Mass Balance Technique
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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.

Figure 2: Schemes of a waste water sewage treatment plant.

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.

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]
C = 400/15 = 26.7 mg/L

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


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 []

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





  ©Copyright 2003 Carnegie Mellon University
This material is based upon work supported by the National Science Foundation under Grant Number 9653194. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.