Modeling Mass Transport in One Dimension
Why do we consider mass transport?
The starting point for modeling of lakes is the Completely-Mixed Flow Reactor (CMFR).
The equations that describe this system assume complete mixing, i.e. homogeneity
throughout the reactor … or water body. Where this assumption is not met, we subdivide
the system into compartments where the assumption is effectively met within the
compartment. It is then necessary to link those compartments as described by their mass
transport characteristics.
Types of mass transport.
We consider two types of mass transport: advection or movement with the bulk flow of
liquid and diffusion or transport by random movement of molecules (molecular diffusion)
or mixing by eddies (eddy diffusion). Eddy diffusion occurs at a scale several orders of
magnitude greater than molecular diffusion and thus molecular diffusion can be safely
ignored in water column applications.
Quantification of advection.
Advection is movement with the bulk flow; its quantification can be illustrated by
examining the mass balance on an embayment of a lake,
V
dC
 Q  Cin  Q  C
dt
This equation quantifies the loss of mass from the embayment to the main lake by
advection. The application is similar for a single cell within a multi-cell embayment or
estuary,
V
dCi
 Q  Ci 1  Q  C
dt
This equation accommodates input from the ‘upstream’ cell and output to the
‘downstream’ cell.
The level of sophistication in the treatment of advection is a matter of scale. Consider the
difference between simple flushing and the more complex current structure present in
some large systems. Our choice of an approach depends on the nature of the problem. In
many water quality applications, treatment of flushing is sufficient. The more advanced
treatment of currents is a field called hydrodynamic modeling and is particularly useful in
tracking the movement of water masses, e.g. chemical spills.
Quantification of diffusion.
Diffusion is quantified using Fick’s 1st Law, where the diffusive flux across any boundary
is given as the product of the diffusion coefficient and the concentration gradient,
J  E 
dC
dx
where E is the eddy diffusion coefficient (D if one is dealing with molecular diffusion),
with units of m2d-1. The term dC/dx is the concentration gradient and accommodates the
fact that molecules will move from regions of high concentration to regions of low
concentration. At the molecular scale, this may be interpreted as being a result of having
more opportunities for molecular collisions in areas of high concentration. At the eddy
scale, consider the case that mixing is impartial and that there is a tendency then to seek
equality (two box model). The negative sign insures that mass moves in the correct
dimension, e.g. for a mass balance on Box 1, diffusion will be a loss term if dC/dx
(between Box 1 and Box 2) is positive.
Calculating the concentration gradient.
A gradient is defined the rate at which a y-coordinate changes with respect to an xcoordinate. Here, the concentration gradient is the rate of change in concentration with
respect to the distance (mixing length, l) over which the concentrations are measured,
dC C2  C1

dx
l
As the difference in concentrations increases, the gradient, and thus the driving force for
diffusion, increases.
Substituting to Fick’s 1st Law gives us,
J  E 
C2  C1
l
and then to the overall mass balance,
C2  C1
dC
 E 
A
dt
l
V
dC
 J  A and
dt
V
dC  E  A

 (C2  C1 )
dt
l
V
or
V
and re-arranging,
dC
  E ' (C2  C1 )
dt
Note here the difference between the turbulent diffusion coefficient (E, m2d-1) which has
intrinsic properties and the bulk diffusion coefficient (E’, m3d-1) which is derived.