i
-.
•
+
inn
I OF
ORNLP
3135

.
.
:
VAR
:
1
. i
o..
1
.
ent.
-
.
EEEEEE
11:25 | 14 |L6
P7777,5-9779477
S
W
T
.
..how-
een
gemeinsa
*
"*-
*
CFSTI PRICES
ONL,313 jt
. Cont. 670616--/
Applications of the Mixing Equation to the Displacement
of Pollution from Lakes
H.C. $
l; MX
JUN 27 1967
R. H. Rainey*
MASTER
A11 of us are well aware of the problems associated with the pollution
of the lakes and rivers in the highly populated areas of the world and of
some of the programs being proposed to stop this pollution. It is generally
thought that, if the addition of pollutants 18 discontinued, natural forces
would restore the waters to their original condition. Although the effects
of pollution could not be completely reversed, an improveinent in water
quality would nearly always result. The time required for this recovery,
.
fresh water displaces the polluted water. This displacement time would vary
from days (or even a few hours) for fast flowing streams to infinite times
for lakes having no exit. The purpose of this paper is to relate the buildup,
or removal, of the pollution to the flow of water thzough a lake, the volume
of the lake, and the amount of pollution added to the lake.
. I have assumed a highly idealized lake system (see Fig. 1) such that:
(1) the flow rate, R, to and from the lake is the same, (2) the concentration
of pollutants in the streams entering the lake, Cz, 18 constant, and (3) the .
pollutants are added to the lake itself at a constant rate, Q, and are .
distributed so that at any given time the concentration, Ca, 18 uniform
throughout the volume, V, of the lake. By writing a material balance around,
this system and integrating with respect to time, T, the following equation
may be obtained :
Oro C+B)(1-2 m.;
where ca is the concentration of pollutants in the lake at the initial time
.(T = 0).
*Oak Ridge National Laboratory
Research sponsored by the U.S. Atomic Energy Commission under contract with !
the Union Carbide Corporation.
. DISTRIBUTION OE THIS DOCUMENT IS UNLIMITED
W
:
***
*
R1 -21 *
!
WALLE.......... T
W 141WXMW
"
2
.
This evaluation ignores many of the following factors that influence the
concentration of pollutants in the lake: precipitation and redissolution of
solide, bacterial action, photosynthesis, etc. Homogeneous dispersal of the
pollutants in the lake, an assumption that was required in order to make the
calculations, is, of course, physically unrealistic; however, the times calcu-
lated on this basis are a compromise between those calculated by assuming a
VILL
fast purging flow and those calculated by assuming slow diffusion.
The applicability of this equation to the determination of the change in
concentration of materials in a lake system depends on how closely the condi-
tions of the system fit the assumptions that were made in deriving the equation.
But even with this limitation, we have found that the times calculated by this
equation are quite useful. For example, calculations indicate that, if the
pollution of Lake Erle were discontinued, the pollution level of the lake would
decrease to 20% of its present level in about 6 years. Since the water from
Lake Erie flows into Lake Ontario, no program for cleaning up Lake Ontario
alone will be effective. Lake Ontario contains about four times as much water
as Erie but has only about a 20% larger flow rate. If the addition of pollution
to both lakes were discontinued, the pollution level in Ontario would rise
for several years as Ontario continued to receive water from the more highly
contaminated Erie. Then in about 20 years, it would decrease to about 10% of
it.8 present level.
These calculations indicate that, although a generation would be required
for a 90% recovery of the lower lakes, an encouraging improvement in the quality
of the water in Lakes Erie and Ontario could be obtained within a few years.
On the other hand, the situation with Lake Michigan and Lake Superior
18 quite different. Because of their very small flow-to-volume ratio, hundreds
of years would be required for the natural displacement of pollution of these
LEGAL NOTICE
Tais report mas prepared as an account of Government sponsored work. Nolthor the United
Stator, por the Commission, nor any person acting on behalf of the Commission:
A. Makos any warranty or representation, expressed or implied, with respect to the accu-
racy, completanoss, or usohulaous of the information contained in this roport, or that the une
of any information, apparatus, mothod, or process dlaclosed in this report may not Infringe
privately owed rights; or .
B. Assumes any liabilidoo with respaot to the use of, or for damages resulting from the
use of any information, apparatus, method, or process diatoned in the report. :
As used in the above, "person acting on behalf of the Commisolon" Includes any om-
ployee or contractor of the Commission, or employee of such contractor, to the extent that
auch employee or contractor of the Commission, or omployee of such contractor prepares,
disseminates, or provides accen to, any information purtuant to his emaployment or contrao!
with the commission, or his employment with such contractor.
w
2
.
.
...
t--
71m
m

.com
lakes. If the Indians had polluted Lake Superior and had discontinued their
bad habits when Columbus set sail for the New World, the pollution in Lake
Superior would not have decreased to 10% of its maximum. concentration untii
the present time.
The equation presented above also illustrates that, if Lake Superior
were to receive the same amount of waste as Lake Erie, the concentration of
vaste In Superior would eventually be greater.
In conclusion, I would like to point out that, although this simple
equation does not accurately describe the many complicated changes taking
place in a lake system, it can give approximations that demonstrate many
of the problems involved in the control of pollution. Whenever the kinetics
of other mechanisms are known, additional parameters can be added to the
equation to improve these approximations.
=

ORNL-DWG 67-6464
F
:
:
.-
.
1
.
-
Cp - ce +(6, +$ 119-6. May
.
-
.
-.
-
.
LL
i.
.

END

.
.
.
.
:
:
:
M
.
VE
i
i
.
.
A
-
;
.
1
Er..
:
2.
DATE FILMED
18 /18 (67
MEL:
is
5.
:
1
:0
1
..