1. 6. CONVENTIONAL POLLUTANS IN RIVERS
If we could first know where we are and whether we are tending, we could better judge what to do, and how to do it.
- Abraham Lincoln

2. 6.1. INTRODUCTION
The use of aerobic biological treatment for domestic wastewater dates back to the late 19th century, and it has been a standard method in the United States since the 1930s. Wastewater treatment has been directed primarily toward removal of “conventional pollutants” since that time.
Conventional pollutants:
biochemical oxygen demand (BOD);
ammonia-nitrogen;
total suspended solids (TSS);
total and fecal coliform bacteria.

3. BOD - important pollutant because it is a measure of biodegradable organics in water and the oxygen that will be consumed in the process of microbial degradation.
DO - probably the single most important chemical parameter that is required to ensure the ecological health of a receiving water.
It is important to control ammonia because it is toxic, even at low concentrations, to fish and other aquatic biota, and it is an additional oxygen-demanding chemical in aquatic systems via nitrification.
Ammonia is in chemical equilibrium with ammonium ions in natural waters. Ammonium is a weak acid with a pKa of about 9.2 at 25 ℃, so over the pH range of natural waters, , ammonium ion predominates.

4. Total suspended solids: third conventional pollutant because they exacerbate a dissolved oxygen problem by sedimentation and forming an oxygen-demanding sludge deposit ; they cause turbidity in the receiving water and may alter the habitat of aquatic biota; and, perhaps most importantly, they can harbor pathogens (disease-causing  microorganisms).
TSS are controlled in aerobic wastewater treatment plants by sedimentation and thickening processes, and they are the ultimate by-product of what amounts to a bioconversion process of transforming soluble organic material into settleable (and some not so settleable) solids.
Coliform-bacteria (both fecal and total) are used as indicators of pathogens in rivers and streams, and they are sometimes included in consideration of conventional pollutants. They can be efficiently killed by chlorination of the effluent at wastewater treatment plants, but at the risk of formation of toxic chlorinated organic compounds. The natural die-away of coliform bacteria can be modeled as a first-order decay reaction. It is much faster in marine waters than fresh water.

5. Table 6.1. Percent Un-ionized Ammonia as a Function of Water Temperature and pH.

6. Effluent permits on municipal wastewater discharges of BOD and TSS are 30 mg L-1 in effluent-limited water quality segments.
Discharges of ammonia and ammonium are not regulated except in cases of ammonia toxicity (> 2 mg NL-1) or in water-quality-limited segments, where ammonia-nitrogen or D.O. concentrations may be expected to violate water quality standards in the receiving stream.
Situations in which the ammonia-nitrogen or dissolved oxygen water quality standards are expected to be violated require a waste load allocation, that is, a mathematical model of the stream segment with important wastewater discharges included as inputs.
Waste load allocations are performed as a special designation for stream segments that are not in compliance with water quality standards. Mathematical models are necessary to determine allowable loadings for each plant as a part of their permit.
Water quality modeling of rivers goes back to 1925 and the Streeter-Phelps equation for dissolved oxygen concentrations in the Ohio River. There have been many developments since that time that make it possible to model other settings in greater detail and with more realistic processes.

7. New research questions:
Coagulation of particles in natural waters. Coagulation affects the rate of chemical sorption and TSS sedimentation to bottom sediments.
In-place pollutants and sediment oxygen demand (SOD). Many pollutants at the sediment-water interface were deposited historically but continue to affect water quality.
Sorption of toxic organics and metals by suspended solids and macromolecules.
Organics (BOD) degradation. Fate of individual trace organic compounds in rivers and natural waters is needed. One can define the bulk of organic material (BOD) as a primary substrate and trace organics as a secondary substrate.

8. 6.2. MASS BALANCE EQUATION: PLUG-FLOW SYSTEM
Conventional pollutants are discharged to rivers or streams because the current removes them from the source swiftly, and eutrophication is not as great a problem in flowing waters as compared to lakes. We begin by writing a mass balance equation around a control volume, an incremental element (slice) of stream volume (Figure 6.1).  If the current is sufficiently fast, we may model the stream as a plug-flow system.
Figure 6.1. Schematic of flow through a stream and incremental control volume.

9. - written for steady flow (dQ/dt = 0) conditions.
Also assuming a constant cross-sectional area (dA/dx) that also requires a constant velocity and incremental volume
With a constant incremental volume, we may divide through by the control volume (V = AΔx).
Below eq`n result in the steady-state ordinary differential equation for a stream with reactions or written as an integral equation:
± r

10. We may solve equation (an ordinary differential equation) by the method of separation of variables. We move the dependent variable (C) to the left-hand side of the equation and the independent variable to the right-hand side, and then we may integrate to solve for the concentration as a function of stream distance.
Integrating:
Integrating we obtain below eq`ns:
Taking the exponent of both sides, we find:
where: C0 is the concentration at the source (x = 0).

