Page 150 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

6.4 Applications of the Proposed Decoupling Procedure 139

V optimal = φk R l chemical . (6.26)
advection
It should be noted that both l chemical and the optimal flow-rate have clear physi-
advection
cal meanings: The physical meaning of l chemical is that for a given fluid flow rate, a
advection
chemical reaction with a given reaction rate can reach equilibrium once this distance
is traversed by the fluid in the flow direction, within the time scale of chemical equi-
librium. Since l chemical is directly proportional to the fluid flow rate, the greater the
advection
fluid flow rate, the larger l chemical . This means that fast flows require relatively long
advection
distances in the flow direction, beyond which a chemical reaction with a given reac-
tion rate can reach equilibrium. In contrast, the physical meaning of the optimal flow
rate is that for a given l chemical , a chemical reaction with a given reaction rate can
advection
reach equilibrium if the fluid flow rate is within the time scale of chemical equilib-
rium. Since the optimal flow rate is also directly proportional to l chemical , the larger
advection
l chemical in the flow direction, the greater the optimal fluid flow rate. This implies
advection
that a large l chemical requires a relatively fast optimal flow rate, so that chemical
advection
reaction for a given reaction rate can reach equilibrium beyond this large l chemical in
advection
the flow direction.
If the flow paths of two fluids are parallel to each other in a fluid mixing system,
solute diffusion/dispersion normal to the flow direction plays a fundamental role
in promoting chemical reactions between different reactive chemical species. In this
case, a second dimensionless parameter, Z, needs to be defined to express the relative
time scale between the solute diffusion/dispersion process and the chemical reaction
process. Notice that Z is independent of the fluid velocity and so still has meaning
for zero fluid flow.

k R l 2
Z = = (Time Scale for Solute Dispersion/Diffusion)/
D (6.27)
Time Scale for Chemical Reaction,

where D is the solute diffusion/dispersion coefficient; l is the characteristic length
of the controlling process in the system; k R is the reaction rate. Since this dimen-
sionless number expresses the ratio of the solute diffusion/dispersion time scale to
the chemical kinetic time scale, it is unity when the two time scales are equal. In this
situation, the chemical equilibrium length, l chemical of the system can be expressed
diffusion
as follows:

D
chemical
l diffusion = , (6.28)
k R
where l chemical is the chemical equilibrium length due to solute diffusion/dispersion
diffusion
for a given chemical reaction. For a given solute diffusion/dispersion coefficient,
there exists an optimal reaction rate such that the chemical reaction can reach equi-
librium within l chemical determined from Eq. (6.28). Thus, for a given l chemical ,the
diffusion diffusion
optimal
corresponding optimal chemical reaction rate, k R , for which the chemical reac-
tion can reach equilibrium within the given l chemical , is as follows:
diffusion