8.2 Basics of Diffusion
257
It should be noted that the coefficients v (see Sect. 1.2), Ax, Ay and Az
(see Sect. 1.2 and 7.3), and D have the same dimensions. However, their
meanings are different: v is the coefficient of kinematic viscosity and represents
the diffusivity for momentum in laminar flow; Ax, Ay and Az are the coefficients
of turbulent viscosity, and they are measures of the diffusivity for momentum
in turbulent flow; D is a measure of the diffusivity for matter.
In Sect. 8.3.4 we will exploit and expand the random walk model to determine
the concentration of particles released under various environmental conditions.
8.2.3 Fick's Equation of Diffusion
Moving particles transport matter. However, determination of the transport
by monitoring individual particles on a microscopic level is very impractical.
It is much easier to calculate transport when diffusion is examined from a
macroscopic point of view.
On a macroscopic level, a suitable measure is the flux of matter between
areas of different concentrations. Let us consider a plane with area A, located
at x = Xo and separating two neighbouring cells (see Fig. 8.1). We assume that
after some n steps, or after time t = nT, there will be N(xo) particles located
in a cell to the left of the plane, and similarly there will be N(xo +.\) particles
located in the cell to the right of the plane. All particles undergo the same
type of motion as described in the previous section. So, at time t + T, half the
particles at x = Xo can move to the right through the plane, to a cell located
at x = Xo +.\. Similarly, half the particles from the cell at x = Xo +.\ can move
to the left.
Therefore, the net number of particles that cross the plane in the positive x
direction in time tis:
.
.
N(xo)
N(xo+.\)
net number of crossmg particles = - - -
.
2
2
(8.8)
The number of crossing particles usually is presented in a form of a flux density
Ix which is the net number of particles crossing a plane per time and per area.
So, we have:
- [N (xo + .\) - N (xo)]
Ix = -'----'------'---'--'-"2AT
(8.9)
or:
I = _ D [N (xo + .\) _ N (xo)]
.r
.\
A'\
A'\'
(8.10)
where the definition (8.5) of the coefficient of diffusion, D, has been used. For
ocean water, the coefficient D is of the order of 10- 9 m 2 /s (Ozmidov, 1986), and
257
It should be noted that the coefficients v (see Sect. 1.2), Ax, Ay and Az
(see Sect. 1.2 and 7.3), and D have the same dimensions. However, their
meanings are different: v is the coefficient of kinematic viscosity and represents
the diffusivity for momentum in laminar flow; Ax, Ay and Az are the coefficients
of turbulent viscosity, and they are measures of the diffusivity for momentum
in turbulent flow; D is a measure of the diffusivity for matter.
In Sect. 8.3.4 we will exploit and expand the random walk model to determine
the concentration of particles released under various environmental conditions.
8.2.3 Fick's Equation of Diffusion
Moving particles transport matter. However, determination of the transport
by monitoring individual particles on a microscopic level is very impractical.
It is much easier to calculate transport when diffusion is examined from a
macroscopic point of view.
On a macroscopic level, a suitable measure is the flux of matter between
areas of different concentrations. Let us consider a plane with area A, located
at x = Xo and separating two neighbouring cells (see Fig. 8.1). We assume that
after some n steps, or after time t = nT, there will be N(xo) particles located
in a cell to the left of the plane, and similarly there will be N(xo +.\) particles
located in the cell to the right of the plane. All particles undergo the same
type of motion as described in the previous section. So, at time t + T, half the
particles at x = Xo can move to the right through the plane, to a cell located
at x = Xo +.\. Similarly, half the particles from the cell at x = Xo +.\ can move
to the left.
Therefore, the net number of particles that cross the plane in the positive x
direction in time tis:
.
.
N(xo)
N(xo+.\)
net number of crossmg particles = - - -
.
2
2
(8.8)
The number of crossing particles usually is presented in a form of a flux density
Ix which is the net number of particles crossing a plane per time and per area.
So, we have:
- [N (xo + .\) - N (xo)]
Ix = -'----'------'---'--'-"2AT
(8.9)
or:
I = _ D [N (xo + .\) _ N (xo)]
.r
.\
A'\
A'\'
(8.10)
where the definition (8.5) of the coefficient of diffusion, D, has been used. For
ocean water, the coefficient D is of the order of 10- 9 m 2 /s (Ozmidov, 1986), and