Mass Balance Technique
The law
of conservation of matter states that matter is conserved--that is, neither
created nor destroyed. 
Thus, if we know the amount of material that enters a chain of processes,
and keep an account of all the amounts in different paths, we can calculate
quantities of materials that are hard to measure. For example, we can
calculate the amount of material entering the atmosphere if we know the
amounts that went in, the transformations, and the waste streams to land
and water. This method is called the Mass or Material Balance technique.
An
example of a process from everyday life is sewage treatment (Figure 2).
Wastewater is generated in your homes and is collected with the sewer
system and transported to a treatment plant. When asked what happens to
the sewage at the plant, most people say that the pollutants are removed
from the water and the relatively "clean water" is then discharged
to a water body. But what happens to the pollutants that are removed?
In the treatment process, these pollutants are transferred from the water
to the air, and to solid material known as sludge, or biosolids. And,
a small amount remains in the "clean water." These waste products
must be taken care of so that they do not affect the environment. A mass
balance can be used to determine how much pollutant is in each of its
various forms.
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Figure
2: Schemes of a waste water sewage treatment plant.
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Another
example (though historic) is the steel industry in Pittsburgh. The processing
of steel requires cast amounts of water that then need disposal. As a
result, many of the early "steel towns" were along rivers because
they provided both the water and the means for disposal. Prior to the
environmental regulations in the USA, the disposal of the process water
was directly to the rivers. However, one of the earliest regulations was
the Clean Water Act that prohibited such disposal without treatment to
remove the process waste contained in that water. Since such treatment
was expensive, the next option was to use the waste process water as cooling
water since vast quantities of that was also needed. However, this led
to air pollution as the water evaporated and transported the impurities
into the air. After air pollution legislation was passed, the industry
operators needed to remove the waste impurities.
These are
just two examples of the search to find a sink for pollution that does
not exist. Many environmental problems have been caused by neglecting
to think of the pollutants in terms of the conservation of matter and
a mass balance.
A mass balance
is an accounting of a material for a specific system boundary. In other
words, you are keeping track of all sources of the material that enter
the system, all sinks of the material that leave the system, and all storage
of the material within the system. A mass balance can be done for four
scenarios, or combinations of those scenarios as follows:
- Dynamic
(flows change over time)
- Steady
State (flows do not change over time; the system is in equilibrium)
- Conservative
pollutants (the pollutant does not change form over time; no reactions)
- Non-conservative
pollutant (the pollutant changes form over time due to chemical, physical,
or biological reactions)
The dynamic
scenario is the most difficult to model mathematically. For this module,
only the steady state conservative and steady state non-conservative scenarios
are discussed to illustrate how the technique can be applied to environmental
systems.
Steady
State Scenario
The accounting
system to track pollutants is as follows:
input rate = output rate + reaction rate
The reaction
rate is equal to 0 if the pollutant is conservative. The reaction rate
can be + or – if the pollutant is non-conservative.
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EXAMPLE:
Two streams enter a lake in the system shown below. The main stream
has a flow of 10 m3/s, and a chloride concentration of
20 mg/L. The tributary stream has a flow of 5 m3/s and
a chloride concentration of 40 mg/L. What is the chloride concentration
leaving the lake system? Note that chloride is a conservative pollutant.
The answer is obtained by balancing the sinks and sources of pollutants
to the lake system as follows:
| [10
m3/s]*[20 mg/L] |
+ |
[5
m3/s]*[40 mg/L] |
= |
[C
mg/L]*[10 m3/s + 5 m3/s] |
| 200 |
+ |
200 |
= |
C*[15] |
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C |
= |
400/15
= 26.7 mg/L |
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Often
the reaction rate is due to biological degradation also known as a decay
rate. The decay rate is often modeled as a first order reaction, which
means that the amount that decays is proportional to the amount present
at any time. In other words:
Ct = [C0]*e-[k*t]
Therefore
for a steady state non-conservative pollutant, the equation needs and
additional term to account for the decay as follows:
Decay rate = -[k*C*V]
k = reaction rate
C = concentration at time
V = volume of the system modeled
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EXAMPLE:
Assume the lake system has a volume of 10*106 m3,
and the pollutant is non-conservative with a decay rate of 0.2 1/day.
Flow and concentrations in the streams are as in the figure below.
What is the concentration of the pollutant leaving the lake system?
Input
= [5 m3/s]*[10 mg/L] + [0.5 m3/s]*[100
mg/L] = 100 [m3.mg/L.sec]
Output
= [5 m3/s + 0.5 m3/s]* [C] = 5.5 * [C]
Decay
= -[0.2 1/day] * [C] * [10 * 106 m3] = -23.1 * [C]
Input
= Output + Decay
C
= 3.5 mg/L
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