CE5504 – Surface Water Quality Modeling CE5504 Surface Water Quality Modeling Lab 2. Numerical Methods and Population Growth Modeling

Begin with a mass balance on microbial growth CE5504 – Surface Water Quality Modeling

(Mihelcic 1999, Figure 5.4) Exponential growth model CE5504 – Surface Water Quality Modeling

Environmental Resistance CE5504 – Surface Water Quality Modeling (Mihelcic 1999, Figure 5.5)

Logistic growth model 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 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 CE5504 – Surface Water Quality Modeling

The Death (Respiration) Coefficient CE5504 – Surface Water Quality Modeling

Putting It All Together (Batch Reactor) CE5504 – Surface Water Quality Modeling (Mihelcic 1999, Figure 5.10)

Putting It All Together (Completely-Mixed Flow Reactor) CE5504 – Surface Water Quality Modeling

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

The Effect of Step Size CE5504 – Surface Water Quality Modeling (Spain 1982, Figure 5.1)

Code CE5504 – Surface Water Quality Modeling

Code (continued) CE5504 – Surface Water Quality Modeling

Code (continued) CE5504 – Surface Water Quality Modeling

Advanced Numerical Techniques The 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 Techniques The 4 th Order Runge Kutta Method For i = 0 To tmax For j = 1 To 1 / dt k1 = mu * x X1 = x * dt * k1 k2 = mu * X1 X2 = x * 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 Techniques Error Comparison CE5504 – Surface Water Quality Modeling Error (%) for various values of dt at t = 5 days dtEulerHeun4th RK Error criterion: <=0.01 %