11. 6.2.1. Steady Flow Mass Balance in a Stream: Variable Coefficients
Above eq’s are the general forms of the stream equation assuming steady flow (dQ/dt = 0), constant velocity, and general reaction kinetics.
Frequently: flow is not steady or the parameters and coefficients (Q, A, u, k) are variable with distance or time.
Example: suppose that the flowrate and cross-sectional area increase over a stretch of stream in the following manner:
Then the mass balance around the incremental control volume (shown in Figure 6.1) is now:

12. Multiplying through we obtain eq`n:
For constant-volume systems (steady flow), the first term on the left-hand side of above equation is zero (dV/dt = 0), and dividing through by the incremental control volume we obtain:
The third term on the right-hand side of last equation is assumed to be negligible because the product of two incremental changes should be small relative to other terms in the equation.
Final result may be expressed as:

13. Last equation is the general equation for one-dimensional transport in a stream or river (neglecting dispersion).
All coefficients (Q, A, r) may be functions of distance and time, and the resulting equation may be solved analytically, in simple cases, or numerically by the method of characteristics.
Under steady-state conditions, the partial differential equation reduces to an ordinary differential equation:

14. 6.3. STREETER-PHELPS EQUATION
1925: Streeter and Phelps published a seminal work on the dissolved oxygen "sag curve" in the Ohio River.
They were able to demonstrate that the dissolved oxygen decreased with downstream distance due to degradation of soluble organic biochemical oxygen demand (BOD), and they proposed a mathematical equation to describe the phenomenon, which has since become widely known as the Streeter-Phelps equation.
The oxidation of carbonaceous biochemical oxygen demand is generally written as a first-order reaction, although there have been some studies indicating a dependency on oxygen concentration as well as BOD concentration.

15. For a constant-velocity stream and steady-state condition:
L = ultimate BOD concentration, ML-3
= mean velocity, LT-1
kd = first-order deoxygenation rate constant, T-1
For dissolved oxygen, it is possible to write a mass balance equation:
where: C = D.O. concentration, ML-3
ka = first-order reaeration rate constant, T-1
Cs = saturated D.O. concentration, ML-3

16. The two terms on the right-hand side of last equation represent opposing processes in streams, the rate of deoxygenation due to carbonaceous BOD versus the rate of reaeration. Cs – C is the concentration driving force serving to reoxygenate the water from atmospheric oxygen at the air-water interface. It is preferable to write last equation as a D.O. deficit equation (D).
Note that the sign changed on the deoxygenation and reaeration terms because D.O. deficit (D) is equal to the driving force of oxygen reaeration (which is opposite in sign to the D.O. concentration).

17. We need a simultaneous solution of above equations in order to reproduce the Streeter-Phelps equation. It is a set of ordinary differential equations, but they are uncoupled since equation can be solved directly for the BOD concentration (L) with distance, and the equation for L can then be substituted into equation.
The solution of above equation for the BOD concentration is given by equations below:
Substituting equations, we may solve for the D.O. deficit by separation of variables or the integration factor method:

18. Last equation can be rearranged so that all of the terms involving the dependent variable (D) are on the left-hand side of the equation, and the forcing function is on the right-hand side:
Using the integration factor method, this equation is of the form:
with the solution over the interval from 0 to t:
A comparison of above equations allows for the following definitions:

19. The solution to above equation may be determined from the general solution.
Last equation is the final solution for dissolved oxygen deficit versus distance after a point source discharge of BOD in a one-dimensional, steady-state, plug-flow system.
We could have also solved for the D.O. concentration rather than the deficit. The solution is given by equation below by substituting (Cs-C) for D in above equation.

20. Figure 6.3.
Streeter-Phelps classical D.O. sag curve. Top: ultimate BOD concentration decreases exponentially with distance.
Middle: D.O. deficit reaches a maximum point where the deoxygenation rate in the stream equals the reaeration rate.
Bottom: D.O. sag curve is critical at distance xc.

21. 6.3.1. Critical Deficit and Distance (Dc, xc)
The critical deficit and the downstream distances under steady-state conditions can be solved explicitly using above equation.
Solving for the critical deficit (Dc), one arrives at equation:
Substituting above equation into the equation for dissolved oxygen deficit allows one to solve for the critical distance (xc).
If the initial dissolved oxygen deficit is zero at x = 0 (D.O. is saturated, Cs), then above equation simplifies to:

22. Various inputs and parameters have an effect on the D. O
Various inputs and parameters have an effect on the D.O. sag curve and D.O. deficit curve shown schematically in Figure 6.3. The self-purification ratio was defined by Fair and Geyer to be ka/kd, the ratio of the reaeration rate constant to the deoxygenation rate constant. It is an important dimensionless number that strongly affects both Dc and xc.
The initial concentration of ultimate BOD (L0) is directly proportional to the critical deficit concentration; the greater is the initial BOD concentration, the greater will be the dissolved oxygen deficit concentration at xc. L0, the ultimate BOD concentration at x = 0, is equal to the mass discharge rate of ultimate BOD diluted by the flowrate of the river.
If only BOD5 (5-day BOD) concentrations are available, a suitable conversion factor should be used to convert to ultimate BOD concentrations.
where: W = wastewater mass discharge rate, MT-1
Q = flowrate of the river, L3T-1

23. Here, instantaneous mixing of the wastewater discharge in both the lateral dimension (across the width of the stream) and the vertical dimension with depth have been assumed.
If the flowrate of the wastewater discharge cannot be neglected relative to the stream discharge rate and/or if the ultimate BOD concentration of the upstream segment cannot be neglected relative to the BOD concentration below the discharge, then a flow-weighted average concentration should be calculated for according to equation:
where the subscript w refers to the waste discharge and s refers to the stream. The effect of the wastewater discharge as measured by W or L0 is to increase the D.O. deficit curve throughout all x; the critical distance (xc) does not change, but the critical deficit concentration (Dc) changes linearly with W or L0. See Figure 6.4.

24. Temperature plays a role in D. O. reaeration kinetics, D. O
Temperature plays a role in D.O. reaeration kinetics, D.O. solubility (Cs), and deoxygenation (microbial degradation of BOD). The three effects are in opposition.
1. D.O. reaeration rate constant increases with increasing temperature.
2. D.O. solubility decreases with increasing temperature so the driving force (D) for reaeration decreases somewhat.
3. Deoxygenation rate constant increases with increasing temperature.
If activation energy is constant over a small temperature range (not always the case), the Arrhenius equation can be simplified:
where: Ea - the activation energy;
R - the universal gas constant;
T - absolute temperature;
A - the pre-exponential constant.

