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Lecture 6: Feedback Systems of Reactors. CE 498/698 and ERS 685 Principles of Water Quality Modeling. W 1. W 2. Q 01 c 0. Q 12 c 1. Q 23 c 2. Q 21 c 2. k 2 V 2 c 2. k 1 V 1 c 1. Feedback. 1. W 1. 2. W 2. Q 01 c 0. Q 12 c 1. Q 23 c 2. Q 21 c 2. Lake 1:. k 2 V 2 c 2. - PowerPoint PPT Presentation
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CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 1
Lecture 6: Feedback Systems of Reactors
CE 498/698 and ERS 685
Principles of Water Quality Modeling
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 2
FeedbackW1 W2
Q01c0 Q12c1 Q23c2
Q21c2
k1V1c1 k2V2c2
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 3
22111111211
1 cQcVkcQWdtdc
V Lake 1:
W1 W2Q01c0 Q12c1 Q23c2
Q21c2
k1V1c1 k2V2c2
1 2
22122222311222
2 cQcVkcQcQWdt
dcV Lake 2:
0011 cQW
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 4
22111111211
1 cQcVkcQWdtdc
V Lake 1:
22122222311222
2 cQcVkcQcQWdt
dcV Lake 2:
Steady-state: 0dtdc
1212111 Wcaca 2222121 Wcaca and
111211 VkQa
2112 Qa 1221 Qa
23222122 QVkQa
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 5
1313212111 Wcacaca
2323222121 Wcacaca
3333232131 Wcacaca
system parameters loadingsunknowns
LINEAR ALGEBRAIC EQUATIONS
Matrix algebra
WCA
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 6
1313212111 Wcacaca
2323222121 Wcacaca
3333232131 Wcacaca WCA
333231
232221
131211
aaa
aaa
aaa
A
3
2
1
c
c
c
C
3
2
1
W
W
W
W
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 7
Gauss-Jordan methodTo compute the matrix inverse
Identity matrix
33 identity matrix:
100
010
001
I
CCI
augmented matrix:
100
010
001
333231
232221
131211
aaa
aaa
aaa
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 8
Gauss-Jordan methodTo compute the matrix inverse
1) Normalize2) Elimination
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 9
Gauss-Jordan method
100
010
001
333231
232221
131211
aaa
aaa
aaa
1) Normalize
100
010
001
111
333231
232221
11
13
11
12
a
aaa
aaaa
a
aa
Divide by a11
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 10
Gauss-Jordan method2) Elimination
100
010
001
111
333231
232221
11
13
11
12
a
aaa
aaaa
a
aa
00
121
1121
11
1321
11
1221 a
aa
a
aa
aa
a
100
011
001
1
2111
11
333231
2111
132321
11
12222121
11
13
11
12
aa
a
aaa
aa
aaa
aa
aaa
a
a
aa
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 11
Gauss-Jordan method2) Elimination
101
011
001
0
0
1
3111
2111
11
3111
133331
11
1232
2111
132321
11
1222
11
13
11
12
aa
aa
a
aa
aaa
aa
a
aa
aaa
aa
a
a
a
aa
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 12
Gauss-Jordan method1) Normalization
101
011
001
0
0
1
3111
2111
11
3111
133331
11
1232
2111
132321
11
1222
11
13
11
12
aa
aa
a
aa
aaa
aa
a
aa
aaa
aa
a
a
a
aa
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 13
Gauss-Jordan method
Matrix inverse
133
132
131
123
122
121
113
112
111
100
010
001
aaa
aaa
aaa
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 14
Gauss-Jordan method example
4.71102.03.0
3.193.071.0
85.72.01.03
321
321
321
ccc
ccc
ccc
100
010
001
102.03.0
3.071.0
2.01.03augmentedmatrix
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 15
Gauss-Jordan method example
100
010
001
102.03.0
3.071.0
2.01.03
100
010
00333.0
102.03.0
3.071.0
067.0033.01
Divide by 3(normalize)
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 16
Gauss-Jordan method example
100
010
00333.0
102.03.0
3.071.0
067.0033.01
10100.0
01033.0
00333.0
020.10190.00
293.0003.70
067.0033.01
Divide by 7.003(normalize)
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 17
10100.0
0143.0005.0
00333.0
020.10190.00
042.010
067.0033.01
Gauss-Jordan method example
1027.0101.0
0143.0005.0
0005.0333.0
012.1000
042.010
068.001
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 18
Gauss-Jordan method example
100.0003.0010.0
004.0143.0005.0
007.0005.0332.0
100
010
001
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 19
Gauss-Jordan method
• Can also be used to solve for concentrations
3
2
1
333231
232221
131211
W
W
W
aaa
aaa
aaa
3
2
1
100
010
001
c
c
c
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 20
Excel - MINVERSE
1. Enter your [A] matrix
2. Block an area the same size
3. Type =MINVERSE(block location of [A]matrix) and press CNTL+SHIFT+ENTER
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 21
WCA We want to solve for {C}
WACAA 11 Multiply both sides by [A]-1
WAC 1
IAA 1
CCI Definitions of identity matrix
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 22
Homework Problem 6.2(a)
• Use both Gauss-Jordan method and Excel MINVERSE function
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 23
{C} = response{W} = forcing functions[A]-1 = parameters
{response} =[interactions]{forcing functions}
31
3321
3211
313
31
2321
2211
212
31
1321
1211
111
WaWaWac
WaWaWac
WaWaWac
Response of reactor 1
Unit change in loading of reactor 2
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 24
Matrix Multiplication (Box 6.1)# columns in matrix 1 = # rows in matrix 2
3231
2221
1211
333231
232221
131211
bb
bb
bb
aaa
aaa
aaa
BA
312321221121
321322121211311321121111
bababa
babababababa
BA
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 25
Terminology
DIAGONAL
Effect of direct loading
SUPERDIAGONALEffects of d/s loadingson u/s reactors
SUBDIAGONALEffects of u/s loadingson d/s reactors
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 26
Time-variable response for two reactors
22111111211
1 cQcVkcQWdtdc
V
22122222311222
2 cQcVkcQcQWdt
dcV
2221212
2121111
ccdt
dc
ccdtdc
11
1211 k
VQ
1
2112 V
Q
2
1221 V
Q
22
122322 k
V
QQ where
CE 498/698 and ERS 685 (Spring 2004)
Lecture 6 27
Time-variable response for two reactors
General solution if c1=c10 at t = 0 t
st
fsf ececc 111
ts
tf
sf ececc 222
where ’s are functions of ’sc’s are coefficients that depend on eigenvalues and initial concentrations
f = fast eigenvalues = slow eigenvalue
f >>ssee formulason page 111
