Digital Signal Processing Reference
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parameter models are also used for long time studies; where as distributed models are used for short
term transport phenomena.
One spatial variable (1D)
1D model are commonly used in two different situations; models of rivers and deep and narrow lakes.
The justification of the 1D approach has been argued as “It may be questioned to what extent a one-
dimensional model adequately represents a natural system such as a lake or reservoir. From the point
of view of the hydrodynamics, provided that the lake is neither extremely long and narrow, nor
extremely broad and shallow, a one dimensional hydrodynamic approach is satisfactory.” Orlob
(1983a) (Hamilton and SCHL-adow 1997,) see some shortcomings of the 1D modelling approach: “By
contrast, water quality variables, for example nutrient concentrations, exert a neglible effect on the
density distribution, and therefore could potentially display a two- or three-dimensional distribution
despite a 1D density distribution.” This is a reason for using a 3D model, but a little later in the same
text, Hamilton and SCHL-adow (1997) argues for the usage of a 1D model anyway. The reasons are
that a 3D-model is harder to construct, harder to verify and its harder to know all the initial conditions.
Therefore a 1D model can give greater certainty in the results. ”While this is recognized as a
shortcoming of a 1D model, it does not necessarily imply that a multi-dimensional approach would
produce a more correct picture.” It's hard to know all the initial conditions for a 3D-system and it's
hard to verify the results. Therefore a 1D model with greater certainty in the results could be preferred
Two spatial variables (2D)
Shallow waters, like estuaries, where the water quality is homogenous in depth-direction, are modelled
using depth averaged 2D models. For reservoirs there exist 2D vertical-longitudinal hydrodynamic
models.
Three spatial variables (3D)
The 3D model is the ultimate aim of water quality models since it is capable of modelling flow and
temperature profiles in large lakes, where both horizontal and vertical movements are of significance
for the water quality. The two competing numerical techniques for this class of problems are finite
elements (or the closely related finite difference), and approximating the distributed nature with a
series of interconnected well-mixed volumes modelled as a large and sparse collection of ordinary
differential equations.
Compartment models
A common simplification when modelling a distributed system is to view the system as a set of fully
stirred tanks, connected to each other so that the outflow from one tank is the inflow of another. Each
tank, or compartment, contains an ODE, and the models are called compartment or compartmental
models. Compartment models make it possible to model distributed phenomena with ODEs, and avoid
PDEs.
3.3.3.
Surface Water Quality Modelling Requirements
Modelling is one of the main tools in understanding the surface water problems and finding
appropriate solutions to it. Today, surface water quality management has moved well beyond the
urban point-source problem to encompass many other types of pollution. In addition to wastewater, we
now deal with other point sources such as industrial wastes as well as non-point sources such as
agricultural runoff (Chapra, 1997). Over the past 75 years, engineers have developed water quality
models to simulate a wide variety of pollutants in a broad range of receiving waters. In recent years,
these receiving water models are being coupled with models of watersheds, groundwater, bottom
sediments, to provide comprehensive frameworks predicting the impact of human activities on water
quality. As Thomann (1998) terms it, a ''Golden Age'' of water quality modelling is upon us (Chapra,
2003).
For years, the analysis of water quality has concentrated on the dissolved oxygen (DO) and
biochemical oxygen demand (BOD). The balance between DO and BOD concentrations was the result
of two processes: the reaeration of the water column, and the consumption of DO in oxidation of
BOD. Later emphasis has been on extending and refining the Streeter-Phelps formulation by using a
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