25. For deoxygenation and reaeration rate constants:
Theta (θ) for deoxygenation is normally taken to be and for reaeration it is Thus the effect of increasing temperature is greater on deoxygenation compared to reaeration, and this causes the critical deficit to increase and to move upstream as depicted in Figure 6.4b.
Increasing temperature also results in a marked decrease in dissolved oxygen saturation concentration, which decreases the dissolved oxygen concentration (C) by limiting the driving force for reaeration (Cs - C).

26. Figure 6.4.
Effect of waste discharge (W), temperature (T), flowrate (Q), and initial deficit (D0) on the critical dissolved oxygen deficit concentration (Dc).

27. Increasing stream flow has two effects on the dissolved oxygen deficit profile. First, the most important effect is on initial BOD concentration (L0), which decreases proportionately with flowrate: L0 = W/Q.
Second, the flowrate influences reaeration rate constants in complex fashion. Reaeration rate constants are often expressed as a power function of mean velocity and an inverse power function of mean depth.
Generally, the effect of increasing flowrate on mean depth (to the n power) is larger than the effect on mean velocity (to the m power), so the reaeration rate constant decreases with increasing stream flow. This is a somewhat counterintuitive result that is not true in the tailwaters of dam releases and rapids.

28. At any rate, the overriding effect of increased flowrate is on the initial ultimate BOD concentration, L0 at x = 0, and the net effect is to "push" the D.O. deficit curve further downstream and to decrease the critical deficit (Dc) as shown in Figure 6.4c.
Effects of flowrate on stream physical parameters can be summarized by equations:
where Q is the flowrate and a = , b = , c =

29. 6.3.2. Reaeration Rate Constants
O'Conner and Dobbins developed the first model equation for reaeration rate constants in streams in It was based on Danckwert's surface renewal theory, which can be approximated by
kL = controlling liquid-film mass transfer coefficient;
Dm = molecular diffusion coefficient for oxygen in water;
r = surface renewal rate;
l = Prandtl's mixing length (the average distance that the vertical velocity fluctuation travels).
The hyperbolic cotangent portion of equation was approximately equal to 1.0, and it was dropped. Rate of surface renewal is proportional to shear stress at the surface because shear at the surface causes vertical mixing. A definition of surface renewal can be stated as

30. O'Conner and Dobbins argued that, for flume and field studies of the Mississippi River, the average vertical velocity fluctuation was approximately one-tenth of the average longitudinal velocity, and that the vertical mixing length was about one-tenth of the mean depth of the stream. Therefore, for isotropic turbulence,
Substitution of equations yields the O'Conner-Dobbins reaeration formula, which is theoretically based except for the assumptions on vertical velocity fluctuations and mixing length.
For uniform dissolved oxygen concentration with depth, the reaeration rate constant is simply the mass transfer coefficient divided by the mean depth.

31. In English units, the final reaeration rate constant is given by equation:
For a smooth quiescent surface, the mass transfer coefficient of atmospheric oxygen into streams is approximately 0.7 m d-1 Thus a stream of 2 m mean depth would have a reaeration rate constant equal to 0.35 day-1 under quiescent conditions. The presence of ripples or a broken surface would increase the river reaeration rate constant markedly.
There have been numerous other reaeration formulas developed, pertaining to different stream velocities, usually empirical power-function relationships of the form
Of these formulas, the Churchill-Elmore-Buckingham has been the most applied, especially for large rivers (Table 6.2).

32. Table 6.2.
Stream Reaeration Formulas

33. 6.4. MODIFICATIONS TO STREETER-PHELPS EQUATION
When Streeter and Phelps (1925) developed their equation for dissolved oxygen concentrations in the Ohio River, the principal dissolved oxygen (D.O.) sink was carbonaceous biochemical oxygen demand (CBOD), and reaeration was the primary source of D.O. But several other sources and sinks of dissolved oxygen are common in streams and rivers:

34. Nitrogenous biochemical oxygen demand (NBOD) is analogous to CBOD, except that it emanates from reduced nitrogen species, especially ammonium.
Often, ultimate NBOD is assumed to be the stoichiometric equivalent of the ammonium concentration. We can estimate the mass equivalency as 64/14 = 4.57 mg NBOD per mg NH4+-N on a stoichiometric basis:
Thus, if the ammonium concentration in the stream is 2.0 mg NH4+ as N per liter, the concentration of NBOD would be equal to 9.14 mg L-1 NBOD.
Nitrifying bacteria are ubiquitous in flowing waters; they are often periphytic, attached to rocks, or they can be planktonic.

35. Sediment oxygen demand is a major sink for dissolved oxygen in heavily polluted streams and rivers. A consequence of particulate BOD and algae sedimenting to the stream bed, the organic "ooze" can create a major oxygen demand on the overlying water.
In areas of intense pollution, the sediment oxygen demand (SOD) is in the range of g m-2 d-1. For natural streams and rovers with small wastewater discharges, the SOD is typically g m-2 d-1.
Sediment oxygen demand is measured with a benthic respirometer, a box that is lowered onto the sediment with a flange for a tight fit, a D.O. probe, and a small stirrer.
Dissolved oxygen concentrations are recorded continuously on shipboard to determine the sediment oxygen demand (decrease in dissolved oxygen concentration within the respirometer) per time and unit area of the box.

