MIXING OF DISSOLVED CONSTITUENTS

3.1 MIXING OF DISSOLVED CONSTITUENTS

Di ffusion and dispersion are the processes by which a tracer spreads within a fluid. Diffusion is the random advection of tracer molecules on scales smaller than some de fined length scale. At small (microscopic) length scales, tracers di ffuse primarily through Brownian motion of the tracer molecules, whereas at larger scales, tracers are di ffused by random macroscopic variations in the fluid velocity. In cases where the random macroscopic variations in veloc- ity are caused by turbulence, the di ffusion process is called turbulent diffusion. Where spa- tial variations in the macroscopic velocity are responsible for the mixing of a tracer, the process is called dispersion. It is common practice to use the terms diffusion and dispersion interchangeably to describe the larger-scale mixing of contaminants in natural water bodies. In open waters, spatial variations in the macroscopic velocity are usually associated with shear flow and shoreline geometry, whereas in ground waters, macroscopic (seepage) veloc- ity variations are caused by the complex pore geometry and spatial variations in hydraulic conductivity. The case of turbulent di ffusion in an aqueous environment with a steady uni- form mean flow is illustrated in Figure 3.1. In this case, the ambient velocity field consists of

a large-scale mean flow with small-scale fluctuations about the mean. The fluctuations occur over length scales smaller than the width of the plume. Regardless of the mechanism respon- sible for the spatial variations in velocity, whenever these velocity variations are either truly random or spatially uncorrelated over a de fined mixing scale, the mixing process is described by Fick’s law (Fick, 1855), which can be stated in the generalized form

Water-Quality Engineering in Natural Systems, by David A. Chin Copyright © 2006 John Wiley & Sons, Inc.

92 FATE AND TRANSPORT IN AQUATIC SYSTEMS

FIGURE 3.1 Turbulent di ffusion of tracer particles in uniform flow. (From NOAA, 2005a.)

where q d i is the dispersive mass flux [M/L 2 T] 1 in the x i -direction, D ij is the dispersion coe fficient tensor, and c is the tracer concentration. In cases where the dispersion coe fficient varies with direction, the dispersion process is called anisotropic; in cases where the dispersion coe fficient is independent of direction, the dispersion process is called isotropic. Hence, for isotropic dispersion, Fick’s law is given by

∂ c q d i ⫽ ⫺D ᎏ ᎏ

Whereas the Fickian relation given by Equation 3.2 parameterizes the mixing e ffect of velocity variations with correlation length scales smaller than some de fined mixing scale or support scale, tracer molecules are also advected by larger-scale fluid motions. The mass flux associated with the larger-scale (advective) fluid motions is given by

q a i ⫽V i c (3.3) where q a i is the advective tracer mass flux [M/L 2 T] in the x i -direction and V i is the large-

scale fluid velocity in the x i -direction. Since tracers are transported simultaneously by both advection and dispersion, the total flux of a tracer within a fluid is the sum of the advec- tive and di ffusive fluxes given by

q i ⫽q a i ⫹q d

i ⫽V i c⫺Dᎏ ᎏ c ∂ (3.4) x

where q i is the tracer flux in the x i -direction. Equation 3.4 can also be written in vector form as

(3.5) where q is the flux vector and V is the large-scale fluid velocity. The expression of the

q ⫽ Vc ⫺ D ∇c

tracer flux in terms of an advective and diffusive component must generally be associated

1 Dimensional units are shown in brackets throughout.

MIXING OF DISSOLVED CONSTITUENTS

qn

dA

Control volume

Ambient flow field

FIGURE 3.2 Control volume in a fluid transporting a tracer. (From Chin, David A., Water-Resources Engineering. Copyright © 2000. Reprinted by permission of Pearson Education, Inc., Upper Saddle River, NJ.)

with a length scale, L, that is a measure of the averaging volume used to estimate the advective velocity, V, and the di ffusion coefficient, D. The main distinction between advection and di ffusion or dispersion is that advection is associated with a net movement of the center of mass of a tracer, whereas di ffusion and dispersion are associated with spreading about the center of mass.

Consider the finite control volume shown in Figure 3.2, where this control volume is contained within the fluid transporting the tracer. In accordance with the law of conserva- tion of mass, the net flux [M/T] of tracer mass into the control volume is equal to the rate of change of tracer mass [M/T] within the control volume. The law of conservation of mass can be put in the form

ᎏ ∂ ᎏ 冕 c dV ⫹ 冕 q ⋅ n dA ⫽ 冕 S

V m dV ∂ (3.6) t where V is the volume of the control volume, c is the tracer concentration, S is the surface

area of the control volume, q is the flux vector (Equation 3.5), n is the unit outward nor- mal to the control volume, and S m is the mass flux per unit volume originating within the control volume. Equation 3.6 can be simpli fied using the divergence theorem, which relates a surface integral to a volume integral by the relation

冕 q · n dA ⫽ 冕 ∇ · q dV

Combining Equations 3.6 and 3.7 leads to the result

ᎏ ᎏ 冕 c dV ⫹ ∇ · q dV ⫽ S dV (3.8)

94 FATE AND TRANSPORT IN AQUATIC SYSTEMS

Since the control volume is fixed in space and time, the derivative of the volume integral with respect to time is equal to the volume integral of the derivative with respect to time, and Equation 3.8 can be written in the form

冕 ᎏ ᎏ⫹ ∇ · q ⫺S m dV ⫽ 0 V (3.9) 冢 ∂ t 冣

This equation requires that the integral of the quantity in parentheses must be zero for any arbitrary control volume, and this can only be true if the integrand itself is zero. Following this logic, Equation 3.9 requires that

ᎏ ∂ c ᎏ⫹ ∇·q⫺S m ⫽0

This equation can be combined with the expression for the mass flux given by Equation

3.5 and written in the expanded form ᎏ ∂ ` c ᎏ⫹ ∇ · (Vc ⫺ D ∇c) ⫽ S

which simpli fies to

ᎏ ∂ ᎏ⫹V· c ∇c ⫹ c(∇ · V) ⫽ D ∇ 2 c⫹S

This equation applies to all tracers in all fluids. In the case of incompressible fluids, which is typical of the water environment, conservation of fluid mass requires that

(3.13) and combining Equations 3.12 and 3.13 yields the following di ffusion equation for incom-

∇· V ⫽ 0

pressible fluids with isotropic dispersion:

ᎏ ᎏ ⫹V · ∇c ⫽ D ∇ 2 c⫹S

In cases where there are no external sources or sinks of tracer mass (a conservative tracer), S m is zero and Equation 3.14 becomes

ᎏ ᎏ ⫹V · ∇c ⫽ D ∇ 2 c (3.15)

If the dispersion coe fficient, D, is anisotropic, the principal components of the dispersion coe fficient can be written as D i , and the di ffusion equation becomes

ᎏ 冱 ∂ V i D ᎏ⫹S

i⫽ 1 ∂ x i 冱 i⫽ 1 i ∂ x 2 i

PROPERTIES OF THE DIFFUSION EQUATION

where x i are the principal directions of the dispersion coe fficient tensor. Equation 3.16 is the most commonly used relationship describing the mixing of contaminants in aquatic envi- ronments, and it is known as either the advection–dispersion or the advection–diffusion equation , with the former term being more appropriate in most cases.