CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 1
Lecture 7: Computer Methods for Well-Mixed Reactors
CE 498/698 and ERS 685
Principles of Water Quality Modeling
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 2
Modeling Tradeoff
more realism
mo
re c
om
ple
xity
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 3
Simplifying Assumptions
• Idealized loading curves• Q, k, V are constant
• First-order reactions
What if these don’t apply????
Computers and numerical methods
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 4
Completely Mixed Lake Model
VtW
cdtdc
c
VtW
dtdc
Hv
kVQ where
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 5
Euler’s MethodCh. 25 in Chapra and Canale
c
t
h
forward difference:h
cc
tt
cc
tc
dt
dc ii
ii
iii
1
1
1
+
ti
ciconc. at present ti
+
ti+1
ci+1conc. at future ti+1
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 6
Euler’s MethodCh. 19 in Chapra and Canale
fwd difference:h
cc
dt
dc iii 1
hc
V
tWcc i
iii
1or
i
iiii cV
tW
h
cc
dt
dc 1
hctfcc iiii ,1 or
where dtdc
cVtW
ctf ii
iii ,
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 7
Example 7.1Given: 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 = 50106 g yr-1
Simulate concentration from t = 0 to 20 yr using timestep of 1 year
1-6
5
yr 35.0525.0
2.01010
Hv
kVQ
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 8
Example 7.1At ti = 0, ci = 15 mg L-1 and W(ti) = 50106 g yr-1
1-6
6
1 L mg 75.590.11535.010
1050151
cci
ci for next computation
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 9
Two equations:
22122
12111
,,
,,
kcctfdtdc
kcctfdtdc
Euler’s Method
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 10
Heun’s methodCh. 25 in Chapra and Canale
c
t
+
ti
ci+
ti+1
ci+1
hctfccctfdt
dciiiiii
i ,, 01
slope 1 (predictor)
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 11
Heun’s method
c
t
+
ti
ci+
ti+1
ci+1
011
1 , iii ctfdt
dc
slope 2
c0i+1
2
,,
22 slope 1 slope 0
11 iiii ctfctf
dtdc
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 12
Heun’s method
c
t
+
ti
ci+
ti+1
ci+1c0i+1
2
,,
22 slope 1 slope 0
11 iiii ctfctf
dtdc
h
ctfctfcc iiiiii 2
,, 011
1
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 13
Examplewithout iteration
1-yr 35.0From previous calcs:
75.441535.010
105015,0, 6
6
fctf ii
175.4415101 cci h
0875.2975.5935.010
105075.59,1, 6
60
11 fctf ii
91875.5112
0875.2975.44151 c
At ti = 0, ci = 15 mg L-1 and W(ti) = 50106 g yr-1
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 14
4th-order Runge-Kutta
general form of RK methods: hcc ii 1
slope estimate
Euler: hkcc ii 11
Heun: hkkcc ii
211 21
4th-order RK: hkkkkcc ii
43211 2261
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 15
4th-order Runge-Kutta
where
hkkkkcc ii
43211 2261
34
23
12
1
,
21
,21
21
,21
,
hkchtfk
hkchtfk
hkchtfk
ctfk
ii
ii
ii
ii
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 16
Spreadsheet ApplicationsExample: Euler’s 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)
=
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 17
Spreadsheet ApplicationsExample: Euler’s 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)
= kH
v
V
Q
kH
v
V
Q
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 18
Spreadsheet ApplicationsExample: Euler’s 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) ci+1
0 W(t0) Initial conc.
1 W(t1)
kH
v
V
Q
i
i cV
tW
i
i cV
tW
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 19
Spreadsheet ApplicationsExample: Euler’s 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) ci+1
0 W(t0) Initial conc. =ci+slope*h
1 W(t1)
kH
v
V
Q
i
i cV
tW
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 20
Spreadsheet ApplicationsExample: Euler’s 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) ci+1
0 W(t0) Initial conc. =ci+slope*h
1 W(t1) =ci+1
kH
v
V
Q
i
i cV
tW
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 21
Spreadsheet ApplicationsHeun’s method
ti W(ti) ci Slope 1 (s1)
c0i+1 Slope 2
(s2)ci+1
0 W(t0) Initial conc. =ci+s1*h =ci+0.5(s1+s2)*h
1 W(t1) =ci+1
i
i cV
tW
1
1i
i cV
tW
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 22
Spreadsheet ApplicationsHeun’s method
ti W(ti) ci Slope 1 (s1)
c0i+1 Slope 2
(s2)ci+1
0 W(t0) Initial conc. =ci+s1*h =ci+0.5(s1+s2)*h
1 W(t1) =ci+1
i
i cV
tW
01
1i
i cV
tW
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 23
Spreadsheet ApplicationsHeun’s method
ti W(ti) ci Slope 1 (s1)
c0i+1 Slope 2
(s2)ci+1
0 W(t0) Initial conc. =ci+s1*h =ci+0.5(s1+s2)*h
1 W(t1) =ci+1
i
i cV
tW
1
1i
i cV
tW
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 24
Major Homework #1
Parameters from Example 5.3
Eigenvalues
Calculations
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 25
Major Homework #1
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
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 26
Major Homework #1
Eigenvalues
11 yr-1
22 yr-1
CE 498/698 and ERS 485 (Spring 2004)
Lecture 7 27
Major Homework #1
Year Time ci,1 ci,2 ci,3 ci,4 ci,5 k1,1 k2,1 k3,1 k4,1 ci+1,1 k1,2 k2,2
1963 0
=prev+h
given