36. Net primary production is defined as gross primary production minus respiration. Biomass of algae predominates in most streams and rivers compared to fish, higher plants, and macroinvertebrates.
Therefore dissolved oxygen models include a term for net primary production by algae and phytoplankton. Usually the dissolved oxygen model is applied under steady-state conditions, and the net primary production is considered as a daily average value.
Photosynthesis can be considered as varying sinusoidally during the photoperiod, while respiration occurs continuously. Peak concentrations of oxygen in a stream are expected to occur at midday, but if one wants to measure the worst case conditions, they would occur just before sunrise when algal blooms have consumed significant concentrations of dissolved oxygen during night respiration.
Respiration amounts to an average of 2.5% of gross photosynthesis, so the net effect of phytoplankton is to supply dissolved oxygen to the stream prior to death and sedimentation.

37. D = dissolved oxygen deficit concentration, mg L-1
Background BOD is indicative of nonpoint source pollution to streams and rivers. A continuous input of CBOD occurs due to runoff from agricultural land, stormwater discharges, and highway runoff.
These background concentrations of BOD cause a general background D.O. deficit under steady-state conditions. For a plug-flow stream, one can write the mass balance equation for D.O. deficit, D:
D  = dissolved oxygen deficit concentration, mg L-1
x = longitudinal distance, km
u = mean velocity, km d-1
ka = reaeration rate constant. day-1
kd = deoxygenation rate constant, day-1
L = carbonaceous ultimate BOD, mg L-1
Under steady-state conditions and a continuous background concentration of CBOD, the background D.O. deficit concentration is a constant.

38. One may think of Db as a kind of safety factor that water quality engineers apply in waste load allocations to assure a conservative estimate of the stream's assimilative capacity.
In addition, background D.O. deficits of some magnitude are almost always present in streams as evidenced by the lack of saturated D.O. concentrations far from waste discharges.
Table 6.3 gives ranges of model parameters that can be used in dissolved oxygen models for streams and rivers.
Coliform bacteria (both focal and total) are considered as conventional pollutants, and their downstream concentrations are estimated by the following simple exponential equation:
C  is the coliform concentration in colony forming units, cfu mL-1 or most probable number, mpn mL-1; C0 is the initial concentration of coliform bacteria at x = 0; and k is the first-order die-away rate constant, day-1.

39. Table 6.3. Method of Determination and Range of Dissolved Oxygen Model Rate Constants at 20℃ with Typical Temperature Correction Factors for Rivers.

40. Table 6.3. (Continuation).

41. Sedimentation of CBOD
Streeter-Phelps equations imply that CBOD is entirely soluble and that loss of CBOD in the stream is only by deoxygenation (oxidation of soluble organics).
However, particulate CBOD is discharged at wastewater treatment plants as well. Domestic treated wastewater is permitted to contain 30 mg L-1 of CBOD and 30 mg L-1 of total suspended solids (TSS).
The total suspended solids contain organic matter that may be exerted in a BOD test. If CBOD is in particulate form, it will settle out of the water column.
We may modify the Streeter-Phelps equations slightly to include the possibility of CBOD sedimentation.

42. u = mean velocity, LT-1 L = carbonaceous BOD, ML-3
kr = total CBOD loss rate constant, T-1
    = ks + kd
L0 = initial CBOD concentration at x = 0, ML-3
x = longitudinal distance, L
D = D.O. deficit, ML-3
kd = CBOD deoxygenation rate constant, T-1
ka = reaeration rate constant, T-1

43. Above equation is the mass balance equation for a plug-flow stream.
By substituting for L in above equation, we can solve equation by the integration factor method.
It is similar to the Streeter-Phelps equation except that kr replaces kd in two places.
The shape of the D.O. deficit curve (the D.O. "sag" curve) is similar to Streeter-Phelps , but the CBOD concentration in the stream decreases more quickly near the discharge point due to sedimentation.
Figure 6.5 shows how to determine kr and kd from a semilog plot of carbonaceous BOD (L) versus travel time (x/u). Two distinctly different slopes can be estimated that define the rate constants.

44. Figure 6.5.
Semilog plot of CBOD concentra-tion in a stream versus travel time in days. Two slopes are evident: kr, includes sedimentation of particulate matter and deoxygenation; kd is for deoxygena-tion alone.

45. 6.4.2. Principle of Superposition
Modifications to the Streeter-Phelps steady-state dissolved oxygen model include consideration of NBOD, sediment oxygen demand, net primary production, and background D.O. deficit (nonpoint source runoff of BOD).
These additional sources and sinks of dissolved oxygen can be solved individually, and their solutions can be summed to yield the overall solution.
Because the dissolved oxygen mass balance equation is a linear differential equation, we can invoke the principle of superposition to obtain the solution (i.e., summation of the various solutions for each source and sink).
Figure 6.6 : the schematic of the D.O. model.
There we can see how each source and sink term contributes to the D.O. deficit concentration (D).

46. Figure 6.6.
Schematic illustration of D.O. deficit (D), carbonaceous BOD (L), and nitrogenous BOD (N) with nitrogenous deoxyge-nation (kn), carbonaceous deoxygenation (kd), reaeration (ka), sedimentation (ks) of CBOD, net photosynthesis (P-R), and sediment oxygen demand (S).
If respiration is greater than photosynthesis (nighttime conditions), then net photosynthesis is negative, and the term (R - P) contributes to the D.O. deficit. During daylight hours, primary production of oxygen is greater than respiration and dissolved oxygen is created. Using a steady-state approach implies that a daily average (P - R) value is used in the model. Of course, critical (worst case) conditions occur before sunrise when R > P.

