CE5504 – Surface Water Quality Modeling
CE5504 Surface Water Quality Modeling
Lab 2. Numerical Methods and Population Growth Modeling
dcVdt
dcVdt
Begin with a mass balance on microbial growth
CE5504 – Surface Water Quality Modeling
indXV Q X Q X Reactiondt
dXV V k Xdt
dX k Xdt
(Mihelcic 1999, Figure 5.4)
Exponential growth model
CE5504 – Surface Water Quality Modeling
dX Xdt
( )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
KXK X eX
(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 S Xdt 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
XYS
1dS dXdt Y dt
The Death (Respiration) Coefficient
CE5504 – Surface Water Quality Modeling
ddX k Xdt
Putting It All Together(Batch Reactor)
CE5504 – Surface Water Quality Modeling
max
max
1
1 1
ds
ds
dX X S k Xdt K K S
dS X S k Xdt Y K K S
(Mihelcic 1999, Figure 5.10)
Putting It All Together(Completely-Mixed Flow Reactor)
CE5504 – Surface Water Quality Modeling
max
max
1
1 1
in ds
in ds
dX Q Q X SX X k Xdt V V K K S
dS Q Q X SS S k Xdt V V Y K K S
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
dX Xdt
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 tdX X dtX X dX
Next t
max0For t to t Step dtdX X dtX X dX
Next t
Code (continued)
CE5504 – Surface Water Quality Modeling
max0
,
For t to t Step dtdX X dtX X dXPrint t X
Next t
Code (continued)
CE5504 – Surface Water Quality Modeling
max01/
,
For t to tFor k dtdX X dtX X dX
Next kPrint t X
Next t
Advanced Numerical TechniquesThe Heun’s Method
CE5504 – Surface Water Quality Modeling
For i = 0 To tmax
For j = 1 To 1 / dt
k1 = mu * x X1 = x + dt * k1
k2 = mu * X1
x = x + (k1 + k2) / 2 * dt Next j Next i
CE5504 – Surface Water Quality Modeling
Advanced Numerical TechniquesThe 4th Order Runge Kutta Method
For i = 0 To tmax
For j = 1 To 1 / dt
k1 = mu * x X1 = x + 0.5 * dt * k1
k2 = mu * X1
X2 = x + 0.5 * dt * k2 k3 = mu * X2 X3 = x + dt * k3 k4 = mu * X3 x = x + (k1 + 2 * k2 + 2 * k3 + k4) / 6 * dt Next j Next i
Advanced Numerical TechniquesError Comparison
CE5504 – Surface Water Quality Modeling
Error (%) for various values of dt at t = 5 days
dt Euler Heun 4th RK
1 98.90 85.81 23.70
0.5 95.35 56.70 3.60
0.1 58.68 5.58 0.01
0.01 9.40 0.07
0.001 0.99 0.001
0.0001 0.10
0.00001 0.02
0.000001 0.001
Error criterion: <=0.01 %
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