Variety Of Water Quality Models Biology Essay


A variety of water quality models are available to determine pollutant transport in a stream. However, the choice of model is of paramount importance, as different models represent different river processes. There are a number of processes that occur in rivers; namely advection, diffusion, dispersion, re-aeration and decay. The model parameters therefore need to be calibrated in order to reflect what processes are occurring in the stream, as to adequately reflect the transport of solute. J.L. Martin and S.C. McCutcheon (1998) describes the flow of a given river reach to be dominated by advection or diffusion from the computation of Peclet's Number. If this dimensionless number is much lower than one, then they consider diffusion to control the flow. Whereas, values much greater than one is controlled by advection and diffusion can be neglected.

The advection process is represented by figure 2.1, the upstream point in the river reach is shown as x1 and hence the downstream point is x2. V1 and V2 represent the upstream and downstream volumes respectively. The volumes remain unchanged, thus V1=V2. Likewise c1 and c2 depicts the upstream and downstream concentration which are also equal.

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Figure 2.2 Diagram depicting the process of diffusion

The diffusion process is represented by figure 2.2. Since the velocity is zero for this process, there are no upstream and downstream points hence both are labelled as x1. The concentration of any pollutant can be represented by the following equation:



C= Concentration of the pollutant (kg/m3)

m= Mass of the pollutant (kg)

V= Volume of the pollutant (m3)

The volume of the pollutant is directly proportional to time for this process, therefore V1 is less than V2. An inversely proportional relationship can be seen in equation (2.2) between concentration and volume of the pollutant, therefore C1 is greater than C2.

Figure 2.3 Diagram depicting the process of dispersion

The process of dispersion, as represented in figure 2.3, is reflected as a combination of advection and diffusion. The pollutant is carried downstream due to the velocity of the river. The upstream and downstream points are reflected by x2 and x1 respectively. The downstream volume V1 is less than that of the upstream volume V2.

The process of re-aeration refers to the absorbing of oxygen by the river, from the atmosphere. Water quality models can convey that the re-aeration process occurs in a river by means of the re-aeration rate. In water quality models, the re-aeration coefficient principal used, is to quantify the process of re-aeration that occurs in dissolved oxygen models. E.C. Tsivoglou (1965) proposed a direct method to determine the stream re-aeration coefficient by use of the radioactive - tracer technique. However, this technique had disadvantages in that the obtaining of licences to use radioactive - tracers in publicly accessible rivers is a tedious process. Rathbun et al, 1975, and Rathbun et al, 1978, substituted the radioactive-trace krypton for a hydrocarbon gas to modify the original technique proposed by E. C. Tsivoglou, 1965.

The two fundamental parameters that must be known are velocity and the dispersion coefficient (DL). These will therefore assist in the estimation of model parameters. Also, there are empirical as well as theoretical methods for calculating velocity. Such methods include the float method, current meters, flow measurement at control structures and remote sensing. The dispersion coefficient (DL) is available through numerous amounts of empirical and theoretical formulae's. The empirical formulas reflect the dispersion coefficient (DL) to be a function of velocity, width of channel, slope and depth of channel. J.L. Martin and S.C. McCutcheon (1998), defines three methods that the dispersion coefficient can be obtained:

It can be estimated from literature of similar rivers and streams.

From a variety of empirical equations.

Dye or tracer studies can be used to estimate it.



The model was proposed by Streeter and Phelps in (1925) and was summarized and released by Phelps (1944):



D= Oxygen deficit (Cs-C)

C= Oxygen concentration

Cs= Oxygen saturation concentration

L= Carbonaceous biochemical oxygen demand

k1= Deoxygenation coefficient

k2= Re-aeration coefficient

T= Time

The integration of equation (2.4), thus produces the oxygen sag equation shown as equation (2.5)

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L0= Ultimate B.O.D

D0= Oxygen deficit

This model is also referred to as the oxygen-sag model or the Streeter-Phelps model. The limitations of this model involve the dispersion that is not taken into account. Therefore, its application would be limited as it does not simulate actual river conditions. An advantage with regard to this model is that re-aeration is taken into account. The parameters in this model that would require optimization are k1 and k2.