47. The overall mass balance equation for D. O
The overall mass balance equation for D.O. deficit in a plug-flow river can be developed from a control volume approach.
Parameters and units are defined in Table 6.2, and H is the mean depth of the river. Assuming steady-state, we can rearrange the equation into the form of a linear ordinary differential equation with forcing functions on the right-hand side of the equation (nonhomogeneous). After substituting for L and N in above equation we have the final equation:
Last equation is a general nonhomogeneous ODE that is solvable by the integration factor method. The overall solution is given by below equation:

48. The first term on the right-hand side of the equation is the die-away of initial deficit due to reaeration; the second term is due to initial carbonaceous BOD (L0); the third term is due to initial nitrogenous BOD (N0); the fourth term is due to sediment oxygen demand; the fifth term is due to (R - P); and the last term is due to background BOD (Lb) caused by nonpoint sources.
Table 6.4 shows the individual solutions for each source and sink term that can be summed to give last equation.
Figure 6.7 shows the graphical solution for each source and sink term that can be summed to give the overall solution.
The solutions for carbonaceous BOD and nitrogenous BOD in the river are given by below equations:
Other sources and sinks such as sediment oxygen demand, net prima-ry production, and background BOD are assumed to be constant within the stretch of the river that is being modeled.

49. Table 6.4. Source and Sink Contributions to the Dissolved Oxygen Deficit Equation and Solution Equation in a Plug-Flow River.

50. Figure 6.7. Sources and sinks of D.O. deficit in a stream

51. 6.5. WASTE LOAD ALLOCATIONS
When water quality standards are expected to be violated even under conditions that all dischargers meet their effluent permits, a waste load allocation must be performed using water quality models.
Waste load allocations are performed for critical conditions (worst case); usually this occurs under low flow (7-day, 1-in-10-year recurrence) in the summer at night (no primary production).
QUAL-2EU simulates dissolved oxygen with all the terms included in Table 6.4, but it also includes eutrophication and nutrients, including organic nitrogen, ammonia, nitrite, nitrate, organic P, phosphate-P, phytoplankton biomass and chlorophyll a, coliform bacteria, and temperature (heat balance). It allows for dispersion in large rivers and includes a Monte Carlo uncertainty analysis option. It can be run under steady-state or dynamic water quality conditions, but the flow regime is steady.

52. River Segmentation
A new river segment is necessary when initial conditions or parameters change in a dissolved oxygen model that is to be used for a waste load allocation.
This process of dividing the river reach into segments of constant coefficients is termed "segmentation." For example, in above equation, changes in the initial concentrations D0, L0, and N0 would require a new stream segment. Initial concentrations may change due to:
Tributary input or confluence.
Wastewater discharge.
Dam or rapids (rapid reaeration).

53. L0 = initial CBOD concentration at x = 0, ML-3
When a wastewater discharge enters the river, a flow-weighted average mass balance of the streams is computed for the new initial conditions, assuming instantaneous complete mixing at x = 0.
L0 = initial CBOD concentration at x = 0, ML-3
W = wastewater discharge, MT-1
Ls = upstream CBOD concentration entering x = 0, ML-3
Q = river flowrate, L3T-1
Qw = wastewater flowrate, L3T-1
If the wastewater mass of CBOD is large compared to upstream sources and if the wastewater flowrate is small relative to the river, above equation reduces to the following:
Similar mass balances are performed for NBOD to obtain N0 at x = 0.

54. C0 = initial mixed D.O. concentration, ML-3
The initial D.O. deficit concentration is somewhat more complicated. The dissolved oxygen concentrations and temperatures should be averaged first, then the new D.O. deficit (D0) is calculated based on the flow-weighted average temperature.
C0 = initial mixed D.O. concentration, ML-3
Q1 = main river flowrate, L3T-1
C1 = main river D.O. concentration entering x = 0, ML-3
Q2 = tributary flowrate, L3T-1
С2 = tributary D.O. concentration entering x = 0, ML-3
T0 = initial mixed temperature,℃
T1 = main river temperature,℃
T2 = tributary temperature,℃
D0 = initial mixed D.O. deficit, ML-3
Csat = saturation concentration of D.O. at T0, ML-3

55. Changes in other parameters that are assumed constant in above equation would require a new segment, such as:
Velocity, u (changes in Q or area).
Sediment oxygen demand, S.
Net primary production, P - R.
Background CBOD due to nonpoint source pollution, Lb.
Reaeration rate constant (changes in velocity or depth).
Figure 6.8 shows how a river has been divided into four segments of constant coefficients and the subsequent model simulation.
Segment 1 was created due to a new waste discharge at x = 0, which affected D0, L0 and N0. Segment 2 was necessary because a small dam at x = 20 km caused reaeration and a step-function change in D0.
Segment 3 was created due to another waste discharge, which affected D0, L0 and N0 at x = 35 km (distance variable is reset to zero).
Segment 4 was created due to the confluence of the river with a large tributary, which changed D0, L0, N0, u and ka. At the beginning of segment 4, x = 45 km , the x variable was once again reset to x = 0 and model above equation was reapplied.

56. Figure 6.8. Segmentation of a river into four segments with constant coefficients for dissolved oxygen modeling.

57. Diurnal Variations
Waste load allocations for small rivers are often complicated by algal productivity and diurnal swings in dissolved oxygen concentrations during critical conditions (7- day, 1-in-10-year low flow).
In these cases, it is necessary to solve the time-variable mass balance differential equation. Time-variable (or dynamic) solutions are necessary because water quality standards are written in terms of a daily average minimum allowable D.O. concentration (5 mg L-1) and a minimum concentration that cannot be violated at any time (4 mg L-1).
Thus it is necessary to know the minimum D.O. concentration in the diurnal cycle at the critical point in the stream. Normally it occurs at sunrise after an entire night of respiration without photosynthesis. Daily average D.O., concentrations can then be computed from the mean of the daily fluctuations.

