Environmental Engineering Reference
In-Depth Information
10
6
s, to be read in the plot produced by the following sequence
10
C? Answer: 1.1
of commands:
Also for the constant heat flux boundary condition:
l
@T
@x
ðx ¼
0
; tÞ¼j
0
(4.6)
with a constant flux
j
0
an analytical solution exists (Baehr and Stephan
1994
). It is
given by:
p
Dt
2
j
0
l
x
2
p
Tðx; tÞ¼T
0
þ
ierfc
(4.7)
Dt
with the integral error function
1
p
exp( -
2
ðxÞ¼
x
Þx
ðÞ
ierfc
erfc
(4.8)
, but can easily be
computed by use of (
4.8
). A corresponding m-file is included in the accompanying
software under the name
'ierfc.m'
.
The integral error function is not specified in MATLAB
®
4.2 A Simple Numerical Model
In Chaps. 2 and 3 it is shown that processes and fundamental laws can be formulated
in form of differential equations. Above in this chapter it was shown that a solution
for a differential equation could be given by an explicit formula. With reference to
mathematical analysis, functions given in formulae, as in (
4.4
), are called analytical
solutions.
In fact there are analytical solutions for relatively few situations, compared
to the immense complexity, which can be represented in differential equations.
For that reason an alternative approach has gained importance, in which the
mathematical algorithm on the computer yields an approximation for the solution.
The mathematical discipline dealing with these approximations is numerics. The
methods used in numerics are called numerical methods, and the solutions are
called
numerical solutions
- in contrast to analytical solutions.
As an example for the numerical method a simple procedure is presented, which
delivers an approximate solution for the transport equation Fig.
4.3
that was devel-
oped above. For reason of simplicity the demonstration covers the 1D situation.
