Lecture 7: Computer Methods for Well-Mixed Reactors

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Lecture 7 Computer Methods for Well-Mixed
Reactors

• CE 498/698 and ERS 685
• Principles of Water Quality Modeling

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Modeling Tradeoff
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Simplifying Assumptions

• Idealized loading curves
• Q, k, V are constant
• First-order reactions

What if these dont apply????
Computers and numerical methods
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Completely Mixed Lake Model
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Eulers Method
Ch. 25 in Chapra and Canale
c
t
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Eulers Method
Ch. 19 in Chapra and Canale
fwd difference
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Example 7.1
Given
Q 10000 m3 yr-1 V 106 m3 Z 5 m
k 0.2 yr-1 v
0.25 m yr-1 c0 15 mg L-1 At
t 0, step loading 50?106 g yr-1
Simulate concentration from t 0 to 20 yr using
timestep of 1 year
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Example 7.1
At ti 0, ci 15 mg L-1 and W(ti) 50?106 g
yr-1
ci for next computation
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Eulers Method
Two equations
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Heuns method
Ch. 25 in Chapra and Canale
c
t
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Heuns method
c
t
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Heuns method
c
t
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Example
without iteration
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4th-order Runge-Kutta
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4th-order Runge-Kutta
where
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Spreadsheet Applications
Example Eulers method for Example 7.1
Q 10000 m3 yr-3
V 1000000 m3
Z 5 m
v 0.25 m yr-1
k 0.2 yr-1
h yr (timestep)
?




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Spreadsheet Applications
Example Eulers method for Example 7.1
Q 10000 m3 yr-3
V 1000000 m3
H 5 m
v 0.25 m yr-1
k 0.2 yr-1
h yr (timestep)
?




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Spreadsheet Applications
Example Eulers method for Example 7.1
Q 10000 m3 yr-3
V 1000000 m3
Z 5 m
v 0.25 m yr-1
k 0.2 yr-1
h yr (timestep)
?

ti W(ti) ci Slope (dc/dt) ci1
0 W(t0) Initial conc.
1 W(t1)
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Spreadsheet Applications
Example Eulers method for Example 7.1
Q 10000 m3 yr-3
V 1000000 m3
Z 5 m
v 0.25 m yr-1
k 0.2 yr-1
h yr (timestep)
?

ti W(ti) ci Slope (dc/dt) ci1
0 W(t0) Initial conc. cislopeh
1 W(t1)
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Spreadsheet Applications
Example Eulers method for Example 7.1
Q 10000 m3 yr-3
V 1000000 m3
Z 5 m
v 0.25 m yr-1
k 0.2 yr-1
h yr (timestep)
?

ti W(ti) ci Slope (dc/dt) ci1
0 W(t0) Initial conc. cislopeh
1 W(t1) ci1
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Spreadsheet Applications
Heuns method
ti W(ti) ci Slope 1 (s1) c0i1 Slope 2 (s2) ci1
0 W(t0) Initial conc. cis1h ci0.5(s1s2)h
1 W(t1) ci1
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Spreadsheet Applications
Heuns method
ti W(ti) ci Slope 1 (s1) c0i1 Slope 2 (s2) ci1
0 W(t0) Initial conc. cis1h ci0.5(s1s2)h
1 W(t1) ci1
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Spreadsheet Applications
Heuns method
ti W(ti) ci Slope 1 (s1) c0i1 Slope 2 (s2) ci1
0 W(t0) Initial conc. cis1h ci0.5(s1s2)h
1 W(t1) ci1
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Major Homework 1
Parameters from Example 5.3
Eigenvalues
Calculations
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Major Homework 1
Parameters from Example 5.3 Parameters from Example 5.3 Parameters from Example 5.3 Parameters from Example 5.3 Parameters from Example 5.3 Parameters from Example 5.3
1 2 3 4 5
Depth
Area
Volume
Outflow
Loading

Settling
Reaction rate
Timestep, h
One value for all 5 lakes
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Major Homework 1
Eigenvalues Eigenvalues Eigenvalues

?11 yr-1
?22 yr-1







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Major Homework 1
Year Time ci,1 ci,2 ci,3 ci,4 ci,5 k1,1 k2,1 k3,1 k4,1 ci1,1 k1,2 k2,2
1963 0
prevh
given