58. C = D.O. concentration, ML-3
The mass balance equations are written in terms of D.O. concentra-tion (rather than D.O. deficit) because the standards are in terms of concentrations necessary to protect fish and aquatic biota.
C = D.O. concentration, ML-3
t = time, T
Q = flowrate, L3T-1
A = cross-sectional area, L2
x = longitudinal distance, L
ka = reaeration rate constant, T-1
Cs = saturation D.O. concentration, ML-3
kd = CBOD deoxygenation rate constant, T-1
kn = NBOD deoxygenation rate constant, T-1
L(x) = CBOD concentration with distance, ML-3
N(x) = NBOD concentration with distance, ML-3
P(x, t) = primary production as a function of x and t, ML-3T-1
R(x) = plant respiration with distance, ML-3T-1
S(x) = sediment oxygen demand with distance, ML-2T-1
H = mean depth, L

59. Last equation can be solved numerically using the method of characteristics. In essence, the method follows a parcel of fluid down the river at constant velocity, computing variations in time. We obtain the concentration profile in space and time by recovering the results at each spatial increment (Δx) and time step (Δt).
In order to get started, one needs a concentration profile with distance at t = 0, and a diurnal cycle of concentrations at the head end of the river segment (x = 0). The diurnal cycle is repeated at x = 0 until a periodic solution is obtained throughout the segment. The solution is representative of worst case (critical) conditions.
The model should be calibrated and verified with independent sets of field data that are roughly representative of low-flow conditions. Then the actual waste load allocation can be performed under mandated conditions (7-day, 1-in-10-year low flow) to obtain the allowable discharge (s) to the river that do not violate water quality standards.
Field survey data are very sensitive to cloud cover, which reduces primary production, lowering the average D.O. concentration throughout the day. Also, estimates of P(x, t) and R(x) are difficult to obtain.

60. Last equation requires a forcing function for primary production in each segment of the river. The rate of photosynthetic oxygen production, P(t), may be represented as a half-cycle sine wave:
Pm = maximum rate of primary oxygen production, mg L-1 d-1
ts = time at which the source begins, days
f = fraction of the day over which the source is active (usually 1/2 day)
In order to extend the periodic function of primary production for more than 1 day, a Fourier series is used:
There is a diurnal variation in oxygen concentration at the head end of the mend, which can be handled in a manner similar to above equation.

61. 6.6. UNCERTAINTY ANALYSIS
In waste load allocations and, for that most environmental modeling, government regulators would like to know more than the average concentrations that one expects for a particular pollutant. They would also like to know how certain we are of the answer.
In statistical terms, this means that the modeler should provide the decision-maker with the best estimate of the pollutant concentration (the mean) and the variation expected (the standard deviation).
This allows the decision-maker more information to decide whether to act. Under random sampling, the expected value of a variable is its mean, and the measure of its variation is its variance (the standard deviation squared).

62. Because environmental models are simplified approximations of reality, they contain errors. In using models and field data, we embrace errors of several types. Errors propagate through models to cause uncertainty in the final result. This section is about "error analysis" or the equivalent term, uncertainty analysis.
Models include the following types of error:
Model error.
Errors in the state variables (dependent variables and initial  conditions).
Errors in the input data used to drive the model.
Parameter error (rate constants, coefficients, and independent variables).

63. The first type of error, model error, is the incorrect formulation of the model, that is, the difference between the environment and the differential equations in the model. Processes may have been omitted or included improperly; more state variables may be needed to describe the ecosystem; and unforeseen reactions may become important in different ranges of parameter space.
In all cases, the only indication of model error is that the field data do not match model predictions. Model error is the most difficult type to consider in uncertainly analyses, but one approach is to use several different models for the same application.
By "flexing" the models in simulations of a wide variety of field conditions, it helps to accentuate the differences among the models. If a model performs better under certain circumstances, one can discover the cause of model error.
Other errors in state variables, input data, and parameters are best addressed through the following methods: sensitivity analysis, first-order analysis, or Monte Carlo simulations.

64. Figure 6.10.
Diurnal dissolved oxygen data from Station 2A at x = 1.6 mi.
Data from Station 2 at x = 2.7 mi.
Data from Station 3 at x = 6.8 mi.

65. Table 6.5. Model Input Data for Calibration, South Fork Shenandoah River

66. First-Order Analysis
First-order analysis is based on the approximation of variation in the model-dependent variable(s) with variation in the independent variable(s) (or parameters). We measure the variation in the parameters by their statistical moments (the variance).
Consider a function or a set of equations where y is the dependent variable and x is the independent variable:
If the function is smooth and well behaved, and provided that the variance of x is not too large, the expected value of is approximately equal to the value of the function with the expected value of x:
We can write a Taylor series expansion of f(x) to approximate the function:
If we "linearize" last equation by using only the first two terms of the equation, we can consider the expansion of the function about its mean:

67. The variance of f(x) is
where S is the sample standard deviation and S2 is the sample variance of the function about any point from the mean, .
For a multivariate relationship, we must consider the covariation among independent variables:
where ρxixj is the correlation coefficient in a linear least-squares regression between xj and xi variables.
Above equation provides the basis for first-order uncertainty analysis.
First-order uncertainty analysis has the disadvantages of being an approximation to the real solution - it may be erroneous for highly nonlinear equations due to the variance of higher order terms.
Nevertheless, it has been shown to be quite an effective predictor of error in many environmental models, and it preserves the covariance structure of the model parameters properly.

