Environmental Engineering Reference
In-Depth Information
When the proportionality factor is denoted by
, the same relation is expressed
a
by the equation
@c
@t
¼ ac
(1.12)
which is a differential equation for the population
c
as a function of time
t
.With
(
1.12
) the first task in modeling is already performed. The conceptual model,
the proportionality relationship, is expressed as a differential equation. In this
book differential equations are derived for various different processes in differ-
ent environmental compartments. The user is led from a conceptual model
concerning processes to the mathematical formulation of one or more differential
equations.
This task is completed with the formulation of the initial condition: at time
t ¼
0
the population has the value
c
0
, or:
cðt ¼
0
Þ¼c
0
(1.13)
The second step of modeling is the solution of the differential equation under
consideration of the boundary condition. There are several different means to do
that. For simple equations the solution can be written explicitly in a formula, here:
cðtÞ¼c
0
exp
ðatÞ
(1.14)
the
formula can be evaluated and plotted directly. The following commands need to be
given in the command window (Fig.
1.6
):
The given exponential function fulfils both requirements. In MATLAB
®
The concentration obviously increases by a factor of 2.8 during the time period
of length 1. Before continuing let us have a short view on the commands given
above. The first two commands specify the parameters
and
c
0
. The third command
defines the vector
t
, containing 11 elements: 0, 0.1, 0.2,
a
, 1. Note that
MATLAB
®
prints the vector into the command window when the semicolon at
the end of the line is omitted. The fourth command initiates several tasks. At first
the vector
t
is multiplied by the parameter value of
alpha
. The result of this scalar
multiplication is again a vector (see Sect.
1.2
). As a next task, the exponential
function is calculated for that vector. The command for the exponential function is:
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