FUNDAMENTALS OF FATE AND TRANSPORT - Water Quality Engineering in Natural Systems - page 27

3

FUNDAMENTALS OF FATE AND TRANSPORT

3.1 INTRODUCTION velocity field, the flow is turbulent, and mixing is due to

both turbulent and molecular diffusion. In this case,
turbulent diffusion causes the tracer to spread over a
much larger area that in the case where the flow is
laminar, since turbulent diffusion is generally much
greater than molecular diffusion. In Figure 3.1c, the
velocity is not spatially uniform, the flow is turbulent,
and mixing is due to dispersion, turbulent diffusion, and
molecular diffusion. Note that the spatial variation in
velocity causes different parts of the tracer area to move
with different velocities, thereby “tearing” the tracer
area apart; this is the dispersion process.

The dispersal of a tracer in a fluid is called mixing , which
occurs as a result of different particles of the tracer
moving along different paths, which occurs when differ-
ent tracer particles move at different velocities. The
velocity of an individual tracer particle has both micro-
scopic and macroscopic components. The microscopic
component is associated with the random movement of
the tracer molecule called Brownian motion , and the
macroscopic component is associated with the motion
of the fluid continuum containing the tracer particle. In
cases where the (continuum) fluid flow is turbulent, the
macroscopic velocity at any point in space can be further
divided into a local (time-averaged) velocity plus
random velocity fluctuations associated with the turbu-
lent flow. Mixing due the molecular-scale random veloc-
ity fluctuations is called molecular diffusion , and mixing
due to random turbulent macroscopic velocity fluctua-
tions is called turbulent diffusion . Additional mixing can
be caused by spatial variations of the local (time-
averaged) velocity, and mixing by this mechanism is
called dispersion . Mixing characteristics in an aqueous
environment are significantly influenced by the spatial
and temporal characteristics of the ambient velocity
field, as well as the size of the region containing tracer
particles. Some common mixing characteristics are illus-
trated in Figure 3.1. In Figure 3.1a, the fluid has a spa-
tially uniform velocity field, the flow is not turbulent,
and mixing is due only molecular diffusion. Note that a
spatially uniform and laminar fluid velocity does not
cause mixing beyond molecular diffusion. In Figure
3.1b, the fluid has a spatially uniform (time averaged)

3.2 THE ADVECTION-DIFFUSION EQUATION

Mixing by both molecular and turbulent diffusion are
fundamentally similar to each other in that they are
both associated with the random movement of particles,
with molecular-scale motion causing molecular diffu-
sion and macroscopic-scale fluid motion causing turbu-
lent diffusion. As a consequence of this similarity, both
molecular and turbulent diffusion are commonly
assumed to be described by Fick's law (Fick, 1855),
which can be expressed in the generalized form

∂
∂

c
x

q

d

= − D

(3.1)

i ij

j

where q i d is the tracer mass flux (ML−2T−1)* −2 T −1 )* due to dif-
fusion in the x i direction, D ij is the diffusion coefficient

* Dimensional units are shown in brackets throughout the text.