Environmental Engineering Reference
In-Depth Information
Chapter 2
Fundamentals of Modeling, Principles
and MATLAB
®
2.1 Model Types
As the term of 'model' has a wide variety of meanings, clearly a very narrow type of
models is addressed in this topic. Even within the special field of 'environmental
models' only the sub-class of
deterministic models
is treated; statistical models do
not appear, although statistics plays a significant role in environmental sciences and
technology.
In deterministic models all variables and parameters are functions of indepen-
dent space and time variables. The
independent variables
are referred to by the
usual notation
x
,
y
,
z
and
t
. In most models, especially in the relatively simple
examples presented here, it is sufficient to formulate the problem considering only a
subset of these four variables.
Depending on the number of space dimensions, one speaks of 0D, 1D, 2D or 3D
models. 0D models have no space dependency, only a time dependency. Ecological
models concerning populations of biological species in an environmental compart-
ment are in their majority of that type. As
t
is the only independent variable, the
analytical formulation leads to
ordinary differential equations
. These are differen-
tial equations, which depend on one variable only; in contrast to
partial differential
equations
, where there are at least two independent variables.
Models with no time dependency are denoted as steady,
steady state
or station-
ary. The corresponding terms for time dependent simulations are: unsteady or
transient
. A steady state is approached in real systems, if the internal processes
have time enough to adjust to constant outer conditions. It is a necessary condition
for steady state that exterior processes or parameters do not change in time.
Otherwise steady conditions cannot be reached.
1D models include one space dimension only. Models for the soil compartment
are mostly 1D, as the changes in vertical direction are of concern: seepage to the
groundwater table or evaporation to the ground surface. Processes in rivers (image a
water level peak or a pollutant plume moving downstream) can be regarded in 1D
under certain conditions. Water from surface water bodies infiltrating into aquifers
