CE5504 – Surface Water Quality Modeling CE5504 Surface Water Quality Modeling Lab 1. Numerical...
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E5504 – Surface Water Quality Modeling CE5504 Surface Water Quality Modeling b 1. Numerical Methods and Population Growth Modeli dc V dt dc V dt
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Transcript of CE5504 – Surface Water Quality Modeling CE5504 Surface Water Quality Modeling Lab 1. Numerical...
CE5504 – Surface Water Quality Modeling
CE5504 Surface Water Quality Modeling
Lab 1. Numerical Methods and Population Growth Modeling
dcVdt
dc
Vdt
Begin with a mass balance on microbial growth
CE5504 – Surface Water Quality Modeling
in
dXV Q X Q X Reactiondt
dXV V k Xdt
dXk X
dt
(Mihelcic 1999, Figure 5.4)
Exponential growth model
CE5504 – Surface Water Quality Modeling
dXX
dt
( )0
ttX X e
( )0
ttX X e
Environmental Resistance
CE5504 – Surface Water Quality Modeling (Mihelcic 1999, Figure 5.5)
Logistic growth model
CE5504 – Surface Water Quality Modeling
max 1dX X
Xdt K
max0
0
1t
t
KX
K Xe
X
(Mihelcic 1999, Figure 5.7)
Example: carry capacity effects
CE5504 – Surface Water Quality Modeling (Mihelcic 1999, Figure 5.6)
Monod Model
CE5504 – Surface Water Quality Modeling
maxs
dX SX
dt K S
(Mihelcic 1999, Figure 5.8)
Example: resource competition
CE5504 – Surface Water Quality Modeling (Mihelcic 1999, Figure 5.9)
The Yield Coefficient
CE5504 – Surface Water Quality Modeling
XY
S
1dS dX
dt Y dt
The Death (Respiration) Coefficient
CE5504 – Surface Water Quality Modeling
d
dXk X
dt
Putting It All Together
CE5504 – Surface Water Quality Modeling
max
max
1
11
ds
ds
dX X Sk X
dt K K S
dS X Sk X
dt Y K K S
(Mihelcic 1999, Figure 5.10)
Numerical Integration
CE5504 – Surface Water Quality Modeling
non-idealized loading functions
variable parameters
multi-segment systems
non-linear kinetics
The Euler Method
CE5504 – Surface Water Quality Modeling
dXX
dt
dX X dt
new oldX X dX
The Effect of Step Size
CE5504 – Surface Water Quality Modeling (Spain 1982, Figure 5.1)
Code
CE5504 – Surface Water Quality Modeling
max0For t to t
dX X dt
X X dX
Next t
max0For t to t Step dt
dX X dt
X X dX
Next t
Code (continued)
CE5504 – Surface Water Quality Modeling
max0
,
For t to t Step dt
dX X dt
X X dX
Print t X
Next t
Code (continued)
CE5504 – Surface Water Quality Modeling
max0
1/
,
For t to t
For k dt
dX X dt
X X dX
Next k
Print t X
Next t