The mixing cell model is a one dimensional model containing cells with unique hydrologic, lithologic and sorptive properties (Rood, 2005). Ordinary differential equations can be used to describe each cell's water and solute balance. The mixing cell model is also referred to as the compartment model (Whicker and Shultz, 1982), box model and tanks-in-series model (Shanahan and Harleman, 1984; Rao and Hathaway, 1989). This model reflects the contaminant inventories as a function of time. Each cell could differ in vertical dimensions, bulk density, hydraulic characteristics (moisture content, porosity, etc). Codell et al (1983) used the following equation to represent a single contaminant with first order decay :



c = Solute concentration (kg/m3)

D = Dispersion coefficient (m2/s)

A = Cross sectional area perpendicular to flow (m2)

Rd = Retardation coefficient (Unitless)

q = Specific discharge or Darcy velocity (m/s)

λ = First order decay constant (t-1)

Figure 2.4 Diagram depicting the mixing cells model

A river reach is comprised of a combination of thoroughly mixing cells as illustrated in figure 2.4. The output concentration from the first cell will form the input concentration for the next cell and so forth. The limitation of the cells in the series model is that the advection process is not represented. Hence, this model does not accurately portray the processes within a river reach. This model only has one model parameter that will be required to be optimized. The model parameter being time (T), which can be computed from following equation:



T= Time (s)

V= Volume (m3)

Q= Flow (m3/s)

The volume and flow of each cell can be divided by the area of the cell and hence equation (2.5) can be reflected as:



T=Time (s)

Δx=Length of each cell (m)

V=Velocity (m/s)


This model was introduced by Beer and Young (1983), where a single parameter Tr (aggregate dead-zone residence time) can be used to portray the dispersive process that occurs in a river reach. The aggregate dead-zone model has problems in predicting the short term impacts; this is due to the uncertainty in the number of connections needed to represent the process of advection and dispersion. A mass balance for a solute in a fully mixed aggregate dead zone can be seen in equation (2.6):



V= Active mixing volume

U(t)= Input concentration of water quality determined

X(t) = Output concentration of water quality

Q= Flow

k = Decay or sedimentation rate coefficient

The aggregate dead-zone model can represent the process of advection, and the transformed equation takes the form:




= The time for the leading portion of the cloud of solute to be advected through the river reach

U(t)= Input concentration of water quality determined

X(t) = Output concentration of water quality

k = Decay or sedimentation rate coefficient

Young and Wallis (1993) proposed the following equation to reflect the process of advection, dispersion and proportional decay for a general pollutant in a river reach:




= The time for the leading portion of the cloud of solute to be advected through the river reach

U(t)= Input concentration of water quality determined

X(t) = Output concentration of water quality

k = Decay or sedimentation rate coefficient

= Mass decay of the solute

The aggregate dead-zone model works on the assumption that only a portion of the river reach will be fully mixed. The model is accurate when used to reproduce tracer responses for natural rivers (Wallis, 1994; Rutherford, 1994). The aggregate dead-zone model requires two model parameters to be solved for namely α and T1.

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Figure 2.5 Diagram depicting the Aggregate dead zone model

The aggregate dead-zone model can be described in figure 2.5. A section of a river reach is broken up into a plug flow zone and a thoroughly mixing unit. The time for the plug flow zone is equivalent to α and the time for the thoroughly mixing zone is equivalent to T1.


The hybrid cells in series model, addresses the limitation of the aggregate dead-zone model and the cells in series model. This model also depicts similarities with the ADE model in that adsorption and desorption, growth and decay, and transient storage can be reflected (Kumarasamy M, 2007).

Figure 2.6 Diagram depicting the Hybrid cells in series model

The hybrid cells in series model can be illustrated by figure 2.6. The model consists of a single plug flow zone and two thoroughly mixing cells. There are three parameters that are required to be solved; namely α, T1 and T2. This model is able to adapt to a number of different processes that occur in any given river reach and hence it is much more advantageous than the other models mentioned.


All the above models have model parameters that require solving. The techniques used to solve these parameters vary. The technique that will be looked at will be that of least square regression.


Linear regression is used to develop a line that best represents the data. This method is able to determine the gradient and intercept for a line, that minimizes the sum of the squares of the vertical distances, between the data and the line. Linear regression can be depicted by the equation in the form of (Page A, 1987):


Where :

x = independent variable

y = dependent variable

b = gradient of the line

a = y-intercept


The technique of non-linear regression is more complex. As a result, a curve may be fitted by minimizing the sum of the squares of the vertical distance between the curve and a set of data points. A Data set, which reflects the relationship between the modelled data, is curved. This can be done by creating new variables that are non-linear functions of the old variables (Baker, S.L). This process usually involves taking logs on either side.

The equation can be transformed into the linear form

and could now be depicted as a straight line graph (Baker, S.L). Non-linear regression can be applied to any equation that reflects Y as a function of X and more parameters (Page A, 1987). This technique is more accurate and is more applicable for the water quality models reviewed than that of linear regression.


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