q η is independent of c η , equation (0.29) is
linear.
In general, in case of a surface reaction of the component η, the kinetics of
the reaction have to be described . If this component is not in competition
with the other components, one speaks of adsorption. The kinetic equation
thus takes the general form
∂ t s η (x, t)= k η f η (x, c η (x, t),s η (x, t)) (0.30)
with a rate parameter k η for the sorbed concentration s η [kg/kg], which is
given in reference to the mass of the solid matrix. Here, the components
in sorbed form are considered spatially immobile. The conservation of the
total mass of the component undergoing sorption gives
Q (2) = − b ∂ t s η (0.31)
η
with the bulk density b = s (1−φ), where s denotes the density of the solid
phase. With (0.30), (0.31) we have a system consisting of an instationary
partial and an ordinary differential equation (with x ∈ Ω as parameter). A
widespread model by Langmuir reads
f η = k a c η (s η − s η ) − k d s η
with constants k a ,k d that depend upon the temperature (among other
factors), and a saturation concentration s η (cf. for example [24]). If we
assume f η = f η (x, c η ) for simplicity, we get a scalar nonlinear equation in
c η ,
∂ t (Θc η )+ ∇· (qc η − ΘD∇c η )+ b k η f η (·,c η )= Q (1) , (0.32)
η