68. Monte Carlo Analysis
Monte Carlo analysis is a “brute force” solution of the problem of uncertainty analysis in environmental model. One uses a random number generator to select parameter values from a known or suspected distribution (probability density function). After the proper number of parameters have been randomly selected (synthetic sampling), the model is run and the results of the simulation are saved.
Then the process is repeated over and over until a statistically significant number of simulations (large number of realizations) have been recorded. From the distribution of results, a mean predicted value and a standard deviation can be calculated that reflects the combined uncertainty of all parameter errors.
The advantages of Monte Carlo simulation are that it does not require linearization of the model equations and it is, in theory, nonparametric.
Monte Carlo analysis is computationally expensive for large models, and it doesn't preserve the covariance structure among the parameters (unless one performs the synthetic sampling in a special manner to obtain a realistic set of outcomes).

69. Parameter distributions may be determined by multiple samples collected in the field, determined in microcosm experiments in the laboratory, or from expert judgment.
Typical distribution of parameters are normal distribution, lognormal, “hat” function and trapezoid function.
Normal distributions are thought to arise from "additive" processes in the environment.
Perhaps the most common distribution of parameters in the environment is the lognormal distribution. Below equation is the pdf for a lognarmal distribution:

70. Monte Carlo analysis offers some advantages over first-order analysis.
Steps in the analysis and computer program for Monte Carlo include the following:
1. Determine the uncertainty distributions (pdf) for each parameter, input function, and variable that you want to analyze.
2. Sample each of the parameter distributions using a random number generator that takes into account the probability of each value. Most computer operating systems and some spreadsheets have a random number generator that you can call to synthetically sample your parameter distributions prior to each model simulation.
3. Use the set of parameter values selected as input for the first simulation. Run the model and save the output to tape or file.
4. Repeat steps 2 and 3 a large number of times, usually , or until the statistical output no longer changes by repeated realizations of the model.
5. Sort the stored output data and plot the output as the mean value plus or minus the probability range that is desired.

71. Table 6.6. A Comparison of Methods for Uncertainty Analysis

72. 6. 7. DISSOLVED OXYGEN IN LARGE RIVERS AND ESTUARIES 6. 7. 1 BOD-D. O
6.7. DISSOLVED OXYGEN IN LARGE RIVERS      AND ESTUARIES BOD-D.O. Deficit, Steady State
The control volume approach can be used to write mass balance equations for BOD and D.O. (or D.O. deficit) in a large river or estuary that has considerable longitudinal mixing. For BOD and D.O. deficit, we may consider the following reactions in the simplest case:
Within the control volume:
The mass balance equation for BOD:

73. Above equation can be expanded and terms canceled as shown below:
Rearranging, we obtain the proper equation for 1-D transport and decay of BOD in a large river or estuary.
A more general form equation is given by equation:
The D.O. deficit equation is developed in a similar manner to above equations.
Under steady-state conditions and constant coefficients above eq`ns can be simplified and solved for the BOD-D.O. deficit in a large river:

74. The analytical solution to above equation is that of a second-order, linear, ordinary differential equation with constant coefficients. It is a homogeneous equation (with-out forcing functions) of the general algebraic quadratic form:
The general solution is of the form:
In order to determine the integration constants, A and B, for the general equation, we must use upstream and downstream boundary conditions. Figure 6.13 is a schematic of the problem.
We will divide the analytical solution into two part:
(1) an upstream segment above the wastewater discharge, W;
(2) a downstream segment below the discharge stretching all the way to the ocean, for the case of a coastal plain estuary.

75. For the upstream solution to above equation, where x is negative above the wastewater discharge, we have the following boundary conditions:
BC 1: at x = -∞, L = 0
BC 2: at  x = 0, L = L0
Based on the first boundary condition the integration constant B in above equation must be equal to zero because the root g is always positive and j is always negative. Otherwise, the second term in equation would be infinite since both j and x are negative at x = -∞. The first term is zero at x = -∞ because e-∞ = 0.
From the second BC we know that the integration constant A is equal to L0.
The solution to the upstream segment for BOD is

76. Similarly, the solution to equation for the downstream segment is based on the boundary conditions.
BC 1: at x = -∞, L = 0;
BC 2: at  x = 0, L = L0;
The solution
One problem remains: the concentration of BOD at the source of the wastewater discharge is not known. Unlike the case of a plug-flow stream, we cannot assume that the concentration is equal to the mass input rate of BOD divided by the flowrate (W/Q). Now we must consider a mass balance equation around an infinitesimal slice of the estuary at x = 0.
The mass of material entering the upstream elemental slice at the discharge point (x = 0) is QL0, where the waste discharge (W) enters. The mass flux entering the upstream element is given by Fick's first law analogy.

77. Evaluating the derivatives dL1/dx and dL2/dx at the discharge point gives:
Substituting above equations, we have:
Rearranging equation, we can substitute the definition of j and g.
Recognizing that the flowrate is equal to the mean freshwater velocity times the cross-sectional area, the final solution for the BOD concentration at the discharge point:

78. Now we must solve above equation for the D. O
Now we must solve above equation for the D.O. deficit concentration in the large river or estuary. The analytic solution technique is similar to that for BOD except that we have a second-order, linear, nonhomogeneous ordinary differential equation.
The solution(s) are of the general form:
We solve above equation with the general solution and boundary conditions to determine A and B in the upstream and downstream directions from the discharge point.

