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Modelling water quality

Dissolved oxygen (DO) is one of the most important indicators for the quality of natural water systems as fish and other aquatic animal species require oxygen. Streams must have a minimum of 2 mg/L DO to maintain higher life forms, most game fish require 4 mg/L. When biodegradable organics such as sewage are discharged into a stream, microorganisms convert the organics into new cells and oxidized waste products. During this process, DO is consumed, resulting in the biochemical oxygen demand (BOD, the amount of oxygen required by aquatic decomposers), another important indicator for water quality. 

The oxygen-sag model was first introduced by Streeter and Phelps in 1925 to predict the oxygen concentration and deficit in a stream caused by the discharge of sewage. Consider a stream with constant flow rate and cross-sectional area (and hence velocity), and consider an element of the stream moving with the flow. The change of BOD in this element as function of time, t, can be described by a first order reaction:

                                                                                                                                                                                               (1)

where L is BOD (mg/L) and k₁ (d^-1) is the rate coefficient of biochemical decomposition of organic matter. Note, the oxygen concentration does not appear here (the reaction is first order only in L) and we have assumed there is no diffusion of oxygen or L between different elements of the stream (these approximations are needed to make the model tractable). Since BOD depletion is directly related to the oxygen consumption (i.e. we do not take any other oxygen depletion mechanisms into account), the resulting change of the oxygen concentration due to the biochemical reaction is:

                                                                                  dC_de           dL

 k₁L            (2) dt   dt

where C is the oxygen concentration (mg/L) and the index de stands for de-oxygenation. The main process replenishing the oxygen content in a stream is reaeration, i.e. the uptake of oxygen from the atmosphere across the water surface due to the turbulent motion of water and molecular diffusion. The driving force for reaeration is the difference between the oxygen saturation concentration, C_s (mg/L), and the actual oxygen concentration in the water, C, which is called the oxygen deficit, D = Cs – C:

^dCre  k₂ C_s C k₂D    (3) dt

where k₂ is the atmospheric oxygen dissolution rate coefficient i.e. the re-oxygenation constant (d^-1). Factors affecting k₂ include the stream turbulence, the surface area of the stream, the water depth and the temperature. The resulting balance for the oxygen concentration in the water can thus be written as:

                                                                   dC     dCre           dCde

                                                                                                k₁L k₂D                                                             (4)

                                                                    dt        dt         dt

and the change in the oxygen deficit is given by

                                                                                                                                                                                               (5)

Assuming that the stream flow rate and the flow rate of the waste water effluent are constant and that the mixing of sewage water and the stream water happens instantaneously across the full cross-section of the stream (i.e. all dispersion effects are neglected) a simple mass balance at the mixing points yields for the BOD

                                                                                           L Q  L     Q

                                                                             L0  R                                                                                                                                              R    SW                                                                                                                                                     SW     (6)

QR QSW

and for the oxygen concentration

                                                                                           C Q C      Q

                                                                            C0  R                                                                                                                                             R    SW                                                                                                                                                     SW     (7)

QR QSW

where Q (m³/d) is the flow rate and the indices R and SW stand for river and sewage, respectively. Using equation (7), the oxygen deficit at the mixing point can be calculated:

                                                                                         D₀  C_s C₀                                                                            (8)

The temperature dependence of the rate coefficients and the saturation concentration can be described by equations (9) – (11):

Note that all three equations require that the temperature is given in °C and that k₁(T₀) means k₁ at temperature T₀.

Integrating equation (5) and determining the maximum of the resulting function gives the maximum oxygen deficit, D_cr, and the corresponding time, t_cr:

^L₀ek1tcr

                                                                                       D_cr                                                                          (12)

f

                                                                                  1              f 1D0 

tcr  ln f 1  (13) k₁f 1   L0 

where f = k₂/k₁.

Tasks:

An unpolluted river (L_R = 0) flows with a flow rate of Q_R = 3.8 m³/s and a stream velocity of v = 0.65 m/s. The water temperature is 20 °C. The water is saturated with oxygen and the reaeration constant for this stream at T₀ = 20 °C is k₂ = 0.9 d^-1. Due to an accident, untreated sewage with a flow rate of Q_SW = 0.27 m³/s and a BOD of L_SW = 600 mg/L is discharged into the river. The oxygen concentration of the sewage stream, C_SW, is zero and the rate coefficient of biochemical decomposition of organic matter in this stream at 20 °C is k₁ = 0.3 d^-1. 

a)      Present the profiles of L, C and D as function of the distance from the discharge point. How far downstream does the maximum oxygen deficit occur and does it present any threat to fish?

b)      In a separate plot present the oxygen concentration profile as function of distance from the discharge point for T = 10 °C, 20 °C, 25 °C. Also, calculate the critical oxygen deficit and the corresponding distance. Comment on how these properties change with temperature.

c)      Due to draught and flooding, the flow rate of water in the river can vary. To study the influence of the flow rate on the water quality, calculate the critical oxygen deficit and the corresponding distance from the discharge point for Q_R = 0.6 m³/s, 3.8 m³/s, 18.7 m³/s. Assume that the crosssection of the river remains constant. Comment on the effect. (Note, you do not need to calculate the concentration profiles for C).

d)      The oxygen-sag model as presented by Streeter and Phelps is the simplest model to model the water quality but it neglects important phenomena that influence the oxygen content of the water. The extended BOD model also takes into account the sedimentation of biodegradable organic matter, benthic oxygen demand (i.e. BOD due to the decay of organic matter in sludge deposits) and the net creation of oxygen by aquatic plants.

For this model, equation (1) becomes:

                                                                                                                                                                                               (14)

where k₃ is rate coefficient for BOD removal by sedimentation (k₃ = 0.3 d^-1) and B is the benthic oxygen demand (B = 1.5 mg/(L d)). To take the net production of oxygen due to photosynthesis into account, equation (5) has to be modified to 

                                                                                                                                                                                               (15)

where P is the rate of oxygen addition to the water by the photosynthetic activity of aquatic plants and algae. Here, P = 0.6 mg/(L d). Compare the oxygen concentration profiles as function of distance from the discharge point for the oxygen sag-model and the extended BOD model and explain your observations. Use the same function file as for the other tasks to answer question d).

e)      What other factors should be taken into account in more realistic models?

For this hand-in exercise it is required that you create a script file from which a MATLAB report can be produced. A worksheet on publishing reports plus several videos is available on Learn in the folder for the hand-in exercise. Your published report should contain the code as well as your comments, discussion and plots. Note that only your script file needs to be published as a report, not the function file, but you must submit your script and function file. 

References:

Streeter N. W., Phelps E. B. (1925); A Study of the Pollutions and Natural Purification of the Ohio River. Public Health Bulletin NO. 146. U.S. Public Health Service

The deadline for the hand-in exercise is Monday, 5/12/2016, 16:00.

Please submit the following files in the drop-box on Learn:

•      Your script file formatted in such a way that a report in MATLAB can be produced as a PDF file. Do not submit the PDF report itself. We will use your MATLAB files to create the report.

•      Your function file containing the rhs of the ODEs. Note that you only need one function file to describe both models.

If you experience any problems accessing Learn, please e-mail your files directly to ... before the deadline.

Marking scheme (detailed marking criteria are available on Learn)