Environmental Engineering Reference
In-Depth Information
use condition ( 10.8 ). However,
the derivative to be used looks a bit more
complicated:
@c 2
@ l 2 ¼c 20 t exp
ð l 2 tÞþc 10 gl 1
l 2 l 1
1
l 2 l 1
t exp
ð l 2
ð
ð l 1
ð l 2
Þ
exp
exp
(10.21)
For the implementation of this formula use the following function
function f = myfun2(lambda2);
global tfit c2fit c0 c02 lambda1 g
t = tfit;
c1 = exp(-lambda1*t);
c2 = exp(-lambda2*t);
dl = lambda2-lambda1;
clambda2 = (g*lambda1*c0/dl-c02)*t.*c2 - (g*lambda1*c0/dl/dl)*(c1-c2);
c2 = c02*c2+g*lambda1*c0*(c1-c2)/dl;
More on explicit solutions for decay chains is noted by Yuan and Kernan ( 2007 ).
Bauer et al. ( 2001 ) as well as Guerrero et al .( 2009 ) focus on the transport of decay
chains.
10.5 Transport Parameter Fitting
The described algorithm can also be applied for the estimation of transport
parameters. Here this is demonstrated for 1D transport as described by the Ogata-
Banks solution:
þ exp
erfc
c in
2
x vt
2
v
D x
x þ vt
2
cðx; tÞ¼c 0 þ
erfc
p
p
(10.22)
Dt
Dt
The situation is examined in which the velocity v is the most unknown para-
meter. Such a situation can be met quite often in the description of environmental
systems.
In order to apply the procedure introduced above, the derivative of the solution
due to velocity is needed. The derivative of the complementary error function is
given by:
@
@x
2
p
x 2
erfc
ðxÞ¼
p
exp
(10.23)
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