79. Evaluation of D0 is accomplished via a mass balance (inputs = outputs) around an infinitesimally thin slice at the discharge point. Assuming that the discharge is saturated with dissolved oxygen, we have inputs and outputs of D.O. deficit due to dispersion and advection. Reactions do not have enough time to affect the D.O. deficit balance over an infinitesimal slice.
inputs
outputs
Because the advection terms on both sides of above equation cancel out, the equation tells us that the mass flux of D.O. deficit into the discharge point is equal to the mass flux out from the discharge point - it is a smooth D.O. deficit profile even though there is a cusp at x = 0 in the BOD profile (Figure 6.13).
The final analytic solution for the D.O. deficit concentration is given by equation:

80. Figure 6. 13. Biochemical oxygen demand (BOD) and dissolved oxygen (D
Figure Biochemical oxygen demand (BOD) and dissolved oxygen (D.O.) concentration in an estuary, plut flow with dispersion model.

81. 6.7.2. Estimation of Parameters and Circulation in Estuaries
Circulation in estuaries is density-driven by saline intrusion of bottom waters. Neglecting Coriolis forces, we may simplify the problem to that of two-dimensional circulation as depicted in Figure 6.15.
Saline waters intrude below the plane of no-net motion, and there is a reversal in current direction with depth in the estuary (Figure 6.15).
Continuity requires vertical mixing upward of bottom waters, which increases dispersion, as does the vertical shear velocity. Ebb and flood velocities at any time can be as much as five times the net (tidal-averaged) longitudinal velocity, u (Figure 6.16).
where u = net (tidal-averaged) velocity due to freshwater flow, LT-1; Qf  = freshwater discharge, L3T-1; A  = average cross-sectional area .

82. Figure 6.15. Schematic diagram of two-dimensional estuarine circulation.

83. Stage
Figure Tidal cycle of an estuary showing (a) normalized stage, (b) ebb and flood velocity, and (c) salinity concentrations.
Ebb (+) and Flood (-) Velocity
Salinity
Tidal Cycle, f

84. Usually, u is estimated in one of two ways:
It is difficult to actually measure the net nontidal velocity due to freshwater discharge given the dynamics of an estuary.
Usually, u is estimated in one of two ways:
(1) using last equation, the freshwater flowrate is estimated from the last gaging station on the river, and the average cross-sectional area is estimated from bathymetric maps of the channel;
(2) from release of a fluorescent dye (e.g., Rhodamine WT) that can be measured sensitively with a fluorometer.
It is recommended to release the dye at high water slack tide (stage is a maximum in Figure 6.16) or at low water slack tide (stage is a minimum) and to make all measurements at high water or low water slack tide.
The net nontidal velocity is based on the distance from where the dye was released to the maximum concentration at a later tidal cycle.
where xp is the distance from the release point to the maximum concentration, and tn is the time since release of the dye (n, number of tidal cycles).

85. There are no simple formulas to estimate the longitudinal dispersion coefficient in an estuary. It can be obtained from:
Salinity data.
Spatial distribution of dye (tracer) experiments.
Temporal distribution of dye (tracer) experiments.
We begin with the use of salinity data to estimate E. Salinity is nature's own tracer of the longitudinal dispersion in an estuary.
A mass balance differential equation for salinity, a conservative substance, it steady-state is
Two boundary conditions are necessary to solve the second-order ordinary differential equation:
Where S is the salinity concentration and S0 is the source concentration (the ocean) at x = 0. The solution to above equation is for negative values of distance, moving from the mouth to the upstream river.

86. A second method to obtain longitudinal dispersion coefficients is from dye (tracer) studies. Either an impulse input (slug) of dye or a continuous input can be released from a bridge at an upstream location. The dye is followed with boats and a fluorometer over several tidal cycles. The mass balance partial differential equation that describes an impulse discharge to a 1-D estuary is
Under steady flow conditions (tidal-averaged conditions) and a relatively constant cross-sectional area, above equation can be simplified to a partial differential equation (PDE) with constant coefficients.
C = concentration of dye, ML-3; M = mass of dye injected, M; A = average cross-sectional area of the estuary, L2; E = longitudinal dispersion coefficient, L2T-1; t = time since release of the dye; x  = longitudinal distance with x = 0 being the location where the dye was released and, downstream distance is positive; u = net nontidal velocity, LT-1; k = sum of the first-order reaction rate coefficients for disappearance of the dye, T-1.

87. Theoretically, the spatial distribution of dye concentrations should form Gaussian (normal) profiles after an initial mixing period.
Figure 6.18 shows the results of a tracer experiment performed in a physical model of the Delaware Estuary.
The standard deviation of the spatial distributions are related mathematically to the dispersion coefficient
The degradation or disappearance of the dye tracer can be estimated by integrating the mass under each concentration profile and by plotting lnM versus time. The slope of the semilog plot yields the first-order degradation rate constant, k.
where: Mt is the integrated mass at any tidal cycle.
In Figure 6.18, the peak concentration (Cpx) of dye for each tidal cycle provides a method to measure the net nontidal velocity.
Cpx = peak concentration in a spatial distribution of dye at a given tidal cycle, ML-3; φ = distance of the dye measurement from the peak concentration, L, φ = x – ut.

88. Figure Dye concentration (mgL-1) in the Delaware Estuary at consecutive high water slack tide intervals after impulse input of a dye tracer.

89. We have tried to estimate transport characteristics for large rivers and estuaries to solve BOD-D.O. modeling problems.
The reaeration constant is one of the  most difficult reaction rate constants to estimate due to the dynamic nature of estuaries.
Empirical and fundamental equations are not available.
To summarize our estimates of transport and reaction parameters in estuaries, the following ranges apply:
Large rivers have no tidal action and no salinity intrusion; tidal rivers display tidal action (changes in stage) but little saline intrusion; and estuaries have both tidal action and salinity profiles.
Models similar to those in this section can be applied to other water quality problems in estuaries such as toxic metals and organic chemicals.