This problem involves two related, but separate, quantiites: the amount of oxygen required to oxidize organic (waste) matter present in a body of water and the amount of DO available in the water to accomplish this task. The following sections explain the differential equation that governs each of these quantities.
DO and BOD
Dissolved oxygen (DO) is the amount of molecular oxygen dissolved in water and is one of the most important criteria in determining natural water quality. DO also affects wastewater treatment processes. The water's carrying capacity for DO, also known as the DO saturation level, depends on teh temperature of the water. DO saturation levels for different water temperatures can be measured with a DO meter; the values shown below are typical. Note that cooler water contains a greater concentration of dissolved oxygen than warmer water.
Biochemical oxygen demand (BOD) is the amount of oxygen required to oxidize organic matter that is biochemically present in water and is, therefore, an indirect measure of organic water contamination. The greater the BOD, the greater the oxygen depletion in a stream or lake. It is a measure of waste strength insofar as it measures the oxygen-consuming property of waste in terms of oxygen that is biologically consumed. The BOD (in mg/L) after days is
where is the dilution bottle volume, is the sample volume, and is the sample dilution. (For example, the sample dilution would be 30 to 1 for a 10-mL wastewater sample placed inside a 300-mL bottle filled with dilution water.) The BOD is usually measured under controlled conditions, such as a temperature T=20C and darkness (to prevent oxygen-producing algae). and are the levels of dissolved oxygen in the sample bottle at the outset and after days, respectively.
The rate of BOD consumption at each instant of time is proportional to the BOD remaining in the water supply at that time. That is, if is the remaining BOD at time , then
where the deoxygenation rate, >0, (with units of 1/time) depends on a number of factors, including the number and type of microorganisms and the water temperature. Since is positive, the organic contaminants decay exponentially with time. You will learn to use Maple to solve this differential equation in Chapter 6 (see the ftp site). The solution is
where is the BOD remaining at the outset ( ). Let denote the amount of oxygen consumed through time in mg/L. Assuming this is the only process affecting the oxygen content of the water sample, the total amount of oxygenwill be conserved. That is, for all >=0, =constant. Note that the constant can be determined, by measurement, at any instant of time; , the ultimate BOD, is the total amount of BOD (waste) available for consumption (at ). It is also the total amount of consumed oxygen when all the waste is depleted ( -> ).
The DO Sag Curve and the Streeter-Phelps Equation
The Streeter-Phelps equation accurately models the amount of DO in a stream after wastewater is discharge into it. This model follows the pollutant downstream as it travels at the stream velocity. When a pollutant is introduced into a water source, the DO typically decreases to a minimum before gradually recovering to the saturation level. The plot of the DO as a function of time is called the DO sag curve . There are two competing processes in this interaction: reaeration and deoxygenation . Reaeration adds molecular oxygen to the stream from the atmosphere (up to the saturation point); deoxygenation depletes the oxygen. Only the biochemically degradable microorganisms responsible for BOD are considered in the present analysis.
Let denote the oxygenation rate (per day), the reaeration rate (per day), the oxygen deficit in the stream (the difference between the saturation and the actual DO level), and the stream BOD remaining at time . (Note that and are the same quantities discussed in the first part of this application.) The Streeter-Phelps model states that the rate of change of the stream oxygen deficit, , increases in direct proportion to the stream BOD remaining, with proportionality constant given by the deoxygenation rate , since the BOD is an indirect measure of organic water contamination itself. On the other hand, the rate of change of the oxygen deficit decreases in direct proportion to the deficit at time , with proportionality constant given by the reaeration rate . The Streeter-Phelps equation for the oxygen deficit represents both of these interacting processes:
The solution to the Streeter-Phelps equation is
where is the oxygen deficit when the pollutant first enters the stream ( ). Suppose =0.4/day, =2.0/day, =54.8 mg/L, and the initial DO level is 2.2 mg/L at a stream temperature of 21C.
The minimum of the DO sag cureve, which occurs at the sag time , is the time when the oxygen deficit is greatest (minimum DO) and represents the time of greatest stress to fish in the stream. Since the pollutant is flowing downstream at the stream velocity, it is necessary to identify both when and where the minimum is attained.