CE5504 Surface Water Quality Modeling
dc
V

dt
Lab 2. Numerical Methods and Population Growth Modeling
CE5504 – Surface Water Quality Modeling
Begin with a mass balance on microbial growth
dX
V
 Q  X in  Q  X  Reaction
dt
dX
V
V k  X
dt
dX
kX
dt
CE5504 – Surface Water Quality Modeling
X t  X 0  e(  t )
Exponential growth model
dX
 X
dt
Xt  X0 e
CE5504 – Surface Water Quality Modeling
(  t )
(Mihelcic 1999, Figure 5.4)
Environmental Resistance
CE5504 – Surface Water Quality Modeling
(Mihelcic 1999, Figure 5.5)
Logistic growth model
dX
 X
 max  1    X
dt
 K
Xt 
K
 K  X 0    max t  
1  

e
X
0



CE5504 – Surface Water Quality Modeling
(Mihelcic 1999, Figure 5.7)
Example: carry capacity effects
CE5504 – Surface Water Quality Modeling
(Mihelcic 1999, Figure 5.6)
Monod Model
 S 
dX
 max  
 X
dt
 Ks  S 
CE5504 – Surface Water Quality Modeling
(Mihelcic 1999, Figure 5.8)
Example: resource competition
CE5504 – Surface Water Quality Modeling
(Mihelcic 1999, Figure 5.9)
The Yield Coefficient
X
Y
S
dS
1 dX
 
dt
Y dt
CE5504 – Surface Water Quality Modeling
The Death (Respiration) Coefficient
dX
  kd  X
dt
CE5504 – Surface Water Quality Modeling
Putting It All Together
(Batch Reactor)

dX 
 X  S 
  max  1    
  kd   X
dt 
 K   Ks  S 


dS
1 
 X  S 
    max  1    
  kd   X
dt
Y 
 K   Ks  S 

CE5504 – Surface Water Quality Modeling
(Mihelcic 1999, Figure 5.10)
Putting It All Together
(Completely-Mixed Flow Reactor)


dX Q
Q
 X  S 
  X in   X   max  1    
  kd   X
dt V
V
 K   Ks  S 


dS Q
Q
1 
 X
  Sin   S    max  1 
dt V
V
Y 
 K
CE5504 – Surface Water Quality Modeling

  S 
  kd   X

  Ks  S 

Numerical Integration
 non-idealized loading functions
 variable parameters
 multi-segment systems
 non-linear kinetics
CE5504 – Surface Water Quality Modeling
The Euler Method
dX
 X
dt
dX    X  dt
X new  X old  dX
CE5504 – Surface Water Quality Modeling
The Effect of Step Size
CE5504 – Surface Water Quality Modeling
(Spain 1982, Figure 5.1)
Code
For t  0 to tmax
dX    X  dt
X  X  dX
Next t
For t  0 to tmax Step dt
dX    X  dt
X  X  dX
Next t
CE5504 – Surface Water Quality Modeling
Code (continued)
For t  0 to tmax Step dt
dX    X  dt
X  X  dX
Print t , X
Next t
CE5504 – Surface Water Quality Modeling
Code (continued)
For t  0 to t max
For k  1/ dt
dX    X  dt
X  X  dX
Next k
Print t , X
Next t
CE5504 – Surface Water Quality Modeling
Advanced Numerical Techniques
The Heun’s Method
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 Techniques
The 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
CE5504 – Surface Water Quality Modeling
Advanced Numerical Techniques
Error Comparison
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 %
CE5504 – Surface Water Quality Modeling