Geochemical systems - Geochemistry: An Introduction - page 117

Inlet

Q Q

C in

C i

Water

Outlet

M P

C i C i

sediment

Sediments

Figure 6.1 Simple box model (a single reservoir) typical of a lake. M : mass of water in the lake, Q :water

inflow

= outflow, P : sedimentation rate.

and, as usual, we will introduce a partition coefficient D i to allow for fractionation
of the element between solid and water, such that C sediment = D i C i . Rearranging this

gives:

C in

d C i
d t

τ H

C i + = (6.2)

+ β

i

+ β

i

1 1

i

PD i where Q is the reac-

tivity of the element relative to sedimentation. This reactivity is zero for an inert tracer, such
as a coloring agent or chlorine ion, and increases with the solid/liquid partition coefficient
of the element. The term in front of the derivative is the residence time

τ H = M / Q is the time to renew the water in the lake and β = /

i )
of element i in the lake, i.e. the mean time that atoms of i spend in the lake before being
removed (see Appendix G ).
The form of (6.2) is extremely instructive. Time is now an integral part of the balance,
contrary to the distillation equations mentioned earlier. If the concentration of i upstream is
zero, we obtain by integration the change in a lake initially at concentration C i (0), possibly
by accidental spillage of a pollutant at t

i

τ = τ H / (1 + β

= 0:

i

C i ( t ) C i (0)e − t /τ = (6.3)

The lake is said to relax from its initial state described by the concentration of the ele-
ment at t

i , which is
shorter for more reactive elements and less than the time it takes to renew the lake water.
The residence time

= 0. The measurement of relaxation rate is the residence time τ

i indicates how quickly a disturbance in the balance of element i
is damped out by the lake. The corresponding expression for non-zero C in

τ

is given in

Appendix G .
When t

, relaxation is complete, the derivative in (6.2) vanishes and a constant con-
centration is reached. This state is known as steady state. The simple relation C i (

τ

∞ ) =

C in /

i ) shows that the right-hand side is the concentration when the initial condition
is fully relaxed ( Fig. 6.2 ) . This is called the forcing term. The forcing term, here the input
from upstream, shows in which direction the system is heading. Again it can be seen that
the final concentration of the lake is lower when the element is more reactive.

(1 + β